Large Deformation Plasticity of Amorphous Solids, with 非晶态固体大变形塑性,和

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材料科学与工程专业英语Unit2ClassificationofMaterials译文

Unit 2 Classification of MaterialsSolid materials have been conveniently grouped into three basic classifications: metals, ceramics, and polymers. This scheme is based primarily on chemical makeup and atomic structure, and most materials fall into one distinct grouping or another, although there are some intermediates. In addition, there are three other groups of important engineering materials —composites, semiconductors, and biomaterials.译文：译文：固体材料被便利的分为三个基本的类型：金属，陶瓷和聚合物。

固体材料被便利的分为三个基本的类型：金属，陶瓷和聚合物。

这个分类是首先基于这个分类是首先基于化学组成和原子结构来分的，化学组成和原子结构来分的，大多数材料落在明显的一个类别里面，大多数材料落在明显的一个类别里面，大多数材料落在明显的一个类别里面，尽管有许多中间品。

尽管有许多中间品。

除此之外，此之外，有三类其他重要的工程材料－复合材料，半导体材料和生物材料。

有三类其他重要的工程材料－复合材料，半导体材料和生物材料。

Composites consist of combinations of two or more different materials, whereas semiconductors are utilized because of their unusual electrical characteristics; biomaterials are implanted into the human body. A brief explanation of the material types and representative characteristics is offered next.译文：复合材料由两种或者两种以上不同的材料组成，然而半导体由于它们非同寻常的电学性质而得到使用；生物材料被移植进入人类的身体中。

ch3-plastic deformation and recrystallize-塑性变形及再结晶双语

Elastic Plastic
σy
Y
Yield(屈服) point: Y Where strain deviates from being proportional to stress (the proportional limit) A measure of resistance to plastic deformation Yield strength(强度): σy Permanent strain= 0.002
Torsion Shear
PREVIOUS NEXT RETURN OUTLINE RETURN
3.2 Slipping and Twinning Two forms in plastic deformation Slipping(滑移)
Permanent deformation (also called plastic deformation) of a single crystal subjected to a shear stress: (a) structure before deformation; and (b) permanent deformation by slip. The size of the b/a ratio influences the magnitude of the shear stress required to cause slip.
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3.2 Slipping and Twinning
Example: Determine the slip system for the (111) plane in a fcc crystal and sketch the result.

机械工程材料ch04-Plastic deformation and strengthening

Mechanical Engineering Materials
(Bilingual Course)
Chapter Four
Contents
Plastic Deformation of Pure Metals Plastic Deformation of Alloys Microstructure and Properties Changes During Plastic Deformation
⑵ 晶格与晶粒发生扭曲，产生内应力及亚晶粒
金属经大量的塑性变形后，由于位错密度的增大和位错间
的交互作用，使位错分布不均匀。大量位错聚集在局部地区，
并将原晶粒分割成许多位向略有差异的小晶块，即亚晶粒。
碎晶
Microstructure change during Plastic Deformation
For a single-crystal specimen
that is stressed in tension,
deformation will be as in
Figure, where slip occurs along
a number of equivalent and
most favorably oriented planes and directions at various
1、多晶体的塑性变形实质是：处于软位向的晶粒发
生塑性变形，在此过程中，
晶向发生转动，带动周围晶粒转动。
Plastic deformation of polycrystalline metals
2.晶粒间位向差阻碍滑移
在多晶体中晶粒间有位向差，使变形不能同时进行。当一个晶粒发生塑性变形时，周围的晶粒如不发生塑性变形，则必须要产生弹性变形来与之协调配合，

A constitutive model for the anisotropic elastic-plastic deformation of paper and paperboard

A constitutive model for the anisotropicelastic–plastic deformation of paper and paperboardQingxi S.Xia,Mary C.Boyce *,David M.ParksDepartment of Mechanical Engineering,Centre of Materials Science and Technology,Massachusetts Institute of Technology,77Massachusetts Avenue,Cambridge,MA 02139-4307,USAReceived 20June 2001;received in revised form 19February 2002AbstractA three-dimensional,anisotropic constitutive model is presented to model the in-plane elastic–plastic deformation of paper and paperboard.The proposed initial yield surface is directly constructed from internal state variables comprising the yield strengths measured in various loading directions and the corresponding ratios of plastic strain components.An associated ﬂow rule is used to model the plastic ﬂow of the material.Anisotropic strain hardening of yield strengths is introduced to model the evolution in the yield surface with strain.A procedure for identifying the needed material properties is provided.The constitutive model is found to capture major features of the highly anisotropic elastic–plastic behavior of paper and paperboard.Furthermore,with material properties ﬁtted to experimental data in one set of loading directions,the model predicts the behavior of other loading states well.Ó2002Published by Elsevier Science Ltd.Keywords:Paper;Paperboard;Plasticity;Yield;Yield surface;Constitutive behavior;Anisotropic1.IntroductionPaper and paperboard are two of the most commonly utilized materials in nearly every industry.Paper is formed by draining a suspension of ﬁbers in a ﬂuid through a ﬁlter screen to form a sheet of pulp ﬁbers.Paperboard is in general composed of several pulp ﬁber sheets bonded by starch or adhesive material,and is usually a multi-layered structure.Schematics of typical paper and paperboard macrostructure and micro-structure are shown in Fig.1,which also depicts the common nomenclature for the three orthogonal di-rections of paper and paperboard.‘‘MD’’refers to the machine (rolling)direction,and ‘‘CD’’refers to the cross or transverse direction.The machine and cross directions form the plane of the structure,and ZD refers to the out-of-plane (or through-thickness)direction.Due to the continuous nature of the paper-making process,ﬁbers are primarily oriented in the plane;furthermore,within the plane,ﬁbers are more highly oriented in the MD than the CD.In this paper,to simplify notation,the 1-direction is used to represent the MD,the 2-direction for ZD and the 3-direction forCD.International Journal of Solids and Structures 39(2002)4053–/locate/ijsolstr*Corresponding author.Tel.:+1-617-253-2342;fax:+1-617-258-8742.E-mail address:mcboyce@ (M.C.Boyce).0020-7683/02/$-see front matter Ó2002Published by Elsevier Science Ltd.PII:S 0020-7683(02)00238-XThe preferential ﬁber orientation results in highly anisotropic mechanical behavior,including aniso-tropic elasticity,initial yielding,strain hardening and tensile failure strength:•Anisotropic elastic constants for paper and paperboard have been measured by several investigators (e.g.,Mann et al.,1981;Castegnade et al.,1989;Persson,1991;Koubaa and Koran,1995).Their data show that the through-thickness moduli are at least two orders of magnitude less than the in-plane mo-duli.In-plane data of Persson (1991)and Stenberg et al.(2001a)show that moduli in the MD are 2–4times greater than those of the CD.•Persson’s (1991)data on paperboard also shows that the initial yield strength ($proportional limit)in the through-thickness direction is two orders of magnitude lower than the in-plane initial yield strength values.Stenberg’s data (Stenberg et al.(2001a,b);Stenberg,2001)on multi-layer paperboard and single-layer pulp shows similar results.Within the plane,these data show that the initial yield strength of paper and paperboard in the MD is typically greater than that in the CD by a factor of 2–4.Stenberg’s data (Stenberg,2001)also show an asymmetry in the initial yield strength for in-plane tension and compres-sion in both the machine and cross directions.•The in-plane tensile stress–strain curves of Persson (1991)and Stenberg (2001)show substantial strain hardening in which the yield strength increased by more than a factor of two after a strain of less than 5%.The in-plane strain hardening is also highly anisotropic.The percentage strain hardening achieved in MD tension is greater than that obtained in CD tension.•The Persson (1991)and Stenberg (2001)data also show that the in-plane tensile failure strength in the MD is greater than that of the CD by a factor of 2–4.•Biaxial failure stress loci (Gunderson,1983;deRuvo et al.,1980;Fellers et al.,1981)show substantially diﬀerent failure strengths in machine and cross directions.These data also show that failure tends to be dominated by one or the other of these two directions when subjected to combined loading in both di-rections.Some material models have been proposed to describe the mechanical behavior of paperboard.These models fall into roughly three categories:network models,laminate models,and anisotropic models of the yield surface and/or the failure surface.Perkins and Sinha (1992)and Sinha and Perkins (1995)described a micromechanically based network model for the in-plane constitutive behavior of paper.A meso-element was constructed to represent theFig.1.Schematics of paper and paperboard macrostructure and microstructure.4054Q.S.Xia et al./International Journal of Solids and Structures 39(2002)4053–4071Q.S.Xia et al./International Journal of Solids and Structures39(2002)4053–40714055 microstructure of theﬁbrous paper network.The mechanical response of the meso-element depends on the ﬁber properties and properties of the inter-ﬁber bonds.They found the inelastic behavior of the inter-ﬁber bonds to play a crucial role in the overall in-plane inelastic behavior of paper.Stahl and Cramer(1998)also developed a network model for low densityﬁbrous work models can incorporate microlevel mechanisms,such as inter-ﬁber interaction and bonding.While these models begin to elucidate the un-derlying mechanisms of deformation,they do not provide a continuum-level description of paper or paperboard.Page and Schulgasser(1989)and Schulgasser and Page(1988)developed models of paperboard based on classical laminate theory.While this type of model can predict the elastic response well,it was not extended to capture the anisotropic yielding and subsequent strain hardening response.Gunderson(1983),Gunderson et al.(1986),deRuvo et al.(1980)and Fellers et al.(1981)each used the Tsai–Wu quadratic yield condition to model the failure loci they obtained experimentally.The quadratic nature of this type of model has many shortcomings when applied to paper and paperboard.Experimental data showed the biaxial failure locus to be distinctly non-quadratic.Arramon et al.(2000)developed a multidimensional anisotropic strength criterion based on Kelvin modes that captures the non-quadratic failure envelope.They applied the model to form a strength envelope for paperboard by constructing tensile and compressive modal bounds.However,these eﬀorts only acted to studyﬁnal failure and did not attempt to study initiation of yield or subsequent strain hardening.In this research,a general three-dimensional constitutive model of the anisotropic elastic–plastic be-havior of paper and paperboard is proposed.The initial elastic behavior is modelled to be linear and or-thotropic.The onset of plasticﬂow is captured by a non-quadratic yield surface.The yield surface is taken to evolve anisotropically with a scalar measure of plastic strain,with plasticﬂow modelled using an as-sociatedﬂow rule.The model is detailed in the following sections,and numerical results are compared to experimental data.2.Experimentally observed behavior2.1.Elastic–plastic behavior of TRIPLEX paperboardDepending onﬁber type,ﬁber density and the chemical/mechanical treatment,the elastic–plastic be-havior of diﬀerent types of paper and paperboard diﬀers in detail.However,general characteristics of the response remain similar.In this contribution,the anisotropic elastic–plastic behavior of paper and paper-board is illustrated using TRIPLEX e1paperboard as an exemplar material.TRIPLEX e is aﬁve-layer paperboard:the three-layer core is constructed from mechanically processed softwood pulp(commonly referred to as‘‘mechanical’’pulp),and two outer layers(sandwiching the core)are constructed from bleached kraft pulp(commonly referred to as‘‘chemical’’pulp).Stenberg et al.(2001a,b)and Stenberg (2001)conducted an extensive experimental investigation documenting the stress–strain behavior of TRIPLEX e.Note that the outer chemical pulp layers are typically stiﬀer and stronger than the inner mechanical layers;however,these layers cannot be separately produced for individual evaluation.There-fore,the behavior of TRIPLEX e material will be presented in terms of its eﬀective composite behavior.In principle,testing and modeling can be applied separately for each distinct paper lamina.The experimental results of Stenberg et al.(2001a,b)and Stenberg(2001)are reviewed below.1TRIPLEX e is a trademark of STORA-ENSO,Finland and Sweden.2.1.1.In-plane behaviorThe in-plane uniaxial tensile stress–strain curves for the MD,the CD and an orientation 45°from the MD are plotted together in Fig.2.These stress–strain curves clearly depict the anisotropic in-plane elastic,initial yield and strain hardening behavior.There is a factor of 2–3diﬀerence in the modulus and initial yield strength between MD and CD.Hardening achieved in MD (ﬂow strength increases by 300%over a strain of 2%)is higher than that in CD (ﬂow strength increases by 200%over a strain of 5%).MD–CD shear properties are deduced from the 45°test result.In-plane lateral strain (CD)vs.axial strain data for MD-tension and similar data for CD-tension were also recorded by Stenberg (2001),corresponding to their in-plane stress strain curves.Upon subtracting the respective elastic strain components,the lateral plastic strains for both the MD and CD tension cases are computed and shown vs.the respective axial plastic strains in Fig.3.These two curves indicate that for both test orientations,the ratio between lateral plastic strain and axial plastic strain is nearly constant until ﬁnal fracture.This data provides information for later construction of the plastic ﬂowrule.4056Q.S.Xia et al./International Journal of Solids and Structures 39(2002)4053–4071Tensile loading/unloading/reloading data (Persson,1991;Stenberg,2001)show that after various amounts of plastic strain,upon unloading,the elastic tensile modulus is nearly unaﬀected by plastic strain,consistent with traditional models of elasto-plasticity.Fig.4shows the in-plane compression stress–strain curves for the MD and CD directions.Note that global specimen buckling was constrained in these tests.These data show that compressive yield is an-isotropic.Furthermore,a comparison of Figs.2and 4shows a yield strength diﬀerence between tension and compression,with the compressive yield strengths being smaller than those in tension by 65%and 25%,for MD and CD,respectively.The anisotropic in-plane elastic–plastic properties obtained from these tests are summarized in Table 1.2.1.2.Out-of-plane behaviorStenberg et al.(2001a,b)experimentally obtained the out-of-plane stress–strain behavior of paperboard using a modiﬁed Arcan design (Arcan et al.,1978).Nominal stress–strain curves were obtained for TRI-PLEX e under various through-thickness loading conditions.Representative ZD tensile and through-thickness shear (ZD–MD)stress–strain curves obtained by Stenberg et al.(2001a,b)are shown in Fig.5.The stress measure is force per unit initial cross-sectional area;the x -ordinate is the nominal strain,deﬁned as the relative normal/shear separation of the top and bottom surfaces of the laminate,divided by the initial laminate thickness.In the tensile curve,at the earliest stage ofdeformation,the stress increases linearly with strain,exhibiting a composite modulus of E 0ZD ¼18:0MPa.The stress–strain relation shows a small amount of pre-peak non-linearity before reaching a peak stress of 0.4MPa.After the peak,the stress–strain curve exhibits pronounced softening.Features similar to those of the through-thickness tensile curve are observed for the through-thickness shear curve.Thecomposite Table 1Experimental results of uniaxial tensile tests (Stenberg,2001)Elastic modulus(GPa)Poisson’s ratio Tensile yield strength (MPa)Plastic strain ratio,d p ?=d p k Compressive yield strength (MPa)MD5.60.3712.0)0.57.3CD2.00.12 6.5)0.133 5.045° 4.18.0Q.S.Xia et al./International Journal of Solids and Structures 39(2002)4053–40714057transverse shear modulus is observed to be G 12¼34MPa,and the peak shear stress is 1.1MPa.The through-thickness ZD–CD shear stress–strain curve shows similar features.Through-thickness tensile/shear testing conducted within a scanning electron microscope (Dunn,2000)on the same material reveals the nucleation of multiple inter-laminar microcracks near the peak stress,followed by their growth and coalescence,resulting in the observed softening.Similar results have been obtained for tests involving combined through-thickness tension and shear (Dunn,2000;Stenberg et al.(2001a,b)).Therefore,the observed peak stress and subsequent softening for through-thickness loading are due to delamination of the paperboard and will not be further considered here.This inelastic through-thickness behavior is modelled in a separate related study (Xia et al.,in preparation).It is also observed that the amount of lateral (in-plane)strain generated during the through-thickness tensile loading is negligible,indicating that Poisson’s ratios m 21and m 23are close to zero.Through-thickness compression loading/unloading stress–strain curves were also obtained by Stenberg (2001).Up to a nominal compressive strain of 3%,the compressive stress increases linearly with strain.With larger strains,the stress starts to increase exponentially with strain.The data also show that only a small amount of permanent deformation is remained after unloading from a peak strain level of more than 20%,indicating non-linear elastic ZD compressive behavior up to moderately large strains.These obser-vations of through-thickness compressive behavior will be incorporated into the modeling work.The anisotropic linear elastic out-of-plane properties are summarized in Table 2.3.Constitutive modelA three-dimensional,ﬁnite deformation constitutive model for paper and paperboard pulp layers is proposed.The model will take the in-plane behavior to be elastic–plastic and the out-of-plane behavior to be elastic.Due to the assumption of elastic out-of-plane behavior,the application of the model alone will be limited to predominant in-plane loading.However,when the model is combined with interlaminardeco-Table 2Elastic out-of-plane properties (Stenberg et al.(2001a,b))Elastic modulusE 0ZD (MPa)Poisson’s ratio m 21Poisson’s ratio m 23Shear modulus G 12(MPa)Shear modulus G 23(MPa)Stiﬀening constant a 18.0)0.0055)0.003534.026.0 5.44058Q.S.Xia et al./International Journal of Solids and Structures 39(2002)4053–4071hesion models(Xia et al.,in preparation),a general-purpose tool is achieved for modeling behavior under signiﬁcant out-of-plane loading,such as occurs during converting processes and the in-service behavior ofa broad class of paper and paperboard products.3.1.Stress–strain relationshipFirst,the total deformation gradient F at a material point within a lamina is multiplicatively decom-posed into an elastic part and a plastic part:F¼F e F p;ð1Þwhere F p represents the accumulation of inelastic deformation.Although the maximum in-plane strain level in traditional applications of paper sheets is small,we adopt the presentﬁnite deformation formulation so that the model can be easily applied to applications such as paperboard converting processes,which generally involveﬁnite rotations of paperboard layers and may exhibit moderately large,but highly lo-calized in-plane strains.The evolution of F p is given by_F p¼L p F p;ð2Þwhere L p is the plastic velocity gradient,which will be deﬁned by theﬂow rule.The elastic strain is obtained by using the elastic Green strain measure:E e¼12F eT F eÂÀI Ã;ð3Þwhere I is the second-order identity tensor.The second Piola–Kirchoﬀstress measure,T,relative to the plastically deformed conﬁguration F p,is then calculated using the linear relationship: T¼C E e½ ;ð4Þwhere C is the fourth-order stiﬀness tensor,which is taken to be orthotropic.Values of the components of C are deﬁned by the orthotropic elastic moduli.To model the through-thickness non-linear elastic com-pressive stress–strain relationship,the through-thickness engineering elastic constant,E ZD,is taken to be an exponential function of the ZD strain under compression as follows:E ZD¼E0ZD expðÀaE e22Þ;ð5Þwhere E0ZD is the initial elastic modulus in ZD,E e22is the ZD elastic Green strain component,and a is aconstant determined byﬁtting the compressive through-thickness stress–strain curve;its value is listed in Table2.The stiﬀness tensor C under ZD compression is in turn determined assuming constant Poisson’s ratios.The Cauchy stress,T,is calculated from its relation to the second Piola–Kirchoﬀstress by T¼ðdet F eÞF eÀ1TF eÀT:ð6Þ3.2.Yield conditionThe through-thickness strengths(tensile,compressive,shear)of paper and paperboard materials are typically two orders of magnitude lower than those observed in the plane.Therefore,the through-thickness stress components play little role in the inelastic deformation and failure of paperboard under in-plane loading.2Furthermore,from investigation of the mechanisms of deformation and failure of paperboard2Under signiﬁcant in-plane compression or very large through-thickness compressive strain,this may not be strictly true.However, for the present,these scenarios will not be considered further.Q.S.Xia et al./International Journal of Solids and Structures39(2002)4053–40714059under through-thickness loading (when through-thickness compression is not dominant),it is clear that the majority of through-thickness inelastic deformation occurs in the form of interlaminar microcracking and delamination of the discrete pulp layers,as opposed to inelastic through-thickness deformation distributed quasi-homogeneously within laminae.Thus we can assume that only the in-plane stress components will drive the in-plane inelastic deformation of the pulp layers.Additionally,in classical metal plasticity and in plasticity-based models of yielding in polymers,the deviatoric stress is taken to drive yield because the underlying deformation mechanisms are governed by shear (for example,dislocation glide in crystalline metals).However,in the case of paper,there is no evidence that yield and subsequent plastic ﬂow are driven by deviatoric stress.Micromechanically,yielding is governed by various forms of inter-ﬁber interactions.Based on these considerations,the total stress will be taken to drive the yield condition described in this research.In order to experimentally deﬁne an in-plane yield surface for paper,multi-axial data is required.Al-though the anisotropy of the yield surface is well-recognized,due to numerous studies of the uniaxial behavior in diﬀerent directions,such as that reported in Fig.2,a literature search reveals virtually no data on the initial and evolving multi-axial yield surfaces of paper and paperboard.However,several researchers have obtained biaxial failure surfaces of paperboard under combinations of MD and CD normal stress.Fig.6shows a representative biaxial failure locus (deRuvo et al.,1980).Similar data have been reported in work by Fellers et al.(1981)and Gunderson (1983).One common feature observed in these failure loci is that the data points tend to form prominent sub-groups,with each sub-group lying on a nearly straight line.For example,for low,but non-zero values of MD stress,failure occurs at nearly the CD uniaxial tensile failure strength;similarly,for low,but non-zero values of CD stress,failure occurs at roughly the MD uniaxial tensile failure strength.This observation suggests that the experimental biaxial failure locus can be well captured by a set of straight lines in two-dimensional biaxial stress space and can be generalized to planes in three-dimensional space.Karaﬁllis and Boyce (1993)and Arramon et al.(2000)developed yield surface and failure surface models,respectively,which capture this non-quadratic feature.In stress space,these lines or planes can be deﬁned by their minimum distance to the origin,together with their corre-sponding normal directions.Given that a comprehensive set of experimental data is generally unavailable (and indeed,is a challenging task to obtain)to determine the full surface,we hereby assume that the yield surface exhibits the same characteristic features observed in the failure surface.Therefore,the yield surface is taken to be constructed of N sub-surfaces,where N K is the normal to the K th such surface,deﬁned in the material coordinates formed by MD,CD and ZD.S K is the equivalent strength of the K th sub-surface,deﬁned by the distance from the origin to each sub-surface,following its normal direction.Thus,the following form of yield criterion isproposed:Fig.6.Biaxial failure surfaces (deRuvo et al.,1980).4060Q.S.Xia et al./International Journal of Solids and Structures 39(2002)4053–4071fðT;S KÞ¼X NK¼1vKTÁN KS K!2kÀ1;ð7Þwhere v K is the switching controller withv K ¼1;if TÁNi>0;0;otherwise;&ð8ÞT is the second Piola–Kirchoﬀstress measure relative to the F p conﬁguration,and2k is an even integer.N K is the normal of the K th sub-surface,deﬁned relative to the material coordinates.A schematic of a four sub-surface system for biaxial loading,with zero in-plane shear stress,is shown in Fig.7.The normals and corresponding sub-surface strengths are illustrated.The parameter2k is taken to be equal to or larger than4,indicating a non-quadratic yield surface.Fig.8shows the eﬀect of diﬀerent2k values in controlling the shape of the yield surface in the biaxial stress plane for this simpliﬁed four sub-surface system.Higher2k-values give rise to sharper corners between adjacent sub-surfaces and reduce the curvature over increasing central portions of each sub-surface.A schematic of a six sub-surface yield surface,with non-zero in-plane shear stress T13,is shown in Fig.9.Thisﬁgure graphically illustratestheFig.8.Eﬀect of constant2k on the shape of yield surface.Q.S.Xia et al./International Journal of Solids and Structures39(2002)4053–40714061normals and corresponding equivalent strengths of the sub-surfaces.For this yield surface,the six normals are taken to be of the following form:N K¼X3i;j¼1N Kije i e j;K¼I;...;VIð9ÞandN K ji ¼N Kij;K¼I;...;VI;ð10Þwhere e i are the basis vectors for the material coordinates formed by the MD,CD and ZD.Here,the sub-surface normal index K ranges over the six Roman numerals I–VI.Because out-of-plane stress components are assumed to have no eﬀect on the plastic deformation within a lamina,components of N K involving the2-direction(i.e.,the ZD direction)are set to be zero.This results in elastic through-thickness behavior,as proposed.Determination of each non-zero entry of these matrices will be discussed in Section3.3.3.3.Flow ruleThe plasticﬂow rule is deﬁned asL p¼D p¼_ c K;ð11Þwhere L p is the plasticﬂow rate and D p is the symmetric part of L p.For paper and paperboard,the in-plane plastic strains(even at failure)are small;therefore we take the skew part of L p to vanish,or W p¼0,as a simpliﬁcation.K is theﬂow direction,and_ c is the magnitude of the plastic stretching rate.K is a second-order tensor with unit magnitude:K¼b K=k b K k;ð12Þwherek b K k¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃb KÁb Kp¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃX3i;j¼1b K ij b K ijv uu t:ð13ÞFig.9.Schematic of yield surface for general in-plane loading,showing six sub-surfaces.4062Q.S.Xia et al./International Journal of Solids and Structures39(2002)4053–4071In Fig.3,the in-plane lateral plastic strain vs.axial plastic strain data showed that the ratio between these two plastic strain components is nearly constant for both the MD and CD simple tension cases.These ratios are taken to deﬁne the normal directions of the two respective sub-surfaces of the tensile quadrant of the biaxial yield surface,in the absence of shear stress.Thus,the plasticﬂow of the material is taken to follow an associatedﬂow rule:b K¼ð14ÞWith the yield condition deﬁned in Eq.(7),b K can be further calculated asb K¼o T ¼X NK¼12k K2kÀ1KvKN KS K;ð15ÞwhereK K¼TÁN KS K:ð16ÞFor the six sub-surface yield surface shown in Fig.9,assuming an associatedﬂow rule,the sub-surface normals N K,K¼I;...;VI are deﬁned using the corresponding plastic strain ratios.For example,for sub-surface I(one)in Fig.9,the plastic strain ratio from the MD simple tension test is nearly constant at)0.5. The two non-zero components of N I are then determined by solving the following two equations: N I3311¼À0:5ð17Þand(to make a unit normal)ðN I11Þ2þðN I33Þ2¼1;ð18Þwhich gives N I11¼2=ﬃﬃﬃ5pand N I33¼À1=ﬃﬃﬃ5p.Similarly,the plastic strain ratio from the CD simple tension testis nearly constant atÀ2=15,giving N II11¼À2=ﬃﬃﬃﬃﬃﬃﬃﬃ229pand N II33¼15=ﬃﬃﬃﬃﬃﬃﬃﬃ229p.With appropriate experimentaldata,similar calculations can determine the normals of each of the sub-surfaces.However,currently,there is no experimental data for the plastic strain ratios for compression in either the machine or cross directions. For the four sub-surface biaxial yield surface shown in Fig.7,the normals for the two sub-surfaces in the compressive quadrants,IV and V,are assumed to have normal directions parallel to those of corresponding tensile sub-surfaces,I and II,respectively,as seen in theﬁgure,but with generally diﬀering strength levels. For the normal of the sub-surface representing yielding under positive pure shear stress(T13¼T31¼0;T ij¼0otherwise),the two non-zero components are N III13and N III31.Due to the symmetry of shear stress,N III 13¼N III31ð19ÞandðN III13Þ2þðN III31Þ2¼1:ð20ÞThus we have N III13¼N III31¼ﬃﬃﬃ2p=2,and the normal of the other sub-surface representing yielding undernegative pure shear stresses is taken to be N VI¼ÀN III.In summary,the components of the normals for the six sub-surfaces determined in the material coordinates are given in Table3.3.4.Strain hardening functionsTo capture the anisotropic strain hardening observed for the in-plane behavior,the equivalent strengths, S K,of each sub-surface are taken to evolve with the accumulated equivalent plastic strain,i.e.。

