Self-consistent modeling of large plastic deformation, texture and morphology evolution in semi-crystalline polymers

A self-consistent model for semi-crystalline polymers is proposed to study their constitutive behavior, texture and morphology evolution during large plastic deformation. The material is considered as an aggregate of composite inclusions, each representing a stack of crystalline lamellae with their adjacent amorphous layers. The deformation within the inclusions is volume-averaged over the phases. The interlamellar shear is modeled as an additional slip system with a slip direction depending on the inclusion's stress. Hardening of the amorphous phase due to molecular orientation and, eventually, coarse slip, is introduced via Arruda-Boyce hardening law for the corresponding plastic resistance. The morphology evolution is accounted for through the change of shape of the inclusions under the applied deformation gradient. The overall behavior is obtained via a viscoplastic tangent self-consistent scheme. The model is applied to high density polyethylene (HDPE). The stress-strain response, texture and morphology changes are simulated under different modes of straining and compared to experimental data as well as to the predictions of other models.     

A crystal plasticity model that incorporates stresses and strains due to slip gradients

This work is concerned with incorporating the kinematic and stress effects of excess dislocations in a constitutive model for the elastoplastic behavior of crystalline materials. The foundation of the model is a three term multiplicative decomposition of the deformation gradient in which the two classical terms of plastic and elastic deformation are included along with an additional term for long range strain due to the collective effects of excess dislocations. The long range strain is obtained from an assumed density of Volterra edge dislocations and is directly related to gradients in slip. A new material parameter emerges which is the size the region about a continuum point that contributes to long range strains.Using Hookean elasticity, the stress at a point is linearly related to the sum of the elastic plus the long range strain fields. However, the driving force for slip is postulated to be due only to the elastic stress so that the long range stress is a back stress in the constitutive relationship for plastic deformation. A consistent balance of the total deformation rate with the three proposed mechanisms of deformation leads to a set of differential equations that can be solved for the elastic stress, rotation and pressure which then implicitly defines the material state and equilibrium stress. Results from the simulation of a tapered tensile specimen demonstrate that the constitutive model exhibits isotropic and kinematic type hardening effects as well as changes in the pattern of plastic deformation and necking when compared to a material without slip gradient effects.     

采用瓣叶复合材料的生物心脏瓣膜的力学分析

采用微观组织结构分析及宏观复合材料分析结合的方法,分析了猪主动脉瓣的非线性复合材料性质,提出了一种适用于猪主动脉瓣的非线性复合材料本构模型,用提出的非线性复合材料本构模型,对闭合承载状态下的等厚度与变厚度几何模型的猪主动脉瓣的应力分布及变形进行了有限元数值模拟,发现：与各向同性瓣叶相比,单向增强复合材料的瓣叶不但具有较强的承载能力,而且具有较大的柔软性。     

The crack tip zone shielding effect for ductile particle reinforced brittle materials

The crack tip zone shielding effect for the ductile particle reinforced brittle materials is analyzed by using a micromechanics constitutive theory. The theory is developed here to determine the elastoplastic constitutive behavior of the composite. The elastoplastic particles, with isotropic or kinematical hardening, are uniformly dispersed in the brittle elastic matrix. The method proposed is based on the Mori-Tanaka's concept of average stress in the composite. The macroscopic yielding condition and the incremental stress strain relation of the composite during plastic deformation are explicity given in terms of the macroscopioc applied stress and the microstructural parameters of the composite such as the volume fraction and yield stress of ductile particles, elastic constants of the two phases, etc. Finally, the contribution of the plastic deformation in the particles near a crack tip to the toughening of the composite is evaluated. The project supported by National Natural Science Foundation of China     

Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements

This paper is concerned with the multiscale simulation of plastic deformation of metallic specimens using physically-based models that take into account their polycrystalline microstructure and the directionality of deformation mechanisms acting at single-crystal level. A polycrystal model based on self-consistent homogenization of single-crystal viscoplastic behavior is used to provide a texture-sensitive constitutive response of each material point, within a boundary problem solved with finite elements (FE) at the macroscale. The resulting constitutive behavior is that of an elasto-viscoplastic material, implemented in the implicit FE code ABAQUS. The widely-used viscoplastic selfconsistent (VPSC) formulation for polycrystal deformation has been implemented inside a user-defined material (UMAT) subroutine, providing the relationship between stress and plastic strain-rate response. Each integration point of the FE model is considered as a polycrystal with a given initial texture that evolves with deformation. The viscoplastic compliance tensor computed internally in the polycrystal model is in turn used for the minimization of a suitable-designed residual, as well as in the construction of the elasto-viscoplastic tangent stiffness matrix required by the implicit FE scheme.Uniaxial tension and simple shear of an FCC polycrystal have been used to benchmark the accuracy of the proposed implicit scheme and the correct treatment of rotations for prediction of texture evolution. In addition, two applications are presented to illustrate the potential of the multiscale strategy: a simulation of rolling of an FCC plate, in which the model predicts the development of different textures through the thickness of the plate; and the deformation under 4-point bending of textured HCP bars, in which the model captures the dimensional changes associated with different orientations of the dominant texture component with respect to the bending plane.     

Plastic deformation anisotropy and work-hardening of composite materials

An energy-balance method is applied to discuss plastic anisotropy and work-hardening rate of a composite material. It is found that aligned fibers introduce a strongly anisotropic mode of plastic deformation while randomly-oriented inclusions produce isotropic plastic deformation. The hardening rate due to randomly-oriented inclusions is shown to be independent of the inclusion shape, and to be equal to that due to spherical inclusions when the elastic constants of the inclusions are the same as those of the matrix.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

Micromechanics, macromechanics and constitutive modeling of the elasto-viscoplastic deformation of rubber-toughened glassy polymers

Under certain conditions, such as sufficiently low temperatures, high loading rates and/or highly triaxial stress states, glassy polymers display an unfavorable characteristic—brittleness. A technique used for reducing the brittleness (increasing the fracture toughness) of these materials is rubber toughening. While there is significant qualitative understanding of the mechanical behavior of rubber-toughened polymers, quantitative modeling tools for the large-strain deformation of rubber-toughened glassy polymers are largely lacking.In this paper, we develop a suite of numerical tools to investigate the mechanical behavior of rubber-toughened glassy polymers, with emphasis on rubber-toughened polycarbonate. The rubber particles are modeled as voids in view of their deformation-induced cavitation early during deformation. A three-dimensional micromechanical model of the heterogeneous microstructure is developed to study the effects of initial rubber particle (void) volume fraction on the underlying elasto-viscoplastic deformation mechanisms in the material, and how these mechanisms influence the macroscopic response of the material. A continuum-level constitutive model is developed for the large-strain elasto-viscoplastic deformation of porous glassy polymers, and it is calibrated against micromechanical modeling results for porous polycarbonate. The constitutive model can be used to study various boundary value problems involving rubber-toughened (porous) glassy polymers. As an example, the case of an axisymmetric notched bar is simulated for the case of polycarbonate with varying levels of initial porosity. The quality of the constitutive model calibration is assessed using a multi-scale modeling approach.     

A two-phase elastoplastic model for unidirectionally-reinforced materials

A complete analytical formulation for the elastoplastic behaviour of a composite material comprising one single array of reinforcing inclusions perfectly bonded to the matrix is developed in this paper. Fundamental relationships establish the link between the total stress and strain variables, and those pertaining to the individual constituents (matrix and reinforcement) regarded as superposed continuous phases. Assuming that each constituent behaves as an elastic perfectly plastic material, the constitutive equations governing the evolution of the reinforced material as a whole are derived. They reveal a hardening phenomenon arising from the non-compatibility between matrix and reinforcement plastic strains. It is shown in particular that the obtained constitutive law falls within the formalism of generalized standard plasticity: the reinforcement residual stress plays the role of a hardening parameter which controls the evolution of the yield surface, while the associated kinematic variable is the plastic strain discrepancy between matrix and reinforcement phases.Owing to its inherent simplicity, the model is easily amenable to a numerical treatment for structural analysis. It is shown in particular how the classical iterative algorithm can be modified accordingly, and an illustrative application is finally presented in the field of civil engineering.     

Corotational rates in constitutive modeling of elastic-plastic deformation

The principal axes technique is used to develop a new hypoelastic constitutive model for an isotropic elastic solid in finite deformation. The new model is shown to produce solutions that are independent of the choice of objective stress rate. In addition, the new model is found to be equivalent to the isotropic finite elastic model; this is essential if both models describe the same material.

