气泡在液体中运动过程的数值模拟

本文用数值方法预测气泡在液体中的百定常运动。运用位标函数进行界面的隐含跟踪并且与有限体积法相结合构成一种可行的计算方法。本文方法允许在界面处存在很大的物性差，而且较容易将表面张力引入控制方程。我们对气液两相流中单个气泡的运动进行了计算，得到了与实验结果符合很好的数值结果。     

Nonlinear dependence of viscosity in modeling the rate-dependent response of natural and high damping rubbers in compression and shear: Experimental identification and numerical verification

The rate-dependent behavior of filled natural rubber (NR) and high damping rubber (HDR) is investigated in compression and shear regimes. In order to describe the viscosity-induced rate-dependent effects, a constitutive model of finite strain viscoelasticity founded on the basis of the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts is proposed. The total stress is decomposed into an equilibrium stress and a viscosity-induced overstress by following the concept of the Zener model. To identify the constitutive equation for the viscosity from direct experimental observations, an analytical scheme that ascertains the fundamental relation between the inelastic strain rate and the overstress tensor of the Mandel type by evaluating simple relaxation test results is proposed. Evaluation of the experimental results using the proposed analytical scheme confirms the necessity of considering both the current overstress and the current deformation as variables to describe the evolution of the rate-dependent phenomena. Based on this experimentally based motivation, an evolution equation using power laws is proposed to represent the effects of internal variables on viscosity phenomena. The proposed evolution equation has been incorporated in the finite strain viscoelasticity model in a thermodynamically consistent way. Simulation results for simple relaxation tests, multi-step relaxation tests and monotonic tests at different strain rates using the developed model show an encouraging correlation with the experiments conducted on HDR and NR in both compression and shear regimes. Finally, an approach to extend the proposed evolution equation for rate-dependent cyclic processes is proposed. The simulation results are critically compared with the experiments.     

A finite strain kinematic hardening constitutive model based on Hencky strain: General framework, solution algorithm and application to shape memory alloys

The logarithmic or Hencky strain measure is a favored measure of strain due to its remarkable properties in large deformation problems. Compared with other strain measures, e.g., the commonly used Green-Lagrange measure, logarithmic strain is a more physical measure of strain. In this paper, we present a Hencky-based phenomenological finite strain kinematic hardening, non-associated constitutive model, developed within the framework of irreversible thermodynamics with internal variables. The derivation is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, and on the use of the isotropic property of the Helmholtz strain energy function. We also use the fact that the corotational rate of the Eulerian Hencky strain associated with the so-called logarithmic spin is equal to the strain rate tensor (symmetric part of the velocity gradient tensor). Satisfying the second law of thermodynamics in the Clausius-Duhem inequality form, we derive a thermodynamically-consistent constitutive model in a Lagrangian form. In comparison with the available finite strain models in which the unsymmetric Mandel stress appears in the equations, the proposed constitutive model includes only symmetric variables. Introducing a logarithmic mapping, we also present an appropriate form of the proposed constitutive equations in the time-discrete frame. We then apply the developed constitutive model to shape memory alloys and propose a well-defined, non-singular definition for model variables. In addition, we present a nucleation-completion condition in constructing the solution algorithm. We finally solve several boundary value problems to demonstrate the proposed model features as well as the numerical counterpart capabilities.     

On phase transformations in shape memory alloy materials and large deformation generalized plasticity

A new version of rate-independent generalized plasticity, suitable for the derivation of general thermomechanical constitutive laws for materials undergoing phase transformations, is proposed within a finite deformation framework. More specifically, by assuming an additive decomposition of the finite strain tensor into elastic and inelastic (transformation induced) parts and by considering the fractions of the various material phases as internal variables, a multi-phase formulation of the theory is developed. The concepts presented are applied for the derivation of a three-dimensional thermomechanical model for shape memory alloy materials. The ability of the model in simulating several patterns of the extremely complex behavior of these materials, under both monotonic and cyclic loadings, is assessed by representative numerical examples.     

