A generalised Modified Cam clay model for clay and sand incorporating kinematic hardening and bounding surface plasticity

This paper proposes a simple non-associated Modified Cam clay model suitable for clay and sand. The yield surface is taken to be that of Modified Cam clay, which is a simple ellipse. The modified model reduces the amount of shear strain predicted, and for clay requires no new parameters because the flow rule uses a well established empirical result. For sand, the critical state frictional dissipation constant is required in addition to the stress ratio at the peak of the yield surface. This permits realistic modelling of the undrained behaviour of sand in states looser and denser than critical. The model resembles more sophisticated models with yield surfaces of more complex shapes, but is much simpler. More realistic behaviour could be obtained by assuming a yield surface with the same form as the potential if required. The model is suitable for incorporating kinematic hardening for the modelling of cyclic loading of clay. In addition, bounding surface plasticity can be included to distinguish between compacted and overconsolidated sand. The contribution in this paper is therefore to provide a generalised simple model based on Modified Cam clay.The authors are grateful to Mr C.D. Khong for discussions on the bounding surface formulation of the CASM model.     

A mixture model for the compaction of saturated sand

Free draining water saturated sands and dry sands compact progressively under cyclic shear loading, and the rate of compaction increases as the shear strain amplitude increases, independent of the confining stress magnitude. As the sand grains are relatively incompressible, this compaction is essentially an irreversible porosity decrease induced by rearrangement of the granular structure. We present a constitutive model of differential type for porosity variation which has the minimal ingredients necessary to reflect the observed cyclic loading phenomena, and determine the associated material functions of a particular form by correlation with cyclic loading data. A hypoelastic shear response is also correlated to cyclic loading data to complete an isotropic constitutive model. Pore pressure generation under cyclic loading in undrained conditions is evaluated to illustrate the predictions of the model.     

Simulation of pseudoelastic behaviour of SMA under cyclic loading

A simple modification to an existing constitutive model for SMA materials is proposed to include the effects of loading frequency. The proposed model is based on the behaviour of experimental results of SMA wires subjected to cyclic strains. Numerical simulations based on the modified constitutive model show good qualitative agreement with experimental results. The influence of loading frequency, temperature, static strain offset and strain amplitude on pseudoelastic hysteresis loops and energy dissipation is studied. Dynamic response of a single degree of freedom system with an SMA element is examined by using the modified constitutive model and Newmark’s time integration method. Basic features of the displacement and SMA element stress time histories are studied. The frequency–response curve of a SDOF is investigated.     

On the problem of cavity expansion in unsaturated soils

The problem of cavity expansion in unsaturated soils is investigated. A unified constitutive model formulated in a critical state framework using the concepts of effective stress and bounding surface plasticity theory is adopted. Consideration is given to the effects of suction and particle crushing in the definition of the critical state and the evolution of the bounding surface. The model accurately captures stress-strain behaviour for a range of load paths encompassing that experienced by soils during cavity expansion. Specifically, the similarity technique is used to solve the cavity expansion problem in speswhite kaolin and Kurnell sand. Eight governing equations are defined and solved simultaneously as an initial value problem including an equilibrium equation for stresses around the cavity. Cylindrical and spherical cavities are considered, as are constant suction and constant moisture content conditions. Substantial differences in the stress-strain response of saturated and unsaturated soils surrounding expanding cavities are observed. The paper highlights the major influence of suction and the importance of accounting for this when using cavity expansion theory to interpret results of the cone penetration and pressuremeter tests.     

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.     

Stress integration procedures for a biaxial isotropic material model of biological membranes and for hysteretic models of muscle fibres and surfactant

Stress calculation for a biaxial isotropic material model of a biological membrane and for hysteretic models of muscle fibres and surfactant is presented in the paper. The non‐linear elastic membrane model is defined by uniaxial and biaxial stress–stretch relations, while the hysteretic models of tissue fibres and surfactant are described by the stress–stretch and surface tension–surfactant area ratio constitutive relationships, respectively. The conditions when tissue is or is not covered by surfactant are considered. It is assumed that the material is subjected to cyclic loading. Quasi‐static and steady conditions are considered. The models are implemented in large strain finite element incremented‐iterative analysis of shell deformations. Numerical examples demonstrate characteristics of the computational procedures and structural response of biological membranes when subjected to cyclic loading. Hysteretic response of biological membranes subjected to cyclic loading is caused by hysteresis of fibres and hysteresis of surfactant. The hysteretic effects may play an important role in the physiology of human body. Copyright © 2006 John Wiley & Sons, Ltd.     

