The structure of the Bousfield lattice
Strained tetragonal states and Bain paths in metals
a r X i v :m t r l -t h /9609008v 1 25 S e p 1996Strained tetragonal states and Bain paths in metalsP.Alippi 1,P.M.Marcus 1,2,and M.Scheffler 11Fritz-Haber-Institut der Max-Planck-Gesellschaft,Faradayweg 4-6,D-14195Berlin-Dahlem,Germany2IBM Research Center,Yorktown Heights,N.Y.10598,USA(February 6,2008)Paths of tetragonal states between two phases of a material,such as bcc and fcc,are called Bain paths.Two simple Bain paths can be defined in terms of special imposed stresses,one of which applies directly to strained epitaxial films.Each path goes far into the range of nonlinear elasticity and reaches a range of structural parameters in which the structure is inherently unstable.In this paper we identify and analyze the general properties of these paths by density functional theory.Special examples include vanadium,cobalt and copper,and the epitaxial path is used to identify an epitaxial film as related uniquely to a bulk phase.PACS numbers:64.70.Kb,61.50.Ks,68.56.Eq,62.20.DcPseudomorphic epitaxy of a cubic or tetragonal (001)film typically results in a strained tetragonal structure.If the stresses on the tetragonal state vanish and also the state corresponds to a local minimum of energy with respect to tetragonal deformations,the structure will be called a tetragonal phase .Such a phase will be stable or metastable depending on whether it has the lowest energy compared to other minima.Frequently metals have two tetragonal phases;sometimes both are cubic,e.g.,bcc and fcc Na and Rb [1,2];sometimes one is cubic and the other phase is tetragonal,e.g.,Ti and V [2,3].They can,of course,also have phases with other structures,e.g.,Ti also has a hcp phase.Many paths can go from one tetragonal phase to the other.If the geometries along such a path have tetrag-onal symmetry and if they connect bcc and fcc phases,the paths have been called Bain paths [4].A purpose of the present work is to define and discuss a partic-ular Bain path which will be called the epitaxial Bain path (EBP).The EBP is produced by isotropic stress or strain in the (001)plane of tetragonal phases accompa-nied by vanishing stress perpendicular to the plane,such as pseudomorphic epitaxy produces on an (001)cubic or tetragonal film.Epitaxy provides a valuable means of stabilizing metastable phases and of putting phases under very large strains,both tensile and compressive in the plane of the epitaxy.The EBP of a material identifies the phase that has been strained,checks quantitatively the elastic behavior,which can be highly nonlinear,and predicts which phase of the material will form on a given substrate.Thus,in order to understand the properties of epitaxial films and new materials,the knowledge of the EBP is indispensable.A different Bain path has long been discussed,partic-ularly by Milstein [1],in which uniaxial stress is applied to a tetragonal state along the [001]axis accompanied by zero stress in the (001)plane;this path is conveniently called the uniaxial Bain path (UBP).We compare the two paths,EBP and UBP,which are both physically realizable.We show that both have the same lowestpossible maximum energy or barrier energy of all Bain paths between the two tetragonal phases.However the EBP has a special value in relating strained tetragonal structures produced by epitaxy to particular tetragonal phases.Depending on where the tetragonal structure lies on the EBP,we shall show that a clear choice can be made which classifies the structure as a strained form of a particular phase.Thus for materials with both a bcc and a fcc phase,the bulk (i.e.,the interior)of an epitax-ial film can be uniquely classified as a strained bcc phase or a strained fcc phase.Experimental geometry analyses of the bulk structure are frequently made by low-energy electron diffraction of ultrathin epitaxial films,which can then be placed on the EBP in the tetragonal plane to identify the phase of which it is the strained form.We employ density functional theory together with the full-potential linearized augmented plane wave method [5].An important advantage of a first-principles calculation,i.e.,a calculation with no adjustable parame-ters,is that it applies just as reliably to highly strained or even unstable geometries,as it does to the unconstrained ground state,i.e.,to the stable phase.The same relia-bility cannot be asserted for empirically adjusted poten-tials,such as the empirical pseudopotential model used in Ref.[1],which is adjusted to reproduce ground-state properties of only certain structures.The first-principles calculation provides a check on the validity of empirical potentials far from the structural ground state and of linear elasticity theory.The calculation proceeds by evaluating the total energy E (a,c )in eV/atom as a function of the tetragonal lat-tice constants a and c .Contour lines of constant E are plotted in Fig.1for vanadium on the (a/a bcc -V/V bcc )plane,where V =ca 2/2is the volume per atom.The contour lines show clearly two minima,one at the bcc structure and one at a tetragonal structure,and between them a saddle point at the fcc structure.Thus vanadium has two fairly deep and well-separated minima and so is a good choice to illustrate the tetragonal paths between minima [6].Since E has an extremum at the cubic struc-tures [3,7],the saddle point will occur exactly at the fcc structure.AlsoplottedinFig.1aretheEBP and the UBP.The EBP is calculated from the minima of E (c )at constant a ,which corresponds to the epitaxial situation,i.e.,a is held fixed and c adjusts to minimize E (c )and thus makes the out-of-plane stress,i.e.,the stress along [001],vanish.Conversely,the UBP calculation fixes c and lets a adjust to minimize E (a )making the in-plane stress vanish.Both paths go through the minima and the saddle point,hence the maximum energy on each path is the same.The saddle point is called in Ref.[1]“a special un-stressed tetragonal state ...at a local energy maxi-mum”[8].Obviously any path on the tetragonal plane that does not go through the saddle point must go through states of higher energy than the saddle point,since it will cross contours in either the upper or lower central sectors formed by the saddle-point contours at E =0.29eV/atom.Two common paths that do not go through the saddle point,hence have higher maximum energies than the EBP or UBP,are : 1.the constant volume paths at the volumes of the minima,which differ from the saddle-point volume.These paths have been called volume conserving Bain paths [2,3];2.the hy-drostatic pressure path,which has equal in-plane and out-of-plane stress components.Figure 2plots E vs.a along the EBP and UBP.Also indicated is the range from a =2.58˚A (c/a =1.6)to a =2.66˚A (c/a =1.2)for which we find along the EPB the tetragonal structure to be inherently unstable [9].In this range the slope of the EBP changes sign,and the stability condition˜c 11˜c 33−˜c 213≥0(1)is violated.Here ˜c ij are calculated from ˜c 11=(a 2/V )∂2E (a,c )/∂a 2,˜c 13=(ac/V )∂2E (a,c )/∂a∂c ,˜c 33=(c 2/V )∂2E (a,c )/∂c 2for the strained lattice [10].Tetragonal states outside the range of Eq.(1)cannot be stabilized with any applied stresses.When a path on the tetragonal plane reaches the boundaries of this unstable region,the tetragonal crystal becomes unstable and an epitaxial film will not be able to form many layers at the bulk spacing.Epitaxial films strained to a state near the unstable region should show large changes in the phonon spectra as elastic constants weaken,which would be in-teresting to study.One consequence of the existence of an unstable region is that a clear conceptual distinction can be made be-tween strained states of the phase at lower c/a (for vana-dium the bcc phase)and strained states of the larger c/a phase (for vanadium at c/a =1.83).If there was a con-tinuous path of allowed states between the two phases,there would not be a clear criterion for regarding an ob-served strained state as strained from a particular one of the two phases.A second consequence of the unstable region is that thestrained system never gets to the saddle point,but trans-forms to another strained phase at an energy at or below the energy of the state where the path strikes the bound-ary of the unstable region.Thus for the EBP and UBP the breakdown energy is at least 0.05eV/atom lower than the saddle-point energy,and it is lower on the UBP.The EBP and UBP can be found for other metals by similar calculations;for Cu and Co our results are shown in Fig.3and Fig.4.The topology of the E (a,c )sur-face of cobalt is similar to the one of vanadium,with the two minima and the saddle point corresponding to the structures where the EBP and the UBP intersect.In this case,however,the stable cubic phase is face-centered (c/a =√2(and a >a fcc );we also show that the film is too far from the bcc region of the EBP to be regarded as strained bcc.Thus,we conclude that the properties of such Cu/Pd(001)film are not bulk like,but instead stabilizedby the(still ultrathin)film properties.In Fig.4(as in Fig.3)the epitaxial paths from linear elasticity theory are also drawn for fcc Cu(and Co). Epitaxial strain on a cubic(001)surface in the linear elastic approximation says normal stress vanishes,hence δc/c=−(2ν/(1−ν))δa/a,whereνis the Poisson ra-tio.A comparison with the calculated EBP in bothfig-ures shows how the linear approximation fails to be valid forδa/a equals to few percent.In the analysis of the Cu/Pd(100)film,it is evident that thefilm is already far from the linear elastic regime.Generalizations of this calculation of physically real-izable paths between phases are possible with existing stein[19]has considered the instability of tetragonal structures with respect to orthorhombic defor-mations and found such instabilities on the UBP outside the range of c/a between the two minima;these insta-bilities probably exist on the EBP also,but have not as yet been studied.It looks practicable to explore with present codes even more general paths of homogeneous strain between tetragonal phases that make more gen-eral deformations of the unit mesh than orthorhombic deformations.Possibly a path of deformed lattices could be found with lower maximum energies,as the path of trigonal lattices of Ref.[7].It would also be possible to find epitaxial paths of tetragonal states for slabs of afi-nite number of layers,which would include relaxations of surface layers.In summary,paths of tetragonal states between two tetragonal phases include two particularly simple cases, namely,those produced by epitaxial(isotropic biaxial) stress and by uniaxial stress.Both cases correspond to maintaining certain surfaces stress-free and both have minimal maximum energies at a saddle point of energy that both pass through.Along these paths a range of in-stability exists,which separates states that can be clearly regarded as strained from one or the other of the two phases.The epitaxial path permits definite identifica-tion of the equilibrium phases of strained epitaxialfilms and could usefully be found for all metals with tetragonal phases.0.76 0.82 0.88 0.941.00 1.06a/a bccV /V b c cFIG.1.Contour lines of constant E in eV/atom for vana-dium on the tetragonal plane showing a bcc minimum (thezero of E )with a bcc =2.93˚A and V bcc =12.6˚A 3(c/a =1),a minimum at a =2.41˚A (c/a =1.83)and V =12.8˚A 3,anda saddle point at a =2.65˚A (c/a =(2)1。
