Classical Plasticity Theory Research Articles

STABILITY CONSIDERATIONS FOR ELASTIC-PLASTIC PROCESSES WITH DAMAGE

Classical theory of plasticity is fairly complete with flow rules, convexity of yield surfaces, extremum principles, and the uniqueness theorem. For the strain-hardening plasticity, Drucker’s postulates are established and proven based on the plastic-work and energy principles. Plasticity models have been further applied to heterogeneous and cementitious materials with certain degrees of success. In this paper, the stability statements of strain-hardening and strain-softening processes in concrete are examined by utilizing thermodynamic potential functions in the stress space and by applying Euler’s theorem of homogenous functions. It is shown that by specifying a strain-hardening parameter to account for the plastic strains and a damage parameter to represent the effect of microcracking, the dissipation inequality can be used to establish the Drucker’s stability postulate for the plastic flow within the framework of the internal variable theory of thermodynamics. Using the same approach and assuming uncoupling between plastic flow and microcracking, the formation leads to a softening stability statement for damage processes in concrete.

Elastoplastic source model for microseismicity and acoustic emission

SUMMARYThe microseismic events can often be characterized by a complex non-double couple source mechanism. Recent laboratory studies recording the acoustic emission during rock deformation help connecting the components of the seismic moment tensor with the failure process. In this complementary contribution, we offer a mathematical model which can further clarify these connections. We derive the seismic moment tensor based on classical continuum mechanics and plasticity theory. The moment tensor density can be represented by the product of elastic stiffness tensor and the plastic strain tensor. This representation of seismic sources has several useful properties: (i) it accounts for incipient faulting as a microseismicity source mechanism, (ii) it does not require a pre-defined fracture geometry, (iii) it accounts for both shear and volumetric source mechanisms, (iv) it is valid for general heterogeneous and anisotropic rocks and (v) it is consistent with elasto-plastic geomechanical simulators. We illustrate the new approach using 2-D numerical examples of seismicity associated with cylindrical openings, analogous to wellbore, tunnel or fluid-rich conduit and provide a simple analytic expression of the moment density tensor. We compare our simulation results with previously published data from laboratory and field experiments. We consider four special cases corresponding to ‘dry’ elastically homogeneous and elastically heterogeneous isotropic rocks, ‘dry’ transversely isotropic rocks and ‘wet’ isotropic rocks. The model highlights theoretical links between stress state, geomechanical parameters and conventional representations of the moment tensor such as Hudson source type parameters.

Application of a nonlinear kinematic-isotropic material model for the prediction of residual stress relaxation under a cyclic load

The effect of residual stress in the fatigue analysis of a formed products must be considered with great care. The relaxation of residual stress under a cyclic load is a transient phenomenon that is described in this paper using classic plasticity theory by employing a combined, non-linear, kinematic and isotropic hardening material model. Results were compared with experimental observations by means of contour method and digital-image-correlation method on specimens made of S235JR steel, with focus on near surface state. This research showed reasonable prediction of initial residual stress state and an over-prediction of strain ratcheting as cyclic load progresses.

On standard gradient plasticity and visco-plasticity

This paper is devoted to the standard theory of gradient plasticity and visco-plasticity, cf. Gudmundson (2004), Fleck and Willis (2009) and the quoted references. Within the framework of the generalized standard materials and the generalized principle of virtual work, our attention is focussed on the mathematical basis of the theory in order to underline the basic assumptions, general results and principal difficulties compared to the classical theory of plasticity. Gradient terms can be introduced in the expression of the energy potential or the dissipation potential. Various general results of the literature on the constitutive modeling and on the response of a solid under a given loading are revisited in this unified description. The quasi-static response is governed by a variational inequality which is an alternative form of Biot equation. A full discussion of this evolution equation is given in terms of the energy and dissipation potentials of the solid. The rate problem and the rate variational principle are considered and compared to the incremental problem and the incremental variational principle which result from an implicit discretization of the evolution equations. In particular, rate and incremental minimum principles are related to a stability condition.

