Large Deformation Plasticity of Amorphous Solids, with 非晶态固体大变形塑性,和
合集下载
相关主题
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
[2] Boyce, M. C., Arruda, E. M., 1990. “An experimental and analytical investigation of the large strain compressive and tensile response of glassy polymers.” Polymer Engineering and Science 30 (20), 12881298.
Smooth time-dependent rigid transformations of the Eulerian Space:
y (X ,t) y * (X ,t) Q ( t)y (X ,t) q ( t)
Principle of Relativity: relation of the two motions is equivalent Q(t) Relative motion of two observers Eulerian bases
Kinematics – Multiplicative Decomposition of the Deformation Gradient II
LLeFeLpFe1 LeF eFe1 DeWe LpF pFp1 DpWp
De symLe We skwLe Dp symLp Wp skwLp
dP
P
P
Macroscopic Force Balance
diTv fb 0
• Internal energy Wint is invariant under all changes
in frame
Wi*nt Wint
– Microforce Balance
Wint TDeJ1TPDPdV t
来自百度文库
s ~s (
)
P
g
o
s s cv
1
P
= evolution of shear resistance (captures strain softening)
To be objective (in general): g* Qg
G* QGQT
F QF Fe QFe Le QLeQ TQ Q T
The relaxed and reference configurations are invariant to the transformations of the Eulerian Space
JTD eTPD P0
• Constitutive framework: ˆ ( F e , F P )
Free energy, stress, and internal variables are a
T Tˆ ( F e , F P )
function of deformation. T P Tˆ P ( F e , F P , D P , )
ˆ ( F e , F P )
T Tˆ ( F e , F P )
T P Tˆ P ( F e , F P , D P , ) h i ( F e , F P , D P , )
(C e, B P ) T F eT (C e , B P )F eT T P T P (C e , B P , D P , ) h i (C e , B P , D P , )
Motivation – Examples of Materials
• Amorphous Solids –
– polymeric and metallic glasses (i.e. Polycarbonate) – Rubber degradation
• Biomaterials
– Soft Collageneous Biological Tissue (i.e. cartilage, cervical tissue, skin, tendon, etc.)
Constitutive Theory – Thermodynamic Restrictions and Flow Rule
C e CeB PBP
Plug into dissipation inequality
T2J1Fe(CCee,BP)Fe TP2syom (CBeP ,BP)YP
Te ReT TRe
T ReTeReT = Cauchy Stress
Constitutive Equations
• FLOW RULE for Plastic STRETCHING
material parameters
Constitutive prescription
FP DPFP FPX,01
Kinematics – Multiplicative Decomposition of the
Deformation Gradient
F p X
FeX
dxFeXdl Segment of the “current
configuration”
d l FpXdX
Segment of the configuration”
Fy(X,t) Deformation Gradient
F FeFp
Decomposition of deformation gradient into its elastic and plastic components (Kroner-Lee)
vy(X,t) Velocity tensor LgrvadF F1 Velocity Gradient
• Conditions of Plastic Flow
– Incompressible
det F p 1 tr L p 0
JdeFtdeFte
– Irrotational
Wp 0
Lp Dp
F p DpFp
Principle of Objectivity Principle of Material Frame Indifference
“relaxed
“Relaxed Configuration”: Intermediate configuration created by elastically unloading the current configuration and relieving the part of all stresses.
h i ( F e , F P , D P , )
Constitutive Theory – Framework
• Frame Indifference
– Euclidean Space
– Amorphous Solids: material are invariant under all rotations of the Relaxed and Reference Configuration
P
Dissipation Inequality and Constitutive Framework
• 2nd Law of Thermodynamics: The temporal increase in free energy ψ of any part P be less than or equal to the power expended on P
FP DP
De QDeQT W e Q W eQ TQ Q T
Principal of Virtual Power
• External expenditure of power = internal energy
Wext t(n)v~dA fbv~dVt W int TL ~eJ1TPD ~PdtV
Large Deformation Plasticity of Amorphous Solids, with Application
and Implementation into Abaqus
Kristin M. Myers January 11, 2019 Plasticity ES 246 - Harvard
DSebfaincek: 2sym o(CBeP,BP)
YP(C e,BP,D P, )SoSback
S02syoC m eTe 2syo m C e( C C ee ,B P)
Constitutive Equations
material parameters
• Free Energy
e P
Constitutive prescription
eGEe o21/2KtrEe2
P (P)
• Equations for Stress
Te2GEeoKtrEe1 Te Stress conjugate to Ee
References: [1] Anand, L., Gurtin, M.E., 2019. “A theory of amorphous solids undergoing large deformations, with application to polymeric glasses.” International Journal of Solids and Structures 40, 1465-1487.
– Engineering Collagen Scaffolds (i.e. skin, nerve, tendon etc.)
Material Characteristics: 1. Large stretches – elastic & inelastic 2. Highly non-linear relationships between stress/strain 3. Time-Dependent; viscoplasticity 4. Strain hardening or softening after initial yield 5. Non-linearity of tension & compression behavior (Bauschinger effect)
Energy dissipated per unit volume (in the relaxed configuration) must be purely dissipatative. Dissipative FLOW STRESS:
YP(C e,BP,D P,)D P0
FLOW RULE:
[3] Lubliner, J. Plasticity Theory. 1990. Macmillan Publishing Company. (Chapter 8)
[4] Abaqus 6.5-4 Documentation “Getting Started with ABAQUS/EXPLICIT.” Hibbitt, Karlsson & Sorensen,INC.