A constitutive model of cold drawing in polycarbonates

This paper presents a set of constitutive equations to model cold-drawing (necking) in polycarbonates (PC). The model is based on a representation of cold drawing as a double glass transition, i.e., a transition from a glass into a rubbery state, when a certain yield surface in the stress space is reached, and a transition back to the glassy state upon unloading or when a certain molecular orientation (draw ratio) is achieved. The stretching process in the rubbery state is modeled by a hyperelastic extension of the J₂-flow theory to the finite strain range. An appropriate yield surface and an associative flow rule (defined via the Kuhn–Tucker optimality conditions) are presented to simulate this process in polycarbonates. The isochoric constraint during double glass transition is treated via an exact multiplicative decomposition of the deformation gradient into volume preserving and spherical parts. Numerical constitutive integration algorithm is based on an operator splitting technique where constraint/consistency during inelastic deformation is enforced via return mapping algorithm. Numerical results are presented to demonstrate the correspondence with the experimental data.     

On free energy-based formulations for kinematic hardening and the decomposition F = fpfe

Within the framework of linear plasticity, based on additive decomposition of the linear strain tensor, kinematical hardening can be described by means of extended potentials. The method is elegant and avoids the need for evolution equations. The extension of small strain formulations to the finite strain case, which is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, proved not straight forward. Specifically, the symmetry of the resulting back stress remained elusive. In this paper, a free energy-based formulation incorporating the effect of kinematic hardening is proposed. The formulation is able to reproduce symmetric expressions for the back stress while incorporating the multiplicative decomposition of the deformation gradient. Kinematic hardening is combined with isotropic hardening where an associative flow rule and von Mises yield criterion are applied. It is shown that the symmetry of the back stress is strongly related to its treatment as a truly spatial tensor, where contraction operations are to be conducted using the current metric. The latter depends naturally on the deformation gradient itself. Various numerical examples are presented.     

A finite deformation formulation of the 3-parameter viscoelastic fluid

The paper addresses, with reference to the 3-parameter fluid, a method to extend constitutive laws for viscoelastic fluids from small to finite deformations. It relies upon the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts. Also care is taken that the second law of thermodynamics is satisfied in every admissible process. To demonstrate the capabilities of the model obtained, simple shear and uniaxial extensional flow are discussed. It turns out that essential effects of material behaviour, which have been observed experimentally, can be predicted by the model.     

Characterization of yielding behavior of polycrystalline metals with single crystal plasticity based on representative characteristic length

Single crystal plasticity based on a representative characteristic length is proposed and introduced into a homogenization approach based on finite element analyses, which are applied to characterization of distinctive yielding behaviors of polycrystalline metals, yield-point elongation, and grain size strengthening. The computational manner for an implicit stress update is derived with the framework of a standard multi-surface plasticity at finite strain, where the evolution of the characteristic lengths are numerically converted from the accumulated slips of all of slip systems by exploiting the mathematical feature of the characteristic length as the intermediate function of the plastic internal variables. Furthermore, a constitutive model for a single crystal reproduces the stress–strain curve divided into three parts. Using two-scale finite element analysis, the macroscopic stress–strain response with yield-point elongation under a situation of low dislocation density is reproduced. Finally, the grain size effect on the yield strength is analyzed with modeling of the grain boundary in the context of the proposed constitutive model and is discussed from both macroscopic and microscopic views.     