The new hypoelastic model is combined with an isotropic flow rule to form an elastic-plastic rate constitutive equation. Use of the principal axes technique ensures that the stress tensor is coaxial with the elastic stretch tensor and that solutions do not depend on the choice of objective stress rate. The flow rule of von Mises and a parabolic hardening law are used to provide an example of application of the new theory. A solution is obtained for the prescribed deformation of simple rectilinear shear of an isotropic elastic and isotropic elastic-plastic material.     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

Time-dependent behavior of passive skeletal muscle

An isotropic three-dimensional nonlinear viscoelastic model is developed to simulate the time-dependent behavior of passive skeletal muscle. The development of the model is stimulated by experimental data that characterize the response during simple uniaxial stress cyclic loading and unloading. Of particular interest is the rate-dependent response, the recovery of muscle properties from the preconditioned to the unconditioned state and stress relaxation at constant stretch during loading and unloading. The model considers the material to be a composite of a nonlinear hyperelastic component in parallel with a nonlinear dissipative component. The strain energy and the corresponding stress measures are separated additively into hyperelastic and dissipative parts. In contrast to standard nonlinear inelastic models, here the dissipative component is modeled using an evolution equation that combines rate-independent and rate-dependent responses smoothly with no finite elastic range. Large deformation evolution equations for the distortional deformations in the elastic and in the dissipative component are presented. A robust, strongly objective numerical integration algorithm is used to model rate-dependent and rate-independent inelastic responses. The constitutive formulation is specialized to simulate the experimental data. The nonlinear viscoelastic model accurately represents the time-dependent passive response of skeletal muscle.     

Mean field modelling of the plastic behaviour of co-continuous dual-phase alloys with strong morphological anisotropy

This work addresses the plastic flow properties of a composite material in which the reinforcing phase is continuous and cannot be suitably represented by isolated ellipsoidal inclusions. The dual-phase metal under consideration is composed of a network of Inconel-601 fibres infiltrated by pure aluminium. Hence, both phases exhibit elastic–plastic behaviour and are continuous in the three dimensions of space. The fibre network presents a large morphological anisotropy that is reflected in the mechanical response of the composite. The modelling is based on Eshelby’s equivalent inclusion theory. Strain partitioning between the phases is computed incrementally based on tangent operators derived from the isotropic response of individual phases. Assessment of the model relies on extensive experimental data. Uniaxial tensile tests, involving measurement of the Lankford coefficient, have been performed at various temperatures on samples containing different volume fractions of fibres. Measurement of the phase stresses by neutron diffraction supplements the information provided by the macroscopic stress–strain curves. It is demonstrated that predictions are valid only when the micro–macro averaging scheme accounts for the co-continuous character of the constitutive phases.     

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

半结晶聚合物损伤演化的实验表征与数值模拟

损伤本构模型对研究材料的断裂失效行为有重要意义, 但聚合物材料损伤演化的定量表征实验研究相对匮乏. 通过4种高密度聚乙烯(high density polythylene, HDPE)缺口圆棒试样的单轴拉伸实验获得了各类试样的载荷-位移曲线和真应力-应变曲线, 采用实验和有限元模拟相结合的方法确定了HDPE材料不同应力状态下的本构关系, 并建立了缺口半径与应力三轴度之间的关系;采用两阶段实验法定量描述了4种HDPE试样单轴拉伸过程中的弹性模量变化, 并建立了基于弹性模量衰减的损伤演化方程, 结合中断实验和扫描电子显微镜分析了应力状态对HDPE材料微观结构演化的影响. 结果表明缺口半径越小, 应力三轴度越大, 损伤起始越早、演化越快; 微观表现为: 高应力三轴度促进孔洞的萌生和发展, 但抑制纤维状结构的产生;基于实验和有限元模拟获得的断裂应变、应力三轴度、损伤演化方程等信息提出了一种适用于聚合物的损伤模型参数确定方法, 最后将本文获得的本构关系和损伤模型用于HDPE平板的冲压成形模拟, 模拟结果与实验结果吻合良好.      