part 2 Plastic deformation and Strain hardening

钨单晶受到不同方向外部切应力作用时滑移方向的变化。
Slip of low-carbon steel (polycrystalline )
低碳钢（多晶）的滑移
Slip
• Slip is the displacement of a portion of a crystal relative to another portion with the crystal structure of both portions remaining unchanged. 滑移是晶体的一部分移动到另一部分的位置，并且在移动过程两部分晶体的结构保持不变。
Slip of metal crystals a –Zn, b –Cd, c-Sn, d-Bi
金属晶体的滑移
Variation of slip orientation in deformed tungsten single crystal at a different
direction of external shear stress
Carbon steel in the elastic region
碳素钢在弹性区的变化
Linear elasticity and subsequent plasticity
比例弹性变形和紧接的塑性变形
Unstable creep in annealed copper
铜退火后的蠕变
Plastic deformation塑性变形
• The deformation which is independent of time and is retained upon stress release is called plastic deformation. 塑性变形是不随时间而变化，应力去除后仍然保留的形变。

A plasticity and anisotropic damage model for plain concrete

A plasticity and anisotropic damage model forplain concreteUmit Cicekli,George Z.Voyiadjis *,Rashid K.Abu Al-RubDepartment of Civil and Environmental Engineering,Louisiana State University,CEBA 3508-B,Baton Rouge,LA 70803,USAReceived 23April 2006;received in ﬁnal revised form 29October 2006Available online 15March 2007AbstractA plastic-damage constitutive model for plain concrete is developed in this work.Anisotropic damage with a plasticity yield criterion and a damage criterion are introduced to be able to ade-quately describe the plastic and damage behavior of concrete.Moreover,in order to account for dif-ferent eﬀects under tensile and compressive loadings,two damage criteria are used:one for compression and a second for tension such that the total stress is decomposed into tensile and com-pressive components.Stiﬀness recovery caused by crack opening/closing is also incorporated.The strain equivalence hypothesis is used in deriving the constitutive equations such that the strains in the eﬀective (undamaged)and damaged conﬁgurations are set equal.This leads to a decoupled algo-rithm for the eﬀective stress computation and the damage evolution.It is also shown that the pro-posed constitutive relations comply with the laws of thermodynamics.A detailed numerical algorithm is coded using the user subroutine UMAT and then implemented in the advanced ﬁnite element program ABAQUS.The numerical simulations are shown for uniaxial and biaxial tension and compression.The results show very good correlation with the experimental data.Ó2007Elsevier Ltd.All rights reserved.Keywords:Damage mechanics;Isotropic hardening;Anisotropic damage0749-6419/$-see front matter Ó2007Elsevier Ltd.All rights reserved.doi:10.1016/j.ijplas.2007.03.006*Corresponding author.Tel.:+12255788668;fax:+12255789176.E-mail addresses:voyiadjis@ (G.Z.Voyiadjis),rabual1@ (R.K.AbuAl-Rub).International Journal of Plasticity 23(2007)1874–1900U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001875 1.IntroductionConcrete is a widely used material in numerous civil engineering structures.Due to its ability to be cast on site it allows to be used in diﬀerent shapes in structures:arc,ellipsoid, etc.This increases the demand for use of concrete in structures.Therefore,it is crucial to understand the mechanical behavior of concrete under diﬀerent loadings such as compres-sion and tension,for uniaxial,biaxial,and triaxial loadings.Moreover,challenges in designing complex concrete structures have prompted the structural engineer to acquire a sound understanding of the mechanical behavior of concrete.One of the most important characteristics of concrete is its low tensile strength,particularly at low-conﬁning pres-sures,which results in tensile cracking at a very low stress compared with compressive stresses.The tensile cracking reduces the stiﬀness of concrete structural components. Therefore,the use of continuum damage mechanics is necessary to accurately model the degradation in the mechanical properties of concrete.However,the concrete material undergoes also some irreversible(plastic)deformations during unloading such that the continuum damage theories cannot be used alone,particularly at high-conﬁning pressures. Therefore,the nonlinear material behavior of concrete can be attributed to two distinct material mechanical processes:damage(micro-cracks,micro-cavities,nucleation and coa-lescence,decohesions,grain boundary cracks,and cleavage in regions of high stress con-centration)and plasticity,which its mechanism in concrete is not completely understood up-to-date.These two degradation phenomena may be described best by theories of con-tinuum damage mechanics and plasticity.Therefore,a model that accounts for both plas-ticity and damage is necessary.In this work,a coupled plastic-damage model is thus formulated.Plasticity theories have been used successfully in modeling the behavior of metals where the dominant mode of internal rearrangement is the slip process.Although the mathemat-ical theory of plasticity is thoroughly established,its potential usefulness for representing a wide variety of material behavior has not been yet fully explored.There are many research-ers who have used plasticity alone to characterize the concrete behavior(e.g.Chen and Chen,1975;William and Warnke,1975;Bazant,1978;Dragon and Mroz,1979;Schreyer, 1983;Chen and Buyukozturk,1985;Onate et al.,1988;Voyiadjis and Abu-Lebdeh,1994; Karabinis and Kiousis,1994;Este and Willam,1994;Menetrey and Willam,1995;Grassl et al.,2002).The main characteristic of these models is a plasticity yield surface that includes pressure sensitivity,path sensitivity,non-associativeﬂow rule,and work or strain hardening.However,these works failed to address the degradation of the material stiﬀness due to micro-cracking.On the other hand,others have used the continuum damage theory alone to model the material nonlinear behavior such that the mechanical eﬀect of the pro-gressive micro-cracking and strain softening are represented by a set of internal state vari-ables which act on the elastic behavior(i.e.decrease of the stiﬀness)at the macroscopic level (e.g.Loland,1980;Ortiz and Popov,1982;Krajcinovic,1983,1985;Resende and Martin, 1984;Simo and Ju,1987a,b;Mazars and Pijaudier-Cabot,1989;Lubarda et al.,1994). However,there are several facets of concrete behavior(e.g.irreversible deformations, inelastic volumetric expansion in compression,and crack opening/closure eﬀects)that can-not be represented by this method,just as plasticity,by itself,is insuﬃcient.Since both micro-cracking and irreversible deformations are contributing to the nonlinear response of concrete,a constitutive model should address equally the two physically distinct modes of irreversible changes and should satisfy the basic postulates of thermodynamics.1876U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900 Combinations of plasticity and damage are usually based on isotropic hardening com-bined with either isotropic(scalar)or anisotropic(tensor)damage.Isotropic damage is widely used due to its simplicity such that diﬀerent types of combinations with plasticity models have been proposed in the literature.One type of combination relies on stress-based plasticity formulated in the eﬀective(undamaged)space(e.g.Yazdani and Schreyer,1990; Lee and Fenves,1998;Gatuingt and Pijaudier-Cabot,2002;Jason et al.,2004;Wu et al., 2006),where the eﬀective stress is deﬁned as the average micro-scale stress acting on the undamaged material between micro-defects.Another type is based on stress-based plastic-ity in the nominal(damaged)stress space(e.g.Bazant and Kim,1979;Ortiz,1985;Lubliner et al.,1989;Imran and Pantazopoulu,2001;Ananiev and Ozbolt,2004;Kratzig and Poll-ing,2004;Menzel et al.,2005;Bru¨nig and Ricci,2005),where the nominal stress is deﬁned as the macro-scale stress acting on both damaged and undamaged material.However,it is shown by Abu Al-Rub and Voyiadjis(2004)and Voyiadjis et al.(2003,2004)that coupled plastic-damage models formulated in the eﬀective space are numerically more stable and attractive.On the other hand,for better characterization of the concrete damage behavior, anisotropic damage eﬀects,i.e.diﬀerent micro-cracking in diﬀerent directions,should be characterized.However,anisotropic damage in concrete is complex and a combination with plasticity and the application to structural analysis is straightforward(e.g.Yazdani and Schreyer,1990;Abu-Lebdeh and Voyiadjis,1993;Voyiadjis and Kattan,1999;Carol et al.,2001;Hansen et al.,2001),and,therefore,it has been avoided by many authors.Consequently,with inspiration from all the previous works,a coupled anisotropic dam-age and plasticity constitutive model that can be used to predict the concrete distinct behavior in tension and compression is formulated here within the basic principles of ther-modynamics.The proposed model includes important aspects of the concrete nonlinear behavior.The model considers diﬀerent responses of concrete under tension and compres-sion,the eﬀect of stiﬀness degradation,and the stiﬀness recovery due to crack closure dur-ing cyclic loading.The yield criterion that has been proposed by Lubliner et al.(1989)and later modiﬁed by Lee and Fenves(1998)is adopted.Pertinent computational aspects con-cerning the algorithmic aspects and numerical implementation of the proposed constitu-tive model in the well-knownﬁnite element code ABAQUS(2003)are presented.Some numerical applications of the model to experimental tests of concrete specimens under dif-ferent uniaxial and biaxial tension and compression loadings are provided to validate and demonstrate the capability of the proposed model.2.Modeling anisotropic damage in concreteIn the current literature,damage in materials can be represented in many forms such as speciﬁc void and crack surfaces,speciﬁc crack and void volumes,the spacing between cracks or voids,scalar representation of damage,and general tensorial representation of damage.Generally,the physical interpretation of the damage variable is introduced as the speciﬁc damaged surface area(Kachonov,1958),where two cases are considered:iso-tropic(scalar)damage and anisotropic(tensor)damage density of micro-cracks and micro-voids.However,for accurate interpretation of damage in concrete,one should con-sider the anisotropic damage case.This is attributed to the evolution of micro-cracks in concrete whereas damage in metals can be satisfactorily represented by a scalar damage variable(isotropic damage)for evolution of voids.Therefore,for more reliable represen-tation of concrete damage anisotropic damage is considered in this study.The eﬀective(undamaged)conﬁguration is used in this study in formulating the damage constitutive equations.That is,the damaged material is modeled using the constitutive laws of the eﬀective undamaged material in which the Cauchy stress tensor,r ij,can be replaced by the eﬀective stress tensor, r ij(Cordebois and Sidoroﬀ,1979;Murakami and Ohno,1981;Voyiadjis and Kattan,1999):r ij¼M ijkl r klð1Þwhere M ijkl is the fourth-order damage eﬀect tensor that is used to make the stress tensor symmetrical.There are diﬀerent deﬁnitions for the tensor M ijkl that could be used to sym-metrize r ij(see Voyiadjis and Park,1997;Voyiadjis and Kattan,1999).In this work the deﬁnition that is presented by Abu Al-Rub and Voyiadjis(2003)is adopted:M ijkl¼2½ðd ijÀu ijÞd klþd ijðd klÀu klÞ À1ð2Þwhere d ij is the Kronecker delta and u ij is the second-order damage tensor whose evolution will be deﬁned later and it takes into consideration diﬀerent evolution of damage in diﬀer-ent directions.In the subsequence of this paper,the superimposed dash designates a var-iable in the undamaged conﬁguration.The transformation from the eﬀective(undamaged)conﬁguration to the damaged one can be done by utilizing either the strain equivalence or strain energy equivalence hypoth-eses(see Voyiadjis and Kattan,1999).However,in this work the strain equivalence hypothesis is adopted for simplicity,which basically states that the strains in the damaged conﬁguration and the strains in the undamaged(eﬀective)conﬁguration are equal.There-fore,the total strain tensor e ij is set equal to the corresponding eﬀective tensor e ij(i.e.e ij¼ e ijÞ,which can be decomposed into an elastic strain e eij (= e eijÞand a plastic straine p ij(= e p ijÞsuch that:e ij¼e eij þe p ij¼ e eijþ e p ij¼ e ijð3ÞIt is noteworthy that the physical nature of plastic(irreversible)deformations in con-crete is not well-founded until now.Whereas the physical nature of plastic strain in metals is well-understood and can be attributed to the generation and motion of dislocations along slip planes.Therefore,in metals any additional permanent strains due to micro-cracking and void growth can be classiﬁed as a damage strain.These damage strains are shown by Abu Al-Rub and Voyiadjis(2003)and Voyiadjis et al.(2003,2004)to be minimal in metals and can be simply neglected.Therefore,the plastic strain in Eq.(3) incorporates all types of irreversible deformations whether they are due to tensile micro-cracking,breaking of internal bonds during shear loading,and/or compressive con-solidation during the collapse of the micro-porous structure of the cement matrix.In the current work,it is assumed that plasticity is due to damage evolution such that damage occurs before any plastic deformations.However,this assumption needs to be validated by conducting microscopic experimental characterization of concrete damage.Using the generalized Hook’s law,the eﬀective stress is given as follows: r ij¼E ijkl e eklð4Þwhere E ijkl is the fourth-order undamaged elastic stiﬀness tensor.For isotropic linear-elas-tic materials,E ijkl is given byE ijkl¼2GI dijkl þKI ijklð5ÞU.Cicekli et al./International Journal of Plasticity23(2007)1874–19001877where I dijkl ¼I ijklÀ13d ij d kl is the deviatoric part of the fourth-order identity tensorI ijkl¼12ðd ik d jlþd il d jkÞ,and G¼E=2ð1þmÞand K¼E=3ð1À2mÞare the eﬀective shearand bulk moduli,respectively,with E being the Young’s modulus and m is the Poisson’s ratio which are obtained from the stress–strain diagram in the eﬀective conﬁguration.Similarly,in the damaged conﬁguration the stress–strain relationship in Eq.(4)can be expressed by:r ij¼E ijkl e eklð6Þsuch that one can express the elastic strain from Eqs.(4)and(5)by the following relation:e e ij ¼EÀ1ijklr kl¼EÀ1ijklr klð7Þwhere EÀ1ijkl is the inverse(or compliance tensor)of the fourth-order damaged elastic tensorE ijkl,which are a function of the damage variable u ij.By substituting Eq.(1)into Eq.(7),one can express the damaged elasticity tensor E ijkl in terms of the corresponding undamaged elasticity tensor E ijkl by the following relation:E ijkl¼MÀ1ijmnE mnklð8ÞMoreover,combining Eqs.(3)and(7),the total strain e ij can be written in the following form:e ij¼EÀ1ijkl r klþe p ij¼EÀ1ijklr klþe p ijð9ÞBy taking the time derivative of Eq.(3),the rate of the total strain,_e ij,can be written as _e ij¼_e eijþ_e p ijð10Þwhere_e eij and_e p ij are the rate of the elastic and plastic strain tensors,respectively.Analogous to Eq.(9),one can write the following relation in the eﬀective conﬁguration:_e ij¼EÀ1ijkl _ rklþ_e p ijð11ÞHowever,since E ijkl is a function of u ij,a similar relation as Eq.(11)cannot be used. Therefore,by taking the time derivative of Eq.(9),one can write_e ij in the damaged con-ﬁguration as follows:_e ij¼EÀ1ijkl _r klþ_EÀ1ijklr klþ_e p ijð12ÞConcrete has distinct behavior in tension and compression.Therefore,in order to ade-quately characterize the damage in concrete due to tensile,compressive,and/or cyclic loadings the Cauchy stress tensor(nominal or eﬀective)is decomposed into a positive and negative parts using the spectral decomposition technique(e.g.Simo and Ju, 1987a,b;Krajcinovic,1996).Hereafter,the superscripts‘‘+”and‘‘À”designate,respec-tively,tensile and compressive entities.Therefore,r ij and r ij can be decomposed as follows:r ij¼rþij þrÀij; r ij¼ rþijþ rÀijð13Þwhere rþij is the tension part and rÀijis the compression part of the stress state.The stress tensors rþij and rÀijcan be related to r ij byrþkl ¼Pþklpqr pqð14ÞrÀkl ¼½I klpqÀPþijpqr pq¼PÀklpqr pqð15Þ1878U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900such that Pþijkl þPÀijkl¼I ijkl.The fourth-order projection tensors Pþijkland PÀijklare deﬁnedas follows:Pþijpq ¼X3k¼1Hð^rðkÞÞnðkÞi nðkÞj nðkÞpnðkÞq;PÀklpq¼I klpqÀPþijpqð16Þwhere Hð^ rðkÞÞdenotes the Heaviside step function computed at k th principal stress^rðkÞof r ij and nðkÞi is the k th corresponding unit principal direction.In the subsequent develop-ment,the superscript hat designates a principal value.Based on the decomposition in Eq.(13),one can assume that the expression in Eq.(1) to be valid for both tension and compression,however,with decoupled damage evolution in tension and compression such that:rþij ¼Mþijklrþkl; rÀij¼MÀijklrÀklð17Þwhere Mþijkl is the tensile damage eﬀect tensor and MÀijklis the corresponding compressivedamage eﬀect tensor which can be expressed using Eq.(2)in a decoupled form as a func-tion of the tensile and compressive damage variables,uþij and uÀij,respectively,as follows:Mþijkl ¼2½ðd ijÀuþijÞd klþd ijðd klÀuþklÞ À1;MÀijkl¼2½ðd ijÀuÀijÞd klþd ijðd klÀuÀklÞ À1ð18ÞNow,by substituting Eq.(17)into Eq.(13)2,one can express the eﬀective stress tensor as the decomposition of the fourth-order damage eﬀect tensor for tension and compression such that:r ij¼Mþijkl rþklþMÀijklrÀklð19ÞBy substituting Eqs.(14)and(15)into Eq.(19)and comparing the result with Eq.(1), one can obtain the following relation for the damage eﬀect tensor such that:M ijpq¼Mþijkl PþklpqþMÀijklPÀklpqð20ÞUsing Eq.(16)2,the above equation can be rewritten as follows:M ijpq¼Mþijkl ÀMÀijklPþklpq þMÀijpqð21ÞOne should notice the following:M ijkl¼Mþijkl þMÀijklð22Þoru ij¼uþij þuÀijð23ÞIt is also noteworthy that the relation in Eq.(21)enhances a coupling between tensileand compressive damage through the fourth-order projection tensor Pþijkl .Moreover,forisotropic damage,Eq.(20)can be written as follows:M ijkl¼Pþijkl1ÀuþþPÀijkl1ÀuÀð24ÞIt can be concluded from the above expression that by adopting the decomposition of the scalar damage variable u into a positive u+part and a negative uÀpart still enhances adamage anisotropy through the spectral decomposition tensors Pþijkl and PÀijkl.However,this anisotropy is weak as compared to the anisotropic damage eﬀect tensor presented in Eq.(21).U.Cicekli et al./International Journal of Plasticity23(2007)1874–190018793.Elasto-plastic-damage modelIn this section,the concrete plasticity yield criterion of Lubliner et al.(1989)which was later modiﬁed by Lee and Fenves(1998)is adopted for both monotonic and cyclic load-ings.The phenomenological concrete model of Lubliner et al.(1989)and Lee and Fenves (1998)is formulated based on isotropic(scalar)stiﬀness degradation.Moreover,this model adopts one loading surface that couples plasticity to isotropic damage through the eﬀective plastic strain.However,in this work the model of Lee and Fenves(1998)is extended for anisotropic damage and by adopting three loading surfaces:one for plastic-ity,one for tensile damage,and one for compressive damage.The plasticity and the com-pressive damage loading surfaces are more dominate in case of shear loading and compressive crushing(i.e.modes II and III cracking)whereas the tensile damage loading surface is dominant in case of mode I cracking.The presentation in the following sections can be used for either isotropic or anisotropic damage since the second-order damage tensor u ij degenerates to the scalar damage vari-able in case of uniaxial loading.3.1.Uniaxial loadingIn the uniaxial loading,the elastic stiﬀness degradation variables are assumed asincreasing functions of the equivalent plastic strains eþeq and eÀeqwith eþeqbeing the tensileequivalent plastic strain and eÀeq being the compressive equivalent plastic strain.It shouldbe noted that the material behavior is controlled by both plasticity and damage so that, one cannot be considered without the other(see Fig.1).For uniaxial tensile and compressive loading, rþij and rÀijare given as(Lee and Fenves,1998)rþ¼ð1ÀuþÞE eþe¼ð1ÀuþÞEðeþÀeþpÞð25ÞrÀ¼ð1ÀuÀÞE eÀe¼ð1ÀuÀÞEðeÀÀeÀpÞð26ÞThe rate of the equivalent(eﬀective)plastic strains in compression and tension,eÀep and eþep,are,respectively,given as follows in case of uniaxial loading:1880U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900_eþeq ¼_e p11;_eÀeq¼À_e p11ð27Þsuch thateÀeq ¼Z t_eÀeqd t;eþeq¼Z t_eþeqd tð28ÞPropagation of cracks under uniaxial loading is in the transverse direction to the stress direction.Therefore,the nucleation and propagation of cracks cause a reduction of the capacity of the load-carrying area,which causes an increase in the eﬀective stress.This has little eﬀect during compressive loading since cracks run parallel to the loading direc-tion.However,under a large compressive stress which causes crushing of the material,the eﬀective load-carrying area is also considerably reduced.This explains the distinct behav-ior of concrete in tension and compression as shown in Fig.2.It can be noted from Fig.2that during unloading from any point on the strain soften-ing path(i.e.post peak behavior)of the stress–strain curve,the material response seems to be weakened since the elastic stiﬀness of the material is degraded due to damage evolution. Furthermore,it can be noticed from Fig.2a and b that the degradation of the elastic stiﬀ-ness of the material is much diﬀerent in tension than in compression,which is more obvi-ous as the plastic strain increases.Therefore,for uniaxial loading,the damage variable can be presented by two independent damage variables u+and uÀ.Moreover,it can be noted that for tensile loading,damage and plasticity are initiated when the equivalent appliedstress reaches the uniaxial tensile strength fþ0as shown in Fig.2a whereas under compres-sive loading,damage is initiated earlier than plasticity.Once the equivalent applied stressreaches fÀ0(i.e.when nonlinear behavior starts)damage is initiated,whereas plasticityoccurs once fÀu is reached.Therefore,generally fþ¼fþufor tensile loading,but this isnot true for compressive loading(i.e.fÀ0¼fÀuÞ.However,one may obtain fÀ%fÀuin caseof ultra-high strength concrete.3.2.Multiaxial loadingThe evolution equations for the hardening variables are extended now to multiaxial loadings.The eﬀective plastic strain for multiaxial loading is given as follows(Lubliner et al.,1989;Lee and Fenves,1998):U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001881_e þeq ¼r ð^ r ij Þ^_e p maxð29Þ_e Àeq ¼Àð1Àr ð^ r ij ÞÞ^_e p min ð30Þwhere ^_e p max and ^_e p min are the maximum and minimum principal values of the plastic strain tensor _e p ij such that ^_e p 1>^_e p 2>^_e p 3where ^_e p max ¼^_e p 1and ^_e p min ¼^_ep 3.Eqs.(29)and (30)can be written in tensor format as follows:_j p i ¼H ij ^_e p jð31Þor equivalently _e þeq 0_e Àeq8><>:9>=>;¼H þ0000000H À264375^_e p 1^_e p 2^_e p 38><>:9>=>;ð32ÞwhereH þ¼r ð^ rij Þð33ÞH À¼Àð1Àr ð^ r ij ÞÞð34ÞThe dimensionless parameter r ð^ rij Þis a weight factor depending on principal stresses and is deﬁned as follows (Lubliner et al.,1989):r ð^ r ij Þ¼P 3k¼1h ^ r k i P k ¼1j ^ r kj ð35Þwhere h i is the Macauley bracket,and presented as h x i ¼1ðj x j þx Þ,k ¼1;2;3.Note that r ð^ rij Þ¼r ð^r ij Þ.Moreover,depending on the value of r ð^r ij Þ,–in case of uniaxial tension ^ r k P 0and r ð^ r ij Þ¼1,–in case of uniaxial compression ^ rk 60and r ð^ r ij Þ¼03.3.Cyclic loadingIt is more diﬃcult to address the concrete damage behavior under cyclic loading;i.e.transition from tension to compression or vise versa such that one would expect that under cyclic loading crack opening and closure may occur and,therefore,it is a challenging task to address such situations especially for anisotropic damage evolution.Experimentally,it is shown that under cyclic loading the material goes through some recovery of the elastic stiﬀness as the load changes sign during the loading process.This eﬀect becomes more sig-niﬁcant particularly when the load changes sign during the transition from tension to com-pression such that some tensile cracks tend to close and as a result elastic stiﬀness recovery occurs during compressive loading.However,in case of transition from compression to tension one may thus expect that smaller stiﬀness recovery or even no recovery at all may occur.This could be attributed to the fast opening of the pre-existing cracks that had formed during the previous tensile loading.These re-opened cracks along with the new cracks formed during the compression will cause further reduction of the elastic stiﬀness that the body had during the ﬁrst transition from tension to compression.The1882U.Cicekli et al./International Journal of Plasticity 23(2007)1874–1900consideration of stiﬀness recovery eﬀect due to crack opening/closing is therefore impor-tant in deﬁning the concrete behavior under cyclic loading.Eq.(21)does not incorporate the elastic stiﬀness recovery phenomenon as well as it does not incorporate any coupling between tensile damage and compressive damage and,therefore,the formulation of Lee and Fenves(1998)for cyclic loading is extended here for the anisotropic damage case.Lee and Fenves(1998)deﬁned the following isotropic damage relation that couples both tension and compression eﬀects as well as the elastic stiﬀness recovery during transi-tion from tension to compression loading such that:u¼1Àð1Às uþÞð1ÀuÀÞð36Þwhere sð06s61Þis a function of stress state and is deﬁned as follows: sð^ r ijÞ¼s0þð1Às0Þrð^ r ijÞð37Þwhere06s061is a constant.Any value between zero and one results in partial recovery of the elastic stiﬀness.Based on Eqs.(36)and(37):(a)when all principal stresses are positive then r=1and s=1such that Eq.(36)becomesu¼1Àð1ÀuþÞð1ÀuÀÞð38Þwhich implies no stiﬀness recovery during the transition from compression to tension since s is absent.(b)when all principal stresses are negative then r=0and s¼s0such that Eq.(36)becomesu¼1Àð1Às0uþÞð1ÀuÀÞð39Þwhich implies full elastic stiﬀness recovery when s0¼0and no recovery when s0¼1.In the following two approaches are proposed for extending the Lee and Fenves(1998)model to the anisotropic damage case.Theﬁrst approach is by multiplying uþij in Eq.(18)1by the stiﬀness recovery factor s:Mþijkl ¼2½ðd ijÀs uþijÞd klþd ijðd klÀs uþklÞ À1ð40Þsuch that the above expression replaces Mþijkl in Eq.(21)to give the total damage eﬀecttensor.Another approach to enhance coupling between tensile damage and compressive dam-age as well as in order to incorporate the elastic stiﬀness recovery during cyclic loading for the anisotropic damage case is by rewriting Eq.(36)in a tensor format as follows:u ij¼d ijÀðd ikÀs uþik Þðd jkÀuÀjkÞð41Þwhich can be substituted back into Eq.(2)to get theﬁnal form of the damage eﬀect tensor, which is shown next.It is noteworthy that in case of full elastic stiﬀness recovery(i.e.s=0),Eq.(41)reducesto u ij¼uÀij and in case of no stiﬀness recovery(i.e.s=1),Eq.(41)takes the form ofu ij ¼uÀijþuþikÀuþikuÀjksuch that both uþijand uÀijare coupled.This means that duringthe transition from tension to compression some cracks are closed or partially closed which could result in partial recovery of the material stiﬀness(i.e.s>0)in the absence U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001883。

