Modeling of nonlinear viscoelasticity at large deformations

A constitutive model of finite strain viscoelasticity, based on the multiplicative decomposition of the deformation gradient tensor into elastic and inelastic parts, is presented. The nonlinear response of rubbers, manifested by the rate effect, cycling loading and stress relaxation tests was captured through the introduction of two internal variables, namely the constitutive spin and the back stress tensor. These parameters, widely used in plasticity, are applied in this work to model the nonlinear viscoelastic behaviour of rubbers. The experimental results, obtained elsewhere, related with shear deformation in monotonic and cyclic loading, as well as stress-relaxation, were simulated with a good accuracy.     

Two‐point constitutive equations and integration algorithms for isotropic‐hardening rate‐independent elastoplastic materials in large deformation

This paper presents alternative forms of hyperelastic–plastic constitutive equations and their integration algorithms for isotropic‐hardening materials at large strain, which are established in two‐point tensor field, namely between the first Piola–Kirchhoff stress tensor and deformation gradient. The eigenvalue problems for symmetric and non‐symmetric tensors are applied to kinematics of multiplicative plasticity, which imply the transformation relationships of eigenvectors in current, intermediate and initial configurations. Based on the principle of plastic maximum dissipation, the two‐point hyperelastic stress–strain relationships and the evolution equations are achieved, in which it is considered that the plastic spin vanishes for isotropic plasticity. On the computational side, the exponential algorithm is used to integrate the plastic evolution equation. The return‐mapping procedure in principal axes, with respect to logarithmic elastic strain, possesses the same structure as infinitesimal deformation theory. Then, the theory of derivatives of non‐symmetric tensor functions is applied to derive the two‐point closed‐form consistent tangent modulus, which is useful for Newton's iterative solution of boundary value problem. Finally, the numerical simulation illustrates the application of the proposed formulations. Copyright © 2008 John Wiley & Sons, Ltd.     

A plasticity model for multiaxial cyclic loading and ratchetting

Summary The nonlinear behavior of metals when subjected to monotonic and cyclic non-proportional loading is modeled using the proposed hardening rule. The model is based on the Chaboche [1], [2] and Voyiadjis and Sivakumar [3], [4] models incorporating the bounding surface concept. The evolution of the backstress is governed by the deviatoric stress rate direction, the plastic strain rate, the backstress, and the proximity of the yield surface from the bounding surface. In order to ensure uniqueness of the solution, nesting of the yield surface with the bounding surface is ensured. The prediction of the model in uniaxial cyclic loading is compared with the experimental results obtained by Chaboche [1], [2]. The behavior of the model in multiaxial stress space is tested by comparing it with the experimental results in axial and torsional loadings performed by Shiratori et al. [5] for different stress trajectories. The amount of hardening of the material is tested for different complex stress paths. The model gives a very satisfactory result under uniaxial, cyclic and biaxial non-proportional loadings. Ratchetting is also illustrated using a non-proportional loading history.     

Nonlinear response in glass fibre non-crimp fabric reinforced vinylester composites

This paper addresses the nonlinear stress-strain response in glass fibre non-crimp fabric reinforced vinylester composite laminates subjected to in-plane tensile loading. The nonlinearity is shown to be a combination of brittle and plastic failure. It is argued that the shift from plastic to brittle behaviour in the vinylester is caused by the state of stress triaxiality caused by the interaction between fibre and vinylester. A model combining damage and plasticity is calibrated and evaluated using data from extensive experimental testing. The onset of damage is predicted using the Puck failure criterion, and the evolution of damage is calibrated from the observed softening in plies loaded in transverse tension. Shear loading beyond linear elastic response is observed to result in irreversible strains. A yield criterion is implemented for shear deformation. A strain hardening law is fitted to the stress-strain response observed in shear loaded plies. Experimental results from a selection of laminates with different layups are used to verify the numerical models. A complete set of model parameters for predicting elastic behaviour, strength and post failure softening is presented for glass fibre non-crimped fabric reinforced vinylester. The predicted behaviour from using these model parameters are shown to be in good agreement with experimental results.     