Modelling inelastic behaviour of orthotropic metals in a unique alignment of deviatoric plane within the stress space

A finite strain constitutive model to predict the deformation behaviour of orthotropic metals is developed in this paper. The important features of this constitutive model are the multiplicative decomposition of the deformation gradient and a new Mandel stress tensor combined with the new stress tensor decomposition generalized into deviatoric and spherical parts. The elastic free energy function and the yield function are defined within an invariant theory by means of the structural tensors. The Hill’s yield criterion is adopted to characterize plastic orthotropy, and the thermally micromechanical-based model, Mechanical Threshold Model (MTS) is used as a referential curve to control the yield surface expansion using an isotropic plastic hardening assumption. The model complexity is further extended by coupling the formulation with the shock equation of state (EOS). The proposed formulation is integrated in the isoclinic configuration and allows for a unique treatment for elastic and plastic anisotropy. The effects of elastic anisotropy are taken into account through the stress tensor decomposition and plastic anisotropy through yield surface defined in the generalized deviatoric plane perpendicular to the generalized pressure. The proposed formulation of this work is implemented into the Lawrence Livermore National Laboratory-DYNA3D code by the modification of several subroutines in the code. The capability of the new constitutive model to capture strain rate and temperature sensitivity is then validated. The final part of this process is a comparison of the results generated by the proposed constitutive model against the available experimental data from both the Plate Impact test and Taylor Cylinder Impact test. A good agreement between experimental and simulation is obtained in each test.     

A structural multi-mechanism constitutive equation for cerebral arterial tissue

A structural multi-mechanism constitutive equation is developed to describe the nonlinear, anisotropic, inelastic mechanical behavior of cerebral arterial tissue. Elastin and collagen fibers are treated as separate components (mechanisms) of the artery. Elastin is responsible for load bearing at low strain levels while the collagen mechanism is recruited for load bearing at higher strain levels. This work builds on an earlier model in which both the elastin and collagen mechanisms are represented by isotropic response functions [Wulandana, R., Robertson, A.M., 2005. An inelastic multi-mechanism constitutive equation for cerebral arterial tissue. Biomech. Model. Mechan. 4 (4), 235–248]. Here, the anisotropic material response of the wall is introduced through the collagen mechanism which is composed of helically distributed families of fibers. The orientation of these families is described using either a finite number of fiber orientations or a fiber distribution function. The fiber orientation or dispersion function can be prescribed directly from arterial histology data, or, taking a phenomenological approach, based on data fitting from bi-axial measurements. The activation of the collagen mechanism is specified using a new fiber strain based activation criterion. The multi-mechanism constitutive equation is applied to the simple case of cylindrical inflation and material constants are determined based on available inelastic experimental data for cerebral arteries. While the proposed model captures all features of this inelastic data, there is a pressing need for further experiments to refine the model.     

Simulation of strain-induced anisotropy for polymers with weighting functions

The alignment of polymer chains is a well-known microstructural evolution effect due to straining of polymers. This has a drastic influence on the macroscopic properties of the initially isotropic material, such as a pronounced strength in the loading direction of stretched films. For modeling the effect of strain-induced anisotropy, a macroscopic constitutive model is developed in this paper. Within a thermodynamic framework, an additive decomposition of the logarithmic Hencky strain tensor into elastic and inelastic parts is used to formulate the constitutive equations. As a key idea, weighting functions are introduced to represent a strain-softening/hardening effect to account for induced anisotropy. These functions represent the ratio between the total strain rate (representing the actual loading direction) and a structural tensor (representing the stretched polymer chains). In this way, we introduce material parameters as a sum of weighted direction-related quantities. The numerical implementation of the resulting set of constitutive is used to identify material parameters based on experimental data, exhibiting strain-induced anisotropy. In the finite-element examples, we simulate the cold-forming of amorphous thermoplastic films below the glass transition temperature subjected to different re-loading directions.     