FEM simulation of viscous properties for granular materials considering the loading rate effect

The deformation and strength characteristics of sandy soils as a kind of granular materials are very complex. The experimental results show that when the strain rate suddenly changes in monotonic loading (ML) case, the stress–strain curve of sandy soils changes sharply and then gradually converges into the original inferred one that would be obtained by continuous ML at constant strain rate after having exhibited clear yielding. Similar behaviors are also observed when ML is restarted at a constant strain rate following a creep loading or stress relaxation stage. An elasto-viscoplastic constitutive model for granular materials is developed, which consists of three components. One of the most important features of the model is that it can take into account the effects of loading rate due to viscous properties on the stress–strain behavior. The stress ratio-axial strain–time relations from four drained plain strain compression (PSC) tests on the saturated Toyoura sand are successfully simulated by the finite element method (FEM) code incorporating the proposed constitutive model. It is shown that the FEM code can simulate the viscous behaviors of sand accurately under arbitrary loading history.     

Large-strain dynamic cavity expansion in a granular material

The dynamic problem of the symmetric expansion of a cylindrical or spherical cavity in a granular medium is considered. The constitutive behaviour of the material is governed by a hypoplasticity relation for granular soils capable of describing both monotonic and cyclic deformation. The problem is solved numerically by a finite-difference technique. A nonreflecting boundary condition used at the outer boundary of the computational domain makes it possible to model a continuous multi-cycle loading on the cavity wall. The solution is illustrated by numerical examples. Possible geomechanical applications to the modelling of the vibratory compaction and penetration in granular soils are discussed.     

Large-strain dynamic cavity expansion in a granular material

The dynamic problem of the symmetric expansion of a cylindrical or spherical cavity in a granular medium is considered. The constitutive behaviour of the material is governed by a hypoplasticity relation for granular soils capable of describing both monotonic and cyclic deformation. The problem is solved numerically by a finite-difference technique. A nonreflecting boundary condition used at the outer boundary of the computational domain makes it possible to model a continuous multi-cycle loading on the cavity wall. The solution is illustrated by numerical examples. Possible geomechanical applications to the modelling of the vibratory compaction and penetration in granular soils are discussed.     

混凝土拉-压疲劳损伤模型及其验证

基于连续损伤力学理论,提出了混凝土单轴拉-压疲劳损伤模型。模型中采用了拉和压两个边界面。加载面、边界面方程均以损伤能量释放率表示。在能量释放率空间内,由加载面与初始损伤面、边界面之间的位置描述损伤状态。通过建立累积损伤与相应循环损伤能量释放率阈值之间的关系,确定了疲劳加载中极限断裂面尺寸的变化规律,由此模拟混凝土在循环荷载作用下的刚度退化过程。结合作者完成的疲劳试验结果,确定了理论模型中的计算参数。经比较,理论模型预测的应力-应变数值、疲劳寿命和试验结果吻合较好。     

Phenomenological formulation of viscoplastic constitutive equation for polyethylene by taking into account strain recovery during unloading

A viscoplastic constitutive equation for polyethylene that properly describes significant strain recovery during unloading was proposed. The constitutive equation was formulated by combining the kinematic hardening creep theory of Malinin and Khadjinsky with the nonlinear kinematic hardening rule of Armstrong and Frederick. In order to describe the strain recovery, the nonlinear kinematic hardening rule was modified. First, a loading surface was defined in a viscoplastic strain space. A loading–unloading criterion was then introduced using the loading surface. Moreover, a new parameter was defined by the relationship between the loading surface and the current state of the viscoplastic strain, and the evolution equation of back stress was modified using this parameter, which has some value only during unloading. Experimental results for polyethylene were simulated by using the modified constitutive equations, and cyclic inelastic deformation in both uniaxial and biaxial states of stress was predicted. Finally, the validity of the above-described modification was verified, and the features of the constitutive equation and the deformation were discussed.     