Effect of crack curvature radius on the stress fields under nonlinear deformation

The influence of crack curvature radius and plastic properties of material on the crack tip fields is evaluated by conventional mechanism-based strain gradient (CMSG) plasticity theory, and classical plasticity theory. Stress fields in single edge tension specimen are investigated numerically for a wide range of the crack curvature radius variations. The strain hardening exponent varied from N=0.075 to N=0.4. The local areas of the influence of the crack tip curvature radius are established for the strain gradient plasticity and classical plasticity. The couple effects of material length scale parameter and plastic properties of material on crack tip fields are established for strain gradient plasticity. The boundaries of CMSG plasticity dominated zone are determined.

Formulation of a new elastoviscoplastic model for time‐dependent behavior of clay

AbstractThis paper presents a generalized, elastoviscoplastic constitutive model, MIT‐SR, that is capable of describing a wide range of time‐dependent characteristics observed in clays from creep to strain‐rate‐dependent shear behavior. The key component of the proposed model is a novel evolution equation that attributes viscoplastic deformations to a state variable,Ra, referred to as internal strain rate, which represents the perturbation of the clay particle assembly due to historical straining. This state variable is driven by external straining actions (under compression or shear), which can be intrinsically linked to the loading step in classical plasticity theory, and decays with time representing a fading memory process. The proposed framework can be used to extend existing time‐independent elastoplastic models. In this case, MIT‐SR is built upon a prior elastoplastic model (MIT‐S1), which uses 3‐D stress‐space surfaces and hardening laws to represent anisotropic effective stress‐strain‐strength properties, and a paraelastic approach to describe nonlinear hysteretic behavior at small strains. The paper highlights the versatility of the proposed MIT‐SR model in representing a wide range of time‐dependent characteristics for normally consolidation behavior and undrained shear behavior. By varying a strain‐rate sensitivity parameter,β, the model can capture a full spectrum from temporary material response to changing strain rate, to isotache‐type behavior where the normal consolidation and critical state lines are functions of the applied steady strain rate. The paper also showcases the model prediction for undrained creep and undrained relaxation behavior, and its promising capability in describing rate‐effects under cyclic direct simple shear.

Comparative assessment of backstress models using high-energy X-ray diffraction microscopy experiments and crystal plasticity finite element simulations

Crystal plasticity (CP) models have been evolving since their inception. Advanced experimental characterization methods have contributed significantly to assess the performance and subsequent improvement of many empirical relations in CP, which were directly adopted from classical plasticity theories of solids at the macro-scale. In this research, high energy X-ray diffraction microscopy (HEDM) has been used to track the stress-state of individual grains within a polycrystalline aggregate of a Nickel-base superalloy subjected to cyclic loading. Using path-dependent, mesoscopic stress-states from the HEDM experiment, the performance of two kinematic hardening models, in the context of CP, has been assessed. One of the models is an empirical Armstrong-Frederick equation, and the other is a geometrically necessary dislocation (GND)-based phenomenological model. The results suggest that the GND-based model is capable of capturing the cyclic crystal plasticity response. The present validation efforts are expected to take CP models one step closer towards their implementation in modern engineering workflow.

A micro plasticity model for pure bending analysis of curved beam-like MEMS devices

Computational models are investigated to analyze plane strain pure bending of rectangular cross sectional curved microbeams in elastic and partially plastic states. A generalization of classical von-mises deformation theory of plasticity is adopted to include size effect, by considering intrinsic material length scale parameter, in the modeling. Assuming linear and power-law hardening model, the governing nonlinear equation in elastic-plastic zones are discretized by generalized differential quadrature (GDQ) method and are solved by direct iteration approach. The numerical calculations of proposed model are validated with published literature. Semi-analytical solution is obtained from the special case of curved microbeam with elastic-rigid plastic constitutive model and linear hardening law. The influences of initial curvature and size effect on yielding initiations at the inner and outer surfaces of microstructure are evaluated. Also, distributions of stress components along the microbeam thickness are studied, for different values of the couple moments, at elastic limits and beyond.