Numerical analysis and elastic–plastic deformation behavior of anisotropically damaged solids

The present paper is concerned with the numerical modelling of the large elastic–plastic deformation behavior and localization prediction of ductile metals which are sensitive to hydrostatic stress and anisotropically damaged. The model is based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. The formulation relies on a multiplicative decomposition of the metric transformation tensor into elastic and damaged-plastic parts. Furthermore, undamaged configurations are introduced which are related to the damaged configurations via associated metric transformations which allow for the interpretation as damage tensors. Strain rates are shown to be additively decomposed into elastic, plastic and damage strain rate tensors. Moreover, based on the standard dissipative material approach the constitutive framework is completed by different stress tensors, a yield criterion and a separate damage condition as well as corresponding potential functions. The evolution laws for plastic and damage strain rates are discussed in some detail. Estimates of the stress and strain histories are obtained via an explicit integration procedure which employs an inelastic (damage-plastic) predictor followed by an elastic corrector step. Numerical simulations of the elastic–plastic deformation behavior of damaged solids demonstrate the efficiency of the formulation. A variety of large strain elastic–plastic-damage problems including severe localization is presented, and the influence of different model parameters on the deformation and localization prediction of ductile metals is discussed.     

An incremental variational formulation of dissipative magnetostriction at the macroscopic continuum level

This paper outlines a new variational-based modeling and computational implementation of macroscopic continuum magneto-mechanics involving non-linear, inelastic material behavior, with a special focus on dissipative magnetostriction. It is based on a constitutive variational principle that optimizes a generalized incremental work function with respect to the internal state variables. In an incremental setting at finite time steps, this variational problem defines a quasi-hyper-magnetoelastic potential for the stresses and the magnetic induction, and incorporates energy storage as well as dissipative mechanisms. The existence of this potential further allows the incremental boundary-value problem of quasi-static inelastic magneto-mechanics to be recast into a principle of stationary incremental energy. The second focus of this paper is on the careful construction of the energy storage and dissipation functions for the model problem of hysteretic magnetostriction at the macroscopic level. It is then demonstrated that the proposed model is capable of predicting the ferromagnetic and field-induced strain hysteresis curves characteristic of magnetostrictive material response in good agreement with experiments. The numerical solution of the coupled non-linear boundary-value problem is based on a monolithic multi-field finite element implementation. As a consequence of the proposed incremental variational principle, the discretization of the multi-field problem appears in a compact symmetric format. In this sense, the proposed formulation provides a canonical framework for the simulation of boundary-value-problems in dissipative magnetostriction at the macro-level. The performance of the proposed algorithm is tested by application to relevant numerical examples.     

Study of plastic/viscoplastic models with various inelastic mechanisms

This paper describes a general framework for the development of plastic or viscoplastic constitutive equations. As the applications are focused on cyclic loadings, only small strains are considered, with an additive decomposition of the total strain into a thermo-elastic part, and several inelastic parts, the evolution of which is determined by several plastic or viscoplastic criteria. Quadratic or linear (crystallographic) criteria could be used, so that the approach is able to describe the contribution of several physical levels, or deformation mechanism, to the inelastic behavior. The present work is restricted to the case of quadratic criteria, and specially to the study of the various interactions which can be introduced between the mechanisms. The most important case is the coupling between kinematic hardening variables which allows to describe: (1) either normal rate sensitivity or inverse rate sensitivity; (2) plasticity-creep interaction; (3) ratcheting for high mean stress but either adaptation or plastic shakedown for lower mean stress.     

On the coupling of anisotropic damage and plasticity models for ductile materials

In this contribution various aspects of an anisotropic damage model coupled to plasticity are considered. The model is formulated within the thermodynamic framework and implements a strong coupling between plasticity and damage. The constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material. The damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The model considers different interaction mechanisms between damage and plasticity defects in such a way that two-isotropic and two-kinematic hardening evolution equations are derived, one of each for the plasticity and the other for the damage. An additive decomposition of the total strain into elastic and inelastic parts is adopted in this work. The elastic part is further decomposed into two portions, one is due to the elastic distortion of the material grains and the other is due to the crack closure and void contraction. The inelastic part is also decomposed into two portions, one is due to nucleation and propagation of dislocations and the other is due to the lack of crack closure and void contraction. Uniaxial tension tests with unloadings have been used to investigate the damage growth in high strength steel. A good agreement between the experimental results and the model is obtained.     

