Elastoplastic Coupling Research Articles

Plastically-driven variation of elastic stiffness in green bodies during powder compaction: Part I. Experiments and elastoplastic coupling

Cold compaction of ceramic powders is driven by plastic strain, during which the elastic stiffness of the material progressively increases from values typical of granular matter to those representative of a fully dense solid. This increase of stiffness strongly affects the mechanical behaviour of the green body and is crucial in the modelling of forming processes for ceramics. A protocol for ultrasonic experimental investigation (via P and S waves transmission) is proposed to quantify the elastic constants (Young modulus and Poisson's ratio) as functions of the forming pressure. Experimental results performed in uniaxial strain allow for the introduction of laws that describe the variation of the elastic constants during densification. These laws are motivated in terms of elastoplastic coupling through the simulation of an isostatic pressure compaction process of alumina powder. A micromechanical explanation of the stiffening of elastic properties during densification is deferred to Part II of this study.

Elastoplastic coupling for thermo-elasto-plasticity at high temperature

The coupling of elastic and plastic deformation is introduced in a framework allowing for the inclusion of thermal effects. In this context where high temperatures may be present, many laws governing materials behaviour become nonlinear functions of both temperature and strain. The understanding of this is crucial to model the behaviour of refractories, sintering processes, rocks at great depth and geo-energy. Two equivalent formulations are presented, one in terms of isothermal elastic quantities and the other in terms of adiabatic. In the former approach a Helmholtz free energy density need not be introduced and the contributions to the variations of elastic stiffness of both plastic flow and temperature can be clearly separated and hence a full explanation is given. A simple example using available experiments on sintered alumina shows the capability of the constitutive framework to describe the behaviour of refractories at high temperature.

Finite-strain formulation and FE implementation of a constitutive model for powder compaction

A finite-strain formulation is developed, implemented and tested for a constitutive model capable of describing the transition from granular to fully dense state during cold forming of ceramic powder. This constitutive model (as well as many others employed for geomaterials) embodies a number of features, such as pressure-sensitive yielding, complex hardening rules and elastoplastic coupling, posing considerable problems in a finite-strain formulation and numerical implementation. A number of strategies are proposed to overcome the related problems, in particular, a neo-Hookean type of modification to the elastic potential and the adoption of the second Piola–Kirchhoff stress referred to the intermediate configuration to describe yielding. An incremental scheme compatible with the formulation for elastoplastic coupling at finite strain is also developed, and the corresponding constitutive update problem is solved by applying a return mapping algorithm.

A Constitutive Model with Effect of Temperature for Frozen Soil

To model the stress-strain relation of frozen soil under different temperatures, an elasto-plastic constitutive model coupling with temperature variable was proposed. Under axisymmetric condition, elastic strain was calculated by the K-G model coupling with temperature. The plastic strain was calculated by using the DP yield criterion, associated flow rule and isotropic hardening law. All of the elastic and plastic parameters are related to the temperature variable. The simulated results show that the proposed model can predict the deformation behavior of frozen soil under different temperatures.

Elastoplastic coupling to model cold ceramic powder compaction

The simulation of industrial processes involving cold compaction of powders allows for the optimization of the production of both traditional and advanced ceramics. The capabilities of a constitutive model previously proposed by the authors are explored to simulate simple forming processes, both in the small and in the large strain formulation. The model is based on the concept of elastoplastic coupling providing a relation between density changes and variation of elastic properties and has been tailored to describe the transition between a granular ceramic powder and a dense green body. Finite element simulations have been compared with experiments on an alumina ready-to-press powder and an aluminum silicate spray-dried granulate. The simulations show that it is possible to take into account friction at the die wall and to predict the state of residual stress, density distribution and elastic properties in the green body at the end of the forming process.

The Numerical Simulations for the Shot Peening Process

To develop a generalized numerical model by the elasto-plastic formulation coupling with dynamic finite element method for the shot peening process of metal alloys is the main goal of this paper. The analysis code is based on the finite deformation theory; the optimum numerical technique and the relevant parameters of forming process. In this study, the simulations of single and multi shot forming process were performed. The effects of impact velocity and strain rate, size of workpiece, and the residual stress were observed and discussed in details. After comparing with the corresponding papers, the universal model for metal shot peening analyses will be developed completely. The most significant advantage of this paper is the development of the precise analysis technology to suit for the future necessity of the manufacturing processes. The suggested models and techniques are helpful in solving of impact problems or metal forming processes.