《翻译的语言学派》
总 论
西方翻译的两大翻译学派——语言学派和文艺学派贯穿了整个西方翻译史。翻译的语言学派又被称作“翻译科学派”。1959年雅可布逊发表他的著名论文《翻译的语言观》开始到1972年结束。西方译论的一大特点即与语言学同步发展 。
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一、布拉格学派与雅可布逊
成立：1926年10月6日，布拉格语方学会（The Linguistic Circle of Prague）召开第一次会议，布拉格卡罗林大学的英语语言和文学教授维伦·马泰休斯宣布了该学会的成立，也标志着布拉格语言学派的诞生。布拉格语言学派是继索绪尔之后最有影响的学派，其突出的贡献是创建了音位学．由于强调语言的交际功能和语言成分的区分功能，所以又常被称作功能主义者或功能语法学派。
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(4)语意走失的四个方面： a.原文内容涉及到本国特有的自然环境、社会制度、文化习俗，译文意思就必然走失； b.每一种语言都有自己的证明音、语法、词汇体系和运用方式，各种语言对世界上各种事物和概念的分类方法也不同。各种语言的词句很难在文体、感情色彩、抽象程度、评价尺度等四个方面完全对应；
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卡的“等值”翻译理论的意义
（一）从某一侧面反映翻译的本质在于确立“等值”关系；（二）等值关系确立并非静态地而是动态地把握；（三）对于从翻译学角度探讨双语转换机制的建立具有借鉴作用；（四）区别了翻译和转换两个概念。
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纽马克简介
彼得·纽马克（Peter Newmark ,1916-）是英国著名的翻译理论家和翻译教育家。他后来提出了著名的“交际翻译”和“语义翻译”法，20世纪90年代又提出“关联翻译法”，标志着他的翻译理论渐趋系统和完善。 纽马克的作品：论文 《翻译问题探讨》《交际性和语义性翻译》《翻译理论和翻译技巧》《翻译理念经与方法的某些问题》《专业翻译教学》《著作：翻译问题探索》《翻译教程》《论翻译》《翻译短译》
固体物理英语
固体物理英语固体物理基本词汇（汉英对照）一画一维晶格 One-dimensional crystal lattice一维单原子链 One-dimensional monatomic chain 一维双原子链 One-dimensional diatomic chain 一维复式格子One-dimensional compound lattice 二画二维晶格 Two-dimensional crystal lattice二度轴 Twofold axis二度对称轴 Twofold axis of symmetry几何结构因子 Geometrical structure factor三画三斜晶系 Triclinic system三方晶系 Trigonal system三斜晶系 Triclinic system刃位错 Edge dislocation小角晶界 Low angle grain boundary马德隆常数 Madelung constant四画元素晶体 Element crystal元素的电负性 Electronegativities of elements元素的电离能 Ionization energies of the elements 元素的结合能 Cohesive energies of the elements 六方密堆积 Hexagonal close-packed六方晶系 Hexagonal system反演 Inversion分子晶体 Molecular Crystal切变模量 Shear module双原子链 Diatomic linear chain介电常数 Dielectric constant化学势 Chemical potential内能 Internal energy分布函数 Distribution function夫伦克耳缺陷 Frenkel defect比热 Specific heat中子散射 Neutron scattering五画布喇菲格子 Bravais lattice布洛赫函数 Bloch function布洛赫定理 Bloch theorem布拉格反射 Bragg reflection布里渊区 Brillouin zone布里渊区边界 Brillouin zone boundary 布里渊散射 Brillouin scattering正格子 Direct lattice正交晶系 Orthorhombic crystal system正则振动 Normal vibration正则坐标 Normal coordinates立方晶系 Cubic crystal system立方密堆积 Cubic close-packed四方晶系 Tetragonal crystal system对称操作 Symmetry operation对称群 Symmetric group正交化平面波 Orthogonalized plane wave电子-晶格相互作用 Electron-lattice interaction 电子热容量 Electronic heat capacity电阻率 Electrical resistivity电导率 Conductivity电子亲合势 Electron affinity电子气的动能 Kinetic energy of electron gas 电子气的压力 Pressure of electron gas电子分布函数 Electron distribution function 电负性 Electronegativity电磁声子 Electromagnetic phonon功函数 Work function长程力 Long-range force立方晶系 Cubic system平面波方法 plane wave method平移对称性 Translation symmetry平移对称操作 Translation symmetry operator 平移不变性 Translation invariance石墨结构 Graphite structure闪锌矿结构 Blende structure六画负电性 Electronegativity共价结合 Covalent binding共价键 Covalent bond共价晶体 Covalent crystals共价键的饱和 Saturation of covalent bonds 光学模 Optical modes光学支 Optical branch光散射 Light scattering红外吸收 Infrared absorption压缩系数 Compressibility扩散系数 Diffusion coefficient扩散的激活能 Activation energy of diffusion 共价晶体 Covalent Crystal价带 Valence band导带 Conduction band自扩散 Self-diffusion有效质量 Effective mass有效电荷 Effective charges弛豫时间 Relaxation time弛豫时间近似 Relaxation-time approximation扩展能区图式 Extended zone scheme自由电子模型 Free electron model自由能 Free energy杂化轨道 Hybrid orbit七画纯金属 Ideal metal体心立方 Body-centered cubic体心四方布喇菲格子 Body-centered tetragonal Bravais lattices 卤化碱晶体 Alkali-halide crystal劳厄衍射 Laue diffraction间隙原子 Interstitial atom间隙式扩散 Interstitial diffusion肖特基缺陷 Schottky defect位错 Dislocation滑移 Slip晶界 Grain boundaries伯格斯矢量 Burgers vector杜隆-珀替定律 Dulong-Petit’s law粉末衍射 Powder diffraction里查孙-杜师曼方程 Richardson-Dushman equation 克利斯托夫方程 Christofell equation克利斯托夫模量Christofell module位移极化 Displacement polarization声子 Phonon声学支 Acoustic branch应力 Stress 应变 Strain切应力 Shear stress切应变 Shear strain八画周期性重复单元 Periodic repeated unit底心正交格子 Base-centered orthorhombic lattice 底心单斜格工 Base-centered monoclinic lattices 单斜晶系 Monoclinic crystal system金刚石结构 Diamond structure金属的结合能 Cohesive energy of metals金属晶体 Metallic Crystal转动轴 Rotation axes转动-反演轴 Rotation-inversion axes转动晶体法 Rotating crystal method空间群 Space group空位 Vacancy范德瓦耳斯相互作用 Van der Waals interaction 金属性结合 Metallic binding单斜晶系 Monoclinic system单电子近似 Single-erection approximation极化声子 Polarization phonon拉曼散射 Raman scattering态密度 Density of states铁电软模 Ferroelectrics soft mode空穴 Hole万尼尔函数 Wannier function平移矢量 Translation vector非谐效应 Anharmonic effect周期性边界条件 Periodic boundary condition九画玻尔兹曼方程 Boltzman equation点群 Point groups迪. 哈斯－范. 阿耳芬效应 De Hass－Van Alphen effect胡克定律Hooke’s law氢键 Hydrogen bond亲合势 Affinity重迭排斥能 Overlap repulsive energy结合能 Cohesive energy玻恩-卡门边界条件 Born-Karman boundary condition费密-狄喇克分布函数 Fermi-Dirac distribution function费密电子气的简并性 Degeneracy of free electron Fermi gas 费密 Fermi费密能 Fermi energy费密能级 Fermi level费密球 Fermi sphere费密面 Fermi surface费密温度 Fermi temperature费密速度 Fermi velocity费密半径 Fermi radius恢复力常数 Constant of restorable force绝热近似 Adiabatic approximation十画原胞 Primitive cell原胞基矢 Primitive vectors倒格子 Reciprocal lattice倒格子原胞 Primitive cell of the reciprocal lattice 倒格子空间 Reciprocal space倒格点 Reciprocal lattice point倒格子基矢Primitive translation vectors of the reciprocal lattice倒格矢 Reciprocal lattice vector倒逆散射 Umklapp scattering粉末法 Powder method原子散射因子 Atomic scattering factor配位数 Coordination number原子和离子半径 Atomic and ionic radii原子轨道线性组合 Linear combination of atomic orbits离子晶体的结合能 Cohesive energy of inert crystals离解能 Dissociation energy离子键 Ionic bond离子晶体 Ionic Crystal离子性导电 Ionic conduction洛伦兹比 Lorenz ratio魏德曼－佛兰兹比 Weidemann－Franz ratio 缺陷的迁移 Migration of defects缺陷的浓度 Concentrations of lattice defects 爱因斯坦 Einstein爱因斯坦频率 Einstein frequency爱因斯坦温度 Einstein temperature格波 Lattice wave格林爱森常数 Gruneisen constant索末菲理论 Sommerfeld theory热电子发射 Thermionic emission热容量 Heat capacity热导率 Thermal conductivity热膨胀 Thermal expansion能带 Energy band能隙 Energy gap能带的简约能区图式 Reduced zone scheme of energy band 能带的周期能区图式 Repeated zone scheme of energy band 能带的扩展能区图式 Extended zone scheme of energy band 配分函数 Partition function准粒子 Quasi- particle准动量 Quasi- momentum准自由电子近似 Nearly free electron approximation十一画第一布里渊区 First Brillouin zone密堆积 Close-packing密勒指数 Miller indices接触电势差 Contact potential difference基元 Basis基矢 Basis vector弹性形变 Elastic deformation排斥能Repulsive energy弹性波 Elastic wave弹性应变张量 Elastic strain tensor弹性劲度常数 Elastic stiffness constant弹性顺度常数 Elastic compliance constant 弹性模量 Elastic module弹性动力学方程 Elastic-dynamics equation 弹性散射 Elastic scattering十二画等能面 Constant energy surface晶体 Crystal晶体结构 Crystal structure晶体缺陷 Crystal defect晶体衍射 Crystal diffraction晶列 Crystal array晶面 Crystal plane晶面指数 Crystal plane indices晶带 Crystal band晶向 direction晶格 lattice晶格常数 Lattice constant晶格周期势 Lattice-periodic potential 晶格周期性 Lattice-periodicity晶胞 Cell, Unit cell晶面间距 Interplanar spacing晶系 Crystal system晶体 Crystal晶体点群 Crystallographic point groups晶格振动 Latticevibration晶格散射 Lattice scattering散射 Scattering等能面 surface of constant energy十三画隋性气体晶体的结合能 Cohesive energy of inert gas crystals 滑移 Slip滑移面 Slip plane简单立方晶格 Simple cubic lattice简单晶格 Simple lattice简单单斜格子 Simple monoclinic lattice简单四方格子 Simple tetragonal lattice简单正交格子 Simple orthorhombic lattice简谐近似 Harmonic approximation简正坐标 Normal coordinates简正振动 Normal vibration简正模 Normal modes简约波矢 Reduced wave vector简约布里渊区 Reduced Brillouin zone禁带 Forbidden band紧束缚方法 Tight－binding method零点振动能 Zero-point vibration energy 雷纳德－琼斯势 Lenard－Jones potential 满带 Filled band十四画磁致电阻 Magnetoresistance模式密度 Density of modes漂移速度 Drift velocity漂移迁移率 Drift mobility十五至十七画德拜 Debye德拜近似 Debye approximation德拜截止频率 Debye cut-off frequency 德拜温度 Debye temperature霍耳效应 Hall effect螺位错 Screw dislocation赝势 Pseudopotential。
武汉大学生命科学学院2007至2008学年第一学期分子生物学期末考试试题A
武汉大学生命科学学院2007至2008学年第一学期分子生物学期末考试试题A武汉大学生命科学学院2007－2008学年第一学期期末考试《分子生物学》 A 试卷Final exam of Molecular Biology Course (Fall 2007)年级(Grade) ______ 专业(Major) ________姓名 Name _______ 学号(Student ID)_________PART I: DESCRIPTION (2 points each)Your answer should describe what each item is and how it functions in the cell. Diagrams, structure and sequence information could be included in your answer, as necessary.1. Trombone model2. TFIID3. Telomerase4. RNA editing5. Tandem mass spectrometry6. tRNA synthetase7. Viral-like retrotransposons8. Transcriptional silencing9. Suppression mutation10. Shine-Dalgarno sequencePART II: MULTIPLE/SINGLE CHOICES (2 points each)1. The following molecule is ____1) 2’-deoxyadenosine 5’-phosphate2) 2’-deoxyadenosine3) dAMP4) dATP2. The strictness of the rules for “Waston-Crick”pairing derives from thecomplementarities___ between adenine and thymine and between guanine and cytosine.1) of base stacking2) of shape3) of hydrogen bonding properties4) of size3. Which of the following statements correctly describe the difference between DNA a nd RNA?____1) The major groove of the regular DNA double helical structure is rich in chemicalinformation.。
重庆理工大学材料科学基础双语翻译第3章modified翻译
Fundamentals of Materials Science and Engineering
Parallelism (平行, 对应, 类似) axe（轴） projection(投影) divide v. (除) multiply v.（乘） integer(整数) comma（逗点, 逗号） Enclose v.（装入） bracket（括弧） reduced（简化的）
Fundamentals of Materials Science and Engineering
be devoted to专心于: be dedicated to/ dealing with, being used for only one thing. 专于 Entity实体本质 n. pl. entities [c] something that exists as a single and complete unit 实体、整体
This Chapter will be devoted to （专于） the next level of 的等级 the structure of materials, specifically特别的, to some of the arrangements 安排that may be assumed 假定的 by atoms in the solid state固态. All metals金属, many ceramic materials陶瓷材料, and certain某些 polymers 聚合物form crystalline structures晶体结构 under normal 正常solidification 凝固 conditions条件. For those that do not crystallize使晶体, this longrange atomic order命令 is absent缺少; these noncrystalline非晶体 or amorphous非晶体物质 materials are discussed讨论 briefly 短暂at the end of this chapter.