Stress-fractional model with rotational hardening for anisotropic clay

Anisotropic clays are very common in practical engineering, where their constitutive relations were usually developed based on classical plasticity theory with associated flow rule. However, the softening responses of clays under compression following anisotropic consolidation can be non-associated and state-dependent. Fractional derivative is defined based on an integral form that can consider the material state and nonassociativity from the initial mathematical definition. In this study, a new anisotropic constitutive method for clay is developed by enriching the fractional plasticity with combined isotropic and rotational hardening. The developed model can consider the effect of evolving material anisotropy on the state-dependent non-associated behaviour of clay. To validate the approach, undrained and drained test results of different clays, are simulated and discussed, where a good agreement between the model simulations and test results is observed. It is also found that the material anisotropy keeps evolving with shearing, until reaching the final value at critical state.

A unification of finite deformation J2 Von-Mises plasticity and quantitative dislocation mechanics

We present a framework which unifies classical phenomenological J2 and crystal plasticity theories with quantitative dislocation mechanics. The theory allows the computation of stress fields of arbitrary dislocation distributions and, coupled with minimally modified classical (J2 and crystal plasticity) models for the plastic strain rate of statistical dislocations, results in a versatile model of finite deformation mesoscale plasticity. We demonstrate some capabilities of the framework by solving two outstanding challenge problems in mesoscale plasticity: 1) recover the experimentally observed power-law scaling of stress-strain behavior in constrained simple shear of thin metallic films inferred from micropillar experiments which all strain gradient plasticity models overestimate and fail to predict; 2) predict the finite deformation stress and energy density fields of a sequence of dislocation distributions representing a progressively dense dislocation wall in a finite body, as might arise in the process of polygonization when viewed macroscopically, with one consequence being the demonstration of the inapplicability of current mathematical results based on Γ-convergence for this physically relevant situation. Our calculations in this case expose a possible ‘phase transition’ - like behavior for further theoretical study. We also provide a quantitative solution to the fundamental question of the volume change induced by dislocations in a finite deformation theory, as well as show the massive non-uniqueness in the solution for the (inverse) deformation map of a body inherent in a model of finite strain dislocation mechanics, when approached as a problem in classical finite elasticity.

The role of J3 in the Orientation Analysis of the Adiabatic Shear Band of Ultrafine‐Grained Interstitial‐Free Steel

Within the framework of classical plastic theory, the instability inclination angle of metal is 45 ° under axial compression loading. However, microscopic observations by scanning electron microscopy (SEM) and transmission electron microscopy (TEM) show the ultrafine‐grained (UFG) interstitial‐free (IF) steel inclination angle of adiabatic shear instability is less than 45 °. Herein, a constitutive model that accounts for the effect of the third stress invariant is proposed based on the Johnson–Cook model to investigate the effect of J3 on the adiabatic shear instability inclination angle of UFG IF steel. To give a clear and direct interpretation of this effect, the model is implemented in the commercial software ABAQUS/Explicit by user subroutine VUMAT. Then shear‐compression specimens with different groove angles are used to investigate the influence of J3 on the orientation of adiabatic shear band. Subsequently, dynamic tests of UFG IF steel shear‐compression specimen with different groove angles are conducted through Split‐Hopkinson Pressure Bar. The experimental results show that noncoaxial influence (the effect of J3 or Lode angle) deserves enough attention in the orientation analysis of adiabatic shear instability of UFG IF steel.

A New 3D Empirical Plastic and Damage Model for Simulating the Failure of Concrete Structure

A new plastic–damage constitutive model based on the combination of damage mechanics and classical plastic theory was developed to simulate the failure of concrete. In order to explain different material behaviors of concrete under tensile and compressive loadings, the plastic yield criterion, the different kinematic hardening rule for tension and compressive and the isotropic flow rule were established in the effective stress space. Meanwhile, two different empirical damage evolution equations were adopted: one for compression and the other for tension. A multi-axial damage influence factor was also introduced to fully describe the anisotropic damage of concrete. Finally, the model response was compared with a wide range of experiment results. The results showed that the model could well describe the nonlinear behavior of concrete in a complex stress state.