High strain-rate localization and failure of crystalline materials

A theoretical and a computational model are introduced to study the micromechanical high strain-rate failure mechanisms of shear-strain localization in monocrystalline fcc structures. A theoretical framework for a constitutive model for the dynamic finite plastic deformation of rate-dependent single fcc crystals is developed. The micromechanics of plastic flow are based on a high strain-rate single crystal plasticity model and a visco-plastic power law. The single crystal is subjected to far-field dynamic tensile strain-rates ranging from 100/s to 2000/s. An explicit finite-element model is introduced for the integration of the numerically stiff visco-plastic constitutive relations. The total deformation rate tensor is obtained by the central difference explicit integration of the equations of motion. The plastic deformation rate tensor is obtained from the solution of an initial-value nonlinear problem for the resolved shear stresses. In time intervals when the differential equations for the resolved shear stresses are not numerically stiff, the initial-value problem is integrated by the explicit fifth-order Runge-Kutta adaptive time-step method. In time domains where the propagated error grows and the time-step must be restricted due to stability requirements, which is an indication of numerical stiffness, the initial-value problem is integrated by an A-stable method. To correctly differentiate time-step reductions due to stability, from time-step reductions due to accuracy, a stiffness ratio is defined. The present analysis corroborates experimental observations that high strain-rate shear-strain localization, in rate-dependent crystals, is a function of thermal and geometrical softening, overall strain-rates, strain hardening, strain-rate hardening, and strain-rate sensitivity.     

Phenomenological modelling of rate-dependent plasticity for high strain rate problems

The results of two related theoretical investigations for large-scale computations of elasto-plastic deformations at ultrahigh strain rates are summarized in this paper. The first effort concerns the development of a phenomenological constitutive model for finite deformation elasto-plasticity which includes the effects of thermal softening, strain hardening, rate-dependence, as well as the noncoaxiality of the plastic strain rate and the stress deviator, and the incorporation of this model in a large-scale explicit finite-element code. The second effort involves the investigation of localized deformations and shear banding at high strain rates. It is shown that the constitutive model considered, together with standard quadrilateral finite elements with one point integration (piece-wise constant strain and stress fields), can nicely produce the observed intense localized deformations without recourse to any special elements. The results are illustrated in terms of the shear localization observed in uniaxial extension, and of void collapse under uniaxial compression. In addition, the effect of the noncoaxiality of the plastic strain rate and the stress tensor is included in the model, and its influence on strain localization at high strain rates is investigated.     

Predicting damage and failure under low cycle fatigue in a 9Cr steel

Experimental data have been generated and finite element models developed to examine the low cycle fatigue (LCF) life of a 9Cr (FB2) steel. A novel approach, employing a local ductile damage initiation and failure model, using the hysteresis total stress–strain energy concept combined with element removal, has been employed to predict the failure in the experimental tests. The 9Cr steel was found to exhibit both cyclic softening and nonlinear kinematic hardening behaviour. The finite element analysis of the material's cyclic loading was based on a nonlinear kinematic hardening criterion using the Chaboche constitutive equations. The models’ parameters were calibrated using the experimental test data available. The cyclic softening model in conjunction with the progressive damage evolution model successfully predicted the deformation behaviour and failure times of the experimental tests for the 9Cr steels performed.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part I: an anisotropic ductile elastic-plastic-damage model

During past decades, many material models using the continuum damage mechanics (CDM) approach have been proposed successfully in the small deformation regime to describe inelastic behaviors and fracturing phenomena of a material. For ductile materials, large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper, the ductile damage at finite deformations is modeled as an anisotropic tensor quantity. Then, a fourth-order symmetric stress correction tensor is proposed for computationally efficient and easy implementation in the finite element formulations. Consequently, an explicit form of the fourth-order constitutive equations of the proposed elastic-plastic-damage model is derived. Both isotropic and kinematic hardening effects are included in the formulation. The new constitutive model can predict not only the elastic-plastic behaviors, but also the sequential variations of ductile materials. An evaluation of the constitutive and damage evolution equations is presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Stabilized cyclic stress-strain calculation for metals under multiaxial loading

A simple plasticity model for modeling the stabilized cyclic stress-strain responses is developed to consider the effect of non-proportional additional hardening. In the proposed model, the plastic modulus for uniaxial loading is extended to multiaxial loading by introducing the non-proportionality factor and the additional hardening coefficient. The two introduced factors take into account the effects of non-proportional additional hardening, not only on the shape of the loading path, but also on the material and its microstructure. And then, the basic Armstrong-Frederick nonlinear hardening rule is modified to model the evolution of the back stress. The consistency condition is enforced to obtain the relationship between the back stress and plastic modulus. The proposed model requires only six material constants for estimating the stabilized responses. Comparisons between the test results (30CrNiMo8HH steel, SA 333 Gr.6 steel, and 1 %CrMoV steel) and model predictions show that the proposed model predicts relatively accurate stress responses under both proportional and non-proportional loading paths.     