吉林大学材料科学基础3
2 or R a 4
a
4R
a
☞ FCC unit cell volume VC
VC a 2 R 2
3
3
16R
3
2
C h a p t e r 3 / Structures of Metals and Ceramics
☞ The number of atom in unit cell (n) (晶胞原子数)
C h a p t e r 3 / Structures of Metals and Ceramics
Lattice (晶格, 点阵) The regular geometrical arrangement (规则几何排列) of points in crystal space. It means a three-dimensional array of points coinciding with atom positions (or sphere centers)
C h a p t e r 3 / Structures of Metals and Ceramics
The Body-Centered Cubic (BCC) (体心立方晶格) crystal structure
An aggregate of atoms
C h a p t e r 3 / Structures of Metals and Ceramics
☞ Relationship between a and R a: the cube edge length R: atomic radius (原子半径) a2+a2 = (4R)2
crystalline and noncrystalline between materials having the same composition exist Significant property differences e.g. ceramics polymers
The structure of the hard sphere solid
but one, and computing the volume available to the single unfrozen sphere. FCC, HCP and all other close packing configurations give exactly the same result to this order, proportional to the volume of the Voronoi cell. The next approximation is to consider the effect that the motion of one sphere has on the volume available to its nearest neighbors. To compute this effect, we freeze the (equilibrium) positions of all but two nearest-neighbor spheres and exactly compute the volume, in R6 , of the allowed 2-sphere configurations. When the two nearest-neighbor spheres are in the same layer, the results are the same for FCC and HCP or indeed any layering. However, when the two spheres are in adjacent layers there is slightly more available volume in the HCP case than in the FCC or any other layering. We conclude that the motion of each sphere in the HCP lattice impinges less on the motion of its neighbors than the motion of each sphere in the FCC lattice, and hence that small perturbations of the HCP lattice have more entropy than small perturbations of the FCC lattice. II. Calculations We choose Cartesian (x, y, z ) coordinates such that there are hexagonal layers parallel to the x, y plane. In particular we will call that layer a β -plane which contains sphere centers at the origin O = (0, 0, 0) and the six sites: a=( d = (−
材料科学与工程基础(英文)_南京航空航天大学中国大学mooc课后章节答案期末考试题库2023年
材料科学与工程基础（英文）_南京航空航天大学中国大学mooc课后章节答案期末考试题库2023年1.The driving force for steady-state diffusion is the __________.答案:concentration gradient2.Diffusion coefficient is with the increasing diffusion temperature.答案:exponentially increased;3.Due to , alloys are usually than pure metals of the solvent.答案:solid solution strengthening, stronger;4.The finer the grains, the larger the , and .答案:strength, hardness, toughness;5.With plastic deformation,the increase of dislocationdensity will result in .答案:higher strength;6.In general, Brinell Hardness test is to measure thematerial’s hardness.答案:relatively softer7.Yield strength is corresponding to the occurrenceof deformation.答案:noticeable plastic8.Strain Hardening is also named as .答案:work hardening9.Vacancy diffusion is usually interstitial one.答案:slower than10.Edge and screw dislocations differ in what way?答案:angle between Burgers vector and line direction.11. A ____ may form when impurity atoms are added to a solid, in which case theoriginal crystal structure is retained and no new phases are formed.答案:solid solution12.One explanation for why graphite powder acts so well as a “solid lubricant”is .答案:carbon atoms in graphite are covalently bonded within planar layers but have weaker secondary bonds between layers13.Substitutional atom (impurity) is an example of ______.答案:point defect14.Interstitial solid solution belongs to .答案:finite solid solution;15.The atomic packing factor for FCC is .答案:0.7416.The coordination number of BCC crystal structure is .答案:817.The crystal structure of Cu is ?答案:FCC18.How many atoms does the face centered cubic unit cell contain?答案:Four19.If the electron configuration of Fe is 1s2 2s2 2p6 3s2 3p6 3d6 4s2, then theelectron configurations for the Fe3+ is 1s2 2s2 2p6 3s2 _____.答案:3p6 3d520.Bonds in most metals are referred to as ______.答案:Non-directional21.Covalent bonding occurs as a result of _________ sharing.答案:electron22.Which of the following is NOT an example of primary bonding?答案:Van der Waals23.Atomic weight (A) of an element corresponds to the weighted average of theatomic masses of the atom’s naturally occurring ___________.答案:isotopes24.The point on a phase diagram where the maximum number of allowablephases are in equilibrium is .答案:eutectic point25.Sterling silver (92.5%Ag/7.5%Cu) is an example of ___________.答案:Solid solution26.Engineering stress-strain curve and true stress-strain curve are equal up to .答案:Yeild point27.Among thefollowingtypical transformations of austenite in steels,____________transformation is diffusionless.答案:martensitic28.The heat-treatable aluminum alloy can be strengthened by .答案:Both of above29.In the as-quenched state, martensite is very hard and so brittle that a heattreatment known as must be accomplished sequently.答案:tempering30.During heat treatment of steel, austenite transforms into martensite by .答案:quenching31.Which of the following plane has the highest planar density for fcc.答案:(111)32.Which of the following describes recrystallization?答案:Diffusion dependent with no change in phase composition33.Heating the cold-worked metal progresses in three stages: .答案:recovery, recrystallization, grain growth;34.Strength is increased by making dislocation motion .答案:difficult35.The boundary above which only liquid phase exist is called _________.答案:liquidus36.We have an annealed carbon steel which has hardness of 150HBS. Supposewe know the hardness of Pearlite is 200HBS and the hardness of Ferrite is 80HBS, determine the carbon amount of this steel.答案:0.45%37.The maximum solubility of C in γ-austenite - solid solution is .答案:2.1438.In a plain steel that contains 0.2 percentage carbon, we should expect: .答案:a 25% pearlite and 75% pro-eutectoid ferrite39. A copper-nickel alloy is high-temperature heat treated; the diffusion of Cuinto Ni and Ni into Cu regions is referred to as _____________________.答案:Inter-diffusion40.The phase diagram of Sn-Pb alloy is called .答案:Eutectic phase diagram。
2-D Bravais lattice二维布拉维格子
All sets of primitive vectors shown will work for the 2D lattice
a2 b xˆ c yˆ a1 b xˆ
Example (2D):
Bravais Lattice
All lattices are not Bravais lattices:
R n a1 R n a1 m a2
Where n, m, p = 0, ±1, ±2, ±3, …….
And the vecat1o,rsa,2 , and a3
3D R n a1 m a2 p a3
are called the “primitive lattice vectors” and are said to span the
the same when viewed from
any lattice point)
c
• It can be thought of as a Bravais lattice with a basis consisting of more than just
one atom per lattice point – two atoms in this case. So associated with each point
Example (1D):
lattice. These vectors are not parallel.
Example (2D):
a1 b xˆ
b
c
a2 c yˆ
a1 b xˆ
b
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
y x
integral equation methods in scattering theory
Integral equation methods in scattering theory are a set of mathematical techniques used to analyze the interaction of waves with obstacles. These methods are essential in understanding the behavior of waves in complex media and in particular, in determining the scattering properties of objects.In scattering theory, the interaction of a wave with an obstacle is typically described using integral equations. These equations express the relationship between the scattered field and the incident field, as well as the properties of the obstacle itself. The most common integral equation method in scattering theory is the Lippmann-Schwinger equation.The Lippmann-Schwinger equation is a Fredholm integral equation that relates the scattered field to the incident field and the obstacle's scattering operator. It is derived from the conservation of energy and momentum in the scattering process. The equation provides a means to calculate the scattered field efficiently, given a known incident field and obstacle's scattering operator.Another important integral equation method in scattering theory is the Born approximation. The Born approximation is a perturbative method that approximates the exact solution of the Lippmann-Schwinger equation using a series expansion. It is useful when the obstacle's scattering operator is small compared to the incident field, allowing for an analytical solution of the scattering problem.In addition to these two methods, there are other integral equation techniques that can be used in scattering theory, such as the Rayleigh-Sommerfeld diffraction formula and the Kirchhoff integral formula. These methods are derived from different physical assumptions and are suitable for different types of scattering problems.Integral equation methods in scattering theory have found applications in various fields, including acoustics, electromagnetics, and quantum mechanics. Inacoustics, for example, these methods are used to study the scattering of sound waves by obstacles such as buildings or mountains. In electromagnetics, they are used to analyze the interaction of electromagnetic waves with conducting objects or dielectrics. In quantum mechanics, integral equation methods are used to study the scattering of particles by potentials or potentials.Integral equation methods in scattering theory provide a powerful tool for understanding wave interactions with obstacles. They allow for efficient calculations of scattered fields and provide insights into the physical properties of scattering systems. As such, these methods continue to play a crucial role in various fields of applied mathematics and physics.。
固体物理希尔伯特空间
固体物理希尔伯特空间英文回答：Hilbert space is a fundamental concept in solid state physics. It is a mathematical framework that allows us to describe the quantum mechanical states of particles in a solid. In simple terms, it is a space of all possible states that a particle can occupy.In solid state physics, we often deal with systems that consist of a large number of particles, such as electronsin a crystal lattice. Each particle can be described by a wave function, which is a mathematical function that gives the probability amplitude of finding the particle in a particular state. The wave function lives in a Hilbert space, which is a complex vector space with certain mathematical properties.One of the key properties of a Hilbert space is that it is a complete space, meaning that every sequence of vectorsin the space has a limit that also belongs to the space. This property allows us to define operators, such as the Hamiltonian operator, that act on the wave functions and describe the dynamics of the system.Another important property of a Hilbert space is thatit is a inner product space, which means that it has a notion of distance between vectors. This allows us to define the concept of orthogonality, where two vectors are orthogonal if their inner product is zero. Orthogonal vectors play a crucial role in quantum mechanics, as they represent states that are mutually exclusive.To illustrate the concept of Hilbert space, let's consider an example of a one-dimensional lattice with two sites. Each site can be either occupied or unoccupied by an electron. We can describe the state of the system using a two-dimensional Hilbert space, where the basis vectors represent the states of the two sites. For example, the basis vector |01⟩ represents the state where the firstsite is unoccupied and the second site is occupied.In this example, the Hamiltonian operator woulddescribe the energy of the system and how it evolves over time. By acting on the basis vectors, the Hamiltonian operator would give us the energy of each state and how it changes as the system evolves.中文回答：希尔伯特空间是固体物理中的一个基本概念。
法布里珀罗基模共振英文
法布里珀罗基模共振英文The Fabryperot ResonanceOptics, the study of light and its properties, has been a subject of fascination for scientists and researchers for centuries. One of the fundamental phenomena in optics is the Fabry-Perot resonance, named after the French physicists Charles Fabry and Alfred Perot, who first described it in the late 19th century. This resonance effect has numerous applications in various fields, ranging from telecommunications to quantum physics, and its understanding is crucial in the development of advanced optical technologies.The Fabry-Perot resonance occurs when light is reflected multiple times between two parallel, partially reflective surfaces, known as mirrors. This creates a standing wave pattern within the cavity formed by the mirrors, where the light waves interfere constructively and destructively to produce a series of sharp peaks and valleys in the transmitted and reflected light intensity. The specific wavelengths at which the constructive interference occurs are known as the resonant wavelengths of the Fabry-Perot cavity.The resonant wavelengths of a Fabry-Perot cavity are determined bythe distance between the mirrors, the refractive index of the material within the cavity, and the wavelength of the incident light. When the optical path length, which is the product of the refractive index and the physical distance between the mirrors, is an integer multiple of the wavelength of the incident light, the light waves interfere constructively, resulting in a high-intensity transmission through the cavity. Conversely, when the optical path length is not an integer multiple of the wavelength, the light waves interfere destructively, leading to a low-intensity transmission.The sharpness of the resonant peaks in a Fabry-Perot cavity is determined by the reflectivity of the mirrors. Highly reflective mirrors result in a higher finesse, which is a measure of the ratio of the spacing between the resonant peaks to their width. This high finesse allows for the creation of narrow-linewidth, high-resolution optical filters and laser cavities, which are essential components in various optical systems.One of the key applications of the Fabry-Perot resonance is in the field of optical telecommunications. Fiber-optic communication systems often utilize Fabry-Perot filters to select specific wavelength channels for data transmission, enabling the efficient use of the available bandwidth in fiber-optic networks. These filters can be tuned by adjusting the mirror separation or the refractive index of the cavity, allowing for dynamic wavelength selection andreconfiguration of the communication system.Another important application of the Fabry-Perot resonance is in the field of laser technology. Fabry-Perot cavities are commonly used as the optical resonator in various types of lasers, providing the necessary feedback to sustain the lasing process. The high finesse of the Fabry-Perot cavity allows for the generation of highly monochromatic and coherent light, which is crucial for applications such as spectroscopy, interferometry, and precision metrology.In the realm of quantum physics, the Fabry-Perot resonance plays a crucial role in the study of cavity quantum electrodynamics (cQED). In cQED, atoms or other quantum systems are placed inside a Fabry-Perot cavity, where the strong interaction between the atoms and the confined electromagnetic field can lead to the observation of fascinating quantum phenomena, such as the Purcell effect, vacuum Rabi oscillations, and the generation of nonclassical states of light.Furthermore, the Fabry-Perot resonance has found applications in the field of optical sensing, where it is used to detect small changes in physical parameters, such as displacement, pressure, or temperature. The high sensitivity and stability of Fabry-Perot interferometers make them valuable tools in various sensing and measurement applications, ranging from seismic monitoring to the detection of gravitational waves.The Fabry-Perot resonance is a fundamental concept in optics that has enabled the development of numerous advanced optical technologies. Its versatility and importance in various fields of science and engineering have made it a subject of continuous research and innovation. As the field of optics continues to advance, the Fabry-Perot resonance will undoubtedly play an increasingly crucial role in shaping the future of optical systems and applications.。
英语语言学布拉格学派纯英文全解
1929 ---- the First International Congress of Slavists in Prague. Came up with Outline
1930 ---- International phonetic system representative conferencein Prague (detail on language, especially on phonetic structure).