A CONSTITUTIVE MODEL FOR TRANSFORMATION, REORIENTATION AND PLASTIC DEFORMATION OF SHAPE MEMORY ALLOYS

A constitutive model is developed for the transformation, reorientation and plastic deformation of shape memory alloys (SMAs). It is based on the concept that an SMA is a mixture composed of austenite and martensite, the volume fraction of each phase is transformable with the change of applied thermal-mechanical loading, and the constitutive behavior of the SMA is the combination of the individual behavior of its two phases. The deformation of the martensite is separated into elastic, thermal, reorientation and plastic parts, and that of the austenite is separated into elastic, thermal and plastic parts. Making use of the Tanaka’s transformation rule modified by taking into account the effect of plastic deformation, the constitutive model of the SMA is obtained. The ferroelasticity, pseudoelasticity and shape memory effect of SMA Au-47.5 at.%Cd, and the pseudoelasticity and shape memory effect as well as plastic deformation and its effect of an NiTi SMA, are analyzed and compared with experimental results.     

On internal dissipation inequalities and finite strain inelastic constitutive laws: Theoretical and numerical comparisons

Internal dissipation always occurs in irreversible inelastic deformation processes of materials. The internal dissipation inequalities (specific mathematical forms of the second law of thermodynamics) determine the evolution direction of inelastic processes. Based on different internal dissipation inequalities several finite strain inelastic constitutive laws have been formulated for instance by Simo [Simo, J.C., 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering 99, 61–112]; Simo and Miehe [Simo, J.C., Miehe, C., 1992. Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Computer Methods in Applied Mechanics and Engineering 98, 41–104]; Lion [Lion, A., 1997. A physically based method to represent the thermo-mechanical behavior of elastomers. Acta Mechanica 123, 1–25]; Reese and Govindjee [Reese, S., Govindjee, S., 1998. A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures 35, 3455–3482]; Lin and Schomburg [Lin, R.C., Schomburg, U., 2003. A finite elastic–viscoelastic–elastoplastic material law with damage: theoretical and numerical aspects. Computer Methods in Applied Mechanics and Engineering 192, 1591–1627]; Lin and Brocks [Lin, R.C., Brocks, W., 2004. On a finite strain viscoplastic theory based on a new internal dissipation inequality. International Journal of Plasticity 20, 1281–1311]; and Lin and Brocks [Lin, R.C., Brocks, W., 2005. An extended Chaboche’s viscoplastic law at finite strains: theoretical and numerical aspects. Journal of Materials Science and Technology 21, 145–147]. These constitutive laws are consistent with the second law of thermodynamics. As the internal dissipation inequalities are described in different configurations or coordinate systems, the related constitutive laws are also formulated in the corresponding configurations or coordinate systems. Mathematically, these constitutive laws have very different formulations. Now, a question is whether the constitutive laws provide identical constitutive responses for the same inelastic constitutive problems. In the present work, four types of finite strain viscoelastic and endochronically plastic laws as well as three types of J₂-plasticity laws are formulated based on four types of dissipation inequalities. Then, they are numerically compared for several problems of homogeneous or complex finite deformations. It is demonstrated that for the same inelastic constitutive problem the stress responses are identical for deformation processes without rotations. In the simple shear deformation process with large rotation, the presented viscoelastic and endochronically plastic laws also show almost identical stress responses up to a shear strain of about 100%. The three laws of J₂-plasticity also produce the same shear stresses up to a shear strain of 100%, while different normal stresses are generated even at infinitesimal shear strains. The three J₂-plasticity laws are also compared at three complex finite deformation processes: billet upsetting, cylinder necking and channel forming. For the first two deformation processes similar constitutive responses are obtained, whereas for the third deformation process (with large global rotations) significant differences of constitutive responses can be observed.