Elastoplastic Theory Research Articles

An elasto-plastic theory of dislocation and disclination fields

A linear theory of the elasto-plasticity of crystalline solids based on a continuous representation of crystal defects – dislocations and disclinations – is presented. The model accounts for the translational and rotational aspects of lattice incompatibility, respectively associated with the presence of dislocations and disclinations. The defects content relates to the incompatible plastic strain and curvature tensors. The stress state is described by using the conjugate variables to strain and curvature, i.e., the stress and couple-stress tensors. Defect motion is described by two transport equations. A dynamic interplay between dislocations and disclinations results from a disclination-induced source term in the transport of dislocations. Thermodynamic guidance provides the driving forces conjugate to dislocation and disclination velocity in a continuous context, as well as admissible constitutive relations for the latter. When dislocation and disclination velocity vanish, the model reduces to deWit’s elasto-static theory of crystal defects. It also reduces to Acharya’s linear elasto-plastic theory for dislocation fields when the disclination density is ignored. The theory is intended for use in instances where rotational defects matter, such as grain boundaries. To illustrate its applicability, a finite high-angle tilt boundary is modeled using a disclination dipole and its behavior under tensile loading normal to the boundary is shown.

Modelling the stress–deformation behaviour of municipal solid waste

Conceptual modelling of municipal solid waste landfills is vital for prognosis of the long-term development and the assessment of risk potential. Landfills are considered as heterogeneous porous structures, in which flow and transport processes of gases and liquids are combined with local material degradation and mechanical deformation of the solid skeleton. This contribution deals with a constitutive model focusing on the mechanical deformation behaviour of the disposed waste dividing the solid material into two different fractions. The first fraction includes more grainy particles and has properties comparable with granular materials. The second fraction consists of more fibrous constituents having an important influence on the strength of the composite material. Stress–strain behaviour of the two fractions is modelled according to the theory of finite elasto-plasticity. Settlements generated by mechanical creep are also included in the model. The developed constitutive model is validated and calibrated on results from laboratory experiments applying genetic algorithms. Numerical analyses using the finite-element method show the capability of the presented approach.

Effect of Thermal Load on Mechanical Properties of Ni/YSZ Anode Support Micro Tubular Solid Oxide Fuel Cell

The most commonly used Micro tubular solid oxide fuel cell (MT-SOFC) anode material is a two phase nickel and yttria stabilized zirconia (Ni/YSZ) cermet. And the mechanical stability of anode support layer, in anode-supported electrolyte designs, is very important for large scale applications. During the assembly of stack and normal operation, MT-SOFC is easy to crack under the fuel pressure and thermal loading due to various mechanical properties. In this study, MT-SOFC model was founded on the background of MT-SOFC stack of electric vehicle and was analyzed by finite element method, based on theories of elasto-plasticity, thermo-mechanical coupling. The effect of thermal load was investigated. It concluded that the failure of the micro-tubular cell occurs mainly because of the residual stress due to the mismatch between the coefficients of thermal expansion of the materials of the electrode assembly. The results are important for studying the life and final spallation of MT-SOFC of electric vehicle.

Numerical Prediction of Damage Evolution in Sheet Metal Forming Processes with Nonlinear and Complex Strain Paths

Various thin-walled parts with fairly complex shapes are produced from sheet metals such as automotive panels and other structural parts. In these processes, damage and fracture may be observed on the work piece, and formability plays a fundamental role. Therefore, determination of forming limits and prediction of rupture modes in these operations is very important for process design engineers. In this paper, first, based on plane stress elasto-plasticity and finite strain theories a fully coupled elastic-plastic-damage model is used to predict damage evolution in one sheet metal forming process with nonlinear and complex strain paths. As the plane stress algorithm is valid for thin sheet metals and finite strain theory is recommended for large deformations or rotations, the model is able to quickly predict both deformation and damage behaviour of the parts with nonlinear and complex strain paths. The numerical simulations are compared with experimental tests. Comparison of the numerical and experimental results shows that the proposed damage model is accurate for various forming conditions. Hence, it is concluded that finite element method combined with continuum damage mechanics, can be used as a reliable and rapid tool to predict damage evolution in sheet metal forming processes with nonlinear and complex strain paths.

