Associative coupled thermoplasticity at finite strain with temperature-dependent material parameters

The paper deals with associative coupled thermoplasticity at finite strains. J2 plasticity model is based on hyperelastic formulation with multiplicative decomposition of deformation gradient into elastic and plastic parts. As a novel aspect, temperature dependence of all material parameters is introduced. For such model, variational procedure is carried out and discretization by the mixed finite element method is performed. Consistent linearization is carried out and algorithmic elasto-plastic tangent moduli are given. Verification of algorithm is provided by two examples.     

Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains

This work describes the finite element implementation of a generalised strain gradient and rate-dependent crystallographic formulation for finite strains and general anisothermal conditions based on a multiplicative decomposition of the deformation gradient. The implementation involved the development of both a novel finite element formulation to determine the spatial slip rate gradients at each material point, and an implicit numerical integration scheme at the constitutive level to update the stresses and solution dependent variables. The time-integration procedure uses a Newton–Raphson scheme with a single level of iteration to solve the incremental non-linear equations associated with the non-local constitutive formulation. Closed-form solutions for the relevant fourth-order Jacobian tensors are given. The proposed numerical scheme is formulated in a general form and hence should be applicable to most existing crystallographic models. The crystallographic formulation is then used to investigate the effect of the morphology and volume fraction of the reinforcing phase of a two-phase single crystal on its macroscopic behaviour.     

A fully coupled diffusional-mechanical formulation: numerical implementation,analytical validation,and effects of plasticity on equilibrium

A macroscopic coupled stress-diffusion theory which accounts for the effects of nonlinear material behaviour, based on the framework proposed by Cahn and Larché, is presented and implemented numerically into the finite element method. The numerical implementation is validated against analytical solutions for different boundary valued problems. Particular attention is payed to the open system elastic constants, i.e. those derived at constant diffusion potential, since they enable, under circumstances, the equilibrium composition field for any generic chemical-mechanical coupled problem to be obtained through the solution of an equivalent elastic problem. Finally, the effects of plasticity on the overall equilibrium state of the coupled problem solution are discussed.     

A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling

A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure C_e. The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.     

Shakedown reduced kinematic formulation,separated collapse modes,and numerical implementation

Our shakedown reduced kinematic formulation is developed to solve some typical plane stress problems, using finite element method. Whenever the comparisons are available, our results agree with the available ones in the literature. The advantage of our approach is its simplicity, computational effectiveness, and the separation of collapse modes for possible different treatments. Second-order cone programming developed for kinematic plastic limit analysis is effectively implemented to study the incremental plasticity collapse mode. The approach is ready to be used to solve general shakedown problems, including those for elastic–plastic kinematic hardening materials and under dynamic loading.     

On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F=F_eF_p. A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F_p. Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.     

Thermomechanical modeling of metals at finite strains: First and mixed order finite elements

The aim of this paper is to describe an updated EAS (Enhanced Assumed Strain) finite element formalism developed to model the thermomechanical behavior of metals submitted to large strains. We will also expose the use of mixed order elements (first order mechanical elements strongly coupled with quadratic thermal elements) which, as we will show, is of particular interest for modeling fast processes inducing important temperature gradients. The features of this formalism, used jointly with an Updated Lagrangian approach and an hypoelastic anisothermal constitutive formulation, will be described. Three applications involving finite strains and important thermomechanical couplings will be studied. The results obtained will be compared with the results given by the now classical SRI (Selective Reduced Integration) formalism.     

Improved Carroll's hyperelastic model considering compressibility and its finite element implementation

Acta Mechanica Sinica - In engineering component design, material models are increasingly used in finite element simulations for an expeditious and less costly analysis of the design prototypes. As...     

On strain localization analysis of elastoplastic materials at finite strains

The problem of strain localization into planar bands of rate-independent elastoplastic solids with smooth yield surface and plastic potential is analyzed reconsidering the work of Rice and Rudnicki in 1980. It is shown that strain localization with elastic unloading on one side of the band first becomes possible either at localization in the comparison solid corresponding to the loading branch of the constitutive equation or at the snap-back threshold. The elastic unloading is shown to start from the condition of neutral loading, occurring in fact at the onset of localization. The case of localization with elastic unloading into the band and plastic loading outside that was not considered by Rice and Rudnicki is taken into account.     