Mechanics_of_materials-2014

U
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Shape of the potential defines: • binding energy U = Φ(∞) - Φ(r0) • equilibrium distance Atomic bonds • elastic properties (curvature at ionic equilibrium distance) covalent • heat expansion metallic • max. strength polar, Van-der-Waals, H-bonding
2. Plastic deformation
• Theory of dislocations • Deformation of polycrystals • Deformation of polymers
3. Hardening mechanisms
• • • • • • General theory of hardening Work hardening Solid solution hardening Precipitation hardening Grain boundary strengthening Martensitic hardening of steels
Advanced Chapters in Mechanics of Materials
Prof. Dr. Alexander Hartmaier Email: Alexander.Hartmaier@rub.de Micromechanical and Macroscopic Modelling Interdisciplinary Centre for Advanced Materials Simulation
R = n1 a1 + n2 a 2 + n3 a 3

材料科学基础第八章

The combination of a crystallographic plane and, within that plane, a crystallographic direction along which slip (i.e., dislocation motion) occurs.
SLIP SYSTEMS = slip plane+ slip direction
This discrepancy in mechanical strengths could be explained by a type of linear crystalline defect
2020/5/29
Chapter 8 Deformation and Strengthening
4
mechanism
增殖和消亡等现象。至此位错理论得到广泛承认。
2020/5/29
Chapter 8 Deformation and Strengthening
5
mechanism
8-A DEFORMATION MECHANISMS FOR METALS
Dislocation Theory of Crystal Deformation
Chapter 8 Deformation and Strengthening mechanism
第八章 (固体材料的)变形和强化机制
Why Study Deformation and Strengthening Mechanisms?
the nature of dislocations and the role they play in the plastic deformation process

Mechanical Properties

Normally, the cross section is circular, but rectangular specimens are also used. During testing, deformation is confined to the narrow center region.
Compression tests
NORTH UNIVERSITY OF CHINA
A compression test is conducted in a manner similar to the tensile test, except that the force is compressive and the specimen contracts along the direction of the stress. Previous equations are utilized to compute compressive stress and strain, respectively.
Schematic representation of the apparatus used to conduct tensile stress-strain tests. The specimen is elongated by the moving crosshead; load cell and extensometer（伸长计） measure, respectively, the magnitude of the applied load and the elongation.
College of Materials Science & Engineering
NORTH UNIVERSITY OF CHINA

A Model for Deformation and Segregation of Solid-Liquid Mixtures

LAWRENCE A. LALLI is Senior Technical Specialist, Aluminum Company of America, Alcoa Laboratories, Alcoa Center, PA 15069. Manuscript submitted November 5, 1984. METALLURGICAL TRANSACTIONS A
TA t = ~PAt
~v
Fig. I --Definition of stresses.
In this paper the stresses in the solid will be considered as average quantities over the combined solid-liquid area; consequently, r = o- - ~p [2] From Figure 1, it would appear that a should be equal to the liquid area fraction, g; = AL/Ar. It has been experimentally observed, however, that a is generally close to unity. Over the complete range of liquid area fractions, the expression below has also been suggested: ~s
B. Equilibrium
II.
D E S C R I P T I O N OF T H E M O D E L
A. Definition o f Stresses
In order to account for the deformation and fluid flow, the stresses in the solid and liquid must be considered separately. Figure 1 depicts these stresses which can be written 14 as

The influence of plasticity in hydraulic fracturing

$The influence of plasticity in hydraulic fracturing$

International Journal of Fracture84:61–79,1997.c1997Kluwer Academic Publishers.Printed in the Netherlands.The inﬂuence of plasticity in hydraulic fracturingPANOS PAPANASTASIOUSchlumberger Cambridge Research,High-Cross,Madingley Road,Cambridge,CB30EL,UKe-mail:panos@Received30July1996;accepted in revised form24January1997Abstract.This paper examines the inﬂuence of plasticity in hydraulic fracturing.Fluidﬂow in the fracture is modelled by lubrication theory.Rock deformation is modelled by the Mohr–Coulombﬂow theory of plasticity and the propagation criterion is based on the softening behaviour of rocks.The coupled,nonlinear problem is solved by a combinedﬁnite difference–ﬁnite element scheme.The results show that plastic yielding near the tip of a propagating fracture provides an effective shielding,resulting in an increase of the rock effective fracture toughness by more than an order of magnitude.Higher pressure is needed to propagate an elasto-plastic fracture than an elastic fracture,and the created elasto-plastic fracture is shorter and wider than the elastic fracture of the same volume.Key words:Hydraulic fracturing,plasticity,net pressure,ﬁnite elements.1.IntroductionHydraulic fracturing is one of the techniques used to stimulate the production of oil or gas in low permeability reservoirs.This technique involves pressurization of an isolated zone of the wellbore with pumpedﬂuid in order to initiate and propagate a tensile fracture.Once a fracture is created,sand-like grains(known as proppant)pumped with theﬂuid,pack together in the fracture preventing closure after the hydraulic pressure is relieved.The fracture provides a highly conductive path for theﬂow of hydrocarbons towards the well(Economides and Nolte, 1989).Fracture length and width,ﬂuid design,proppant addition and pumping schedule are optimized using models that incorporate principles ofﬂuidﬂow,rock deformation and fracture mechanics.In practice,a lot of attention is focussed on the prediction of wellbore pressure; wellbore pressure is normally measured during the treatment and is usually the only parameter available to evaluate the operation.The currently-used fracture propagation models are based on linear elasticity and often underestimate the down-hole pressure that is measured in theﬁeld operations.Two main approaches have been proposed in order to explain the discrepancies between models and ﬁeld observations.Theﬁrst approach(Shlyapobersky,1985)says that the value of the fracture toughness that is measured in the laboratory underestimates the in-situ value.The high value of the in-situ fracture toughness,which can be an order of magnitude greater than the laboratory-determined value,can be attributed to a scale effect,to the inﬂuence of conﬁning(in-situ) stress and to micro-cracking and in-elasticity near the crack-tip.However,in the second approach Cleary(1991)pointed out that the high values of observed net pressure(difference between fracturing pressure and minimum in-situ stress)could not be due to a high rock fracture toughness simply because high net pressure is also observed during reopening of pre-existing fractures.He proposed as an alternative that the observed high net pressure is related to a sharp drop ofﬂuid pressure and the existence of a dry region near the crack-tip.More recently,Johnson and Cleary(1991)claimed that elasticity cannot provide the required high62P.Papanastasioupressure drop to match the observed net pressure.They stated that other mechanisms must be introduced in the models and proposed that rock dilation may be the source of high pressure drop near the tip.According to the dilation hypothesis,the rock dilation behind the advancing fracture would constrain the opening which may lead to the sharp pressure gradients(Barr, 1991).In an attempt to explain the observed high net pressures,an elasto-plastic analysis was car-ried out in order to evaluate the effect of inelastic rock behaviour,in particular,the inﬂuence of dilation on the fracture tip behaviour.However,rock dilation is associated with frictional sliding,either along particles or micro-cracks.This deformation process is described appropri-ately by the well-established theory of plasticity,and so the importance of plastic yielding also became a part of the investigation.Theﬁrst results of this work on the inﬂuence of inelastic rock behaviour in hydraulic fracturing were presented in Papanastasiou and Thiercelin(1993). Other related studies in the literature considered either stationary fractures(Wang et al.,1994) or fractures subjected to constant pressure(Van den Hoek et al.,1993).However,the loading response of a propagating crack is completely different from a stationary crack.The problem is strongly nonlinear,characterized by full coupling betweenﬂuidﬂow,rock deformation and fracture propagation.Rock is modelled by the elasto–plastic constitutive equations of the Mohr–Coulombﬂow theory.A cohesive crack model which takes into account the softening behaviour that most rocks exhibit in the post-failure regime is employed as the propagation criterion.Fluidﬂow is modelled by lubrication theory.The assumption of constant propagation velocity,which was used in an earlier analysis(Papanastasiou and Thiercelin,1993),is relaxed here.The ﬂuid-ﬂow equation is solved using the more realistic boundary condition of a givenﬂow rate at the wellbore.The storage of the fracture is also taken into account.The problem is solved numerically by a combinedﬁnite difference/ﬁnite element scheme and meshing/remeshing is employed in order to carry out longer propagations.The new analysis enables us to compare the results,net pressure and fracture dimensions,for elastic and elasto-plastic fractures at the same pumping time.2.Hydraulic fracturing modelFor a clear investigation of the effect of rock plasticity,we analyse here the simple classical plane strain fracture geometry(Figure1).We may argue that the plane strain geometry appropriately describes the behaviour near the fracture tip which is the area of most interest in this problem.We assume that three physical processes govern hydraulic fracture propagation in weak rocks:viscousﬂuidﬂow in the fracture,elasto–plastic deformation caused by the stress concentration due toﬂuid pressure and in-situ stresses,andfracture propagation into the rock formation.Fluid leak-off from the fracture into the formation is not taken into consideration here,i.e.the rock is assumed to be impermeable.Results on the inﬂuence ofﬂuid leak-off and pore pressure in a hydraulic fracture in a poro-elastic medium can be found in Detournay and Cheng(1991) and Lenoach(1995).The inﬂuence of plasticity in hydraulic fracturing63Figure1.Geometry for a plane strain hydraulic fracture.2.1.F LUID FLOWIt is assumed that the fracturingﬂuid is an incompressible uniform Newtonian viscousﬂuid. We use a one-dimensional approximation for the mass balance.The continuity equation which imposes conservation of mass,is0(1) where is the local fracture width and is the localﬂow rate.It is assumed that motion in the fracture occurs only along the-direction with average velocity.Theﬁrst term in Equation (1)represents the storage of the fracture that arises from changes in the fracture width and the second term represents the massﬂow in a cross-section perpendicular to the fracture axis.In Equation(1)we ignore any leak-off from the fracture surface into the rock formation.A second equation is derived from conservation of momentum.The lubrication equation, which relates the pressure gradient to the fracture width for a Newtonianﬂuid of viscosity degenerates to3(2)12where is theﬂuid pressure in the fracture.Theﬂuidﬂow Equations(1)and(2)were supplemented with the boundary conditions of constantﬂow rate at the wellbore and zeroﬂuid pressure at theﬂuid front0064P.PapanastasiouFinally,as an initial solution,the analytic solution which was derived for elastic materials in Desroches et al.(1993)was used.2.2.R OCK DEFORMATIONSome inelastic behaviour,whose extent depends on material and loading conditions,must be present in the vicinity of a crack tip,because of the stress concentration which occurs in this region.When the inelastic zone is small enough,solutions from linear elastic fracture mechanics can be used to analyse the fracture process.To account for large-scale inelastic effects,principles from nonlinear fracture mechanics must be employed.In soft rocks such as clay or poorly consolidated sandstones,it is less likely that the inelastic zone will be small enough to justify the use of solutions based on linear elasticity.In such a case one should use the plasticity theory which properly describe the irreversible deformation due to excessive shear stresses around the fracture tip.In this section we present the basic laws which govern elasto-plastic material behaviour.For a more complete treatment the reader is referred to (Chen and Han,1985).Plastic behaviour is characterized by an irreversible straining which occurs once a certain level of stress is reached.After initial yielding the material behaviour will be partly elastic and partly plastic.In elasto–plastic analysis the strain increment is decomposed into an elastic and a plastic part:(3)The elastic strain increment can be obtained from Hooke’s law:1(4) where the tensor of elastic moduli can be expressed in terms of Young’s modulus and Poisson ratio21212(5)with being the Kronecker delta.Plastic strains are generated when the yield surface0is reached.Among the different yield criteria,the Mohr–Coulomb model adequately describes the pressure-sensitive behaviour of rocks which exhibit dilatancy(increase in porosity)when sheared.The Mohr–Coulomb model is usually employed in cases of compressive shear yielding or failure.Unlike in most cases of classical fracture mechanics,in the hydraulic fracturing problem the stressﬁeld is compressive due to the presence of the in-situ stresses.As will be explained in the next session, the tensile failure along the propagation line will be modelled using a cohesive-type model. The Mohr–Coulomb criterion can be written as11sin1sin32cos1sin0(6)where is the material cohesion,is the angle of internal friction and1and3are the maximum and minimum principal stresses.The inﬂuence of plasticity in hydraulic fracturing65 The plastic strain increments,can be generally expressed by a nonassociatedﬂow rule in the form(7) where is the plastic potential and the scalar function is the so-called plastic multiplier.In hardening plasticity the yield surface is generally a function of stresses and of the hardening parameter(8)A similar form can be taken for the plastic potential,to which the normality principle is applicable(9) This allows for the case of nonassociated plasticity;associated behaviour can be obtained as a special case by making.During plastic deformation0leading to the consistency condition0(10) which assures that during plastic deformation,the subsequent stress and strain states remain on the subsequent yield surface.Equation(10)can be written as0(11)where1(12)In order to compute the plastic strain increments we derive the expression for the plastic multiplier by substituting the Equations(4)and(7)in(3):(13)As mentioned before,the yield surface and plastic potential are generally functions of stresses and the hardening parameter.In cohesion hardening models the friction angle is kept constant and the cohesion is taken to be a function of the effective plastic strain .The hardening parameter(a scalar)is calculated from the principle of plastic power equivalence(14)66P.Papanastasiouwhere the effective stress is connected to the cohesion via2cos1sin(15)Once the scalar function is determined,the plastic strain increment is known from theﬂow rule(7)and the corresponding stress increment can be determined from Equations(11)and(13):1(16) where1is a switch function(see below).This can be written as(17) where the elastic–plastic tangent stiffness tensor is1(18) and(19)is the plastic tangent stiffness tensor and represents the degradation of the instantaneous material stiffness due to plasticﬂow.The incremental stress-strain relationship(16)is valid only for the case of plastic loading00.If the material is elastic0or unloads elastically from an elastic–plastic state,0and0,the elastic stress–strain relationship(4)should be used. Accordingly,the switch function1in Equation(16)is deﬁned such that11if0and00if0or0and0(20)In a stationary crack,plastic yielding takes place in the area near the crack tip,which is characterized by a high stress concentration.The plastic zones near the tip have the so-called ‘rabbit ears’shape.When the fracture propagates the plastic zones unload elastically behind the advancing crack and the new area near the current tip deforms plastically(Papanastasiou and Thiercelin,1993).In summary,the rock mass remote from the fracture may deform elastically,whereas the region near the body of the fracture is initially elastic,then deforms plastically andﬁnally unloads elastically after the fracture has advanced.2.3.F RACTURING PROCESSThe fracture toughness IC is deﬁned as a material parameter for elastic solids and can be calculated in a lab experiment under the condition that no plastic yielding takes place aroundThe inﬂuence of plasticity in hydraulic fracturing67 the tip.A crack will propagate in an elastic solid when the stress intensity factor I reaches a critical value of fracture toughness IC:I IC(21) The stress intensity factor is related to the strain energy release rate I2 II12(22)For linear elastic situations the strain energy release rate equals the path-independent-integral introduced by Rice(1968):I(23) where(24) in which is the strain energy density,is the traction vector,is the displacement vector and an element of the arc along the integration contour.The direction of propagation is along the coordinate axis.The-integral is independent of the actual path chosen,provided that the initial and end points of the contour are on opposite faces of the crack and the contour contains the crack tip.If the paths are chosen to be far away from the tip zone the -integral values are also mesh insensitive and for this reason the-integral constitutes the most robust fracture propagation criterion forﬁnite element analysis in elastic solids.The-integral in elasto–plastic solids was originally employed as a measure of the intensity of stress and deformationﬁelds near the tip of a stationary crack under monotonic loading and it can be used for characterizing crack initiation and limited amounts of crack extension (Kanninen and Popelar,1985).The-integral loses its meaning as a crack-tip parameter and another propagation criterion must be utilized in the problem of hydraulic fracturing because of signiﬁcant elastic unloading which takes place behind the advancing crack.The most robust propagation criterion currently available in nonlinear mechanics is based on constitutive modelling of the cohesive zone(Barenblatt,1962).The cohesive zone is the region ahead of the crack tip in which the continuum description of the material behaviour ceases to be a good description.This region is characterized by micro-cracking around the crack tip and interlocking along a portion of the crack(Labuz et al.,1985).According to Bazant(1986)the main fracture is formed by inter-connection of these micro-cracks.To model the fracture process of rock,an effective crack length is assumed which consists of a length of visible crack and a cohesive zone which is the nonlinear region where a closing stress acts(Figure2)(Ingraffea,1987).The closing stress is determined by the softening behaviour that many rocks exhibit in the post-peak region during a displacement control uniaxial tensile test.A typical stress-displacement curve from a uniaxial tensile test is shown in Figure3. This model implies that normal stress continues to be transferred across a discontinuity which may or may not be visible.It is assumed that this stress transfer is due to grain bridging and interlocking of the three-dimensionality of the crack surfaces.In this separation process,the restraining stress is a function of opening displacement which falls off to zero at a critical opening displacement()(Figure3).By taking the contour along the crack-tip cohesive68P.PapanastasiouFigure2.Representation of the fracturing process of rock.Figure3.Constitutive model for the cohesive.zone,Rice(1968)showed that the value of the-integral,when separation at the crack tip is just taking place,is(25)This means that the area under the stress-displacement curve is also equal to the strain energy release rate IC(Figure3)when the size of the cohesive zone is small compared to the crack length.It is also assumed that because of the softening behaviour the cohesive zone localizes into a narrow band ahead of the visible crack tip.The last assumption is very convenient forﬁnite element analysis where the softening behaviour is modelled by interface elements of zero thickness,lying in the crack plane ahead of the visible crack tip(Hillerborg et al.,1976). The crack path is assumed to be known from symmetry conditions.As the load increases,the interface elements are initially linear and stiff,but after the stress reaches the tensile strength of the rock they carry a decreasing stress depending on the opening displacement(Figure2). When the opening displacement at the actual crack tip reaches the critical value()the stress disappears and the actual crack propagates.We assume here a linear variation betweenThe inﬂuence of plasticity in hydraulic fracturing69 closure stress and opening displacement.In such a case two parameters are needed to describe the constitutive behaviour of the interface elements:the tensile strength of the rock and the rock fracture toughness IC which is related to the strain energy release rate IC via(21) and(22).In this study a two-dimensional,isoparametric,quadratic displacement,6-noded interface element was employed.A consistent isoparametric formulation permits modelling of curved crack surfaces and provides an element that is compatible with the8-noded quadratic displacementﬁnite elements that are used to discretize the internal domain.A linear softening material was assumed in order to keep the number of parameters which are needed to describe the constitutive behaviour to a minimum.putational resultsI NPUT PARAMETERSIn the next two sections computational results using the boundary condition of constantﬂow rate at the wellbore will be presented.Results with constantﬂuid velocity were presented in Papanastasiou and Thiercelin(1993).The implementation of the new boundary condition enables the comparison of elastic and elasto-plastic analyses for the same pumping time.All the results are based on the following parameters,unless otherwise stated:the parameters of theﬂuid-ﬂow equations are theﬂuid viscosity and theﬂow-rate at the wellbore05108MPa sec000001m3sec mthe elastic constants are25GPa02for simplicity,an elastic-perfectly plastic material,thus0in Equations(11)–(19)and associative behaviour was assumed,with3030577MPawhere and are the rock friction angle,dilation angle and cohesion,respectively.the parameters which characterize the fracturing process are the rock fracture toughness IC and the uniaxial tensile strengthIC2MPa m3MPaﬁnally,the computations were performed for a non-hydrostatic stressﬁeld with a ratio 132and values of stresses150MPa325MPa70P.PapanastasiouFigure4.Fracture proﬁles propagating from0.18m(solid line),0.64m(dashed/dotted line)and0.84m(dashed line)initial lengths.For the solution of the coupled problem ofﬂuidﬂow,rock deformation and the fracturing process we adopted a mixed formulation.Coupling occurs only via fracture faces which separate the two domains where different numerical techniques were used.Aﬁnite difference scheme was used for the discretization ofﬂuidﬂow equations in time and space and aﬁnite element method was used for the solution of rock deformation and the fracturing process.A meshing/remeshing scheme was employed in the analysis in order to be able to carry out longer propagations.The meshing/remeshing scheme was based on the following steps:the sensitive area near the tip where high gradients exist was discretized using aﬁne mesh and the region away from the crack was discretized with a coarser mesh.After some propagation steps and before the fracture tip moves into the coarse mesh,theﬁne mesh was shifted near the tip and the information from the old mesh was mapped into the new mesh using higher order interpolation functions(Lagrangian polynomial).With the extension of the fracture the discretization of the farﬁeld was also extended.The efﬁciency of the remeshing scheme was validated by checking the equilibrium of the external and internal forces immediately after remeshing.3.1.E FFECT OF INITIAL FRACTURE LENGTHSince in the current analysis we only model fracture propagation and not fracture initiation we start the propagation by assuming that a fracture of an initial length exists.However, plasticity is path dependent and the assumed initial fracture length may critically inﬂuence the propagation steps which follow.Therefore,theﬁrst computations were aimed at investigating the effect of the initial fracture length on the propagating elasto-plastic fractures.Figure4 shows the width proﬁle of a fracture as a function of the initial length.Three different initial lengths were tested:0.18m,0.64m,and0.84m,and the fracture was propagated up to1.0m. Thisﬁgure shows that the value of the initial fracture length is memorized in the width proﬁle, because there is no plastic zone associated with the initial fracture,but does not affect the proﬁle in the region where the fracture propagates.The effect of the initial fracture length will be observed as a step in the nextﬁgures of elasto-plastic fracture proﬁles.This effect can,Figure5.Width proﬁles for elastic(solid lines)and elasto-plastic(dashed lines)fractures.Figure6.Pressure proﬁles when tip is at1.86m(conﬁning stress25MPa).however,be neglected as the pressure proﬁle is mainly controlled by the near-tip behaviour which is modelled correctly.3.2.S MALL-SCALE YIELDINGFigure5shows the proﬁles of elastic(solid lines)and elasto-plastic(dashed lines)fractures for the same fracture lengths.In this case,the elasto-plastic fracture is almost20percent wider than the elastic fracture.Figure6shows the comparison of the net pressure distribution in the fracture at the last propagation step.In this case30percent higher net pressure is needed to propagate the elasto-plastic fracture.Note that in these computations theﬂuid reaches the cohesive zone but does not penetrate it since we assume that the cohesive zone is impermeable.The fracture dimensions and propagation pressures can also be compared at the same pumping time.Figure7shows the fracture width at the wellbore and Figure8shows the fracture length as a function of time for elastic and elasto–plastic fractures.The elasto–plastic fractures produced after a given pumping time are approximately20percent wider and20percent shorter than the elastic fractures.Figure9and Figure10compare the net pressures asFigure7.Fracture width at wellbore vs time.Figure8.Fracture length vs time. pressure at wellbore vs time. pressure at wellbore vs fracture length.Figure11.History of plastic zones in terms of accumulated shear plastic strain;small-scale yielding.a function of time and as a function of fracture length,respectively.The pressures for elastic and elasto–plastic fractures are relatively close at theﬁrst propagation step,then they initially increase for some propagation steps until the stressﬁeld and plastic zones are fully developed andﬁnally they decrease during propagation.Although the net pressure for the elasto-plastic fracture is30percent higher than the elastic fracture of the same length,it is50percent higher when the comparison is made as a function of time.Note that the apparent discontinuities in some of theﬁgures(Figure9and Figure10for example)are numerical artifacts due to the extension of the discretized far-ﬁeld domain in the remeshing scheme.During fracture propagation the size of the plastic zones initially increases but eventually reaches a constant size.The plastic zones unload elastically behind the advancing crack.Figure 11shows the history of the plastic zones in terms of accumulated plastic shear strain.In these computations the fracture has reached a length of1.20m after having been propagated from an initial length of0.18m.At any instant,the size of the active plastic zones extends no more than0.06m from the fracture tip.Therefore,because the size of the plastic zones is muchsmaller than the fracture length we may conclude that small-scale yielding applies.Figure12.Effective fracture toughness(EFT)vs fracture growth.As mentioned before,the strain energy release rate I cannot be used as a propagation cri-terion when plastic yielding takes place.However,if another propagation criterion appropriate for nonlinear analysis(e.g.,the cohesive model(Barenblatt,1962)),is used,the-integral can be used to calculate the energy dissipated during fracture propagation and to determine an‘effective fracture toughness’(EFT)from(21)to(24).For elasto-plastic material the strain energy density in(24)must be augmented by the plastic component:(26) where the elastic part is given by1(27)2and the plastic work contribution is given by(28)in which and are the effective stress and effective plastic strain deﬁned in Equations(15) and(14),respectively.The value of is calculated by accumulating the incremental plastic work over the strain path.For the elastic fractures the-integral value is independent of crack length whereas for the elasto–plastic fractures it increases initially and eventually reaches an asymptotic value. Figure12shows the EFT,calculated from the-integral,as a function of fracture growth.The initial value of EFT equals the rock fracture toughness IC and dictates the energy required to propagate the elastic fractures.The energy required to propagate the long elasto-plastic fractures is given by the limit value.Higher energy is needed to propagate the elasto–plastic fractures because plastic yielding softens the material surrounding the tip,thereby reducing the level of stress in the direction of propagation and providing an effective shielding.3.3.L ARGE-SCALE YIELDINGBased on experience from classical fracture mechanics,the results may change signiﬁcantlywhen we pass from the case of small-scale yielding,where the different concepts are wellFigure13.Size of plastic zones in terms of plastic shear strain;large-scale yielding.Figure14.Width proﬁles for elastic(solid lines)and elasto-plastic(dashed lines)fractures.deﬁned,to the case of large-scale yielding.The size of the plastic zones increases with the contrast of the magnitude of the in-situ stresses;it is strongly inﬂuenced by the strength of the rock,the elastic modulus and the pumping parameters(ﬂuid viscosity andﬂow rate) (Papanastasiou and Thiercelin,1993).For this reason in the next computation series we increased theﬂuid viscosity andﬂow rate at the wellbore0in order to increase the size of the plastic zones(Papanastasiou and Thiercelin,1993).The computations were performed using the values5106MPa sec00001m3sec mThe isolines of the plastic shear strain are shown in Figure13,for a fracture length of0.83m. The size of the plastic zone is signiﬁcantly greater than in Figure11;we are now dealing with large-scale yielding.Figure14shows the proﬁles of elastic(solid lines)and elasto-plastic (dashed lines)fractures for the same fracture lengths.In this case,the elasto–plastic fractureshave approximately30percent wider openings than the elastic fractures.Figure15shows the。