A procedure of strain-softening model for elasto-plastic analysis of a circular opening considering elasto-plastic coupling

A simple numerical procedure of strain-softening model for the elasto-plastic analysis of a circular opening is presented by modifying the procedure proposed by Lee and Pietruszczak (2008). The proposed procedure has two advantages over that of Lee and Pietruszczak (2008). One is that the elasto-plastic coupling can be considered, the other is that the equilibrium equation and compatibility equation need not be expressed with respect to the normalized radius. Through MATLAB programming, the accuracy and validity of the proposed procedure are demonstrated through some examples. The influence of elastic modulus and Poisson’s ratio evolution on the distribution of radial displacement and hoop stress, ground response curve (GRC) and plastic radius was investigated, and the results show that, with the decrease of elastic modulus evolutional ratio, the radial displacement, hoop stress and plastic radius all increase. With the increase of Poisson’s ratio evolutional ratio, the radial displacement, hoop stress and plastic radius all decrease. As for influence extent, the evolution of elastic modulus has great influence on the radial displacement; and the evolution of Poisson’s ratio has smaller influence on the radial displacement, both of them have very slight influence on the hoop stress near the excavation face but greater on the hoop stress near the interface of elastic and plastic region. And both of them have very slight influence on the plastic radius.

Solid-Gas Coupling Model and Numerical Simulation of Coal Containing Gas Based on Comsol Multiphysic

Solid-gas coupling effect of coal containing gas is studied in order to understand the gas percolation mechanism in coal and rock. On the premise of that porosity and permeability of coal and rock are in dynamic changes and Klinkenberg effect, then seepage mechanics and elastic-plastic mechanics are considered together to established solid-gas coupling model of coal containing gas. With the given fixed solution conditions and parameters, the simulation results of mathematical model is found by the Comsol Multiphysic finite element software. Simulation results are consistent with the stress-strain law, deformation and failure modes of specimen in the experiment. Seepage law obtained in numerical simulation has same trends with experimental data. The elastoplastic solid-gas coupling model of coal containing gas can effectively describe the mechanical percolation characteristics of coal containing gas.

Elasto-plastic coupling analysis of circular openings in elasto-brittle-plastic rock mass

This paper deals with elasto-plastic coupling solutions for the prediction of displacements around circular openings in elasto-brittle-plastic rock mass subject to hydrostatic stress. Both linear Mohr–Coulomb criterion and nonlinear Hoek–Brown criterion on the basis of a non-associated flow rule are considered. The restrictive condition between strength parameters and Young’s modulus in post-failure region for rock materials is theoretically established. Meanwhile, the proposed restrictive condition is validated by FLAC3D code that the advanced elasto-plastic coupling model (EPCM) based on Mohr–Coulomb criterion is merged in. Finally, the elasto-plastic coupling dimensionless displacements of circular opening in elasto-brittle-plastic rock mass are analyzed with different deterioration degree of Young’s modulus using two kinds of rocks. The results show that the deformation of rock mass increases obviously with the decrease of Young’s modulus in the plastic region, but Young’s modulus has little influence on the plastic region radius and stresses distribution form.

Analysis of deformation localization based on proposed theory of ultrasonic wave velocity propagating in plastically deformed solids

The localization of plastic deformation is discussed as “stationary discontinuity” characterized by a vanishing velocity of an acceleration wave derived using the author’s proposed theory of ultrasonic wave velocities propagating in plastically deformed solids. To formulate the proposed theory, the elasto-plastic coupling effect was introduced to consider the elastic stiffness degradation due to the plastic deformation. The driving force of the deformation localization is caused by the yield vertex effect, which introduces a pronounced softening of the shear modulus, and geometrical softening due to double slip caused by lattice rotations. In the present paper, it is examined theoretically and experimentally that the diagonal terms of the introduced elasto-plastic coupling tensor represent a slight hardening followed by a pronounced softening of the elastic modulus induced by the point defect development caused by cross slides among dislocations at multiple slip stages similar to the yield vertex effects. The off-diagonal terms represent geometrical softening induced by lattice rotations such as texture evolution. Then, based on the coincidence of the onset strains between localization and acceleration waves of vanishing velocity, the diagrams of diffuse necking, localized necking and forming limit are analyzed by applying the proposed acoustic tensor, which is based on the generalized Christoffel tensor derived by the author, and solving cut off conditions of the quasi-longitudinal wave to determine the onset strains of deformation localization and localization modes. As a result, diagrams of diffuse necking, localized necking and forming limit were obtained. Moreover, the localization modes were determined and distinguished as the SH-mode, SV-mode, tearing mode and splitting mode.