multilateral opposition 多边对立 It is a more loosely established relationship: /a/ and /i/ for
instence are alike only to the extent that both are vowels, a quality shared by ant other pairs of vowels.
Second, there was an emphasis on the systemic character of language.
Third, language was looked on as functional in another sense, that is, as a tool performing a number of essential functions or tasks for the community using it.
other, what phonetic features they possess, how they can be classified, etc.
Structural Studies of the Ferroelectric Phase Transition in Bi4Ti3O12
Structural Studies of the Ferroelectric Phase Transitionin Bi4Ti3O12Qingdi Zhou and Brendan J.Kennedy*School of Chemistry,The University of Sydney,Sydney,NSW2006AustraliaChristopher J.HowardAustralian Nuclear Science and Technology Organization,Private Mail Bag1,Menai, NSW2234,Australia,and School of Physics,The University of Sydney,Sydney,NSW2006AustraliaReceived July3,2003.Revised Manuscript Received October20,2003A variable-temperature synchrotron X-ray diffraction study of the phase transitions in the ferroelectric n)3Aurivillius oxide Bi4Ti3O12is described.At room temperature the structure of Bi4Ti3O12is orthorhombic in space group B2eb and this continuously transforms to the high-temperature tetragonal I4/mmm structure via an intermediate orthorhombic phase.The possible space groups of this intermediate orthorhombic phase have been identified by using group theory.IntroductionThe Aurivillius oxides are represented by the general formula(Bi2O2)2+(A n-1B n O3n+1)2-where B is a diamag-netic transition metal such as Ti4+or Nb5+and A is an alkali or alkaline earth cation.1The structure of the Aurivillius oxides consists of arrays of Bi2O2and per-ovskite-like A n-1B n O3n+1layers.The ferroelectric prop-erties of such oxides have been known for around50 years,2,3yet the structural origins of their ferroelectricity have only recently been established.4,5Following the pioneering work of Scott and co-work-ers,6the possibility of employing these Aurivillius oxides in ferroelectric memory devices has been extensively studied.A serious barrier to their practical utilization is their poor thermal stability.7-10Formation of thin-film ferroelectric devices involves a sintering step at high temperatures and this invariably degrades the performance of the simpler Aurivillius oxides.Conse-quently,a number of studies of the high-temperature behavior and structures of Aurivillius oxides have been reported.11-14In comparison to the n)2oxides based on SrBi2Ta2O9very little is known about the high-temperature properties of the n)3oxides such as Bi4-Ti3O12,yet such oxides are reported to have ferroelectric properties superior to those of the better-studied n)2 oxides.15Although it has been reported5that Bi4Ti3O12is monoclinic at room temperature,very-high-resolution powder diffraction data suggest that powder samples of Bi4Ti3O12are actually orthorhombic at room temper-ature.16Heating Bi4Ti3O12above670°C is reported to result in an apparently first-order phase transition to a paraelectric tetragonal phase.16,17A significant volume change between the high-temperature and low-temper-ature phases is likely to be detrimental to the stability of any thin films annealed or sintered above the Curie temperature.Conversely,continuous transitions are less likely to adversely influence the properties of the thin films.Hervoches and Lightfoot have demonstrated, using powder neutron diffraction methods,that for Bi4-Ti3O12the high-temperature paraelectric phase is in space group I4/mmm and that the room-temperature orthorhombic phase is in B2eb.16The main structural basis for the ferroelectricity in Bi4Ti3O12is the displacement of the Bi atoms within the perovskite-like layers,along the crystallographic a-axis with respect to the chains of corner-sharing TiO6 octahedra.This corresponds to a[110]displacement referred to the parent I4/mmm.The TiO6octahedra are tilted relative to each other and the tilt system can be*To whom correspondence should be addressed.Phone:61-2-9351-2742.Fax:61-2-9351-3329.E-mail: b.kennedy@.au.(1)Aurivillius,B.Arkiv Kemi1949,1,463.(2)Subbarao,E.C.J.Phys.Chem.Solids1962,23,665.(3)Smolenskii,G.A.;Isupov,V.A.;Agranovskaya,A.I.Sov.Phys. Solid State1961,3,651.(4)Rae,A.D.;Thompson,J.G.;Withers,R.Acta Crystallogr.,Sect. B:Struct.Sci.1992,48,418.(5)Rae,A.D.;Thompson,J.G.;Withers,R.;Willis,A.C.Acta Crystallogr.,Sect.B:Struct.Sci.1990,46,474.(6)Paz de Araujo,C.A.;Cuchlaro,J.D.;McMillan,L.D.;Scott, M.;Scott,J.F.Nature(London)1995,374,627.(7)Shimakawa,Y.;Kubo,Y.;Tauchi,Y.;Kamiyama,T.;Asano,H.; Izumi,F.Appl.Phys.Lett.2000,77,2749.(8)Shimakawa,Y.;Kubo,Y.;Tauchi,Y.;Asano,H.;Kamiyama,T.; Izumi,F.;Hiroi,Z.Appl.Phys.Lett.2001,79,2791.(9)Boulle, A.;Legrand, C.;Guinebretie`re,R.;Mercurio,J.P.; Dauger,A.Thin Solid Films2002,391,42.(10)Aizawa,K.;Tokumitsu,E.;Okamoto,K.;Ishiwara,H.Appl. Phys.Lett.2000,79,2791.(11)Macquart,R.;Kennedy,B.J.;Hunter,B.A.;Howard,C.J.; Shimakawa,Y.Integr.Ferroelectr.2002,44,101-112.(12)Hervoches,C.H.;Irvine,J.T.S.;Lightfoot,P.Phys.Rev.B 2002,64,100102(R).(13)Liu,J.;Zou,G.;Yang,H.;Cui,Q.Solid State Commun.1994, 90,365.(14)Macquart,R.;Kennedy,B.J.;Vogt,T.;Howard,C.J.Phys. Rev B2002,66,212102.(15)Park,B.H.;Kang,B.S.;Bu,S.D.;Noh,T.W.;Lee,J.;Jo,W. Nature(London)1999,401,682.(16)Hervoches,C.H.;Lightfoot,P.Chem.Mater.1999,11,3359.(17)Hirata,T.;Yokokawa,T.Solid State Commun.1997,104,673.5025Chem.Mater.2003,15,5025-502810.1021/cm034580l CCC:$25.00©2003American Chemical SocietyPublished on Web11/21/2003described as a-a-c0in Glazer’s notation.18These two structural features act in concert to lower the symmetry from tetragonal to orthorhombic,however,they are not linked,and there is no reason to suppose that both modes will condense at precisely the same temperature. Rather,it is probable that these modes will condense successively and the two end member phases will be linked by an intermediate ing powder neutron diffraction methods,Macquart and Lightfoot have in-dependently shown that the A21am to I4/mmm transi-tion in the n)2Aurivillius phases SrBi2Ta2O911and Sr0.85Bi2.1Ta2O912proceeds via an intermediate paraelec-tric Amam phase in each case.This Amam phase is also seen in PbBi2M2O9(M)Nb,Ta).19That is,upon cooling from the I4/mmm phase,the tilting of the octahedra occurs before the cation displacement.The same se-quence is reported to occur in the n)4Aurivillius oxide SrBi4Ti4O15.20The aim of the present work is to establish whether the B2eb to I4/mmm transition in Bi4Ti3O12is first order as proposed by Hirata and Yokakowa17or if it actually occurs continuously via an intermediate phase as seen in SrBi2Ta2O9.11To establish this we have investigated the temperature dependence of the structure of Bi4-Ti3O12from room temperature to800°C using high-resolution synchrotron radiation.Near the Curie tem-perature fine temperature intervals have been used to detect the intermediate phase which exists over a very limited temperature range.Experimental SectionThe crystalline sample of Bi4Ti3O12was prepared by the solid-state reaction of stoichiometric quantities of Bi2O3 (99.999%,Aldrich)and TiO2(99.9%,Aldrich).The heating sequence used was700°C/24h and850°C/48h,with intermediate regrinding.The sample was slowly cooled to room temperature in the furnace.The sample purity was established by powder X-ray dif-fraction measurements using Cu K R radiation on a Shimadzu D-6000Diffractometer.Room-and variable-temperature syn-chrotron X-ray powder diffraction patterns were collected on a high-resolution Debye Scherrer diffractometer at beamline 20B,the Australian National Beamline Facility,at the Photon Factory,Japan.21The sample was finely ground and loaded into a0.3-mm quartz capillary that was rotated during the measurements.All measurements were performed under vacuum to minimize air scatter.Data were recorded using two Fuji image plates.Each image plate was20×40cm and each covered40°in2θ.A thin strip(ca.0.5cm wide)was used to record each diffraction pattern so that up to30patterns could be recorded before reading the image plates.The data were collected at a wavelength of0.75Å(calibrated with a NIST Si 640c standard)over the2θrange of5-75°with step size of 0.01°.The patterns were collected in the temperature range of100-800°C in25°C steps or600-803°C in7°C steps, and with30-min counting time at each temperature.Struc-tures were refined by the Rietveld method using the program Rietica.22The positions of the cations were well described in these analyses,however the estimated standard deviations (esds)of the Ti-O bond distances(typically around0.02Å) preclude any detailed discussion of the temperature depen-dence of the bond distances.Results and DiscussionThe published room-temperature structure for Bi4-Ti3O12was used as a starting model in our Rietveldrefinements,and the structural refinement proceededwithout event.The temperature dependence of thelattice parameters and volumes are illustrated in Figure1.All the lattice parameters show a smooth increasedue to thermal expansion as the sample is heated toca.500°C.Above this temperature the cell continuesto expand along both the b-and c-directions,howeverthe a-parameter is essentially constant.This behavioris very similar to that displayed by a number of simplerABO3perovskites23-25and is apparently related to thegradual reduction in distortion resulting from a reduc-tion in the magnitude of the tilting of the BO6octahedraas the temperature is increased.Near675°C there is arapid decrease in the a-parameter although as is clearlyevident from Figure1this is not a discontinuousdecrease but rather a rapid progressive drop.At thesame temperature the c-axis expands rapidly,Figure1.The transition to the tetragonal phase is clearlyevident in the200/020and317/137reflections(near2θ≈15.8and26.7°,respectively,whereλ)0.75Å).As illustrated in Figure2the diffraction pattern recordedat670°C shows obvious splitting of the200/020and317/137reflections that is clearly indicative of orthor-hombic symmetry.This splitting remains clearly visibleto the eye until677°C and can be discerned by profileanalysis at684°C.At691°C no splitting or diagnosticasymmetry of these or other peaks is apparent and itis concluded that the structure is tetragonal.That is,the transition to the tetragonal structure occurs near690°C,which is around15°above the reported ferro-electric Curie temperature for Bi4Ti3O12.