An elasto-plastic damage constitutive theory and its prediction of evolution of subsequent yield surfaces and elastic constants

Based on pair functional potentials, Cauchy–Born rule and slip mechanism, a material model assembling with spring-bundle components, a cubage component and slip components is established to describe the elasto-plastic damage constitutive relation under finite deformation. The expansion/shrink, translation and distortion of yield surfaces can be calculated based on the hardening rule and Bauschinger effect defined on the slip component level. Both kinematic and isotropic hardening are included. Numerical simulations and predictions under tension, torsion, and combined tension–torsion proportional/non-proportional loading are performed to obtain the evolution of subsequent yield surfaces and elastic constants and compare with two sets of experimental data in literature, one for a very low work hardening aluminum alloy Al 6061-T6511, and another for a very high work hardening aluminum alloy annealed 1100 Al. The feature of the yield surface in shape change, which presents a sharp front accompanied by a blunt rear under proportional loading, is described by the latent hardening and Bauschinger effect of slip components. Further, the evolution law of subsequent yield surfaces under different proportional loading paths is investigated in terms of their equivalence. The numerical simulations under non-proportional loading conditions for annealed 1100 Al are performed, and the subsequent yield surfaces exhibit mixed cross effect because the kinematic hardening and isotropic hardening follow different evolution tendency when loading path changes. The results of non-proportional loading demonstrate that the present model has the ability to address the issue of complex loading due to the introduction of state variables on slip components. Moreover, as an elasto-plastic damage constitutive model, the present model can also reflect the variation of elastic constants through damage defined on the spring-bundle components.

Quantitative analysis of rockburst for surrounding rocks and zonal disintegration mechanism in deep tunnels

Rock masses without pre-existing macrocracks can usually be considered as granular materials with only microcracks. During the excavation of the tunnels, microcracks may nucleate, grow and propagate through the rock matrix; secondary microcracks may appear, and discontinuous and incompatible deformation of rock masses may occur. The classical continuum elastoplastic theory is not suitable for analyzing discontinuous and incompatible deformation of rock masses. Based on non-Euclidean model of the discontinuous and incompatible deformation of rock masses, the distribution of stresses in the surrounding rock masses in deep tunnels is fluctuant or wave-like. The stress concentration at the tips of microcracks located in vicinity of stress wave crest is comparatively large, which may lead to the unstable growth and coalescence of secondary microcracks, and consequently the occurrence of fractured zones. On the other hand, the stress concentration at the tips of microcracks located around stress wave trough is relatively small, which may lead to the arrest of microcracks, and thus the non-fractured zones. The alternate appearance of stress wave crest and trough thus may induce the alternate occurrence of fractured and non-fractured zones in deep rock masses. For brittle rocks, the dissipated energy of microcrack growth is small, but the elastic strain energy stored in rock masses may be larger than the dissipated energy growths of pre-existing microcracks and secondary microcracks. The sudden release of the residual elastic strain energy may lead to rockburst. Based on this understanding, the criteria of rockburst are established. Furthermore, the relationship between rockbursts and zonal disintegration in the surrounding rock masses around deep tunnels is studied. The influences of the in-situ stresses and the physico-mechanical parameters on the distribution of rockburst zones and the ejection velocity of rock fragments are investigated in detail.