Comparison Study of Constitutive Models for Overconsolidated Clays

Widely distributed in natural deposits, the overconsolidated (OC) clays have attracted extensive experimental investigations on their mechanical behaviors, especially in the 1960s and 1970s. Based on these results, numerous constitutive models have also been established. These models generally fall into two categories: one based on the classical plasticity theory and the other the bounding surface (BS) plasticity theory, with the latter being more popular and successful. The BS concept and the subloading surface (SS) concept are the two major BS plasticity theories. The features of these two concepts and the representative models based on them are introduced, respectively. The unified hardening (UH) model for OC clays is also based on the BS plasticity theory but distinguishes itself from other models by the integration of the reference yield surface, unified hardening parameter, potential failure stress ratio, and transformed stress tensor. Modification is made to the Hvorslev envelop employed in the UH model to improve its capability of describing the behaviors of clays with extremely high overconsolidation ratio in this paper. The comparison among the BS model, SS model, and UH model is performed. Evidence shows that all these three models can characterize the fundamental behaviors of OC clays, such as the stress dilatancy, strain softening and attainment of the critical state. The UH model with the revised Hvorslev envelop has the fewest parameters which are identical to those of the modified Cam-Clay model.

THE FOUNDATIONS OF CONTINUUM DAMAGE MECHANICS

Damage Mechanics has become a useful theory in describing the nonlinear behavior of solids driven by the nucleation and growth of cracks and microcracks. This approach, based on the first principles of mechanics and thermodynamics, has also been combined with classical theories of plasticity to address a wide range of loading applications. In spite of the many different damage mechanics models and representations that are proposed, the foundation of damage mechanics is not well understood or at least not thoroughly published giving rise to the many inaccurate definitions and formulations. The intent of this paper is to provide the background of the continuum damage mechanics outlining the fundamentals on which this field theory is set up. The internal variable theory of continuum thermodynamics is reviewed and is shown that with Legendre transformation technique, various potential functions can be developed for damage mechanics formulation in either stress or strain space. The concept of constrained or neighboring equilibrium state is also introduced and is explained. The paper will conclude with the derivation of the general damage potential and a suggestion is given for the isotropic damage formulation with the resulting uniaxial stress-strain relation.

Dislocation pattern formation in finite deformation crystal plasticity

Stressed dislocation pattern formation in crystal plasticity at finite deformation is demonstrated for the first time. Size effects are also demonstrated within the same mathematical model. The model involves two extra material parameters beyond the requirements of standard classical crystal plasticity theory. The dislocation microstructures shown are decoupled from deformation microstructures, and emerge without any consideration of latent hardening or constitutive assumptions related to cross-slip. Crystal orientation effects on the pattern formation and mechanical response are also demonstrated. The manifest irrelevance of the necessity of a multiplicative decomposition of the deformation gradient, a plastic distortion tensor, and the choice of a reference configuration in our model to describe the micromechanics of plasticity as it arises from the existence and motion of dislocations is demonstrated.

Hysteretic Shell Finite Element

AbstractA hysteretic shell finite element for the nonlinear, static, and dynamic analysis of structures is presented, formulated on the basis of classical theory of plasticity and finite deformatio...