(Some care isrequired when comparing the transition temperaturesreported in various studies because of both samplevariation(induced by the different heating regimesused)and possible variations in the high-temperaturethermometry.)Above691°C the structure has been refined in thetetragonal space group I4/mmm and is as described byHervoches and Lightfoot.16That the paraelectric phaseis tetragonal in I4/mmm well above the ferroelectricCurie temperature was confirmed from the high-resolu-tion powder neutron diffraction data by Hervoches andLightfoot from both the cell metric and the absence ofany superlattice reflections indicative of TiO6tilting. The thermal expansion of the c-axis both above and below the T c is relatively linear and can be well-fitted to a simple linear equation c)5.277×10-4T+32.725 for T<677°C,and c)5.116×10-4T+32.725for T> 691°C.Examining the temperature dependence of the long c-axis we observed a large difference between the(18)Glazer,A.M.Acta Crystallogr.B1972,28,3384.(19)Macquart,R.;Kennedy,B.J.;Hunter,B.A.;Howard,C.J.J. Phys.:Condens.Matter2002,14,7955.(20)Hervoches,C.H.;Snedden,A.;Riggs,R.;Kilcoyne,S.H.; Manuel,P.;Lightfoot,P.J.Solid State Chem.2002,164,280.(21)Sabine,T.M.;Kennedy,B.J.;Garrett,R.F.;Foran,G.J.; Cookson,D.J.J.Appl.Crystallogr.1995,28,513.(22)Howard C.J.;Hunter,B.A.A Computer Program for Rietveld Analysis of X-ray and Neutron Powder Diffraction Patterns;Lucas Heights Research Laboratories:New South Wales,Australia,1998; pp1-27.(23)Howard,C.J.;Knight,K.S.;Kisi,E.H.;Kennedy,B.J.J. Phys.:Condens.Matter2000,12,L677.(24)Kennedy,B.J.;Howard,C.J.;Thorogood,G.J.;Hester,J.R. J.Solid State Chem.2001,161,106.(25)Kennedy,B.J.;Howard,C.J.;Chakoumakos,B.C.J.Phys. C:Condens.Matter1999,11,1479.5026Chem.Mater.,Vol.15,No.26,2003Zhou et al.values of the low-temperature orthorhombic,ferroelec-tric phase and those for the high-temperature tetragonal and paraelectric phase.Clearly the value for the c -parameter at 684°C does not fall into either series.A similar conclusion can be made for both the a -and b -parameters.These temperature-dependent changes in the lattice parameters are more reminiscent of the continuous-phase transitions observed in oxides such as Bi 2PbNb 2O 919than of the more subtle changes observed in the high-temperature second-order incommensurate-to-commensurate phase transition observed in Bi 2-MoO 6.26A feature of many first-order phase transitions is the coexistence of a two phase region.Attempts to fit the pattern at 684°C to a two-phase orthorhombic/tetragonal model with lattice parameters obtained by appropriate linear extrapolation were unsuccessful.It was concluded that a single phase was present at this temperature and this was neither the orthorhombic in B 2eb nor the tetragonal I 4/mmm .(26)Buttrey,D.J.;Vogt,T.;White,B.D.J.Solid State Chem.2000,155,206.(27)/∼stokesh/isotropy.html.Figure 1.Temperature dependence of the lattice parameters and volume for Bi 4Ti 3O 12obtained from Rietveld analysis of variable-temperature synchrotron diffraction data.For ease of comparison the values in the 2× 2×1orthorhombic cells have been reduced to the equivalent tetragonal values.Figure 2.Portions of the Rietveld fits for Bi 4Ti 3O 12showing the temperature dependence of the splitting of the tetragonal 110reflection into the orthorhombic 200/020pair near 2θ)15.8°and of the 127reflection into the 317/137pair near 2θ)26.8°.The 0012reflection is apparent near 2θ)15.6°.Ferroelectric Phase Transition in Bi 4Ti 3O 12Chem.Mater.,Vol.15,No.26,20035027In summary,we observe an orthorhombic structure at 684°C whose lattice parameter clearly distinguishes it from the low-temperature orthorhombic and high-temperature tetragonal phases.(In fact,even if we could not distinguish this as a separate phase,the observed continuity of transition,together with the group theo-retical arguments to follow,would indicate that such an intermediate orthorhombic phase is involved in the transition.)If we assume that all three phases have commensurate structures then it should be possible to identify the space group of the intermediate phase from a group theoretical analysis.To do this we used the program ISOTROPY.27This analysis confirmed that a direct B 2eb to I 4/mmm transition could not be continu-ous.The two modes responsible for the transition were identified as Γ5-describing the cation displacement and X 3+associated with the tilting of the TiO 6octahedra.28Whichever of these modes condenses first,an interme-diate orthorhombic structure based on a 2× 2×1superstructure of the parent I 4/mmm structure is involved,as illustrated in Figure 3,and the successive phase transitions through either of these intermediates are allowed to be continuous.If the initial distortion is cation displacement via the Γ5-mode then a ferroelec-tric orthorhombic structure that lacks any tilting of the TiO 6octahedra is expected.The resulting structure in Fmm 2can continuously transform to the observed room-temperature orthorhombic B 2eb phase through tilting of the TiO 6octahedra via the X 3+mode.(This descrip-tion refers still to the parent structure in I 4/mmm .)Alternatively,the initial distortion may be the introduc-tion of tilting of the TiO 6octahedra resulting in a Cmca phase followed by cation displacement.The available synchrotron diffraction data do not allow us to unequivoc-ally distinguish between these two possibilities;how-ever,by analogy with SrBi 2Ta 2O 9,we favor the latter possibility.11The confirmation of this will require a high-resolution neutron diffraction study.It is illuminating to compare our results with those of Hirata and Yokokawa.17First,Figure 2of their report suggests they recorded very little data in the ferroelec-tric phase with only four temperature points obvious.A somewhat greater number of temperatures were examined (nine)in the paraelectric phase,apparently at 20°C intervals.Crucially,no data appear to have been collected between 475and 675°C,the latter corresponding to the reported ferroelectric Curie tem-perature for Bi 4Ti 3O 12.There is then a relatively large jump in the temperatures used by Hirata and Yokokawa to above 695°C.In our present study we have used relatively coarse temperature increments (25°C)to monitor the general form of the phase transition,but then we have used much finer intervals (7°C)to probe the nature (first order or continuous)of the ferroelectric to paraelectric transition.We conclude that as a result of the relatively coarse temperature intervals used by Hirata and Yokokawa it was not possible for those authors to establish whether the ferroelectric to paraelec-tric transition in Bi 4Ti 3O 12was first order or continuous.The temperature dependence of the lattice parameters established using powder neutron diffraction data de-scribed by Lightfoot and Hervoches 16,29is very similar to that observed here for the data collected in 25°C intervals.On the basis of a similar density of data,Lightfoot and Hervoches 29concluded that the transition is first order.In comparison,our data collected in 7°C intervals using high-resolution synchrotron X-ray meth-ods strongly suggests the transition is continuous,albeit involving an intermediate phase.In conclusion we have identified the existence of an intermediate orthorhombic phase in the solid-state phase transition of the ferroelectric n )3Aurivillius phases Bi 4Ti 3O 12.Two possible orthorhombic phases were iden-tified using group theory,Fmm 2and Cmca ,depending on the sequence in which the two modes responsible for the lowering of symmetry condense.By analogy with the n )2oxide SrBi 2Ta 2O 9,the most likely sequence of transitions is B 2cb 670°C fCmca 695°CfI 4/mmm .Clearly confirming the existence of and establishing the precise structure of the proposed intermediate phase is of considerable interest,and efforts aimed at this are in progress.Acknowledgment.This work performed at the Australian National Beamline Facility was supported by the Australian Synchrotron Research Program,which is funded by the Commonwealth of Australia under the Major National Research Facilities program.B.J.K.acknowledges the support of the Australian Research Council.The assistance of Dr.James Hester at the ANBF is gratefully acknowledged.We thank Dr.P.Lightfoot for bringing ref 29to our attention while this manuscript was under review.CM034580L(28)Miller,S.C.;Love,W.F.Tables of Irreducible Representations of Space Groups and Co-representations of Magnetic Space Groups ;Pruett Press:Boulder,CO,1967.(29)Hervoches,C.H.;Lightfoot,P.Proceedings of CIMTEC,10th International Ceramics Congress ,Florence,Italy,July 2002.Figure 3.Schematic diagram showing the group -subgroup relationships for the n )3Aurivillius oxides.The solid lines show the transitions that are allowed to be continuous.The tilt system for the intermediate phases is given.5028Chem.Mater.,Vol.15,No.26,2003Zhou etal.。
1. Introduction The Cell Structures of Certain Lattices
And out of the ground the Lord God formed every beast of the field, and every fowl of the air; and brought them into Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof. Genesis 2:19 Kaum nennt man die Dinge beim richtigen Namen, so .. verlieren sie ihren gefahrlichen Zauber. Der primitive Mensch benannte alles und jedes falsch. Ein einziger furchtbarer Zauberbann umgab ihn, wo und wann war er .. nicht gefahrdet? Die Wissenschaft hat uns von Aberglauben and Glauben befreit. Sie gebraucht immer die gleichen Namen, mit Vorliebe griechisch-lateinische, und meint damit die wirklichen Dinge. .. .. Missverstandnisse sind unmoglich. ( ∗ ∗ ) Elias Canetti, Die Blendung, Hanser, Munich, 1963, p. 425 Abstract. The most important lattices in Euclidean space of dimension n ≤ 8 are the lattices A n ( n ≥ 2 ), D n ( n ≥ 4 ), E n ( n = 6 , 7 , 8 ) and their duals. In this paper we determine the cell structures of all these lattices and their Voronoi and Delaunay ∗ polytopes in a uniform manner. The results for E ∗ 6 and E 7 simplify recent work of Worley, and also provide what may be new space-filling polytopes in dimensions 6 and 7.