Chapter7rolling

shape rolling is used to make products with
various cross-sections.
轧制是一种通过一组轧辊对长的工件施加压力, 从而减小其厚度或改变其横截面的工艺.除了平板轧制外,还可以用成形轧制来生产各种横截面的产品.
Products made by rolling include: plate, sheet, foil, rod, seamless pipe, and tubing; shapedrolled products, such as I-beams and structural shapes; and bars of various cross-section. Other rolling operation include ring rolling and thread rolling.
some materials which are brittle, cannot be cold worked.
7.2 rolling
7.2.1 principle
Rolling is a process where the metal is compressed between two rotating rolls for reducing its crosssectional area.
若金属的变形温度高于再结晶温度,称为热变形; 若金属的变形温度低于再结晶温度,称为冷变形.
Recrystallisation temperature
The approximate minimum temperature at which complete recrystallisation of a cold worked metal occurs within a specified time．在特定的时间内,金属冷变形后发生完全再结晶的最低温度.

9. A ductile ceramic eutectic composite with high strength at 1,873 K

letters to nature
3. Livingston, W. Radial ﬁlamentary structure in a sunspot umbra. Nature 350, 45–46 (1901). 4. Evershed, J. Radial movement in sun-spots. Mon. Not. R. Astron. Soc. 69, 454–457 (1909). 5. Thomas, J. H. & Weiss, N. N. in Sunspots: Theory and Observations (eds Thomas, J. H. & Weiss, N. A.)
3–59 (Kluwer, Dordrecht, 1992). 6. Ruiz Cobo, B. & del Toro Iniesta, J. C. Inversion of Stokes proﬁles. Astrophys. J. 398, 375–385 (1992). 7. Makita, M. & Kawakami, H. A study of line asymmetry in unipolar sunspots. Publ. Astron. Soc. Jpn. 38,
A ductile ceramic eutectic
composite with high str Nhomakorabeangth
at 1,873 K
Y. Waku, N. Nakagawa, T. Wakamoto, H. Ohtsubo, K. Shimizu & Y. Kohtoku
Ube Research Laboratory, Corporate Research & Development, UBE Industries, Ltd, Ube City, Yamaguchi, 755, Japan