基于微结构演化的V5Cr5Ti合金塑性本构模型

为合理描述V5Cr5Ti合金的塑性变形行为,本文建立了基于微结构演化的塑性本构模型。首先,采用小尺寸试样开展了V5Cr5Ti合金单轴拉伸试验,并对其在不同应变程度下的微结构演化特征进行了分析。研究发现,影响V5Cr5Ti合金塑性变形行为的主要因素是位错密度演化以及团簇状和弥散析出相。据此建立了位错密度演化方程、组分相含量体积分量演化方程,并考虑团簇状和弥散状第二相对V5Cr5Ti合金流动应力的影响,进一步建立了包括非热应力、热激活应力和弥散相强化应力的流动应力关系式。最后,根据隐式应力更新算法对新模型进行了有限元实现,并与实验结果进行比较,验证了新模型的合理性和预测精度。     

基于微结构演化的V5Cr5Ti合金塑性本构模型

为合理描述V5Cr5Ti合金的塑性变形行为,本文建立了基于微结构演化的塑性本构模型。首先,采用小尺寸试样开展了V5Cr5Ti合金单轴拉伸试验,并对其在不同应变程度下的微结构演化特征进行了分析。研究发现,影响V5Cr5Ti合金塑性变形行为的主要因素是位错密度演化以及团簇状和弥散析出相。据此建立了位错密度演化方程、组分相含量体积分量演化方程,并考虑团簇状和弥散状第二相对V5Cr5Ti合金流动应力的影响,进一步建立了包括非热应力、热激活应力和弥散相强化应力的流动应力关系式。最后,根据隐式应力更新算法对新模型进行了有限元实现,并与实验结果进行比较,验证了新模型的合理性和预测精度。     

Constitutive modelling of cyclic plasticity of metal particulate-reinforced brittle matrix composites under compression-compression loading

The Mori-Tanaka approach is used to modelling metal particulate-reinforced brittle matrix composites under cyclic compressive loading. The J₂-flow theory is considered as the relevant physical law of plastic flow in inclusions. Ratchetting of the composite is prevented by the strong constraint exerted by the matrix on the inclusions, even under the assumption of evanescent kinematic hardening. However, the weakening constraint power of the matrix caused by microfracture damage around inclusions is closely coupled with the plasticity of inclusion and leads to ratchetting even when the plastic deformation of inclusions is described by an isotropic hardening rule. A detailed parametric study has revealed that ratchetting is followed by either plastic or elastic shakedown, depending on the load amplitude, composite parameters and the mean length of microcracks.     

A micromechanics-based elastoplastic damage model for granular materials at low confining pressure

This paper is devoted to the formulation of a micromechanics-based constitutive model for granular materials under relatively low confining pressure. The constitutive formulation is performed within the general framework of homogenization for granular materials. However, new rigorous stress localization laws are proposed. Some local constitutive relations are established under the consideration of irreversible thermodynamics. Macroscopic plastic deformation is obtained by considering local plastic sliding in a limit number of families of contact planes. The plastic sliding at each contact plane is described by a non-associated plastic flow rule, taking into account pressure sensitivity and normal dilatancy. Nonlinear elastic deformation related to progressive compaction of contacts is also taken into account. Material softening is described by involving damage process related to degradation of microstructure fabric. The proposed model is applied to some typical granular materials (sands). The numerical predictions are compared with experimental data.     

Modeling rate-independent hysteresis in large deformations of preconditioned soft tissues

A phenomenological model is proposed for characterizing rate-independent hysteresis exhibited by preconditioned soft tissues. The preconditioned tissue is modeled as an isotropic composite of a hyperelastic component and a dissipative (inelastic) component. Specifically, the constitutive equations are hyperelastic in the sense that the stress is determined by derivatives of a strain energy function. Inelasticity of the dissipative component is controlled by a yield function with different functional forms for the hardening variable during deformation loading and unloading. The constitutive equations proposed in this paper are simple. In particular, they depend on only seven material constants: three controlling the response of the elastic component and the remainder controlling the response of the dissipative component. More importantly, the material constants can be determined to match rather general loading and unloading behavior. It is observed that the hysteretic response of the model compares well with experimental data for passive uniaxial loading/unloading of Manduca muscle. Moreover, the present model treats partial loading and reloading of preconditioned tissue as elastic–plastic response, which is different from the treatment of pseudo-elastic models used in the literature.