自然辩证法中心线索
自然辩证法中心线索英文回答：The central thread of dialectical naturalism is the idea that nature is a dynamic, self-organizing system that is constantly evolving and changing. This view of nature is based on the premise that there are no absolute truths or fixed laws of nature, but rather that all things are in a constant state of flux and transformation.Dialectical naturalism also emphasizes the importance of contradiction and conflict as driving forces in natural processes. According to this view, progress and development occur through the resolution of contradictions and the emergence of new forms and structures. This process is often characterized by periods of instability and chaos, as well as moments of sudden insight and breakthrough.Overall, dialectical naturalism provides a holistic and dynamic view of nature that captures its inherentcomplexity and fluidity. It is a philosophy that is deeply rooted in the scientific method and has implications for a wide range of fields, including biology, physics, andsocial science.中文回答：自然辩证法的中心线索是自然是一个动态的自组织系统，不断地进化和变化。
基于梯度点阵结构的浆果变刚度柔性夹持机构设计
华南农业大学学报 Journal of South China Agricultural University 2024, 45(2): 273-279DOI: 10.7671/j.issn.1001-411X.202212021王昱, 陈婉琼, 曾山, 等. 基于梯度点阵结构的浆果变刚度柔性夹持机构设计[J]. 华南农业大学学报, 2024, 45(2): 273-279.WANG Yu, CHEN Wanqiong, ZENG Shan, et al. Design of berry compliant clamping mechanism with variable stiffness based on gradient lattice structure[J]. Journal of South China Agricultural University, 2024, 45(2): 273-279.基于梯度点阵结构的浆果变刚度柔性夹持机构设计王 昱，陈婉琼，曾 山，凡 健，孙 超，姚凯斌(华南农业大学 工程学院/南方农业机械与装备关键技术教育部重点实验室, 广东 广州 510642)摘要: 【目的】根据浆果在采摘、分拣等环节中的夹持性能需求，在机器人末端使用高刚度驱动连接和低刚度无损夹持的夹持机构，有效推进农业机器人在浆果生产领域的推广。
【方法】针对典型浆果−小番茄的夹持工况，引入多层级拓扑优化理论，提出基于梯度点阵结构的变刚度柔性夹持机构设计方法，构建变刚度柔性夹持机构的优化设计模型，实现由单一材料构建的刚度梯度分布的一体化柔性夹持机构设计。
【结果】采用Polyjet增材制造技术，加工获得柔性质量约45 g的夹持机构样件。
对茂名‘千禧’小番茄的夹持试验发现，夹持机构样件输入端的驱动载荷为11.00~14.56 N时，小番茄的压缩率为0.90%~1.91%，机械损伤度为0。
【结论】优化设计的变刚度柔性夹持机构可有效且几乎零损伤地夹持外表层较为脆弱的浆果，为浆果自动化采摘、分拣等环节中夹持装备的设计提供了可行的设计方法。
钛合金的网架结构
Titanium alloy lattice truss structuresDouglas T.Queheillalt *,Haydn N.G.WadleyDepartment of Materials Science and Engineering,University of Virginia,140Chemistry Way,P.O.Box 400745,Charlottesville,VA 22904-4745,United Statesa r t i c l e i n f o Article history:Received 21March 2007Accepted 3September 2008Available online 20September 2008Keywords:Sandwich structures (B)Diffusion bonding (D)Buckling (I)a b s t r a c tA high-temperature forming and diffusion bonding method has been investigated for the fabrication of modified pyramidal lattice core sandwich structures from a titanium alloy.A periodic asymmetric hex-agonal perforation pattern was cut into thin Ti–6Al–4V sheets which were then folded along node rows by a combination of partial low-temperature bending followed by simultaneous hot forming/diffusion bonding to form sandwich panel structures with core relative densities of 1.0–4.1%.The out-of-plane compression and in-plane longitudinal shear properties of these structures were measured and compared with analytical estimates.Premature panel failure by node shear-off fracture was observed during shear testing of some test structures.Node failures were also initiated at stress concentrations at the truss–facesheet interface.A liquid interface diffusion bonding approach has been investigated as a possible approach for reducing this stress concentration and increasing the truss–facesheet interfacial strength.Published by Elsevier Ltd.1.IntroductionLightweight metallic sandwich panel structures consisting of low density cores and solid facesheets are widely used in aerospace applications [1–5].Cellular core structures based upon honeycomb topologies are most often used because of their high compressive and flexural stiffness and strength-to-weight ratios,good vibration damping and low through thickness thermal conductivity [6–8].These honeycomb structures are closed-celled with no access to the core region and are susceptible to internal corrosion [6].Lattice truss structures with tetrahedral,pyramidal and Kagomécell topol-ogies are beginning to be explored as alternate core materials.For example,pyramidal lattice truss structures can be fabricated from ductile aluminum and stainless steel alloys by perforating a metal sheet to form a periodic diamond perforation pattern,followed by a node row folding process [9–14].The folded core can be brazed or laser welded to solid facesheets to form a sandwich structure.The lattice topology,core relative density and parent alloy mechan-ical properties combine to determine the mode of truss deformation and therefore the mechanical response of these structures [9,10].Previous studies with lattices fabricated from high elastic mod-ulus,low strength stainless steels indicate that high core relative density cores fail by yielding of the truss columns [9–11,13,14].For elastic–perfectly plastic materials,the peak out-of-plane com-pressive strength,r pk ,scales linearly with the relative density;r pk ¼R r ys qwhere R is a lattice topology (geometry)dependent scaling factor (for a pyramidal lattice,R ¼sin 2x ,where x is the angle of inclination of the truss),r ys is the yield strength of the so-lid material and the relative density q¼q c =q s is the density of the core,q c ,divided by that of the solid material,q s .Increasing the so-lid materials yield strength enables the lattice strength to be in-creased when yielding is the operative failure mode.Table 1summarizes the specific yield strength for several alloys that have been used to fabricate lattice truss structures.Data for Ti–6Al–4V is also shown.It has the highest specific yield strength and a signifi-cantly higher maximum service temperature than 6061aluminum alloys.A lattice structure fabricated from Ti–6Al–4V might there-fore have useful applications.Here,we explore the use of a modified form of the sheet folding method,developed for highly ductile materials,and used it to fab-ricate modified pyramidal lattice structures from a Ti–6Al–4V al-loy.A combination of cold and hot forming/diffusion bonding was needed to fabricate the sandwich panel structures.The out-of-plane compressive and in-plane longitudinal shear properties have been measured and compared with analytical estimates for truss plastic yielding and buckling.The resulting structures have higher strength-to-weight ratios than similar lattices made from either heat treatable aluminum or stainless steel alloys.Premature panel failure by node flat shear fracture and node truss shear-off fracture was observed during shear testing of some structures.A li-quid interface diffusion bonding technique was investigated as a possible approach for the mitigation of this failure mode.2.Fabrication approach 2.1.Forming limit considerationsPyramidal lattice structures can be fabricated from high ductil-ity materials by a folding process [10–13].However,when this was0261-3069/$-see front matter Published by Elsevier Ltd.doi:10.1016/j.matdes.2008.09.015*Corresponding author.Tel.:+14349825678;fax:+14349825677.E-mail address:dougq@ (D.T.Queheillalt).Materials and Design 30(2009)1966–1975Contents lists available at ScienceDirectMaterials and Designj o u r n a l h o m e p a g e :w w w.e l s e vier.c om/loc ate/matdesattempted here the limited ambient temperature formability of Ti–6Al–4V resulted in sheet fracture on the tensile stress side of the node after only a half of the required bending deflection.This prob-lem is commonly encountered during deformation processes with low ductility alloys[15,16].The minimum bend radius(the radius at which cracks appear on the tensile stressed surface of a bend)for sheet material under plane strain conditions has been analyzed by Yang[17].He shows that the minimum bend radius,R,depends on the sheet thickness,t,and tensile ductility,e t:R¼t50t À1ð1Þwhere e t is the tensile elongation to failure of the solid material in% and is expressed as a function of sheet thickness[17].Fig.1shows the minimum bend radius/sheet thickness ratio predicted by Eq.(1).Typical ambient temperature tensile ductility ranges for various alloys are overlaid[18,19].Fabrication of pyra-midal lattice structures via a folding process that bends a diamond perforated sheet with a point node to create a single layer of trusses is facilitated by R/t values61.5.This requires a minimum tensile elongation of about25%.Titanium alloys have tensile elon-gations in the10–15%range.The corresponding bend radius to thickness ratio then lies in the range2.56R/t64which precludes the fabrication of pyramidal lattices by an ambient temperature bend forming approach.To accommodate the lower ductility of this alloy,the fabrication process was modified in two ways.The pyramidal lattice itself was first modified to create a unit cell structure with a shortflat nodal region.Fig.2a and b shows schematic illustrations of an ideal and the modified unit cell geometries,respectively.Thisflat nodal re-gion then enables the use of a larger bending radius folding oper-ation.It also has the beneficial consequences of increasing the node contact area and thus the fracture strength of the node.How-ever,this is accompanied by a reduction of structural efficiency(by the ratio of the mass of material utilized in the node to the total core mass)[12].The core was partially formed at ambient temper-ature and the folding was completed at the higher temperature used for diffusion bonding the core to solid facesheets.2.2.Panel fabricationThe process consisted of perforating a metal sheet to create a modified two-dimensional periodic diamond perforation pattern, partially folding node rows by a cold bending process and then hot forming/diffusion bonding the pyramidal lattice to facesheets to form the sandwich structures.The periodic truncated-diamond pattern punched into sheets of Ti–6Al–4V is shown in Fig.3a.An example of a partially formed modified pyramidal lattice is shown in Fig.3b.The lattice was bent using a bend radius/thickness ratio of R/t$3,which was sufficient to prevent cracking on the tensile side of the bend region.A close-up of the node region is shown as an inset of Fig.3b.This bent radius region isflattened during the subsequent high-temperature forming/diffusion bonding pro-cess,Fig.2b.Fig.4shows a schematic illustration of the tool assembly for hot forming/diffusion bonding the modified pyramidal sandwich structures to solid facesheets.The room temperature formed cores were placed on tool steel load supporting pins,a bottom Ti–6Al–4V facesheet and tool steel support plate combination,Fig.4a.A sec-ond set of tool steel load supporting pins,a top Ti–6Al–4V face-sheet and tool steel support plate combination,Fig.4b and c,was placed on top of the core forming the stacked assembly.Three of these assemblies were stacked vertically and placed in a diffusion bonding furnace.The node support pins and plates were also cleaned prior to assembly and a light coating of boron nitride ap-plied to act as a release agent between thefixture and sandwich structures.Fig.4d shows a schematic illustration of the hot formed/diffusion bonded cores.For the hot forming/diffusion bonding step,the chamber was evacuated to$10À4Torr and the stacked assembly was heated at 10°C/min to900°C,held for6h with a pressure of3.5MPa applied to the assembly andfinally cooled at10°C/min.After diffusion bonding,the node support pins were removed and compression and shear samples were wire electro-discharge machined from the sandwich panels.Fig.5shows photographs of typical node re-gions for the four relative densities investigated here.Table1Selected mechanical and physical properties of various alloys used for fabrication of lattice truss sandwich structuresMaterial Yield strength,r y(MPa)Density,q s(kg/m3)Maximum use temperature,T(°C)Â103r y/q sAl6061-H0702700$17026Al6061-T62682700$17099304SS1768000$92522Al6XN SS2008000$42025Ti–6Al–4V9004430$420203Fig.1.The minimum bend radius/sheet thickness ratio as a function of tensileelongation for various alloys(all data is for alloys in the fully annealed condition atambient temperature).As the alloy ductility decreases it becomes much moredifficult to form lattice truss structures by bending.D.T.Queheillalt,H.N.G.Wadley/Materials and Design30(2009)1966–197519672.3.Relative densityThe modified pyramidal lattice fabricated here is similar to that made by using expanded aluminum sheets [12].The relative den-sity, q;is the volume fraction of the unit cell occupied by metal.For a unit cell defined by the included truss angle,xq¼ð2l þb Þsin x Ácos x Áðl cos x þffiffiffi2p b Þwtl 2;ð2Þwhere l is the truss member length,b is the length of a node,w and t are the width and thickness of the truss member.If x =45°and w =t (the case here),Eq.