Experimental Results – Polycarbonate
From Boyce and Arruda
COMPRESSION
TENSION
• Large deformation regime • Strain-softening after initial yield • Back stress evolution after yield drop to create strain-hardening
DP
PTeo
BoP 2
P
o
s
p1/m
DP=(magnitude)(DIRECTION)
Effective Stress:
1 2Teo BoP
• Evolution of Internal Variables
s
h o 1
Smooth time-dependent rigid transformations of the Eulerian Space:
y (X ,t) y * (X ,t) Q ( t)y (X ,t) q ( t)
Principle of Relativity: relation of the two motions is equivalent Q(t) Relative motion of two observers Eulerian bases
Kinematics – Multiplicative Decomposition of the Deformation Gradient II
LLeFeLpFe1 LeF eFe1 DeWe LpF pFp1 DpWp
De symLe We skwLe Dp symLp Wp skwLp
dP
P
P
Macroscopic Force Balance
diTv fb 0
• Internal energy Wint is invariant under all changes
in frame
Wi*nt Wint
– Microforce Balance
Wint TDeJ1TPDPdV t
来自百度文库
s ~s (
)
P
g
o
s s cv
1
P
= evolution of shear resistance (captures strain softening)
To be objective (in general): g* Qg
G* QGQT
F QF Fe QFe Le QLeQ TQ Q T
The relaxed and reference configurations are invariant to the transformations of the Eulerian Space
JTD eTPD P0
• Constitutive framework: ˆ ( F e , F P )
Free energy, stress, and internal variables are a
T Tˆ ( F e , F P )
function of deformation. T P Tˆ P ( F e , F P , D P , )
ˆ ( F e , F P )
T Tˆ ( F e , F P )
T P Tˆ P ( F e , F P , D P , ) h i ( F e , F P , D P , )
(C e, B P ) T F eT (C e , B P )F eT T P T P (C e , B P , D P , ) h i (C e , B P , D P , )
Motivation – Examples of Materials
• Amorphous Solids –
– polymeric and metallic glasses (i.e. Polycarbonate) – Rubber degradation
• Biomaterials
– Soft Collageneous Biological Tissue (i.e. cartilage, cervical tissue, skin, tendon, etc.)
Constitutive Theory – Thermodynamic Restrictions and Flow Rule
C e CeB PBP
Plug into dissipation inequality
T2J1Fe(CCee,BP)Fe TP2syom (CBeP ,BP)YP
Te ReT TRe
T ReTeReT = Cauchy Stress
Constitutive Equations
• FLOW RULE for Plastic STRETCHING
material parameters
Constitutive prescription
FP DPFP FPX,01
Kinematics – Multiplicative Decomposition of the
Deformation Gradient
F p X
FeX
dxFeXdl Segment of the “current
configuration”
d l FpXdX
Segment of the configuration”
Fy(X,t) Deformation Gradient
F FeFp
Decomposition of deformation gradient into its elastic and plastic components (Kroner-Lee)
vy(X,t) Velocity tensor LgrvadF F1 Velocity Gradient
• Conditions of Plastic Flow
– Incompressible
det F p 1 tr L p 0
JdeFtdeFte
– Irrotational
Wp 0
Lp Dp
F p DpFp
Principle of Objectivity Principle of Material Frame Indifference
“relaxed
“Relaxed Configuration”: Intermediate configuration created by elastically unloading the current configuration and relieving the part of all stresses.
h i ( F e , F P , D P , )
Constitutive Theory – Framework
• Frame Indifference
– Euclidean Space
– Amorphous Solids: material are invariant under all rotations of the Relaxed and Reference Configuration
P
Dissipation Inequality and Constitutive Framework
• 2nd Law of Thermodynamics: The temporal increase in free energy ψ of any part P be less than or equal to the power expended on P
FP DP
De QDeQT W e Q W eQ TQ Q T
Principal of Virtual Power
• External expenditure of power = internal energy
Wext t(n)v~dA fbv~dVt W int TL ~eJ1TPD ~PdtV
Large Deformation Plasticity of Amorphous Solids, with Application
and Implementation into Abaqus
Kristin M. Myers January 11, 2019 Plasticity ES 246 - Harvard
DSebfaincek: 2sym o(CBeP,BP)
YP(C e,BP,D P, )SoSback
S02syoC m eTe 2syo m C e( C C ee ,B P)
Constitutive Equations
material parameters
• Free Energy
e P
Constitutive prescription
eGEe o21/2KtrEe2
P (P)
• Equations for Stress
Te2GEeoKtrEe1 Te Stress conjugate to Ee
References: [1] Anand, L., Gurtin, M.E., 2019. “A theory of amorphous solids undergoing large deformations, with application to polymeric glasses.” International Journal of Solids and Structures 40, 1465-1487.
– Engineering Collagen Scaffolds (i.e. skin, nerve, tendon etc.)
Material Characteristics: 1. Large stretches – elastic & inelastic 2. Highly non-linear relationships between stress/strain 3. Time-Dependent; viscoplasticity 4. Strain hardening or softening after initial yield 5. Non-linearity of tension & compression behavior (Bauschinger effect)
Energy dissipated per unit volume (in the relaxed configuration) must be purely dissipatative. Dissipative FLOW STRESS:
YP(C e,BP,D P,)D P0
FLOW RULE:
[3] Lubliner, J. Plasticity Theory. 1990. Macmillan Publishing Company. (Chapter 8)
[4] Abaqus 6.5-4 Documentation “Getting Started with ABAQUS/EXPLICIT.” Hibbitt, Karlsson & Sorensen,INC.
Experimental Results – Polycarbonate
From Boyce and Arruda
COMPRESSION
TENSION
• Large deformation regime • Strain-softening after initial yield • Back stress evolution after yield drop to create strain-hardening
DP
PTeo
BoP 2
P
o
s
p1/m
DP=(magnitude)(DIRECTION)
Effective Stress:
1 2Teo BoP
• Evolution of Internal Variables
s
h o 1