An incremental theory of large strain and large displacement problems and its finite element formulation

Summary This paper is concerned with the development of an incremental finite'element theory for the large strain and the large displacement problems, referred to the current configuration of the body. Using the convected coordinate system which is embedded in the body, the incremental representations of strain and stress tensors and the energy relations are presented, and then the general procedure to construct the so-called element stiffness matrix in incremental form is considered. The finite element formulation is developed for a typical constitutive relation and it is shown that some correction matrices, some of which have been omitted in the previous works, are to be added to the element stiffness matrix. Finally the method to assemble the equations of the element to the global system is discussed and a simple finite element model satisfying the compatibility condition is presented.The finite element theory developed in this paper is able to be extended to the problems for the general thermodynamical process of a broad class of nonlinear materials.

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Übersicht Mit Hilfe der Methode der finiten Elemente wird eineZuwachstheorie zurBehandlung von Problemen mit endlicher Verformung abgeleitet. Dabei wird ein im Körper eingebettetes, der momentanen Form angepaßtes Bezugssystem verwendet. Es werden Ausdrücke für die Energie sowie für die Änderungen der Spannungs- und Verformungs-Tensoren abgeleitet und es wird ein Verfahren zur Konstruktion der Steifigkeitsmatrix für ein Element angegeben. Ein typisches Stoffgesetz wird dabei zugrundegelegt. Dabei zeigt es sich, daß einige in früheren Arbeiten vernachlässigte Korrektur-Matrizen zu der Steifigkeits-Matrix des Elementes hinzugefügt werden müssen. Die Möglichkeiten der Zusammenfassung der für die Elemente geltenden Gleichungen zu einem globalen Gleichungssystem werden diskutiert und es wird ein den Verträglichkeitsbedingungen genügendes Elemente-Modell angegeben.Das angegebene Verfahren kann für allgemeine thermodynamische Prozesse in einer breiten Klasse nichtlinearer Materialien erweitert werden.

     

Closed set of the uniqueness conditions and bifurcation criteria in generalized coupled thermoplasticity for small deformations

This paper reports the results of a study into global and local conditions of uniqueness and the criteria excluding the possibility of bifurcation of the equilibrium state for small strains. The conditions and criteria are derived on the basis of an analysis of the problem of uniqueness of a solution involving the basic incremental boundary problem of coupled generalized thermo-elasto-plasticity. This work forms a follow-up of previous research (?loderbach in Bifurcations criteria for equilibrium states in generalized thermoplasticity, IFTR Reports, 1980, Arch Mech 3(35):337–349, 351–367, 1983), but contains a new derivation of global and local criteria excluding a possibility of bifurcation of an equilibrium state regarding a comparison body dependent on the admissible fields of stress rate. The thermal elasto-plastic coupling effects, non-associated laws of plastic flow and influence of plastic strains on thermoplastic properties of a body were taken into account in this work. Thus, the mathematical problem considered here is not a self-conjugated problem.     

A thermodynamics framework and numerical aspects for transformation-induced plasticity at large strains

In this work, we present a macroscopic material model for simulation transformation-induced plasticity, which is an important phenomenon in metal forming processes. The model is formulated within a thermodynamic framework at large strains. In order to account for both, phase transformation and plasticity, yield functions are related to these effects. Then, applying the concept of maximum dissipation evolution equations are obtained for the inelastic strains, the transformation strains, a hardening variable and the volume fraction of martensite. Furthermore the numerical implementation of the constitutive equations into a finite element program is described. In a numerical example we investigate the austenite-to-martensite phase transformation in a shaft subjected to thermo-mechanical loading in a hybrid-forming process.     