(2)reduces toq ¼2ffiffiffi2p ð2l þb Þðl þb Þt l2:ð3ÞSamples were fabricated from sheet with thicknesses t =0.635,0.813,1.016,1.270mm.A nominal truss length of l =10.0mm and a flat nodal region b =6.2mm was used for all samples.Table 2shows a comparison between the predicted and measured relative densities for three panels at each relative density.The measured relative densities,calculated by measuring the mass of the sand-wich panel minus the mass of the facesheets,are reasonably pre-dicted and lie just below that calculated by Eq.(3).The deviation between measured and calculated relative densities is attributed to manufacturing variations which included small variations in the truss inclination angle,the finite bend radius at the nodes which was not accounted for in the theoretical calculations and variations in the core thickness over the length of the panels.There were slight variations in the measured core thickness,due to man-ufacturing defects,and an average core thickness was used for the measured relative density calculations.3.Mechanical property predictionsDeshpande and Fleck [9]have developed approximate analyti-cal expressions for the stiffness and strength of pyramidal lattice truss cores assuming elastic–plastic struts.It was assumed that the truss cores were sandwiched between rigid facesheets and their struts were of sufficiently low aspect ratio,t/l ,that there bending stiffness and strength were negligible compared to their stretching stiffness and strength [9].The collapse strength of a lat-tice truss core is determined by the mechanism of strut failure which depends on the cell geometry,strut material properties and the mode of failure during loading (plastic yielding and elastic or plastic buckling).The flat node region in the modified pyramidal truss structure provides no contribution to the compressive stiff-ness and strength or to the shear stiffness.However,it does con-tribute to the shear strength by providing a larger contact area between the core and facesheet.Kooistra and Wadley [12]intro-duced a truss mass fraction to account for the mass of material occupied by the nodes for similar lattices fabricated from an ex-panded aluminum alloy.Here,we have used a similar analysis and the resulting truss mass fraction for these structures is g ¼2l =ð2l þb Þwhere b is defined in Fig.2b As b approaches zero,g =1(an ideal lattice).For non-ideal lattices,g is an analytical esti-mate of the knock-down in the stiffness and strength properties.For all of the samples tested,g =0.763.Table 3shows theanalyticalFig.2.Schematic illustration of (a)an ideal and (b)the modified pyramidal unitcells.Fig.3.Photographs of (a)the periodic asymmetric hexagonal perforation pattern applied to sheets of a Ti–6Al–4V alloy and (b)the partially formed modified pyramidal lattice.1968 D.T.Queheillalt,H.N.G.Wadley /Materials and Design 30(2009)1966–1975expressions for the compressive and shear stiffness and strength of the modified pyramidal lattice truss sandwich structures.4.Experimental mechanical responseThe lattice truss structures were tested at ambient temperature in compression and shear at a nominal strain rate of 4Â10À2s À1in accordance with ASTM C365and C273,respectively.A laser exten-someter measured the facesheet displacement with a precision of ±0.001mm.The nominal compressive strain was obtained by mon-itoring the displacements of the unconstrained facesheets and the nominal shear strain from the displacements of the shear fixtures.For the shear experiments,the samples were rigidly attached to the shear fixtures by steel machine screws and a leading edge stop.To determine parent alloy properties in the diffusion bonded condition,tensile tests were performed on Ti–6Al–4V samples sub-jected to the same thermal cycle used for fabrication of the diffu-sion bonded sandwich structures.The parent material was adequately approximated as an elastic–plastic solid withYoung’sFig.4.Schematic illustration of the lattice structure,face sheet and tool assembly used for hot forming and diffusion bonding the modified pyramidal lattice sandwich structures.D.T.Queheillalt,H.N.G.Wadley /Materials and Design 30(2009)1966–19751969modulus E s ¼116GPa,0.2%offset yield strength r ys ¼900MPaand a linear hardening modulus E t d r =d e ¼1667MPa .4.1.Out-of-plane compressionThe through thickness compressive stress–strain responses for the Ti–6Al–4V lattice truss structures are shown in Fig.6.Follow-Fig.5.Photographs of the Ti–6Al–4V modified pyramidal lattice truss structuresalong with their corresponding relative densities, q¼1:0%;1:7%;2:6%;4:1%.Table 2Measured and predicted relative densities for the Ti–6Al–4V modified pyramidal lattice truss structures Measure relativedensity, q Average relativedensity, q Predicted relativedensity, q 0.00710.0075±0.00040.01020.00790.00750.01390.0139±0.00120.01670.01500.01270.02600.0259±0.00030.02610.02610.02560.03870.0367±0.00180.04070.03580.0355Table 3Analytical expressions for the compression and shear stiffness and strength ofFor the structures here,=45°and =0.763.Fig.6.Through thickness compressive stress–strain responses for Ti–6Al–4V latticetruss structures with relative densities, q,between 1.0%and 4.1%.1970 D.T.Queheillalt,H.N.G.Wadley /Materials and Design 30(2009)1966–1975ing an initial linear response,gradual core yielding was observedfollowed by a peak in the compressive stress.The lattice trussmembers within the structure plastically buckled shortly afteryielding(at a strain of between2%and4%).Continued loading re-sulted in core softening followed by a stress plateau.The plateaustress of the structure corresponded to$50%of the cores peakstrength.Softening was coincident with buckling and the forma-tion of a plastic hinge near the center of the truss member.Neithernode nor truss fracture was observed during any of the compres-sion experiments.4.2.In-plane shear responseThe longitudinal shear stress–strain responses are shown inFig.7.In the in-plane shear orientation,each unit cell has two trussmembers loaded in compression and two in tension.For somesamples the shear stress–strain response exhibited a reasonablylinear behavior followed by premature failure prior to the onsetof an expected peak load,Fig.7.This has been attributed to failureof the trusses and the truss–node interface.However,two of the q¼4:1%cores did not fail prematurely and exhibited characteris-tics typical of lattice truss based sandwich cores including:elasticbehavior during initial loading,followed by macroscopic yieldingof the core and continued load support until a peak stress wasreached which corresponded to the buckling of the compressedtruss members.At high strain the load carrying capacity of the lat-tice structure decreased consistent with node debonding and ten-sile failure at the truss member–node interface,as shown Fig.8a.Alongitudinal metallographic image of a typical truss core–facesheet cross-section is shown in Fig.8b.It can be seen that someresidual porosity existed at the node–facesheet interface as wellas a sharp stress concentration at the confluence of the trusses,theflat node and the facesheet.5.DiscussionpressionAll the lattices showed a similar crush response consisting of anapproximately linear initial loading followed by a distinct yieldattainment of a peak strength and progressive softening thereafter.The initial elastic compressive response of the lattice trusscores was not perfectly linear.This is appears to arise from smalldifferences in the lengths of the lattice truss members which re-sults in non-uniform load distribution.This was consistent withexperimentally observations of aluminum lattice structures whichsubstantially stiffen once the peak load is reached[20].Approximate analytical expressions for the compressive stiff-ness and strength of the pyramidal lattice truss cores can be devel-oped following Deshpande and Fleck[9]and are summarized bythe equations listed in Table3.The unloading modulus just priorto lattice yield has been measured and plotted in Fig.9a as anon-dimensional compressive stiffness,P¼E c=ðE s qÞagainst the lattice relative density, q.The analytical prediction of the non-dimensional compressive stiffness has a value of0.191and mostof the experimental data lie within±20%of this value.The lowestnormalized stiffness corresponds to the highest relative densityand the knock-down is attributed to truss waviness as seen inFig.5c and d.Fig.9b shows the non-dimensional compressive peakstrength,R¼r pk=ðr y qÞagain plotted against q.The analytical pre-diction of the non-dimensional compressive peak strengths arealso plotted for plastic yielding,and elastic and plastic buckling.For the elastic and plastic buckling cases it was assumed that thetruss members were either built-in at the facesheets,whereupon k=2,or pin-jointed for which k=1.The factor k depends on the rotational stiffness of the plastic hinge at the nodes;k=1for a joint that can freely rotate while k=2forfixed-joints whichcannot Fig.7.Shear stress–strain responses for the Ti–6Al–4V lattice truss structures of different relative densities, q.D.T.Queheillalt,H.N.G.Wadley/Materials and Design30(2009)1966–19751971rotate.Fig.9b shows that the plastic buckling model captures the strength dependence upon relative density for samples between 1.0%and 4.1%.For the lower density samples,the data are reason-ably captured by the k =2approximation as minimal truss wavi-ness was observed at the truss–facesheet interface.Whereas,truss waviness at the truss–facesheet confluence contributes to-wards the apparent decrease in strength coefficient for the higher relative density samples and the data lie closer to the k =1approximation.5.2.ShearFig.10a shows the non-dimensional shear stiffness,C ¼G c =ðE s qÞplotted against q .Again,approximate analytical expres-sions for the shear stiffness and strength of the pyramidal lattice truss cores have been developed and are summarized by the equations listed in Table 3.The predicted non-dimensional shear stiffness is also shown and it can be seen that the measured mod-ulus is about half that predicted.This is thought to be a conse-quence of geometric imperfections in the trusses (truss waviness),core–facesheet misalignment and node debonding–truss tensile failure.The data reported here for the shear stiffness was taken at strains within the linear elastic region where geo-metric defects are most pronounced.Audible failures occurred at the onset of the shear tests and serrations in both the elastic and plastic regions of the stress–strain response were observed and correlated to fracture events and were accompanied by acoustic emissions [9,20].These events also contribute to the low shear stiffness data.The analytical predictions of the shear peak strengths are plot-ted in Fig.10b.For the elastic and plastic buckling cases,it was as-sumed that the truss members were built-in at the facesheets (k =2).In addition,a first-order model for panel failure by node flat shearing was developed for the modified pyramidal lattice struc-tures.Maximum yield stress theory (assuming Tresca yield and plane stress conditions)predicts that the shear yield stress,s y =0.5r y ,where r y is the uniaxial tensile yield stress [21,22].This gives a s y =450MPa for the node failure stress analysis of the mod-ified pyramidal lattice.Assuming a node contact area of b Áw per unit cell,the node flat shear fracture force that can be sustained is s y Áb Áw and the shear strength is calculated by dividing by the unit cell area and is also shown in Fig.10b.This serves as an upper limit prediction assuming a perfect bond exists at the interface.Failure was also observed by shearing of the lattice struts of the pyramidal core at the joints with the facesheets.Again,assuming s y =450MPa for the node shear-off stress analysis the node truss shear fracture strength dictated by the node cross-section area t Áw and is also shown in Fig.10b.Several conclusions can be drawn from the data in Fig.10b.Forthe geometry considered here,and for a relative density q<10%,Fig.8.(a)Photograph showing node debonding and fracture occurring during shear ( q¼1:7%).