正常固结非饱和黏性土的三剪边界面模型研究

将三剪统一强度准则分别与非饱和土的单应力变量理论和双应力变量理论相结合提出了正常固结非饱和黏性土的三剪强度准则,在此基础上采用等量代换法与坐标平移法分别推导出可以反映土体全应力状态效应的三剪破坏应力比。将该三剪破坏应力比引入非饱和黏性土修正剑桥模型的屈服面方程以确定初始边界面,采用径向映射法则确定后续边界面,根据插值函数得到加载面上的塑性模量,据之建立了非饱和黏性土单应力变量和双应力变量下分别采用等量代换法和坐标平移法的4个三剪弹塑性边界面模型并对这4个模型做了单调和循环压缩荷载试验验证。结果表明,所建模型均能较好地反映土的变形特征,但相比之下,双应力变量下建立的模型相对单应力变量下建立的模型更接近试验结果,采用等量代换法建立的模型相对坐标平移法更接近试验结果的特点。真三轴数值模拟结果表明,单调加载时,相同应力条件下中主应力影响系数增大会引起非饱和土强度增大;循环加载时,最小主应力或中间主应力的增大均会引起土体抗剪强度的增大,且荷载振幅越大,非饱和土的轴向应变随之增大。比较4个边界面模型在真三轴应力状态的模拟结果表明,单调加载时,非饱和黏性土在双应力变量下的模拟抗剪强度值相对较大;循环加载时,单应力变量下所得塑性变形相对较大。     

Elastic-plastic analysis of a stationary crack under cyclic loading and effect of overload

A recently proposed elastoplastic constitutive model has been implemented in a finite element code to study crack front behaviour under variable loading. The importance of proper modelling of a material's behaviour becomes evident when a variable loading condition is considered. We present stress, strain and displacement distribution along a stationary crack front for constant amplitude cyclic loading with an overload cycle. The analysis predicts a decreased tensile stress and damage accumulation following an overload.     

A UNIAXIAL CYCLIC PLASTICITY MODEL INCLUDING TRANSIENT MATERIAL BEHAVIOUR

Abstract— A phenomenological uniaxial material model, which is a one-dimensional equivalent to a two-surface multiaxial plasticity model, is presented. The model takes into account the transient effects cyclic hardening/softening and mean stress relaxation by means of exponential relationships that are functions of number of reversals. The parameters describing these relationships are obtained from the strain-life curve and the cyclic stress-strain curve. The hysteresis loops are assumed to be bounded by two parallel, straight lines in tension and compression, i.e. bounding lines. The stress-strain curve approaches these lines with increasing strain through a nonlinearly decreasing plastic modulus. The transient effects are simulated by increasing/decreasing the distance between the lines for cycling hardening/softening at each stress or strain reversal. A positive mean stress yields translation downwards of the bounding lines if the material is subjected to strain control, and translation in the opposite direction if the mean stress is negative. Dynamic creep is simulated when the material is subjected to stress control and mean stress. The test data was obtained from an experimental programme on St52–3N, which is a normalized structural steel. The specimens were loaded in strain or stress control. The comparison between tests and simulations shows that the model described simulates the cyclic transient material behaviour quite well. The main inaccuracy is due to using parallel bounding lines, as the tests show that these lines are not completely parallel for the present steel material.     

A model for the Mullins effect during multicyclic equibiaxial loading

In this paper, we derive a model to describe the cyclic stress softening of a carbon-filled rubber vulcanizate through multiple stress–strain cycles with increasing values of the maximum strain, specializing to equibiaxial loading. Since the carbon-filled rubber vulcanizate is initially isotropic, we can show that following initial equibiaxial loading the material becomes transversely isotropic with preferred direction orthogonal to the plane defined by the equibiaxial loading. This is an example of strain-induced anisotropy. Accordingly, we derive nonlinear transversely isotropic models for the elastic response, stress relaxation, residual strain and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These ideas are then combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation for the cyclic stress softening of a carbon-filled rubber vulcanizate. The model developed includes the effects of hysteresis, stress relaxation, residual strain and creep of residual strain. The model is found to compare extremely well with experimental data.     