A model for the viscoelastic behavior of polymers at finite strains

Summary A constitutive model is derived for the isothermal nonlinear viscoelastic response in polymers, which do not possess the separability property. The model is based on the concept of transient networks, and treats a polymer as a system of nonlinear elastic springs (adaptive links), which break and emerge due to micro-Brownian motion of chains. The breakage and reformation rates for adaptive links are assumed to depend on some strain energy density. The viscoelastic behavior is described by an integral constitutive equation, where the relaxation functions satisfy partial differential equations with coefficients depending on the strain history. Adjustable parameters of the model are found by fitting experimental data for a number of polymers in tension at strains up to 400 per cent. To validate the constitutive relations, we consider loading with different strain rates, determine adjustable parameters at one rate of strains, and compare prediction of the model with observations at another rate of strains. Fair agreement between experimental data and results of numerical simulation is demonstrated when the rates of strains differ by more than a decade. Received 1 July 1997; accepted for publication 7 October 1997     

A 3-phase model for the numerical analysis of semi-crystalline polymer films in finite elastoplastic strains

In this paper, the mechanical behavior of semi-crystalline polymer films in finite elastoplastic strains is investigated. A 3-phase constitutive model has been specially developed in a previous paper and validated for various materials in both uniaxial and biaxial uniform hot drawing. In the present study, the numerical implementation of this 3-phase model in a finite element software is outlined in the perspective of using this model in more general non-uniform cases of complex geometries and/or loadings. In the present case, only polyethylene films at room temperature are considered. First, uniaxial tensile experimental tests are performed so as to calibrate the model parameters. Then, for validation purposes, two series of experimental tests are conducted on tensile specimens with central holes and double edge notched tensile (DENT) specimens. During these tests, digital image correlation is used to analyze the strain (or displacement) field history during loading. Finally, numerical computations are performed with the help of the finite element software including the 3-phase model previously implemented (cohesive elements are also needed for the simulation of the crack propagation in DENT specimens). In both cases, the comparison between the experimental and numerical force–displacement curves, together with the comparisons between the experimental and numerical strain fields at different times, give very satisfactory results.     

Modeling the response of filled elastomers at finite strains by rigid-rod networks

Summary  A constitutive model is developed for the isothermal response of particle-reinforced elastomers at finite strains. An amorphous rubbery polymer is treated as a network of long chains bridged to permanent junctions. A strand between two neighboring junctions is thought of as a sequence of rigid segments connected by bonds. In the stress-free state, a bond may be in one of two stable conformations: flexed and extended. The mechanical energy of a bond in the flexed conformation is treated as a quadratic function of the local strain, whereas that of a bond in the extended conformation is neglected. An explicit expression is developed for the free energy of a network. Stress–strain relations and kinetic equations for the concentrations of bonds in various conformations are derived using the laws of thermodynamics. In the case of small strains, these relations are reduced to the constitutive equation for the standard viscoelastic solid. At finite strains, the governing equations are determined by four adjustable parameters which are found by fitting experimental data in uniaxial tensile, compressive and cyclic tests. Fair agreement is demonstrated between the observations for several filled and unfilled rubbery polymers and the results of numerical simulation. We discuss the effects of the straining state, filler content, crosslink density and temperature on the adjustable constants. Received 3 January 2001; accepted for publication 12 July 2001     

A micro-mechanical model for the response of filled elastomers at finite strains

Constitutive equations are developed for the isothermal response of particle-reinforced elastomers at finite strains. A rubbery polymer is treated as a network of chains bridged by junctions. A strand between two junctions is thought of as a series of inextensible segments linked by bonds. Two stable conformations are ascribed to a bond: flexed and extended. Deformation of a specimen induces transition of bonds from their flexed conformation to the extended conformation. A concept of trapped entanglements is adopted, according to which not all junctions are active in the stress-free state. Under straining, some entanglements are transformed from their passive (dangling) state to the active state, which results in a decrease in the average length of a strand. Stress–strain relations for an elastomer and kinetic equations for the rate of transition of bonds from their flexed conformation to the extended conformation are derived by using the laws of thermodynamics. Simple phenomenological equations are suggested for the evolution of the number of active entanglements. The model is determined by five adjustable parameters which are found by fitting experimental data in uniaxial tensile tests. Fair agreement is demonstrated between the results of numerical simulation and observations for a polysulfide elastomer reinforced with polystyrene particles and two natural rubber vulcanizates with different cross-linkers.