(b)A longitudinal metallographic image of a representative diffusion bonded node–face sheet cross-section.1972 D.T.Queheillalt,H.N.G.Wadley /Materials and Design 30(2009)1966–1975panel failure in shear is predicted to be limited by either yielding by buckling (elastic or plastic)of the truss members assuming no node shear-off.As the relative density increases above 10%,failure is then limited by node flat shear fracture.However,for the geom-etry investigated here,node truss shear-off fracture dominates the behavior over a wide range of relative densities.The quality of the node joints between the core and the facesheet resulted in prema-ture failure of a majority of the panels during shear loading by a combination of node flat shear fracture and node truss shear-off fracture and a majority of the data corresponds approximately with that predicted by node truss shear-off fracture.5.3.Node–facesheet joint design issuesThe design of the core to facesheet interface in honeycomb sandwich panels is of utmost importance.Ultimately,this dic-tates the amount of load which can be transferred from the face-sheets to the core.As shown previously,this is similarly true for lattice based cores.Node bond failure has been identified as a key catastrophic failure mode for sandwich structures,especially titanium based honeycombs [23].Similar node robustness issues have been previously observed during shear loading of lattice truss topologies [9,10,20].When sandwich panels are subjected to shear or bending loads,the node transfers forces from the facesheets to the core members (assuming adequate node bond strength exists)and the topology for a given core relative den-sity dictates the load carrying capacity.When the node–face-sheet interface strength is compromised,from either poor joint design or bonding methods,node bond failure occurs and cata-strophic failure of the sandwich panelresults.Fig.9.Analytical predictions and experimental data for (a)compressive stiffness and (b)compressive strengthcoefficients.Fig.10.Analytical predictions and experimental data for (a)the non-dimensional shear stiffness and (b)the shear strength as a function of relative density.Solid circles (d )indicate samples that failed at the nodes prior to the full peak strength.D.T.Queheillalt,H.N.G.Wadley /Materials and Design 30(2009)1966–19751973Titanium alloy honeycomb sandwich panels have been widely used in aerospace structures.They can be formed by either an expansion or corrugation processes [4,6].Numerous methods have been proposed for joining titanium honeycomb cores to facesheets including:solid-state diffusion bonding [24–27],electron beam welding [28],and transient liquid interface diffusion bonding [29–35].Solid-state diffusion bonding of titanium honeycomb structures is limited because stress-reducing fillets are not formed at the core–facesheet interface.During brazing,liquid interface dif-fusion and activated diffusion bonding processes,a liquid filler me-tal is drawn into the nodes and forms a stress-reducing fillet at the core–facesheet interface.One potential mitigation strategy is to use a combination of hot forming/diffusion bonding,followed by brazing.Fig.11a and b show longitudinal and transverse micrographs of a node/facesheet interface that has been hot formed/diffusion bonded by the process developed here and subsequently brazed.A Ticu-ni Òbraze alloy with a nominal composition of 60.0Ti–25.0Ni–15.0Cu wt.%was used as the braze alloy.The sample was vac-uum brazed ($10À4Torr chamber pressure)at a heating rate of 10°C/min to 550°C,held for 1h (to volatilize the binder),then heated to the brazing temperature of 950°C for 60min prior to furnace cooling at $25°C/min.During brazing the molten al-loy is drawn into the node–facesheet interface.It is seen in Fig.11,that the combination of hot forming/diffusion bonding followed by brazing forms a large filet at the truss–facesheet interface.The presence of this filet reduces the stress concentra-tion and increases the node contact area which is likely to reduce the local stress supported at the parisons with other topologiesTo assess how well the Ti–6Al–4V modified pyramidal lattice trusses compete with other lattice and prismatic topologies in compression,the compressive peak strength,r pk ,versus absolute density,q s ,is shown in Fig.12.It can be seen from Fig.12that all the lattice and prismatic topologies roughly follow the same dependency upon density with the lattices out performing the prismatic topologies.In addition to the topology dependence,the parent materials strength-to-weight ratio affect becomes apparent.Ti–6Al–4V possesses a higher strength-to-weight ratio than 6061aluminum and 304stainless steel and therefore sandwich struc-tures made from it are stronger on a per weight basis.In addition,increasing the truss mass fraction g from 0.76to 1.0would in-crease the specific strength of the Ti–6Al–4V lattice truss sandwich structures shown in Fig.12by $25%.6.SummaryA new method for fabricating a pyramidal lattice truss structure has been developed using a combination of ambient tempera-ture forming and a combination of high-temperature forming/diffusion bonding.The approach was illustrated by fabricating and testing sandwich panels using a Ti–6Al–4V alloy and appears extendable to other alloy systems that exhibit limited ambient temperature formability and are suitable for diffusion bonding.Analytical predictions for a regular pyramidal lattice truss struc-ture have been adapted for modified lattices fabricated here.The stiffness and peak strength of these modified pyramidal lattices depends on three dominant factors:(i)the stress–strain response of the parent alloy,(ii)the truss mass fraction,g ,and(iii)the relative density, q,of the lattice core.The modified pyra-midal lattices stiffness and strengths were shown to be reduced from that of an ideal pyramidal lattice by the truss mass effi-ciency factor g.Fig.11.Longitudinal and transverse metallographic images of a typical node–face sheet cross-section after diffusion bonding and subsequentbrazing.pressive peak strength versus density for various sandwich core topologies.(See above mentioned references for further information.)1974 D.T.Queheillalt,H.N.G.Wadley /Materials and Design 30(2009)1966–1975。
有限元复习资料-缩印版
简答：1.What is the difference between the strong and weak forms of system equations?(强形式、弱形式区别)：强形式：要求强的连续性，可微次数必须等于存在于系统方程中偏微分方程的次数。
弱形式：通常是积分形式，要求较弱的连续性，基于弱形式的公式通常可以得到一组更逼近于真实解的离散的系统方程。
2. What are the conditions that assumed displacement has to satisfy in order to apply the Hamilton’s principle?（应用哈密尔顿原理必须满足的条件）：协调性方程、本质边界条件、初时刻和末时刻的条件。
3.Briefly describe the standard steps involved in the finite element method.（有限元的步骤）：域的离散、位移插值、局部坐标系中有限元方程的形成、坐标转换、整体有限元方程的组装、施加位移约束、求解有限元方程。
4.Do we have to discretize the problem domain in order to apply the Hamilton’s principle? What is the purpose of dividing the problem domain into elements?（必须离散问题域吗？为什么要离散？）：不必须；为了更好地假设位移场的参数。
5. How many DOFs does a 2-nodal, planar truss element have in its local coordinate system, and in the global coordinate system? Why is there a difference in DOFs in these two coordinatesystems?（两节点平面桁架单元在局部和整体坐标系中各有多少个自由度？为什么？）：在局部坐标系中有两个自由度，整体坐标系中有4和自由度。
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1. Introduction In [Bou79a] and [Bou79b], Bousfield introduced an equivalence relation on spectra that has turned out to be extremely important. Given a spectrum E , we define the Bousfield class E of E to be the collection of E -acyclic spectra X , where X is E -acyclic if and only if E ∧ X = 0. Then we say that E and F are Bousfield equivalent if and only if E = F . The notion of Bousfield equivalence, and hence Bousfield class, plays a major role in much of modern stable homotopy theory. We can order the collection of Bousfield classes using reverse inclusion. We then have a partially ordered class associated to the stable homotopy category, and Bousfield and others have investigated properties of this partially ordered class. The nilpotence theorem of Devinatz, Hopkins, and Smith [DHS88], for example, is equivalent to the classification of Bousfield classes of finite spectra [HS]. We recently learned that Ohkawa has proved the surprising result that there is only a set of Bousfield classes [Ohk89]; see also [Str97]. He proves there are at most 2 Bousfield classes, where i = 2 i−1 and 0 = ℵ0 . In light of this result, the authors decided to re-examine the structure of the partially ordered set of Bousfield classes. The goal of this paper is to provide some kind of global understanding of the partially ordered set B of Bousfield classes. Using Ohkawa’s result, we are able to perform certain constructions in B, such as a greatest lower bound operation. We also bring to bear many methods and results from lattice theory; for instance, the sub-partially ordered set DL of B, which consists of all Bousfield classes X for which X = X ∧ X , is a very nice sort of distributive lattice known as a frame. This has some nice consequences, and it also leads to some interesting questions. Much of our understanding of the Bousfield lattice is only conjectural; we hope that the conjectures and their implications are interesting enough to prompt further study of this material. There are several questions we have not addressed. In particular, a frame such as DL has an associated topological space. It would be interesting to understand something about this space, even conjecturally. Jack Morava has asked whether this space has a structure sheaf, probably of stable homotopy categories, associated to it. The stalk at K (n), for example, might be
Date : February 1, 2008. 1991 Mathematics Subject Classification. 55P42, 55P60, 06D10. Research partially supported by a National Science Foundation grant. Research partially supported by National Science Foundation grant DMS-9407459.
THE STRUCTURE OF THE BOUSFIELD LATTICE
arXiv:math/9801103v1 [math.AT] 22 Jan 1998
MARK HOVEY AND JOHN H. PALMIERI Abstract. Using Ohkawa’s theorem that the collection B of Bousfield classes is a set, we perform a number of constructions with Bousfield classes. In particular, we describe a greatest lower bound operator; we also note that a certain subset DL of B is a frame, and we examine some consequences of this observation. We make several conjectures about the structure of B and DL.
12ຫໍສະໝຸດ MARK HOVEY AND JOHN H. PALMIERI
the K (n)-local category. There are also many frame-theoretic properties that DL may or may not have, such as coherence. Here is one of the conjectures that we do discuss. Call a Bousfield class X strange if X < HFp . For instance, the Brown-Comenetz dual of the p-local sphere has such a Bousfield class. By general lattice theory, the inclusion DL ֒→ B has a “right adjoint” r : B − → DL which is a retraction onto DL. One can see that r sends every strange Bousfield class to 0 , and also that r induces a map r′ : B/(strange) − → DL, where B/(strange) is the quotient lattice of B by the ideal of strange Bousfield classes. Conjecture 3.12 states that r′ is an isomorphism; this implies, for example, that E ∧ E = E ∧ E ∧ E for all spectra E . Our other main conjectures are Conjectures 5.1, 6.12, 7.4–7.6, and 9.1. Here is the structure of the paper. In Section 2, we define Bousfield classes and the basic operations one can perform on them: join, smash, meet, and complementation. Next we examine DL and its relation to B; in particular, we note that DL is a frame, and we construct a retraction from B to DL. We also give the conjectured description of this retraction in terms of strange Bousfield classes. We discuss more basic structure in Section 4: we discuss minimal and complemented Bousfield classes, and we recall some properties of BA, the set of complemented Bousfield classes. For example, we recall Bousfield’s observation that BA is a Boolean algebra. In Section 5 we examine spectra X for which there is a finite spectrum F with X ∧ F = 0; we give a conjectured classification of the Bousfield classes of such X . This provides some information about BA. In Section 6, we return to the fact that DL is a frame; this allows us to construct a complete Boolean algebra cBA ⊆ DL which (properly) contains BA, and we give a conjectured description of cBA. Then in the next section, we examine Bousfield classes of spectra X for which X ∧ F = 0 for all finite F . This leads to a discussion of some properties of I , the Brown-Comenetz dual of the p-local sphere, as well as several conjectures about spectra with no finite acyclics. We show that these conjectures are all equivalent, and we discuss some of their consequences. Much of the paper to this point suggests that the set of strange Bousfield classes, those classes of p-local spectra X with X < HFp , is interesting; in Section 8 we examine some examples of such spectra. We end the paper in Section 9 with a discussion of the partially ordered class of localizing subcategories—recall that a subcategory is called localizing if it is thick and is closed under coproducts; the main conjecture is that every localizing subcategory is equal to the class of E -acyclics for some spectrum E . This conjecture has several equivalent formulations, and some deep structural consequences. We work p-locally throughout the paper, except for Section 9, in which we work globally. As in all discussions of Bousfield classes of spectra, we work in the stable homotopy category of spectra, as described for example in [HPS97]. The authors would like to thank Dan Christensen and Neil Strickland for many helpful discussions about Bousfield classes. 2. Basic structure of the Bousfield lattice: ∨, ∧, , and a