最全版高分子物理名词中英对照题库

第一章高分子链的结构aeolotropy各向异性anti-configuration反型；反式构型atacticpolymer无规（立构）聚合物averageroot-mean-square均方根backbone主链backbonemotion主链运动backbonestructure主链结构branchedpolymer支化聚合物carbonchain碳链chainconformation链构象chainelement链单元；链节cis-configuration顺式构型cis-isomer顺式异构体cis-isomerism顺式异构现象cis-stereoisomer顺式立体异构体cis-transisomerism顺-反（式）异构现象cis-transisomerization顺反异构化characterization[1]表征；表征法[2]检定；检定法configuration构型conformation构象covalentbond共价键crosslink交联；交联键crosslinkage交联crosslinkednetwork交联网crosslinkedpolymer交联聚合物differentialthermalanalysis差热解析differentialthermogravimetricanalysis微分热重解析differentialthermogravimetriecurve微分热重曲线degreeofisotacticity全同（立构）规整度degreeoforder有序度degreeofsyndiotacticity间同（立构）规整度degreeoftacticity构型规整度diisotactic双全同立构的directionoforientation取向方向end-to-enddistance尾端距forkchain支链forkgroup支基equitacticpolymer全同间同（立构）等量聚合物erythro-diisotactic叠（同）双重全同立构eutacticity理想的构型规整性degreeofisomerization异构化程度flexibility柔性；柔顺性freeinternalrotation自由内旋转freelyjointedchain自由连接链functionalgroup功能基；官能团Gausschain高斯链Gaussdistribution高斯分布Gaussianchain高斯链Gaussiandistribution高斯分布Gaussiannetwork高斯网络non-GaussianChain非高斯链non-Gaussiandistribution非高斯分布geometricalisomer几何异构体geometricalregularity几何规整度graft接枝物graftblockcopolymer接枝嵌段共聚物graftcopolymer接枝共聚物graftcopolymerization接枝共聚graftomer接枝聚合物graftpolymerization接枝聚合head-tailsequence头尾序次head-to-head头-头接head-to-headpolymer头-头接聚合物head-to-tailpolymer头-尾接聚合物hotsettingresin热固性树脂isotactic全同立构；等规立构isotacticchain全同立构链isotacticconfiguration全同立构型isotacticity全同立构规整度isotacticpolymer全同立构聚合物iso-trans-tactic反式全同立构linearchain直链linearchainpolymer直链聚合物linearcopolymer线形共聚物linearmacromolecule线形大分子linearmolecule线形分子linearpolymer线形聚合物highpolymer高聚物halfpolymer低聚物highlybranchedchain高度支化链heatofcrystallization结晶热flexiblelinearmacromolecule柔性线形大分子flexiblesidegroup柔性侧基interpolymer共聚体longchainbranching长链支化longchainmolecule长链分子holotactic全规整macromolecular大分子的；高分子的macromolecularcompound大分子化合物；高分子化合物macromolecule大分子；高分子mainchain主链mainpolymerchain聚合物主链molecularbond分子键molecularconfiguration分子构型monomer单体mutamer旋光异构体meansquareendtoenddistance均方尾端距nonlinearpolymer非线型聚合物netstructure网状结构networkpolymer网状聚合物networkstructure网状结构opticalisomer旋光异构体；旋光异构物orderedstructure有序结构randomcopolymer无规共聚物polymerchain聚合物链；高分子链polymersinglecrystal聚合物primarystructure一级结构；初级结构regularity规则性；整齐度distance均方根尾端距rotamerism几何异构现象；旋转异构现象rigidchain刚性链rigidmacromolecule刚性大分子straight-chainpolymer直链聚合物secondarystructure二级结构syndiotactic间同立构的syndiotacticpolymer间同立构聚合物stability牢固性starpolymer星形聚合物statisticalcopolymer无规共聚物spatialstructure空间结构sub-chainmotion链段运动transisomerism反式异构(现tertiarystructure三级结构unperturbedchaindimension未受扰分子链尺寸；无扰分子链尺寸valencebond价键valencedistance(=bondlength)键长；键距laevo-configuration左旋构型microstructureofpolymer聚合物的微结构microtacticity微观规整性shortrangestructure近程结构molecularmotion分子运动molecularmodel分子模型第二章高分子的齐聚态结构intermolecularattraction分子间引力hydrogenbond氢键amorphous非晶的；无定形的amorphousbirefringence非晶双折射amorphousmaterial非晶资料；无定形资料amorphousphase非晶相amorphouspolymer非晶态聚合物amorphousregion非晶区；无定形区axialorientation沿轴取向auto-orientationmechanism自取向机理biaxialorientation双轴取向biaxialstretch双轴拉伸crystal结晶；晶体crystalangle晶角crystaldefect晶体弊端crystaldiffraction晶体衍射crystalgrain晶粒crystallattice晶格crystallinecopolymer结晶共聚物crystallineorientation晶体取向crystallinepolymer结晶聚合物crystallineportion结晶部分crystallineregion晶区crystallinestructure晶体结构crystallinity结晶性；结晶度crystallite微晶；晶粒crystallitedimension晶粒尺寸crystallitesizedistribution晶粒大小分布crystallizablepolymer（能）结晶聚合物crystallization结晶（作用）crystallizationrate结晶速率crystallizationtemperature结晶温度crystallize结晶cubicsystem立方晶系chainfolding链折叠chain-foldedlamellae折叠链片晶degreeofcrystallinity结晶度程度degreeoforientation取向度diffraction衍射diffractionangle衍射角diffractionpattern衍射图形；衍射花式diffractometer衍射仪extendedchaincrystal挺直链electronmicroscope电子显微镜expandedmaterial发泡资料;晶体birefringence双折射densitycrystallinity密度结晶度foldedchain折叠链foldedchaincrystal折叠链晶体foldedchainlamellae折叠链晶片foldedchainmodel折叠链模型foldedconfiguration折叠构型foldinglength折叠链长度liquidcrystal液晶lamellae片晶lamellaestructure片晶结构lattice晶格；点阵latticeconstant晶格常数latticeenergy晶格能latticemodel晶格模型latticestructure晶格结构monoclinic单斜（晶）的mono-crystalline单晶的morphologicalstructure形态结构molecularorientation分子取向non-crystallizablepolymer非结晶聚合物one-wayorientation单向取向orientatedpolymer取向聚合物orientatedpolymerization取向聚合orientation取向orthorhombicunitcell斜方晶胞platelets片晶polarize起偏（振）镜；起偏光镜polarizingmicroscope偏（振）光显微镜polaroid（人造）偏振片[物]；起偏振片singlecrystal单晶smallanglescattering小角散射第三章高分子溶液viscometry粘度法gelatination胶凝（作用）；胶凝化gelpoint凝胶点coacervation凝聚coagulationpoint凝固点；凝固点equilibriumswelling平衡溶胀heatofsolution溶解热gel凝胶；冻胶idealsolution理想溶液fractionalprecipitation分级积淀fractionalsolution分级溶解fractionation分级non-idealsolution非理想溶液solubility溶解性；可溶性solubilityparameter溶度参数solublepolymer可溶性聚合物dilutesolution稀溶液dilutesolutiontheory稀溶液理论dilutionfreeenergy稀释自由能dilutionheat稀释热dilutionviscometer稀释粘度计freeenergy自由能freeenergyofmixing混淆自由能entropyofmixing混淆熵polarpolymer极性聚合物第四章高聚物的分子量和分子量分布averagemolecularweight平均分子量Z-averagemolecularweightZ均分子量Z-averagemolecularweightZ均分子量weightaveragemolecularweight重（量平）均分子量apparentmolecularweight表观分子量criticalentanglementmolecularweight临界缠结分子量endeffect尾端效应endgroup端基endgroupmethod端基法[测分子量] intrinsicviscosity特色粘数intrinsicviscositynumber特色粘数narrowmolecularweightdistribution窄分子量分布meanmolecularweight平均分子量molecularweight分子量molecularweightdetermination分子量测定molecularweightdistribution分子量分布molecularweightdistributioncurve分子量分布曲线numberaveragemolecularweight数均分子量scatteringangle散射角relativeviscosity相对粘度viscometricaveragemolecularweight粘均分子量viscosity粘度viscosity-averagemolecularweight粘均分子量viscous粘的；粘性的logarithmicviscositynumber比浓对数粘度kinematicviscosity比密粘度non-polarpolymer非极性聚合物interfacialtension界面张力densitygradientcolumn密度梯度管densitygradientsedimentation密度梯度沉降法densitygradienttube密度梯度管concentrationgradient浓度梯度第五章聚合物的转变与废弛stressrelaxation应力废弛stress-straincurve应力-应变曲线deformability变形性deformation形变；变形deformationband（滑移）形变带continuousstressrelaxation连续应力废弛characteristicrelaxationtime特色废弛时间discreterelaxationtime失散废弛时间epitaxy外延；取向生长glasslikepolymer类玻璃聚合物glasspoint玻璃点glasstemperature玻璃化（转变）温度glasstransition玻璃化转变glasstransitionregion玻璃化转变区glasstransitiontemperature玻璃化转变温度glassycompliance玻璃态柔量glassymodulus玻璃态模量glassypolymer玻璃态聚合物glassystate玻璃态half-crystallizationtime半结晶期halflifeperiod半衰期fractionalfreevolume自由体积分数freevolume自由体积noncrystalline非晶的noncrystallineregion非晶区kineticofcrystallization结晶动力学nuclei[复]核；晶核nucleus（[复]nuclei）核；晶核nucleusformation（晶）核生成（作用）rubberyplateauzone橡胶高弹区rubberystate橡胶态rubberyplateauzone橡胶高弹区rubberystate橡胶态rateofcrystalgrowth晶体生长速率transitionzone转变区compressiondeformation压缩变形compressionmodulus压缩模量heterogeneousnucleation异相成核dimensionlessglasstransition无量纲玻璃化转变thermo-mechanicalcurve热机（械）曲线；温度形变曲线permanentdeformation永久变形non-reversibledeformation不可以逆形变；永久形变thermaldeformation热变形thermaldegradation热降解thermaldilation热膨胀compressionset压缩变形initialmodulus初步模量initialtangentmodulus初步切线模量instantaneouscompliance瞬时柔量instantaneousdeformation瞬时形变instantaneouselasticity瞬时弹性instantaneouselasticrecovery瞬时弹性回复instantaneousmodulus瞬时模量dynamicmechanicaldoubleglasstransition动向力学双重玻璃化转变delayeddeformation延缓形变recrystallization再结晶（作用）relativedeformation相对形变relativeelongation相对伸长thermogravimetriccurve热重（解析）曲线；温度重量曲线第六章橡胶弹性anisotropic各向异性的anisotropy各向异性（现象）blend[1]共混物[2]共混blendingpolymer共混聚合物dynamicelasticity动向弹性delayedelasticity延缓弹性elastic弹性的elasticanisotropy弹性各向异性elasticcompliance弹性柔量elasticconstant弹性常数elasticdeformation弹性形变elasticelongation弹性伸长elasticextension弹性延伸elasticisotropy弹性各向同性elasticity弹性elasticitymodulus弹性模量elasticproperty高弹性elastomer高弹体；弹性体elastomericstate橡胶高弹态；高弹态highelasticdeformation高弹形变highelasticity高弹性highelasticrubber高弹性橡胶longrangeelasticity高弹性idealelasticity理想弹性idealelastomer理想弹性体non-elasticdeformation非弹性形变initialelasticity初弹性；瞬时弹性perfectelasticbody理想弹性体modulusofelasticity弹性模量modulusofrigidity刚性模量temporaryset（高）弹性形变dynamicresilience动向弹性回复dynamicrigidity动向刚度；动向刚性rubberelasticbehavior橡胶弹性行为；高弹行为entropicdeformationmechanism熵变形机理entropy-elasticdeformation熵弹形变entropyelasticity熵弹性entropyspring熵弹簧naturerubber天然橡胶non-Hookeanelasticity非虎克弹性naturalcis-polyisoprene天然顺式聚异戊二烯；天然橡胶naturaldrawratio固有拉伸比；自然拉伸比one-waydrawing单向牵伸draw拉伸drawratio拉伸比Poisson’sratio泊松比第七章聚合物的粘弹性anelasticity滞弹性dynamiccompliance动向柔量dynamiccreep动向蠕变dynamicmechanicalbehavior动向力学行为dynamicmechanicalproperty动向力学性能dynamicmechanicaltest动向力学试验Boltzmannsuperpositionprinciple波尔兹曼叠加原理relaxationphenomenon废弛现象relaxationspectra废弛（时间）谱relaxationtime废弛时间retardationtime推迟时间retardedelasticity推迟弹性superpositionprinciple叠加原理naturetime(=relaxationitme)废弛时间；自然时间non-linearviscoelasticity非线性粘弹性recoverycreep回复蠕变principleofsuperposition叠加原理timeofrelaxation松驰时间time-temperatureequivalentprinciple时-温等效原理time-temperaturesuperpositionprinciple时-温叠加原理creep蠕变creepcompliance蠕变柔量creepcurve蠕变曲线dashpot粘壶dynamicmodulus动向模量dynamics动力学dynamicstate动向dynamicviscoelastometer动向粘弹谱仪dynamicviscosity动向粘度；动力粘度elastoviscometer弹性粘度计elasto-viscoussystem弹粘系统elastoviscouspolymer弹粘性聚合物distributionofretardationtimes推迟时间分布Hookeanelasticity虎克弹性Hookeanspring虎克弹簧immediateset瞬时变形initialcreep初步蠕变mechanicalrelaxation力学废弛viscouselasticity粘弹性stickiness粘性compressionstressrelaxation压缩应力废弛maximumrelaxationtime极大废弛时间distributionofrelaxationtimes废弛时间分布stretch(ing)拉伸tangentofthelossangle耗费角正切mechanicallossfactor力学耗费因子lossangle耗费角lossfactor耗费因子lossmodulus耗费模量losstangent耗费角正切intermittentstressrelaxation中止应力废弛discreteviscoelasticspectra失散粘弹谱lastics[1]塑料[2]弹塑料；弹塑（性）体damping阻尼；减幅dampingfactor阻尼因子distortion扭变；畸变;双定向的第八章聚合物的信服和断裂work-to-break断裂功yield信服yieldstrength信服强度yieldstress信服应力Young’smodulus杨氏模量root-mean-squareend-to-endrupturemechanism断裂机理；破坏机理modulusofrupture(=flexuralstrength)抗弯强度；挠曲强度unnotchedIzodimpactstrength无缺口悬臂梁式抗冲击强度impactstrength冲击强度crack[1]裂缝；龟裂；裂纹[2]裂化craze银纹load-elongationcurve载荷-伸长曲线fissure裂缝；裂隙flexingresistance抗挠性；耐屈挠性flexuralmodulus挠曲模量flexuralstrength挠曲强度fluidresin液态树脂fracture断裂；破裂fractureenergy断裂能fracturemechanism断裂机理fracturesurfaceenergy断裂表面能fracturetoughness断裂韧性fragility脆性；易碎性friability脆性；易碎性friction摩擦frictionaldamping摩擦阻尼fringe条纹elongation伸长extendedlength伸展长度extension伸长；延伸extensionatbreak断裂伸长extensionmodulus拉伸模量extensionratio拉伸比fabricability加工性failure破裂；破坏fatigue疲倦fatiguecracking疲倦龟裂fatiguecurve疲倦曲线fatiguefailure疲倦破坏fatiguestrength疲倦强度fatiguetest疲倦试验fayingsurface接触面bendingmodulus波折（弹性）模量bend(ing)strength波折强度bend(ing)stress波折应力breakingelongation断裂伸长第九章聚合物的流变性rheology流变学rheometer流变仪anti-swelling抗溶胀性anti-thixotropy0非触变性；非摇溶（现象）apparentarea表观面积apparentdensity表观密度apparentfluidity表观流度apparentviscosity表观粘度dieswell(ing)挤出胀大；模口胀大dieswellratio挤出胀大比dilatability膨胀性dilatate膨胀dynamic(=dynamical)[1]动向的[2]动力学的dynamicchemorheology动向化学流变学flowability流动性flowbehavior流动行为；流动特色flowbirefringence流动双折射flowcurve流动曲线flowdiagram流动图flowindex流动指数flowline流线flowtemperature流动温度fluid流体fluidmechanics流体力学hydrodynamics流体力学hydronamicorientation流体力学取向non-recoverableflow不可以逆流动；塑性流动pseudo-plasticfluid假塑性流体pseudo-plasticity假塑性；非宾汉塑料pseudo-viscosity假粘度；非牛顿粘度pureviscousflow纯粘性流动rateofshear剪切（变）速率meltcrystallization熔融结晶meltelasticity熔体弹性meltenpolymer熔融聚合物meltflowindex熔体流动指数meltfracture熔体破裂meltindex熔体指数meltingtemperature熔融温度meltingviscosity熔融粘度meltrheology熔体流变学meltviscosity熔体粘度telescopicflow层流viscousflow粘流dragflow粘性流动Binghambody宾汉体Binghamflow宾汉流动Binghammodel宾汉模型Binghamplasticfluid宾汉塑性流体Bingham’syieldvalue宾汉信服值Binghamviscometer宾汉粘度计extrude挤出；挤压；压出extrudingmachine挤出机extrusion挤出；挤压extrusionmolding挤出成形extrusionswelling挤出胀大capillaryextrusionrheometer毛细管挤出流变计necking(colddrawing)颈缩（冷拉）；细颈现象Newtonianbehavior牛顿行为Newtonianflow牛顿流动[纯粘性流动] Newtonianliquid牛顿液体Newtonianviscosity牛顿粘度non-homogeneous不平均的non-Newtonianbehavior非牛顿行为non-Newtonianliquid非牛顿液体shear剪切；切变shear-banding剪切带sheardeformation剪切形变shearelasticity(=shearmodulus)剪切模量shearstress剪切应力shearviscosity剪切粘度plasticflow塑性流动plasticfluid塑性液体torsionalmoment扭矩torsionalpendulum扭摆torsionalvibrationrheometer扭转振动流变仪smelting熔炼；消融softening消融softeningtemperature消融温度；消融点fusionheat消融热laminarflow层流polymermelt聚合物熔体moltenpolymer熔融聚合物moltenstate熔融状态swellingratio溶胀比；胀大比phaseequilibrium相平衡phasetransition相转变第十章聚合物的电学性能热性能光学性能以及表面与界面性能ageingprocess老化过程ageingproperty老化性能；耐老化性ageingresistance耐老化性；抗老化性ageingtime老化时间；老成时间airpermeability透气性dielectric电介质；介电的dielectricanisotropy介电各向异性dielectricbreakdown介电击穿dielectricconstant介电系数；介电常数dielectricdissipationfactor介电耗费因子；介电耗费角正切dielectricheating介电加热dielectricloss介电耗费dielectriclossfactor介电耗费因子dielectriclosstangent介电耗费（角）正切dielectricproperty介电性质dielectricpolarization介电极化dielectricrelaxation介电废弛dielectricspectra介电谱conductivepolymer导电聚合物flameresistance阻燃性flameretardant阻燃剂flammability可燃性heatendurance耐热性heatexpansion热膨胀heatresistance耐热性high-temperaturestability高温牢固性heat-resistantpolymer耐热聚合物thermalconductivity[1]热导率；导热系数[2]导热性highpolymericpolyelectrolyte高聚物电解质insulatingproperty绝缘性质lowtemperaturebrittleness低温脆性thermoplastic[1]热塑性塑料[2]热塑性的thermoplasticity热塑性thermosetplastic热固性塑料thermosettingplastic热固性塑料thermosettingresin热固性树脂thermostability热牢固性；耐热性translucency半透明性；半透明度transparency[1]透明性[2]透光度triboelectricity静电作用；摩擦生电truemeltingpoint真熔点truestrain真应变truestress真应力Vicatsofteningpoint维卡消融点voltagebreakdown击穿电压wearability耐磨性wearresistance耐磨性weatherability耐候性；耐老化性wettability吸湿性；润湿性solidness硬度；硬性strength强度mechanicalproperty力学性能；机械性能thermalcoefficientofexpansion热膨胀系数gasproofness不透气性；气密性gastight不透气的；气密的anti-static抗静电的breakdownstrength击穿强度doping混淆electricbreakdown电击穿electrodialysis电渗析聚合物名称单词ABSresinABS树脂aldehydepolymer醛类聚合物aidehyderesin缩醛树脂；聚醛树脂aldolcondensation醛醇缩合aldolresin醛醇树脂chemicalfibers化学纤维composite复合资料compositematerial复合资料conjugatefiber组合纤维；复合纤维high-densitypolyethylene高密度聚乙烯最全版高分子物理名词中英比较题库lowdensitypolyethylene低密度聚乙烯lowmolecularpolymer低聚物；低分子量聚合物lowpressureprocessedpolyethylene低压法聚乙烯medium-densitypolyethylene中密度聚乙烯phenol-aldehydeplastics酚醛塑料phenol-aldehyderesin酚醛树脂phenolicplastics酚醛塑料polyacrylonitrile聚丙烯腈polybutadiene聚丁二烯polybutene聚丁烯polycarbonate聚碳酸酯polychlorovinyl聚氯乙烯poly(ethyleneterephthalate)聚对苯二甲酸乙二醇酯poly(methylmethacrylate)聚甲基丙烯酸甲酯poly(1-phenylethylene)聚1-苯基亚乙基；聚苯乙烯polyvinylformal聚乙烯醇缩甲醛poly(phenyleneether)聚苯醚polypropylene聚丙烯syntheticresin合成树脂syntheticrubber合成橡胶synthon合成纤维。