Lateral behavior of concrete under uniaxial compressive cyclic loading

In reinforced concrete structures under seismic loading, concrete is subjected to compressive cyclic stress. Although cyclic stress–strain response has been described before, the cyclic behavior of strains in the direction orthogonal to loading has not been characterized yet. Such behavior can be of great importance for evaluating the efficiency of the confinement under cyclic loading. For this purpose an experimental program on cylindrical specimens of concrete strength from 35 to 80 MPa subjected to uniaxial cyclic compression was carried out. Stress versus longitudinal and lateral strains curves have been obtained both for the hardening and softening branches under monotonic and cyclic loading. Governing parameters of the lateral behavior are identified and correlated to describe the response of the lateral strain. Additionally, an analytical model to obtain the lateral deformations of concrete under cyclic uniaxial compression has been formulated and verified experimentally. Finally, some examples are presented in order to illustrate the applicability of the proposed model and its possible incorporation into a 3D constitutive cyclic model.     

Constitutive equation of plastic deformation of a zn-al alloy under tension-tension strain controlled cyclic loading

A constitutive equation of plastic deformation under tension-tension, strain controlled cyclic loading condition was derived from the transition state theory of rate processes. It was considered that the rate of plastic flow during the (tension-tension) cyclic deformation is controlled by a system of two consecutive energy barriers and that the material structural characteristics remain constant during cyclic deformation. The study revealed that within the stress, time, and temperature range, where the backward activations over the energy barriers are negligibly small, tension-tension, strain controlled cyclic deformation is essentially a stress relaxation process. The theory described well the cyclic deformation behavior of a near eutectoid ZnAl alloy. The constitutive parameters determined from the analysis of stress relaxation and tension-tension, strain controlled cyclic loading experimental results were identical. Consequently, it was recommended that stress relaxation can be used to determine the material structural characteristics which can then be used to predict the tension-tension, strain controlled cyclic deformation behavior of the alloy, using the constitutive equation derived in this report.     

基于SMP破坏准则的土体弹塑性动力本构模型

利用土体的塑性流动理论,提出了用于描述饱和砂土动力反应性质的弹塑性本构模型。土体总的变形由三部分组成:即弹性应变、与体积屈服机制相关的塑性应变和与剪切屈服机制相关的塑性应变。土体在初始加载与卸载和重新加载阶段性质的差别通过采用不同的模型参数加以反映。该模型能够较为准确地描述饱和砂土在单调加载和循环加载条件下的反应性质。     

Experimental and theoretical investigation on mechanical behaviors of TB991 weld sealant under cyclic loading

The mechanical behaviors of TB991 weld sealant under cyclic loading conditions were experimentally investigated. The evolution of relaxation stress, cyclic softening, and dissipated energy was evaluated with the effect of strain amplitude and mean strain. The experimental results showed that the stress–strain response curves of the first loading-unloading and cyclic loading-unloading were significantly different. The phenomenon of stress relaxation and cyclic softening occurred under cyclic strain loading conditions. Furthermore, the relaxation stress and dissipated energy decreased rapidly during the initial cyclic loading and then steadily decreased with the increase of cycle number, while the cyclic softening increased rapidly at first and then steadily. Besides, a viscoelastic constitutive model was proposed which can describe the different shapes of stress–strain curve between the initial loading-unloading and the cyclic loading-unloading and also considers the cyclic stress relaxation and cyclic softening of the materials under cyclic loading condition. Comparisons between the numerical results and the experimental data demonstrated that the proposed model can better describe the mechanical behavior of TB991 weld sealant under cyclic loading conditions.     

Biaxial fatigue for proportional and non‐proportional loading paths

In order to study the use of a local approach to predict crack‐initiation life on notches in mechanical components under multiaxial fatigue conditions, the study of the local cyclic elasto‐plastic behaviour and the selection of an appropriate multiaxial fatigue model are essential steps in fatigue‐life prediction. The evolution of stress–strain fields from the initial state to the stabilized state depends on the material type, loading amplitude and loading paths. A series of biaxial tension–compression tests with static or cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Specimens were made of an alloy steel 42CrMo4 quenched and tempered. The shear stress relaxations of the cyclic tension–compression with a steady torsion angle were observed for various loading levels. Finite element analyses were used to simulate the cyclic behaviour and good agreement was found. Based on the local stabilized cyclic elastic–plastic stress–strain responses, the strain‐based multiaxial fatigue damage parameters were applied and correlated with the experimentally obtained lives. As a comparison, a stress‐invariant‐based approach with the minimum circumscribed ellipse (MCE) approach for evaluating the effective shear stress amplitude was also applied for fatigue life prediction. The comparison showed that both the equivalent strain range and the stress‐invariant parameter with non‐proportional factors correlated well with the experimental results obtained in this study.