Numerical and experimental study of ratcheting in cold expanded plate of Al‐alloy 2024‐T3 in double shear lap joints

In this paper strain ratcheting in cold expanded flat plate of Al‐alloy 2024‐T3 in double shear lap joints was studied both experimentally and numerically. In the experimental part, two types of symmetric strain‐controlled and asymmetric stress‐controlled cyclic tests were performed. Also, the cold expanded double shear lap joints subjected to cyclic stress‐controlled tests. The required parameters for simulating the cyclic plastic behaviour of Al‐alloy 2024‐T3 were obtained on the basis of the experimental responses. In the numerical part, a combination of nonlinear isotropic and nonlinear kinematic hardening model (Chaboche) was implemented in the commercial finite element code of ABAQUS, using the subroutine UMAT written in FORTRAN. The results of simulations give an accurate prediction of ratcheting for all types of loading. The obtained results show that increasing the mean stress increases the strain ratcheting. It is clearly shown that the cold expansion process decreases the magnitude of strain ratcheting remarkably compared with “as drilled” specimens and the decrease is bigger for larger cold expansion sizes. Also, it is shown that the middle plane has the highest amount of ratcheting compared to the pin entrance plane and exit plane of the plate hole.     

Torsional fatigue with axial constant stress of oligo‐crystalline 316L stainless steel thin wire

A series of symmetric torsional fatigue with axial constant stress tests, a kind of multiaxial fatigue test, was conducted on oligo‐crystalline 316L stainless steel thin wire, which was less than 3.5 grains across diameter of 200 μm. The material presents significant cyclic hardening under symmetric torsion cycling, and hardening is more obvious with the increasing shear strain amplitude. However, symmetric torsional cycle with constant axial stresses tests characterize rapid initial hardening and then gradually softening until fatigue failure. The axial stress has a great effect on torsional fatigue life. Fractography observation shows a mixed failure mode combined torsional fatigue with tensile strain because of axial tensile stress. A newly proposed model with axial stress damage parameter is used to predict the torsional fatigue life with constant axial stress of small scale thin wire.     

Transient behaviour of branched polymer melts through planar abrupt and rounded contractions using pom–pom models

In this paper, we study the transient flow of branched polymer melts with contrasting shear and elongational properties in planar 4:1 abrupt and rounded-corner contractions. This includes Single and Double Extended forms of the Pom–Pom model (SXPP and DXPP), comparing the transient behaviour for these two different models. With the DXPP version, the evolution of the molecular-chain backbone stretch (λ) is described by a dynamic equation, whilst in the SXPP form, stretch is an instantaneous algebraic function of the stress tensor (τ). Simulations are performed with a hybrid finite volume/element algorithm. The momentum and continuity equations are solved by a Taylor–Galerkin/pressure-correction finite element method, whilst the constitutive equation is dealt with by a cell-vertex finite volume algorithm. We demonstrate some novel features due to the influence and imposition of realistic transient boundary conditions on evolutionary flow-structure. The different effects of various model parameter choices are also exposed through transient field response in principle stress difference fringe patterns, rates of deformation, first and second normal stress difference, stress and stretch.     

Ermüdungsverhalten von normalisiertem 42CrMo4 unter Torsionsbeanspruchung bei Raumtemperatur

Room Temperature Fatigue Behaviour of a Normalized Steel SAE 4140 in Torsion Cyclic deformation behaviour of a normalized steel SAE 4140 in shear strain-controlled torsion is characterized by cyclic softening and cyclic hardening. If mean shear stresses are superimposed to an alternating shear stress, cycle-dependent creep occurs, and the number of cycles to failure decreases. In shear strain-controlled torsional loading, mean stresses are observed to relax nearly to zero within a few cycles. Fatigue life is not influenced by mean shear strains.     

Development of new cyclic plasticity model for 304LN stainless steel through simulation and experimental investigation

Cyclic plastic deformation characteristics of 304LN stainless steel material have been studied with two proposed cyclic plasticity models. Model MM-I has been proposed to improve the simulation of ratcheting phenomenon and model MM-II has the capability to simulate both cyclic hardening and softening characteristics of the material at various strain ranges. In the present paper, strain controlled simulations are performed with constant, increasing and decreasing strain amplitudes to verify the influences of loading schemes on cyclic plasticity behaviors through simulations and experiments. It is observed that the material 304LN exhibits non Masing characteristics under cyclic plastic deformation. The measured deviation from Masing is well established from the simulation as well as from experiment. Simulation result shows that the assumption of only isotropic hardening is unable to explain the hardening or softening characteristics of the material in low cycle fatigue test. The introduction of memory stress based cyclic hardening coefficient and an exponentially varying ratcheting parameter in the recall term of kinematic hardening rule, have resulted in exceptional improvement in the ratcheting simulation with the proposed model, MM-II. Plastic energy, shape and size of the hysteresis loops are additionally used to verify the nature of cyclic plasticity deformations. Ratcheting test and simulation have been performed to estimate the accumulated plastic strain with different mean and amplitude stresses. In the proposed model MM-I, a new proposition is incorporated for yield stress variation based on the memory stress of loading history along with the evolution of ratcheting parameter with an exponential function of plastic strain. These formulations lead to better realization of ratcheting rate in the transient cycles for all loading schemes. Effect of mean stress on the plastic energy is examined by the simulation model, MM-I. Finally, the micro structural investigation from transmission electronic microscopy is used to correlate the macroscopic and microscopic non Masing behavior of the material.     