Interactive visualization of elasto-plastic behavior through stress paths and yield surfaces in finite element analysis

This paper proposes a new approach for representing elasto-plastic behavior through graphical visualization of the yield surface and the stress path in the principal stress space. The yield surface is defined by the material parameters of a given yield criterion, and the stress path is formed by a number of stress points, each of which is obtained for every increment in the applied force. The superposed images of the yield surface and the stress path can be used as a tool for interpreting the results of finite element analysis based on the assumed elasto-plastic yield theory. The paper describes how to construct the stress path and the yield surface by following procedures ranging from data extraction to graphical rendering. It suggests a framework for the interactive control of visualization and data sampling. Geometric modeling of the yield surface is formulated for various failure criteria that are frequently used in finite element analysis. Examples of stress path and yield surface visualization are presented herein to demonstrate the practical use of this approach. All of the features described in this paper can be implemented easily as a post-processing capability in finite element analysis programs.

An Elasto-plastic Damage Constitutive Theory Based on Pair Functional Potentials and Slip Mechanism

The deformation work rate can be expressed by the time rate of pair functional potentials which describe the energy of materials in terms of atomic bonds and atom embedding interactions. According to Cauchy-Born rule, the relations between the microscopic deformations of atomic bonds and electron gas and macroscopic deformation are established. Further, atomic bonds are grouped according to their directions, and atomic bonds in the same direction are simplified as a spring-bundle component. Atom embedding interactions in unit reference volume are simplified as a cubage component. Consequently, a material model composed of spring-bundle components and a cubage component is established. Since the essence of damage is the decrease and loss of atomic bonding forces, the damage effect can be reflected by the response functions of these two kinds of components. Formulating the mechanical responses of two kinds of components, the corresponding elasto-damage constitutive equations are derived. Considering that slip is the main plastic deformation mechanism of polycrystalline metals, the slip systems of crystal are extended to polycrystalline, and the slip components are proposed to describe the plastic deformation. Based on the decomposition of deformation gradient and combining the plastic response with the elasto-damage one, the elasto-plastic damage constitutive equations are derived. As a result, a material model formulated with spring-bundle components, a cubage component and slip components is established. Different from phenomenological constitutive theories, the mechanical property of materials depends on the property of components rather than that directly obtained on the representative volume element. The effect of finite deformation is taken into account in this model. Parameter calibration procedure and the basic characteristics of this model are discussed.

A two-dimensional homogenized model for a pile of thin elastic sheets with frictional contact

The present paper deals with the computational simulation of a pile of thin sheets. The sheets are not laminated or glued, but they interact by frictional contact. In general, it is not possible to perform a full-scale finite element contact computation for piles containing thousands of sheets; the problem size becomes too large, and numerical solution methods suffer from severe convergence problems due to the large number of strongly coupled contact conditions. In this paper, a macroscopic material model is presented for the two-dimensional case. The pile of sheets is homogenized by introducing an effective anisotropic constitutive law, which is motivated by formulations of the theory of elasto-plasticity. This macroscopic material law models the behavior of a pile of sheets, allowing for no tensile stresses in the direction normal to the sheets and obeying Coulomb’s law of friction in the tangential contact plane. Applying this macroscopic material model, an equivalent homogeneous body can be treated using much coarser discretizations. Computational results for the problems are provided, and a comparison with simplified contact computations is done.

The Study of Elastic-Plastic Fatigue Behavior of Nigerian-Rolled NST 37-2 Steel

The NST 37-2 steel represents about 75% volume of Nigerian-produced steel which is yet to be fully characterized for its fatigue behavior. Thus, its suitability for many applications is questionable. This paper presents a framework based on the theory of elasto-plasticity in order to make appropriate recommendations in this regard. Experimentally, tensile tests were carried out on test specimens to establish the baseline material properties of the steel in annealed, as-rolled, normalized and hardened/tempered conditions. Fatigue tests were then conducted at 60% Su; 70% Su and 80% Su of the test material and fractographic examinations on the test specimens were subsequently carried out. The frequency harmonic fatigue analysis was implemented in the ANSYS software environment for the numerical study. The elastic-plastic material property was described by the von Mises yield criterion, the flow rule of Prandtl-Reuss, and the kinematic hardening rule of Prager. The numerical results indicate with respect to rate-dependence fatigue behavior that the annealed test specimen is most resilient under cyclic deformation as compared with the normalized, hardened/tempered and as-rolled specimens respectively. The experimental and numerical results were found to be in close agreement and based on the general performance, the steel material is recommended for use in low cycle, quasi-static fatigue applications.