Deformation Quantization of Symplectic Fibrations

a r X i v :m a t h /9802070v 1 [m a t h .Q A ] 16 F eb 1998DEFORMATION QUANTIZATION OF SYMPLECTIC FIBRATIONS.OLGA KRAVCHENKOAbstract A symplectic ﬁbration is a ﬁbre bundle in the symplectic category.We ﬁnd the relation between deformation quantization of the base and the ﬁbre,and the total space.We use the weak coupling form of Guillemin,Lerman,Sternberg and ﬁnd the characteristic class of deformation of symplectic ﬁbration.We also prove that the classical moment map could be quantized if there exists an equivariant connection.Along the way we touch upon the general question of quantization with values in a bundle of algebras.We consider Fedosov’s construction of deformation quantization in general.In Appendix we show how to calculate step by step the Fedosov connection,ﬂat sections of the Weyl algebra bundle corresponding to functions and their star–product.AMS classiﬁcation:58F06,81S10,58H15,32G08Key–words:deformation quantization,symplectic ﬁbrations,moment map,Weyl algebra,Fedosov quantization.Address:IRMA,Universit´e Louis Pasteur,7rue Ren´e Descartes,67084,Strasbourg,France Email:ok@12OLGA KRA VCHENKO1.Introduction:statement of the problem and the main theorem.Quantization is a map from functions on a(phase)space to operators on some Hilbert space.It involves a parameter(usually the Planck’s constant h or =h(ˇf∗ˇg−ˇg∗ˇf))={f,g}forˇf andˇg s.t.ǫ{ˇf}=f andǫ{ˇg}=g,4.The structure of the product∗on A / n A for all n≥0is given by bidiﬀerential operators.DeWilde–Lecomte[8]and also Fedosov[11]proved that on any symplectic manifold there exists a quantization.The following idea lies behind the Fedosov construction(see[9]):a Koszul–type resolution is considered for C∞(M)[[ ]].Each term of the resolution has a noncommutative algebraic structure hence providing the algebra of functions with a new noncommutative product.Fedosov constructs such a resolution by using the diﬀerential forms on the manifold with values in the Weyl–algebra bundle.The main step then is toﬁnd a diﬀerential on it which respects the algebra structure.This diﬀerential is called Fedosov connection and is obtained by an iteration procedure from a torsion free symplectic connection on the manifold.DEFORMATION QUANTIZATION OF SYMPLECTIC FIBRATIONS.3Lichnerowitz[23]showed that any connection on a symplectic manifold gives rise to a torsion–free symplectic connection.Hence one can get the Fedosov connection from any connection on a tangent bundle:ﬁrst adding some tensor to make it symplectic(section2.2)and then applying the iteration procedure.We introduce a notion of an F–manifold:Deﬁnition1.2.An F–manifoldΦis the following triple(manifold,deformation of a symplectic form,a connection):Φ=(M,ω,∇),whereω=ω0+ α,andα∈Γ(M,Λ2T∗M)[[ ]],a series in with coeﬃcients being closed2–forms on the manifold.In a recent article[16]a similar object is called a Fedosov manifold,namely,a symplectic manifold together with a symplectic connection.Indeed,these three objects(M,ω0,∇)deﬁne theﬁrst three terms in the∗–product:•Classically,a manifold can be considered as an algebra of functions on it,that is M deﬁnes the structure of the commutative product.•A symplectic form deﬁnes the Poisson structure and hence the term at .•A connection deﬁnes the term at 2(as follows from[23]).It turns out that these three terms determine the higher terms.The deformation quantization theorem([11])can be stated as followsTheorem1.3.(Fedosov)An F–manifold(M,ω,∇)uniquely determines a∗-product on the under-lying manifold M.Deligne[7]and Nest and Tsygan[28]showed that the class of isomorphisms of quantizations of a symplectic manifold M is determined by the class of the formωin H2(M)[[ ]].It is called a characteristic class of deformation.We study the deformation of the twisted products of two F–manifolds(B,ωB,∇B)and(F,σ,∇F). Our question is the following:how to deﬁne the product of two F–manifolds and what the∗–product on the total space is,that is what a“twisted product of quantizations”is.We show under certain assumptions that a twisted product of two F–manifolds is again an F–manifold.So we want to relate the∗–products on these three manifolds.One can regard the product M=B⋉F as aﬁbre bundle M→B over a symplectic base B with a symplecticﬁbre F.Obviously,the product depends on how twisted the symplecticﬁbration M=B⋉F is.This can be described by a connection on M→B which should be compatible with the symplectic form on M.When G,the structure group of the bundle M→B,acts by symplectomorphisms onﬁbres this bundle is a symplecticﬁbration(see[18],see bellow section3).The total space M is symplectic and theﬁbres are symplectic submanifolds of it.We need the action to be Hamiltonian and G to be a ﬁnite dimensional Lie group.The case of B being a cotangent bundle of some diﬀerentiable manifold X wasﬁrst developed by Sternberg[30].His construction describes the movement of a”classical particle”in Yang–Mills4OLGA KRA VCHENKOﬁeld for any gauge group G and any diﬀerentiable X.Weinstein gave a general construction of a symplectic form on M in[32].Here we are mainly interested in the particular case of the quantization with coeﬃcients in the auxiliary bundle ofﬁbrewise quantizations.This leads us to a more general case of quantization with coeﬃcients in some bundle of algebras(some examples of such bundles are considered in[13]). Our case is somewhat diﬀerent from the quantization with values in most other auxiliary bundle and is more diﬃcult to perform.Fibres of the auxiliary bundle obtained from the symplecticﬁbration structure are noncommutative algebras.It is this noncommutativity ofﬁbres which makes the quantization procedure more complicated.However we believe that it is useful to understand its mechanism in order to see that quantization is a fundamental notion like some homology theory and hence it should respect aﬁbre bundle structure.We make a new deﬁnition:Deﬁnition1.4.An F–bundle(with an underlying manifold B)is a tripleΨ=(Φ,A,∇A)where•Φ=(B,ωB,∇B)is an F–manifold,•A is an auxiliary bundle of algebras over B,•∇A is a covariant derivative on A,which respects the algebraic structure on theﬁbres.We construct an F–bundle from a symplecticﬁbration M→B.Eachﬁbre of M→B is an F–manifold,so we can quantize theﬁbres.If the connection∇F is G–invariant we can consider a new bundle A over the base B.This bundle A is a bundle of algebras of quantized functions on ﬁbres.Theﬁbre of A over a point b is the algebra of quantized functions on theﬁbre of M→B at the point b:A b=A (M b).(1)The bundle A is deﬁned by(1).The covariant derivative∇A on the bundle A is determined by the connection on the bundle M→B.The construction of the covariant derivative is carried out in section(5.1)so that it respects the algebraic structure,that is it satisﬁes Leibnitz rule with respect to the product on A.Such an F–bundle corresponds to an F–manifold modeled on M,the total space of the symplectic ﬁbration,and hence provides a quantization of the total space.Our main theorem is:Theorem1.5.Consider a symplecticﬁbration M→B with a standardﬁbre being an F–manifold (F,σ,∇F and the base an F–manifoldΦ=(B,ωB,∇B).An F–bundle(Φ,A,∇A),where A b= A (M b)gives a quantization of the underlying manifold B with values in the auxiliary bundle.This also deﬁnes the quantization of the total space M.The main claim of this theorem is that quantization of the base with values in the auxiliary bundle(1)corresponds to a certain F–manifoldΦ=(M,ω,∇),whereωis a polynomial in starting froma symplectic form on M.DEFORMATION QUANTIZATION OF SYMPLECTIC FIBRATIONS.5To carry out the programﬁrst of all one has to construct a symplectic form on M.There is a one–parameter family of symplectic forms on the total space.The construction involves the notion of weak coupling limit of Guillemin,Lerman and Sternberg[18].The behavior of the∗–product when this parameter tends to zero gives us a way to understand the relation between quantizations of the base and theﬁbre with the quantization of the total space.Our main Theorem(1.5)can be reformulated as a statement about the solutions of two equations given in Theorem(5.6).We construct a twisted Weyl algebra bundle corresponding to F–bundle. Then we prove that1.There exists r,a1-form on the base with values in the twisted Weyl bundle,such that theinitial connection becomesﬂat when one adds r to it.2.For each section a of the auxiliary bundle there exists only one correspondingﬂat section ofthe twisted Weyl bundle.Fedosov’s quantization procedure is discussed in Section2,calculations and examples are given in the Appendix.The classical setup for symplecticﬁbrations is discussed in Section3and the quantization of the moment map is presented in Section4.Main results about the quantization of symplecticﬁbrations are given in Section5,examples are discussed in the last section.Notations.Repeated indices assume summation.Grading andﬁltration of the Weyl algebra bundle are Z–grading and Z–ﬁltration,we do not use the natural Z2–grading on the diﬀerential forms.Let E be a bundle over some manifold M.Then A n(M,E)denotes C∞–sections of n–form bundle with values in the bundle E,A k(M,E)=Γ(B,Λk T∗B⊗E).A n(M)denote the bundle of n–forms on M,andA(M,E)=⊕∞n=0A n(M,E).The term“connection”is used in two senses,for a covariant derivative on any vector bundle, usually denoted by∇and also for a connection on aﬁbre bundle i.e.a splitting of the tangent bundle to the total space of aﬁbre bundle into a sum of a vertical and a horizontal subbundles.Acknowledgments.I am deeply grateful to Ezra Getzler,Richard Melrose,Boris Tsygan and David Vogan for help and encouragement.My thanks also go to Alex Astashkevich,Paul Bressler, David Ellwood,Pavel Etingof,Daniel Grieser,Dima Kaledin,Eugene Lerman,and Andras Szenes for many fruitful discussions.This work started at mathematics department at MIT and completed during my stay at IHES as a European postdoctoral fellow.It is a pleasure to acknowledge the support of these two institutions.2.Generalities on deformation quantization.The subject of this section becomes nowadays fairly standard(see for example an excellent intro-duction to Fedosov quantization[22]).2.1.Weyl algebra of a vector space.Let E be a vector space with a non–degenerate skew-symmetric formω.The algebra of polynomials on E is the algebra of symmetric powers of E∗,S(E∗),and it has a skew-symmetric form on it which6OLGA KRA VCHENKOis dual toω.Let e be a point in E and{e i}denote its linear coordinates in E with respect to some ﬁxed basis.Then{e i}deﬁne a basis in E∗.Letωij be the matrix for the skew-symmetric form on E∗.Let us consider the power series in with values in S(E∗):Deﬁnition2.1.The Weyl algebra W(E∗)of a vector space E∗is an associative algebraW(E∗)=S(E∗)[[ ]]:a(e, )= k≥0a k(e) k,with the product structure given by the Moyal–Vey product:a◦b(e, )=exp{−i∂x k∂[v,a]for any a∈W(E∗)Proof.Indeed,∂2[ωij e j,a].So for any derivation one can get a formula:Da=i2ωij e i De j,a].DEFORMATION QUANTIZATION OF SYMPLECTIC FIBRATIONS.7One can deﬁne two natural operators on the algebra W(E∗)⊗ΛE∗:δandδ∗of degree−1and1 correspondingly.δis the lift of the“identity”operatoru:e i⊗1→1⊗dx iandδ∗is the lift of its inverse.On monomials e i1⊗...⊗e i m⊗dx j1∧...∧dx j n∈W m(E∗)⊗Λn E∗δandδ∗can be written as follows:δ:e i1⊗...⊗e i m⊗dx j1∧...∧dx j n→mk=1e i1⊗... e i k...⊗e i m⊗dx i k∧dx j1∧...∧dx j nδ∗:e i1⊗...⊗e i m⊗dx j1∧...dx j n→nl=1(−1)l e j l⊗e i1⊗...⊗e i m⊗dx j1∧... dx j l...∧dx j n.Let a0be a projection of a∈W(E∗)⊗ΛE∗to gr0W(E∗)⊗Λ0E∗,which is the center of the algebra, i.e.the summands in a which do not contain neither e-s or dx-s.Lemma2.5.Operatorsδandδ∗have the following properties:δa=dx j ∂aωkl e k dx l,a],δ∗a=y jι∂m+n(δδ∗a+δ∗δa)+a0.2.2.Symplectic connections(covariant derivatives).The term symplectic connection in this section in fact must be changed to symplectic covariant de-rivative to avoid confusion with another symplectic connection notion in the next Chapter.However there is already an established practice to call a covariant derivative a connection which we decided to follow here.We hope that one can get used to distinguish one from the other from the context.Let us consider connections on a manifold M.Proposition2.6.Letωbe a skew–symmetric2–form on T M.Thenωmust be closed in order for torsion–free connection∇preserving this form to exist.Proof.The skew–symmetry ofωis the following condition:ω(X,Y)=−ω(Y,X).The connection ∇is torsion–free when∇X Y−∇Y X=[X,Y].Suppose such∇exists.Then it preserves the form ωwhen∇ω=0.This means that for all X,Y,Z∈T M:∇X(ω(Y,Z))=ω(∇X Y,Z)+ω(Y,∇X Z)(3)8OLGA KRA VCHENKOSinceω(Y,Z)is a function∇X(ω(Y,Z))=Xω(Y,Z).Then,Xω(Y,Z)−Yω(X,Z)+Zω(X,Y)=ω(∇X Y,Z)−ω(∇X Z,Y)−ω(∇Y X,Z)+ω(∇Y Z,X)+ω(∇Z X,Y)−ω(∇Z Y,X)=ω([X,Y],Z)−ω([X,Z],Y)+ω([Y,Z],X)which is exactly the condition dω=0.Notice,that in the Riemannian case when the form is symmetric,there is a unique torsion free connection compatible with the form,the Levi–Civita connection.In the case of a skew–symmetric form there are plenty of connections compatible with the form,provided that the form is closed.So the statement of uniqueness of Levi–Civita connection in the Riemannian case is substituted by the requirement for the form to be closed in the skew–symmetric setting.Here we are mostly interested in the case when M is a symplectic manifold,i.e.there is a symplectic formω(a closed and nondegenerate2-form on T M).Deﬁnition2.7.A connection which preserves a symplectic form is called a symplectic connection.Any connection on a symplectic manifold gives rise to a symplectic connection:Proposition2.8.[23][25].Letωbe a closed nondegenerate2–form.Then for every connection∇there exists a three–tensor S,such that˜∇=∇+Sis a connection on T M compatible withω.Then for X,Y∈T Mˆ∇X Y=˜∇X Y−12{(∇Xω)(Y,.)