Finite element simulation of plasticity induced crack closure with different material constitutive models

The study presented in this paper analyses the mechanical effects of material constitutive modelling on the numerical prediction of plasticity induced crack closure. With this aim, an elastoplastic stress analysis of a MT specimen was conducted using an implicit three dimensional finite element program. Two materials were studied: an Aluminium Alloy and a High Strength Steel. Several constitutive models were used to describe their cyclic behaviour, ranging from pure isotropic hardening or pure kinematic hardening models to combined isotropic plus kinematic hardening models. Numerical results showed clear differences in plastic behaviour and crack closure predictions for the different types of mechanical models used to describe the mechanical behaviour of the materials. The mechanisms of opening stress stabilization, usually observed in numerical simulations, are explained in this work by analysing the evolution of plastic deformation along the crack flanks. The same type of plastic deformation stabilization behaviour was observed independently of the hardening model in use.     

Rheological models for large deformations of elastic-viscoplastic materials

Four different rheological models for large deformations of elastic-viscoplastic materials are considered. Each model is hyperelastic with the stress being determined by derivatives of a strain energy function. Model 1 is based on a nonlinear evolution equation for a unimodular elastic distortional deformation tensor and is considered to be the reference model since it includes the special case of a nonlinear isotropic elastic solid. Models 2-4 are based on simplified evolution equations which depend linearly on a deviatoric elastic strain tensor. Equations of this type have been used to model non-Newtonian fluids. The main objective of this paper is to examine the influence of approximations introduced to obtain these simplified evolution equations. Examples of steady-state simple shear and steady-state isochoric extension relative to a rotating coordinate system are used to study the response of the simplified models. In particular, it is shown that these simplified models predict the unphysical result that the shear stress decreases towards zero for large values of the rate of deformation in simple shear.     

基于能量的弹塑性损伤实用本构模型

建立一个实用的弹塑性损伤本构模型。在有效应力空间采用经验公式计算塑性变形,能将模型塑性部分与损伤部分解耦,降低模型的数值处理复杂性,同时大大简化模型塑性应变的计算。结合不可逆热力学理论,基于损伤能量释放率建立损伤准则,损伤能量释放率由修正后的弹性Helmholtz自由能导出,模型中将弹性Helmholtz自由能分解为应力球量部分和应力偏量部分,将其应力球量部分产生的损伤取为零,同时根据应力状态引入折减系数对其应力偏量部分进行修正,使得模型能较为准确的模拟混凝土材料在双轴加载下的本构行为。将应力张量谱分解为正、负两部分以分别定义材料受拉、受压损伤演化,并采用受拉损伤变量、受压损伤变量分别模拟混凝土材料在拉、压加载下的本构特性。引入一个加权损伤变量使得模型能较准确的反映混凝土材料的“拉-压软化效应”。最后该文给出初步试验验证,证明了该文模型的有效性。     

Damage-induced stress-softening and viscoelasticity of limited elastic materials

The rate-dependent behavior of filled natural rubber (NR) is investigated in tensile regime. In order to describe the viscosity-induced rate-dependent effects, a constitutive model of finite strain viscoelasticity is proposed on the basis of the multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts. The total stress is decomposed into an equilibrium stress and a viscosity-induced overstress by following the rheological models of Poynting–Thomson and Zener types. To incorporate the Mullins stress-softening phenomenon into a viscoelastic material, an invariant-based stress-softening function is also proposed. To identify the constitutive equation for the viscosity from direct experimental observations, an analytical scheme is proposed that ascertains the fundamental relation between the viscous strain rate and the overstress tensor with limited elastic parent material model. 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, an evolution equation is proposed to represent the effects of internal variables on viscosity phenomena. The proposed evolution equation has been incorporated into the finite-strain viscoelasticity model in a thermodynamically consistent way.     

Inelastic finite strain analysis of structures subjected to nonproportional loading

The inelastic behaviour of elasto-plastic materials is nonlinear, path-dependent, and is a function of the total plastic strain. For finite strain problems, the total inelastic strain in Lagrangian co-ordinates cannot be decomposed additively. A generalized logarithmic strain which is formulated in ‘updated’ Lagrangian coordinates and obtained by numerical integration of the Lagrangian strain rate is therefore introduced in this paper. By the use of this strain measure, which is additively decomposable, the plasticity model proposed by the authors can be extended to the finite strain range. It is shown that by correlating the generalized plastic modulus in the constitutive relations with the experimental uniaxial true stress-logarithmic strain diagrams, the inelastic behaviour of steel structures subjected to nonproportional loading can be analyzed numerically by using the finite element method.