Formulation and validation of a constitutive model for sands in monotonic shear

A constitutive model for sands in monotonic shear is presented. The model is designed to simulate the behavior of sands in the whole stress and strain range of engineering interest with enough accuracy for practical usage. Material parameters were chosen to be state independent and easy to calibrate using conventional testing procedures. The formulation is based on effective stresses, pressure-dependent hyperelasticity, non-associative elastoplasticity, an isotropic hardening law and Rowe’s stress-dilatancy theory. The implementation of Rowe’s stress-dilatancy theory within the framework of elastoplasticity theory is discussed. It is found that Rowe’s theory produces a volumetric plastic strain rate function that has a discontinuity in its first derivative w.r.t. stress, and a smoothed form is proposed instead. Finally, some experimental tests are simulated and the results are briefly discussed.

Study of NC Cold Tube Bending Spring-Back Characteristics

The unloading spring-back of tubes during its manufacturing process shows a strong nonlinearity, which greatly influences the precision of parts. In this paper, the strain distribution of bending tubes was analyzed based on the elasto-plasticity theory, and the theoretical equation for spring-back of tubes was derived. The numerical simulation model for cold tube-bending process was developed with prediction error of 9% compared with experimental results, indicating high reliability of the model. The 12Cr1MoV and 20G tubes were used to analyze the effects of bending angle, bending radius and bending speed on the spring-back of tubes. The prediction equation of spring-back was built, which shows that the spring-back tendency was in accordance with theoretical analysis results. The simulated results show that the spring-back angle is linearly proportional to the bending angle within a certain range. In addition, it is proportional to the relative bending radius and the bending speed.

Influence of Welding Parameters on Residual Stress in 9% Ni Steel for Low Temperature Service

Based on welding heat transfer theory, heat elastoplasticity theory and phase transformation theory, welding residual stress in multi-layer welded joint of 9% Ni steel for liquefied natural gas (LNG) storage tanks was simulated using SYSWELD finite element analysis software. Taking into account nonlinear relationships between temperature and mechanical properties, together with influence of transformation latent heat on temperature field, double ellipsoid heat source model was applied in the simulation of the three fields (temperature field, microstructure field and stress-strain field). Distribution pattern of welding residual stress in 9%Ni steel welded joint was obtained through numerical simulation. Results showed that heat affected zone (HAZ) showing high-level residual stress was the main stress concentrated part of the welded joint. Longitudinal residual stress appeared to be tensile in the weld and HAZ, and compressive in regions remote from the weld. A match of low heat input and high interpass temperature should be employed to lower residual stress level in HAZ. Simulation results were well consistent with theoretical analysis.

Computational post failure analysis with a second gradient theory

ABSTRACT This article presents a theory of Gradient Elastoplasticity for strain-softening materials, in which internal length scales are present in both the elastic and the elastoplastic regime. A numerical discretisation based on the displacement formulation and a C1 finite element is used. It is shown that the formulation enables robust modelling of the post-peak material behaviour leading to localisation of deformation, spontaneously tracking emergent instabilities and quantitatively capturing details of the final localised failure mechanism.

A method for examining the seismic performance of steel arch deck bridges

To examine the seismic resistance performance of a steel arch bridge during a strong earthquake, an approach is proposed in this paper to determine the ultimate strain of an arch rib using the static elasto-plastic large deformation theory. A steel fixed-end deck arch bridge with a span length of 200 m was used as an example in this study. The ultimate strain of the arch rib was calculated using the elasto-plastic finite element model. The seismic responses of the arch bridge exposed to strong earthquakes in two and three directions were also simulated. Moreover, the effect of earthquake motion in the lateral direction on the elasto-plastic earthquake response was discussed, and the seismic resistance of the structure was analyzed. The results show that steel arch bridges have preferable seismic resistance under strong earthquake conditions. While lateral earthquake motion has some influence on the damage to an arch bridge, it has little effect on the displacement, axial force, and bending moment responses of the arch rib. It is feasible to examine the seismic performance of the arch bridge by ascertaining the out-of-plane and in-plane earthquake responses of the bridge.