}♯,where♯:T∗M→T M is the inverse to♭,given by:♭:T M→T∗Mu♭=ω(u,.)for u∈T M.Symplectic connections form an aﬃne space with the associated vector space A1(M,sp(2n)),Lie algebra sp(2n)valued one–forms on M.DEFORMATION QUANTIZATION OF SYMPLECTIC FIBRATIONS.92.3.Deformation quantization of a symplectic manifold.Let M2n be a symplectic manifold with a symplectic formω.In local coordinates at a point x:ω=ωij dx i∧dx j.The symplectic form on a manifold M deﬁnes a Poisson bracket on functions on M.For any two functions u,v∈C∞(M):{u,v}=ωij ∂u∂x j(4)where(ωij)=(ωij)−1.We can deﬁne the bundle of Weyl algebras W M,with theﬁbre at a point x∈M being the Weyl algebra of T∗x M.Let{e1,...e2n}be2n generators in T∗x M,corresponding to dx i.The formωij deﬁnes pointwise Moyal–Vey product.Theﬁltration and the grading in W M are inherited from W(T∗x M)at each point x∈M.Denote by W i the i-th graded component in W M:W M=⊕i W iA symplectic connection,∇,satisfying(3)can be naturally lifted to act on any symmetric power of the cotangent bundle(by the Leibnitz rule)and since the cotangent bundle T∗M∼=W1we can lift ∇to be an operator on sectionsΓ(M,W i)with values inΓ(M,W i⊗T∗M).By abuse of notations this operator is also called∇.It preserves the grading,in other words it is an operator of degree zero.It is clear that in general this connection is notﬂat:∇2=0.Fedosov’s idea is that for W M bundle one can add to the initial symplectic connection some operators not preserving the grading so that the sum gives aﬂat connection on the Weyl bundle.Theorem2.9.(Fedosov.)There is a unique set of operators r k:Γ(M,W i)→Γ(M,T∗M⊗W i+k) such thatD=−δ+∇+r1+r2+ (5)is aﬂat connection andδ∗r i=0.There is a one-to-one correspondence between formal series in with coeﬃcients in smooth functions C∞(M)and horizontal sections of this connection:Q:C∞(M)[[ ]]→Γﬂat(M,W M).(6) Main idea of the proof is to use the following complex:0→Γ(M,W)δ→A1(M,W)δ→A2(M,W)δ→ (7)This complex is exact sinceδis homotopic to identity byδ∗.An equation for r i for each i>1has the formδ(r i)=function(∇,r1,...,r i−1).(8)However it is not diﬃcult to show that this function is in the kernel ofδhence r i exists.First few steps in the construction of D,itsﬂat sections and∗–product in coordinates are given in the Appendix.10OLGA KRA VCHENKOThe noncommutative structure on the Weyl bundle determines a∗–product on functions by this correspondence,namely for two functions f,g∈C∞(M)[[ ]];f∗g=Q−1(Q(f)◦Q(g)).(9)In fact,the equation D2=0is just the Maurer–Cartan equation for aﬂat connection.One can see the analogy with Kazhdan connection[15]on the algebra of formal vectorﬁelds.Notice that δ=dx i∂[γ,·]◦,(10)whereγ∈Γ(M,T∗M⊗W).Then the equation D2=0becomes:1dΓ+[γ,γ]◦We deﬁne a grading on W B⊗C[[ ]]L as on W B.The operator R L could have a degree if it changes the power of ,and since the degree of is2it could only be even:R L= R L2k,k≤0whereR L2k:A q(B,gr·(L))→A q+2(B,gr·+2k(L)).One should add an extra term in the equation on r i(8)for even i=2k one should add an extra termδ(r2k)=function(∇,r1,...,r2k−1)+R L2k.Then like in Theorem2.9we can consider a complex similar to(7)0→Γ(B,W B⊗C[[ ]]L)δ→A1(B,W B⊗C[[ ]]L)δ→A2(B,W B⊗C[[ ]]L)δ→...withδacting only in W B.This complex is still exact(The⊗–product is over aﬁeld C[[ ]]).Since δR L=0one canﬁnd a preimage of R L and the reasoning is exactly as before.Theﬂat connection and the correspondingﬂat sections are constructed similarly to the way it is described in Appendix.However in the case when L has a Lie algebra structure and R L is an inner action of the form iR L=φαπpr2.For theﬁbre over b∈B,F b=π−1(b),letφα(b)denote the restriction ofφαto F b followed byprojection onto F,φα(b):F b→F.Thenφβα(b)=φβ(b)◦φα(b)−1∈Symp(F,σ)for allα,βand b∈Uα Uβ.Ifπ:M→B is a symplecticﬁbration then eachﬁbre F b carries a symplectic structureσb∈Ω2(F b) deﬁned byσb=φα(b)∗σfor b∈Uα.The form is independent ofαas follows from the deﬁnition.Also,if there is a G–invariant symplectic torsion–free connection∇F on F it deﬁnes a symplectic torsion–free connection∇b on eachﬁbre F b.Deﬁnition3.2.A symplectic formωon the total space M of a symplecticﬁbration is called compatible with theﬁbrationπif eachﬁbre(F b,σb)is a symplectic submanifold of(M,ω),withσb being the restriction ofωto F b.Symplectic connections.Each symplectic form compatible with a symplecticﬁbrationπ:M→B deﬁnes a connection on it,i.e.a choice of splitting of the following short exact sequence of vector bundles:0→V M→T M→π∗T B→0.Here V M is the canonically deﬁned bundle of vertical tangent vectors,i.e.thoseﬁelds which vanish on functions coming from the base.In other words the connection is the splitting of the tangent bundle into the sumΓ:T M=H M⊕V M(12)such thatπ∗T B=H M.The connection(12)is compatible with the symplectic form if at each point x∈M:H x M:={X∈T x M:Ω(X,V)=0for all V∈V x M}.So each symplectic form whose restriction onﬁbres is nondegenerate deﬁnes a compatible connection. Namely,the horizontal subbundle consists of all vectorﬁelds which are perpendicular to the vertical ones with respect to the symplectic form.Ingredients:a connection on a principal bundle and a Hamiltonian action along the ﬁbres.Symplecticﬁbrations are associatedﬁbre bundles to the principal bundles with a structure group being the group of symplectomorphisms of theﬁbre,so we have a principal G–bundle and a symplectic manifold(F,σ)to start with.Let usﬁrst consider a principal G–bundle,i.e.a smooth manifold P with a smooth action P×G−→P which is free and transitive.Then the quotient P/G=B is a manifold.For the principal bundle a connection can be deﬁne by a so called ly, theﬁbres of a vertical subbundle V P are naturally identiﬁed with g under the map:g→V ect(P) given by the inﬁnitesimal action of G on P.X∈g→ˆX∈V ect(P).Hence the horizontal subbundle H P can be described not only as a kernel of the projection operator P r:T P→V P,but also as a kernel of a connection1–form:λ:T P→g.It is a G–invariant form on the principal G–bundle P with values in the Lie algebra g,such thatλ=X,for X∈g.ıˆXNow let G act on a symplectic manifold(F,σ)by symplectomorphisms,i.e.there is a group homomorphismG→Symp(F,σ):g→ψg.The inﬁnitesimal action determines the Lie algebra homomorphismdg→V ect(F,σ):X→ˆX,deﬁned byˆX=where T∈A2(B):T(X,Y)=−P r([X H,Y H]),X,Y∈T B and H V is a Hamiltonian function of a vectorﬁeld V with respect to the formσb,deﬁned byµF.Remark3.5.Notice thatσb is nonzero only on vertical vectors,while H T is nonzero only on horizontal ones.This extra term H T is needed for the form to be closed,ǫmakes the formΩǫto be nondegenerate.Proof.Sketch(for the full proof see[18]or[24]).The main idea is to use the so called Weinstein universal phase space–W=P×g∗.Given a connection on P the space W could be identiﬁed with the vertical subbundle of the cotangent bundleW=P×G T∗G=V∗P.A connection is a splittingΓ:T P=H P⊕V P and V∗P is deﬁned as one–forms which vanish on horizontal vectors:V∗P=(H P)⊥.Hence it has a G-equivariant symplectic form coming from the canonical symplectic form on T∗P.Moreover,the action of the group G on W is Hamiltonian(see for example[1]).The moment mapµW:W→g∗is given by the projectionpr g∗:W→g∗.Then the symplectic reduction of W×F at0value of the moment mapµ=µW+µF is exactly M=P×G F,and the symplectic form on M is inherited from W.The explicit formula is obtained in the following way.Let the connectionΓbe given by a con-nection1–form,λp:T p P→g.It determines a horizontal subbundle in T P by H p P={v∈T p P|λp(v)=0}.V∗P=(H P)⊥is also deﬁned byλ.The connectionλ:T P→g and together with the action ρ:g→V P deﬁne the injectionıλ:V∗P֒→T∗P.By deﬁnition of the connection1–form,this injection is equivariant under the action of G and hence the2–formωλ=ı∗λωcan∈A2(V∗P)is invariant under the action of G.This pull–back of the canonical symplectic form on T∗P gives a closed2–form on V∗P.The canonical1–formαon T∗P is deﬁned as follows.Let(p,s p)be a point in T∗p P,let also v be a tangent vectorﬁeld in the tangent bundleπ:T(T∗P)→T∗P then at the point(p,s p)<α|v>(p,sp )=−<s p|π∗v>p.Then the pullback of the canonical one–form to V∗P is<ı∗λα|v>(p,sp )=−<pr g∗s p|λ(π∗v)>p soωλ=−d<pr g∗|λ>.(17)Now the form on W×F isωλ+σ.We now want the form on the reduced space M=(W×F)//G at the regular value of the moment mapµW+µF=0.SinceµW=pr g∗the form on M becomesωΓ=d<µF|λ>+σ.The one–form <µF |λ>can be rewritten as the Hamiltonian H λ.It should be understood in the following way.The connection 1–form λ:T P →g deﬁnes the connection on the associated bundle M =P ×G F .The horizontal subbundle in T M is the image of H P under the map P ×F →M .By abuse of notation we call the map T M →g also λ.From now on λis a 1–form on M with values in the Lie algebra g .Hence H λis a 1–form on M.Its diﬀerential gives a two form d <µF |λ>=d (H λ).Applied to two vectors V,W ∈T M :d (H λ)(V,W )=V H λ(W )−V H λ(W )−H λ([V,W ])using (15)we get that it is nonzero only on horizontal vectors and gives the Hamiltonian of the curvature.We see also that ωΓrestricted to ﬁbres gives the symplectic form on the ﬁbres.This construction is quite general [18]:The symplectic ﬁbrations with connection constructed this way turn out to include all symplectic ﬁbre bundles with connection for which the holonomy group is a ﬁnite dimensional Lie group.In local coordinates.Let us take a point x ∈M .One can introduce a local frame {f α}of vertical tangent bundle V M and a local frame {e i }in T B at a point b =π(x )of B ,with dual frames {f α}and {e i }.Using the connection we obtain a local frame on the tangent bundle T M =π∗T B ⊕V M at a point x .Then the form can be written as a block matrix:Ωǫ= π∗ωB +ǫ2H T 00ǫ2σb (18)Hence the corresponding Poisson bracket is also a block matrix: (π∗ωB +ǫ2H T )−100ǫ−2σ−1b We see that the Moyal product with respect to this form is a product of those on the base and on the ﬁbres.As for the connection,let Γδβγbe the Christoﬀel symbols of the symplectic connection on T F b preserving the form σb along the ﬁbre F b .This connection can be written in the local coordinates as follows:∇F =d F +i (f ∗g −g ∗f ),for f,g ∈A (F ).Let G be a group acting on F Hamiltonially(Deﬁnition3.3).Let g→C∞(F)be its moment map. There is an induced action of G on A (F).We want to quantize the moment map,namely,get a Lie algebra map from the algebra g to the quantized algebra A (F).However it is possible only up-to a two–cocycle in C∞(F)[[ ]],so we get a projective representation or instead we should consider a central extension of A (F).We could also slightly change a deﬁnition of a quantum moment map. We eliminate the central elements by considering the map into the adjoint representation of A (F), the inner automorphisms Inn A (F).They obviously inherit the Lie algebra structure from A (F), soDeﬁnition4.1.A quantum moment map is a map of Lie algebrasµ∨:g→Inn A (F)with the correspondence principle:lim →0µ∨(X)(f)={H X,f}for X∈g and f∈A (F).Remark4.2.This deﬁnition could be reformulated through an isomorphism of associative algebras:µ:U(g)→A (F),such that on vectorﬁelds it gives:µ(X)∨(f)=[µ(X),f]∗.Proposition4.3.Let G act Hamiltonially on(F,σ)with the Hamiltonian function H X.Let(F,Σ=σ+Σ∞i=1 iσ1,∇)be an F–manifold such that∇is a G–invariant connection and assume that one can deﬁne functions H i X,i=1,2,···as followsιˆσi=dH i X.(20)XLet A (F)be the algebra of quantized functions.Then the quantum moment map is given byµ(X)=H X+Σ∞i=1 i H i X(21) Alsoµ∨:g→Inn A (F),µ(X)∨(f)=[µ(X),f]∗is a homomorphism of Lie algebras:µ∨([X,Y])=[µ∨(X),µ∨(Y)]∗.(22) Moreover,there are no higher terms in :[µ(X),f]∗=[H X,f]∗={H X,f}.(23)Proof.1We are going to prove(21–23)by lifting the Hamiltonian H X to a section of the Weyl algebra bundle,since the∗–product on C∞(F)is deﬁned through the the Weyl algebra bundle(6) and(9).Namely,letΓD–ﬂat(F,W F)be the space ofﬂat sections Fedosov connection D constructed from∇corresponding to the quantization of F with the characteristic classΣ.Then there is a one–to–one correspondenceQ:C∞(F)[[ ]]→ΓD–ﬂat(F,W F)which deﬁnes a product on Ff∗g=Q−1(Q(f)◦Q(g)).The structure of a Lie algebra in Weyl algebra bundle W F is deﬁned by theﬁbrewise commutator[a,b]◦=a◦b−b◦a,for a,b∈Γ(F,W F)Recall that the map H:g→C∞(F)is given by the conditionιˆXσ=dH X,(24) whereˆX is aﬁeld corresponding to X under the map g→V ect(F)(13).Weﬁnd the image of a Hamiltonian H X inΓD–ﬂat(F,W F)generalizing to the case of a deformed symplectic form the proof of Fedosov[13,Propositions5.8.1,2],[14].By construction the Fedosov connection D is G–equivariant since any element of the group G preserves the initial symplectic connection∇.An easy calculation shows also thatﬂat sections of an equivariant connection D are also equivariant.It also means that the Lie derivative of D is zero with respect to any vectorﬁeldˆX:[LˆX,D]◦=0.Since LˆX and DˆX=ıˆXD+DıˆXareﬁrst order derivations commuting with D we canﬁnd ananalogue of the Cartan homotopy formula for the Lie derivative on forms with values in the Weylalgebra bundle.The diﬀerence of LˆX and DˆXcould only be an inner automorphism of the Weylalgebra bundle,we denote it[Q(X),·]◦:LˆXa=(ıX D+DıX)a+[Q(X),a]◦.(25) It is easy to see that the equality(25)is true in Darboux coordinates chosen so that theﬁeldˆX isjust a pure derivation in the direction of only one of the coordinates.Then D=D0=d+δand Q(X)=Q0(X)=central section(X, )−ıXδ.Let D=D0+[∆γ,·]◦be anotherﬂat connection,∆γbeing an equivariant one form in W F.Then since the LˆXdoes not change and commutes with the new D as well,the right hand side of(25)must not change either,so one has to subtractıX∆γfrom Q0(X).We want to show that we could chose a central section(X, )in Q(X)to be equal toµ(X)= H X+ ···so that Q(X)=µ(X)−ıXγbecomes a quantization of the moment map,that is aﬂat section corresponding to H X.Recall that locally we can write a connection D as Df=d f+i[γ,γ]◦[γ,·]◦)+(d+i[γ,{H X+ H1X+···}]◦−i[γ,γ]◦[γ,γ]◦。