Upper Bound Analysis for Collapse Failure of Shield Tunnel Face Excavated in Unsaturated Soils Considering Steady Vertical Flow

Natural soils are mostly in an unsaturated state and the corresponding mechanical properties differ significantly from a saturated one. In traditional stability analysis for a shield tunnel face, the soil mass is typically assumed to be dry or saturated for convenience of analysis. In this work, based on the upper bound theorem of classical plasticity theory and the log-spiral failure mechanism, face stability of the shield tunnel excavated in unsaturated soils under vertical steady flow is studied. The profile of shear strength is determined by virtues of the unified effective stress approach and the analytical solution of matric suction under unsaturated steady flow. On this basis, the analytical expression for the supporting pressure is deduced and the sequential quadratic programming is employed to search for the optimal upper bound solution. Through parametric analysis it is found the fitting parameters of SWCC and the groundwater level affect the supporting pressure notably. Besides, the relevant influence of vertical discharge is also significant when clayey soils are concerned.

Modelling nano-indentation of ion-irradiated FCC single crystals by strain-gradient crystal plasticity theory

In this work, a strain-gradient crystal plasticity theory with irradiation effect is proposed for the surface nano-indentation of ion-irradiated FCC metals. In order to characterize the indentation size effect and irradiation hardening of ion-irradiated materials indented by the Berkovich indenter, the hardening contributions of geometrically necessary dislocations and non-uniformly distributed defects are incorporated into the classical crystal plasticity theory. The constitutive equations are implemented into ABAQUS through the user material subroutine VUMAT to numerically simulate the surface nano-indentation of ion-irradiated single crystal copper. The theoretical model is calibrated by comparing the numerical results with experimental data. Both the simulated force-depth relationship without irradiation effect and hardness-depth relationship with/without irradiation effect can match well with corresponding experimental data. The dominant features as observed in the nano-indentation of ion-irradiated materials can be effectively characterized by the proposed theoretical framework, which include the indentation size effect, depth-dependent irradiation hardening, and unirradiated substrate softening effect. Moreover, the evolution of different hardening mechanisms during the surface nano-indentation is systematically analyzed to help understand the macroscopic deformation behavior.

About the limit state of deformable bodies

The theory of limit state deals with statically determinate condition of solids. In this case the system is closed due to the limit conditions, such properties of matter as viscosity, elasticity, etc. cannot influence the limit state. In other words, being at the limit state the nature of the relationship between stress and strain has no effect on the limit state. The article discusses systems of equations which correspond to the classical theory of plasticity. It is assumed that the components of the velocity vector depend on two spatial coordinates only. The constructed system can be used to describe the torsion of a parallelepiped around the three orthogonal axes. For the constructed system of equations group point symmetries, conservation laws were found. It is shown that the system admits 8-dimensional Lie algebra. On the basis of the symmetry group some classes of invariant solutions of rank 1 were constructed. They depend on the arbitrary functions of one variable. It is shown that these solutions can be used to describe plastic torsion of a parallelepiped around three orthogonal axes. It is shown that the system admits an infinite series of conservation laws.

Classic crystal plasticity theory vs crystal plasticity theory based on strong discontinuities—Theoretical and algorithmic aspects

SummaryThis paper deals with two different approaches suitable for the description of plasticity in single crystals. The first one is the standard approach that is based on a continuous deformation mapping. Plasticity is driven by a classic Schmid‐type relation connecting the shear stresses to the shear strains at a certain slip system. By way of contrast, the second approach is nonstandard. In this novel model, localized plastic deformation at certain slip planes is approximated by a strong discontinuity (discontinuous deformation mapping). Accordingly, a modified Schmid‐type model relating the shear stresses to the shear displacements (displacement jump) is considered in this model. Although both models are indeed different, it is shown that they can be characterized by almost the same set of equations, eg, by a multiplicative decomposition of the deformation gradient into an elastic part and a plastic part. This striking analogy eventually leads to a unifying algorithmic formulation covering both models. Since the set of active slip systems is not known in advance, its determination is of utmost importance. This problem is solved here by using the nonlinear complementarity problem (NCP) as advocated by Fischer and Burmeister. While this idea is not new, it is shown that the NCP problem is well posed, independent of the number of active slip systems. To be more explicit, the tangent matrix in the return‐mapping scheme is regular even for more than five simultaneously active slip systems. Based on this algorithm, texture evolution in a polycrystal is analyzed by means of both models and the results are compared in detail.