Crystal Plasticity Finite Element Theoretical Models and Applications for High Temperature, High Pressure and High Strain-Rate Dynamic Process
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摘要: 对于高温、高压、高应变速率加载条件下的材料冲击变形行为，动态晶体塑性模型能够直接反映晶体中塑性滑移的各向异性及其对温度、压力和微观组织结构的依赖性，因而广泛应用于材料的动态冲击力学响应、微观结构演化以及动态损伤破坏的模拟。本文综述了高压冲击下动态晶体塑性有限元的理论模型，主要包括变形运动学、包含状态方程的超弹性本构模型和晶体塑性本构模型，涉及位错滑移、相变、孪生等塑性变形机制，以及层裂、绝热剪切带等动态破坏方式。Abstract: For shock deformation behavior of materials under high temperature, high pressure and high strain-rate loading conditions, dynamic crystal plasticity models can directly reflect the anisotropy of plastic slip and its dependence of temperature, pressure and microstructure in crystals. In consequence, dynamic crystal plasticity models are widely used in simulations of material impact dynamic response, microstructure evolution and dynamic damage. Theoretical models of dynamic crystal plasticity under high pressure shock loading conditions were reviewed in this paper, mainly including: deformation kinetics, hyperelastic constitutive models incorporating equations of state, and crystal plasticity constitutive models. This paper also covers plastic deformation mechanisms, including dislocation slip, phase transition and twinning; as well as dynamic damage, including spalling and adiabatic shear band.
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图 7 位错平均运动与热激活运动以及拖曳运动的对比[27]
Figure 7. Comparison of the average dislocation velocity with the velocities of thermally-activated and drag-dominated dislocation motions[27]
图 12 铅合金动态晶体塑性有限元模拟结果：(a)层裂形核时的压力，(b)层裂形核时的弹性能密度，(c)经250 m/s冲击加载层裂面附近的等效应力；(d)经350 m/s冲击加载层裂面附近的等效应力[24]
Figure 12. Dynamic crystal plasticity finite element simulation results of lead alloy: (a) pressure of spalling nucleation; (b) elastic energy density of spalling nucleation; (c) equivalent stress near the spalling surface under 250 m/s shock loading; (d) equivalent stress near the spalling surface under 350 m/s shock loading[24]
A1 运动学符号说明
A1. Symbol description of kinematics
Symbols Description F(Fe, Fp, Fθ ) Deformation gradient including elastic, plastic and thermal components L(Le, Lp, Lθ ) Velocity gradient including elastic, plastic and thermal components Re Rotation tensor Ue Right stretch tensor α Thermal expansion coefficient tensor 
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A2 热力学符号说明
A2. Symbol description of thermodynamics
Symbols Description Symbols Description Dint Intrinsic dissipation of the system K0 Bulk modulus at zero pressure ψ Helmholtz free energy K′ Pressure derivative of bulk modulus s Entropy of the system TD Debye temperature T Temperature R Molar gas constant KT Isothermal bulk modulus Mmol Molar mass of the material cV Heat capacity at constant volume kB Boltzmann constant Γ Grüneisen coefficient XTN Variables related to the lattice thermal vibration qn Internal variables for microscopic defects such
as dislocations in materialsXTE Variables related to the electron activation
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A3 塑性本构符号说明
A3. Symbol description of plastic constitution
Symbols Description Symbols Description λ α Mean spacing between obstacles ${{\,\rho}_{\rm{for}}^{\alpha} }$ Forest dislocation density τα Resolved shear stress ${{t_{\rm r}^{\alpha}}}$ The drag-dominated mean transit time between obstacles Qα Activation energy B Viscous drag coefficient gα Slip resistance ${{\dot{\rho }}_{\rm{nuc}}^{\alpha }}$ The nucleation rate ${{g}_{\rm{ath}}^{\alpha} }$ Athermal slip resistance ${{\dot{\rho }}_{\rm{hom}}^{\alpha }}$ The homogeneous nucleation rate hαβ Hardening coefficient ${ {\dot{\rho }}_{\rm{het}}^{\alpha }}$ The heterogeneous nucleation rate ρα Total dislocation density ${{\dot{\rho }}_{\rm{mult}}^{\alpha }}$ The multiplication rate ${{\rho}_{\rm{m}}^{\alpha} }$ Mobile dislocation density ${{\dot{\rho }}_{\rm{trap}}^{\alpha }}$ The trapping rate ${{\rho}_{\rm{i}}^{\alpha} }$ Immobile dislocation density ${{\dot{\rho }}_{\rm{ann}}^{\alpha }}$ The annihilation rate bα Burgers vector da Capture distance of annihilation vα Velocity of mobile dislocations ${{t}_{\rm{w}}^{\alpha}}$ The thermal activation-dominated waiting time at a barrier ${ {\dot{\gamma }}^{\alpha }}$ Slip rate on slip system α
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A4 超弹性本构符号说明
A4. Symbol description of hyper-elastic constitution
Symbols Description Symbols Description I Second-order unit tensor $\widehat{{{E}}^{\rm{e}}}$ Isochoric strain in expanded configuration Ee Elastic Green–Lagrange strain $ \widehat{\widehat{{{E}}^{\rm{e}}}}$ Isochoric strain in configuration I Ce Elastic right Cauchy-Green tensor $ \overline {{{{E}}^{\rm{e}}}}$ Volumetric strain in configuration I $\widehat{{{{F}}}^{\rm{e}}}$ Isochoric part of elastic deformation S Second Piola–Kirchhoff stress $\overline {{{{F}}^{\rm{e}}}}$ Volumetric expansion
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A5 相变、孪晶与动态破坏符号说明
A5. Symbol description of phase transformation, twining and damage
Symbols Description Symbols Description Ftr Deformation gradient of phase transformation $S_{\rm{tw}}^{\beta}$ Twin resistance of twin system vp Volume fraction of the parent phase ρdeb Dislocationdebris density vt Volume fraction of the new phase t dmfp Dislocation mean free path related to the volume fraction of twin vN Volume fraction of all new phases ${\varphi}$ Void volume fraction ft Driving force of phase transformation Fd Volumetricpartofplastic deformation gradient in porous crystal plastic model f β Volume fraction of twin Yr Resistance of damage evolution γtw Characteristic shear strain of twining
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