注塑成型的疑难问题和及建议解决方案.

Injection moulding troubleshooter注塑的疑难问题This easy to use troubleshooting guide gives you initial advice for solving injection moulding difficulties. Please click on the name of your problem in the following list to view a detailed description include photograph and proposals for solution.这种易于使用的疑难问题解决指南，为您提供解决注塑方面各种难题的初步建议。

请在以下清单中点击您遇到的问题，查看具体描述（包括照片）以及关于解决方案的建议：Problem Short descriptionBlack specks 黑色斑点 Punctiform or lamellar deposits on the surface of the thermoplastic moulding.热塑性成型件表面上的点状或片状沉积物。

Charring streaks烧焦痕Charring streaks on the surface of the plastic moulding in the form of silvery or light brown to dark brown discolourations. 塑料成型件表面的炭化条纹，具体形态为银色或浅棕色至深棕色污点。

Cold slug 冷料痕Marks, usually close to the sprue, in the form of a comet's tail.斑纹，通常接近浇道，形状类似彗星的尾部。

Coloured streaks色差痕Differences in colour on the surface of moulded plastic parts.注塑部件表面的颜色差异。

Deformation of pearlite(形变珠光体)

INTRODUCTION E V E N though the i r o n c a r b i d e (cementite) in p e a r litic c a r b o n s t e e l s occupies only o n e - t e n t h to one-fifth of t h e i r v o l u m e , the d e f o r m a t i o n and s t r e n g t h of such s t e e l s a r e profoundly affected by its p r e s e n c e . The work h a r d e n i n g of d r a w n p e a r l i t i c s t e e l w i r e s is an exponential function of t r u e s t r a i n , ~ as c o n t r a s t e d to that of i r o n w i r e , which work h a r d e n s l i n e a r l y . 2-4 At a t r u e s t r a i n * of 4, the p e a r l i t i c wire is 4 to 5 *In thispaper, true strainis definedas e = ln(Ao/A), whereA andAoare the finaland originalcrosssectionalareas. t i m e s s t r o n g e r than f e r r i t i c w i r e . C e l l u l a r c o a l e s cence, the kind of d y n a m i c r e c o v e r y which c o n t r o l s the s u b s t r u c t u r a l r e f i n e m e n t of f e r r i t e in c a r b o n f r e e i r o n d u r i n g cold working, 2'5 a p p e a r s to be a l most completely inhibited in p e a r l i t e . 1 Also, effects of the s c a l e oon s t r e n g t h a r e c o m p l e t e l y d i f f e r e n t . The s t r e n g t h of p e a r l i t e is p r o p o r t i o n a l to the r e c i p r o c a l of the s q u a r e root of the i n t e r l a m e l l a r spacing, ~ w h e r e a s the s t r e n g t h of f e r r i t e is p r o p o r t i o n a l to the r e c i p r o c a l of the l i n e a r - i n t e r c e p t cell size,* at l e a s t for d r a w n w i r e s , z *Thisis not the wholestory,however:see Langfordand Cohen. 6 DEFORMATION AND STRENGTH OF CEMENTITE In o r d e r for p e a r l i t e to work h a r d e n as it does, l a m e l l a r c e m e n t i t e m u s t be ductile,* n o n d e f o r m i n g , *GilSevillanohas recentlyreviewedand presentedevidencefor the plasticity of pearliticcementite.42 h a r d p a r t i c l e s do not b r i n g about e x p o n e n t i a l s t r a i n GEORGE LANGFORDis with the Department of Materials Engineering, Drexel University, Philadelphia, PA 19014. This paper is based on a presentation made at a symposium on "Mechanical-Thermal Processing and Dislocation Substructure Strengthening," held at the Annual Meeting in Las Vegas, Nevada, on February.23, 1976, under the sponsorship of the TMS/IMDHeat Treating Committee. METALLURGICALTRANSACTIONSA

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Wi*nt Wint
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References: [1] Anand, L., Gurtin, M.E., 2019. “A theory of amorphous solids undergoing large deformations, with application to polymeric glasses.” International Journal of Solids and Structures 40, 1465-1487.
[3] Lubliner, J. Plasticity Theory. 1990. Macmillan Publishing Company. (Chapter 8)
[4] Abaqus 6.5-4 Documentation “Getting Started with ABAQUS/EXPLICIT.” Hibbitt, Karlsson & Sorensen,INC.
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Constitutive Theory – Framework
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– Euclidean Space
– Amorphous Solids: material are invariant under all rotations of the Relaxed and Reference Configuration
Smooth time-dependent rigid transformations of the Eulerian Space:
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Principle of Relativity: relation of the two motions is equivalent Q(t) Relative motion of two observers Eulerian bases
Experimental Results – Polycarbonate
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COMPRESSION
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det F p 1 tr L p 0
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Principle of Objectivity Principle of Material Frame Indifference
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Constitutive Theory – Thermodynamic Restrictions and Flow Rule
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Kinematics – Multiplicative Decomposition of the Deformation Gradient II
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Fy(X,t) Deformation Gradient
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vy(X,t) Velocity tensor LgrvadF F1 Velocity Gradient
Te ReT TRe
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Constitutive Equations
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material parameters
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Constitutive Equations
[2] Boyce, M. C., Arruda, E. M., 1990. “An experimental and analytical investigation of the large strain compressive and tensile response of glassy polymers.” Polymer Engineering and Science 30 (20), 12881298.
“relaxed
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material parameters
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Constitutive prescription
eGEe o21/2KtrEe2
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Te2GEeoKtrEe1 Te Stress conjugate to Ee
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FLOW RULE:
Large Deformation Plasticity of Amorphous Solids, with Application
and Implementation into Abaqus
Kristin M. Myers January 11, 2019 Plasticity ES 246 - Harvard
P
Dissipation Inequality and Constitutive Framework
• 2nd Law of Thermodynamics: The temporal increase in free energy ψ of any part P be less than or equal to the power expended on P
– Engineering Collagen Scaffolds (i.e. skin, nerve, tendon etc.)