Plasticity-induced anisotropy in amorphous solids: The Bauschinger effect

Amorphous solids that underwent a strain in one direction such that they responded in a plastic manner "remember" that direction also when relaxed back to a state with zero mean stress. We address the question "what is the order parameter that is responsible for this memory?" and is therefore the reason for the different subsequent responses of the material to strains in different directions. We identify such an order parameter which is readily measurable, we discuss its trajectory along the stress-strain curve, and propose that it and its probability distribution function must form a necessary component of a theory of elastoplasticity.

Limitations in the Constitutive Modeling of Unsaturated Soils and Solutions

Over the past six decades, significant attention has been paid to the elastoplastic behavior of unsaturated soils. In the past two decades alone, elastoplastic theory for unsaturated soils has been established and experimental techniques for measuring the elastoplastic behavior of unsaturated soils have become more sophisticated. However, less effort has been directed at developing the best strategy for constitutive modeling of unsaturated soils. At present, there is no standard method for developing constitutive models for unsaturated soils from experimental data, and owing to the extreme complexity of unsaturated soil behavior, there are limitations in the existing modeling methods. If these limitations are not recognized, misleading results in the constitutive modeling of unsaturated soil behavior may occur. This paper discusses the origins of and possible solutions to these limitations. Experimental data from the recent literature are used to demonstrate the use of existing methods for the constitutive modeling of unsaturated soils and potential associated problems. A modified state-surface approach (MSSA), recently proposed to model the elastoplastic behavior of unsaturated soils under isotropic conditions, was applied to overcome the limitations and develop a constitutive model that can best represent the behavior of unsaturated soil. A comparison of the proposed method and existing methods is discussed, and from this discussion, the capability and effectiveness of the proposed method are evaluated.

On the scale dependence of coefficient of friction in unlubricated sliding contacts

Friction at different force, length, and time scales is of great interest in tribology. The mechanical, chemical, and physical (atomic) interactions, each operating at their own force and length scale, make friction a highly scale dependent event. This work is an attempt to trace important mechanisms of friction on commonly used engineering materials over a normal force scale ranging from nN up to N, and thereby altering the contact size from nm 2 up to mm 2. The relevance of existing theories on friction is verified on different engineering surfaces taking into account mechanical (hard/hard, hard/soft), chemical, and physical aspects of the sliding surfaces. The applicability of Amontons’ law is experimentally investigated. For rough surfaces it was found that the existence of a constant coefficient of friction over a wide force and length scales is only a special case. For a hard/hard tribosystem (like DLC/Si 3N 4), a linear dependence of friction on normal force was observed, whereas a non-linear relationship was more evident on hydrophilic surfaces and hard/soft couples. Irrespective of the material system considered, the dependence of friction on normal force can be altered by modifying the surface roughness or texture of the material couple. In all, changes in the force and length scales bring about significant changes in the governing mechanisms of friction. The experimental findings were in good agreement with recent elasto-plastic and fractal contact mechanical theories on friction.

Effective temperature in elastoplasticity of amorphous solids

An effective temperature $T_{\rm eff}$ which differs from the bath temperature is believed to play an essential role in the theory of elasto-plasticity of amorphous solids. The definition of a measurable $T_{\rm eff}$ in the literature on sheared solids suffers however from being connected to a fluctuation-dissipation theorem which is correct only in equilibrium. Here we introduce a natural definition of $T_{\rm eff}$ based on measurable structural features without recourse to any questionable assumption. The value of $T_{\rm eff}$ is connected, using theory and scaling concepts, to the flow stress and the mean energy that characterize the elasto-plastic flow.

Professor Ioannis Vardoulakis (1949–2009)

Professor Ioannis Vardoulakis (1949–2009)