Hyperelastic modelling of small‐strain stiffness anisotropy of cyclically loaded sand

AbstractExperimental evidence shows that soil stiffness at very small strains is strongly anisotropic and depends on the stress level and void ratio. In particular, stiffness anisotropy varies considerably in sand when subjected to cyclic loading, following the stress cycles applied. To model this behaviour, an innovative hyperelastic formulation based on the elastoplastic coupling is incorporated in a new kinematic hardening elastoplastic model. The proposed hyperelastic–plastic model is the first to be capable of correctly simulating all aspects of the small‐strain behaviour of granular materials subjected to monotonic and cyclic loads. This hyperelastic formulation is generally applicable to any elastoplastic model. Copyright © 2009 John Wiley & Sons, Ltd.

Transient hydrogen diffusion analyses coupled with crack-tip plasticity under cyclic loading

The effect of hydrogen on the material strengths of metals is known as the hydrogen embrittlement, which affects the structural integrity of a hydrogen energy system. In the present paper, we developed a computer program for a transient hydrogen diffusion–elastoplastic coupling analysis by combining an in-house finite element program with a general purpose finite element computer program to analyze hydrogen diffusion. In this program, we use a hypothesis that the hydrogen absorbed in the metal affects the yield stress of the metal. In the present paper, we discuss the effects of the cyclic loading on the hydrogen concentration near the crack tip. An important finding we obtained here is the fact that the hydrogen concentration near the crack tip greatly depends on the loading frequency. This result indicates that the fatigue lives of the components in a hydrogen system depend not only on the number of loading cycles but also on the loading frequency.

A model for stress and plastic strain induced nonlinear, hyperelastic anisotropy in soils

AbstractThe experimental evidence that cohesive and granular soils possess an elastic range in which the elasticity is both nonlinear and anisotropic—with stiffness and directional characteristics strongly dependent on stress and plastic strain (the so‐called ‘stress history’)—is given a formulation based on hyperelasticity. This is accomplished within the framework of elastoplastic coupling, through a new proposal of elastic potentials and a combined use of a plastic‐strain‐dependent fabric tensor and nonlinear elasticity. When used within a simple elastoplastic framework, the proposed model is shown to yield very accurate simulations of the evolution of elastic properties from initial directional stiffening to final isotropic degradation. Within the proposed constitutive framework, it is shown that predictions of shear band formation and evolution become closer to the existing experimental results, when compared to modelling in which elasticity does not evolve. Copyright © 2007 John Wiley & Sons, Ltd.

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part I – small strain formulation

Mechanical densification of granular bodies is a process in which a loose material becomes increasingly cohesive as the applied pressure increases. A constitutive description of this process faces the formidable problem that granular and dense materials have completely different mechanical behaviours (nonlinear elastic properties, yield limit, plastic flow and hardening laws), which must both be, in a sense, included in the formulation. A treatment of this problem is provided here, so that a new phenomenological, elastoplastic constitutive model is formulated, calibrated by experimental data, implemented and tested, that is capable of describing the transition between granular and fully dense states of a given material. The formulation involves a novel use of elastoplastic coupling to describe the dependence of cohesion and elastic properties on the plastic strain. The treatment falls within small strain theory, which is thought to be appropriate in several situations; however, a generalization of the model to large strain is provided in Part II of this paper.

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II – the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of ‘elastoplastic coupling’, in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state. The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear – or linear – dependence of the elastic part of the deformation on the forming pressure in the granular – or dense – state) makes the use of existing large strain formulations awkward, if even possible. Therefore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its ‘total’ and ‘plastic’ work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.

Multiple shear band development and related instabilities in granular materials

A new, small-strain constitutive model, incorporating elastoplastic coupling to describe developing elastic anisotropy, and density as a state variable to capture compaction and dilation, is proposed to simulate the behaviour of granular materials, in particular sand. This developing elastic anisotropy is related to grain reorientation and is shown to be crucial to obtain localisation during strain hardening, as experiments exhibit. Post-localisation analysis is also performed under simplificative assumptions, which evinces a number of features, including softening induced by localisation, size effects and snap-back, all phenomena found in qualitative and quantitative agreement with experiments. No prior model of granular material deformation correctly captures all these behaviours. The post-localisation analysis has revealed a new form of material instability in granular materials, consisting of a saturation mechanism, in which shear bands just formed unload, permitting new bands to form. This phenomenon shares similarities with the mechanics of phase transformation in metal strips and results in a stress oscillation during increasing deformation. The investigation of this mechanism of localised deformation reveals that loose and dense sands behave in qualitatively different ways. In particular, saturation is not persistent in dense sand; rather, after several shear bands form and saturate, this process is terminated by the formation of a differently inclined shear band occurring in the material transformed by previous strain localisation. In this case, the resulting ‘global’ stress–strain curve exhibits a few stress oscillations followed by a strong softening. On the other hand, band saturation is found to be a persistent phenomenon in loose sand, yielding a continuing stress oscillation. This provides a consistent description of specific experimental results.

Quasi-adiabatic shear zones in the lithosphere: numerical and experimental approaches

Continental breakup, or compressive lithosphere scale faulting, requires a physical mechanism for wholesale faulting of the lithosphere. We compared numerical and experimental models for the nucleation of quasi-adiabatic shear bands in polyvinylchloride (PVC) with those in an idealized viscoelastoplastic mantle with olivine rheology. In both materials fault nucleation is caused by elastic stress concentration on pre-existing imperfections, with localized yielding confined to its vicinity. Faulting occurs rapidly after the initial elastic energy in the system is charged sufficiently to cause wholesale yielding. Propagation of the fault, monitored by looking at the dissipation of plastic energy, reveals migration of a sharp, thermal-mechanical “crack”- like instability, which appears in the temperature field as a slightly diffused signal. The initial temperature rise in the crack is subtle but increases suddenly when the plate is severed. This autocatalytic behavior has also been described in ductile polymers, which can be used as mechanical analogues. We suggest that elastoplastic coupling in quasi-adiabatic shear banding is a key for fast (< 1 Ma) nucleation of shear zones. These nonlinear phenomena will be illustrated for both experimental and numerical results by nine movies

A unified framework for compressible plasticity

A framework for compressible elastoplasticity is cultivated to proficiently characterize various loading states of an isotropic solid undergoing a full range of deformation, including both hardening and softening behavior. By employing Legendre's dual transformation, it is shown that Drucker's path independence in the small and the linearity of rate change between stress and strain are characteristics of the additive-decomposition of strain rates. The normality property of the plastic strain rate to a loading function is established on the basis of path independence in the small of the potential functions used in the dual transformation. Work-hardening is a sufficient but not a necessary condition for ensuring stability in the small. Within this framework, a compressible elastoplasticity theory is generalized to cover the behavior of softening and elastoplastic coupling. Analysis results indicate that plastic instability for a void-contained aggregate is independent of the hydrostatic component of a stress tensor.

Uniqueness and localization—II. Coupled elastoplasticity

Uniqueness of the incremental response is investigated for an elastoplastic coupling model with elastic stiffness degradation. The criteria of second order work positiveness and strain localization are used in the form derived by Bigoni and Hueckel (1991. Int. J. Solids Structures28, 197–213). A numerical example shows that in a triaxial test the loss of second order work positiveness occurs at the positive hardening modulus, whereas the condition of strain localization is never fulfilled. Thus shear hands or splits are predicted not to form, despite strong elasticity degradation and non-associativity.

Acoustoelastic theory for finite plastic deformation of solids.

Theoretical modeling of acoustoelastic effects in plastically deformed solids has been accomplished through the use of an elastoplastic coupling strain rate. This coupling strain rate, which remains effective even under subsequent elastic loading, is considered to be generated at the yieldsurface vertex and causes elastic modulus degradation due to the growth of plastic anisotropy. The purpose of the present paper is to formulate precisely a generalized acoustoelastic theory for plastically deformed solids with finite deformation, and moreover, to provide the method of nondestructive evaluation of the plastically deformed state, i. e., yield surface, texture change due to the plastic deformation and the occurrence of the instability associated with the microslip band.