Principle Of Maximum Plastic Dissipation Research Articles

Elasto-plasticity theory for large plastic deformation and its use for the material stiffness determination

AbstractIn this paper, we present a finite elasto-plasticity theory for large plastic deformations. For the elastic part of the model, we use the St. Venant–Kirchhoff elasticity. The plastic part is described by the isomorphy concept, the yield condition is covered by the isotropic $$J_2$$ J 2 theory of (Huber in Czas Techn 22:34,1904; von Mises in Math Phys 4:582–592, 1913) and (Hencky in ZAMM 9:215–220, 1924), and the yield condition uses the principle of maximum plastic dissipation. The numeric of this theory is discussed and finally implemented in a Fortran code to use it as material law in the UMAT subroutine of the finite element program Abaqus. The material law is validated using different test calculations like tensile and shear tests as well as a large deformation simulation compared to the Abaqus internal material law. Further, we apply this material model to determine the effective material stiffness tetrad of large deformed inhomogeneous materials. For these purposes, we additionally present an automated method for determining material stiffnesses of an arbitrary material in Abaqus.

T-Splines for Isogeometric Analysis of the Large Deformation of Elastoplastic Kirchhoff–Love Shells

In this paper, we develop a T-spline-based isogeometric method for the large deformation of Kirchhoff–Love shells considering highly nonlinear elastoplastic materials. The adaptive refinement is implemented, and some relatively complex models are considered by utilizing the superiorities of T-splines. A classical finite strain plastic model combining von Mises yield criteria and the principle of maximum plastic dissipation is carefully explored in the derivation of discrete isogeometric formulations under the total Lagrangian framework. The Bézier extraction scheme is embedded into a unified framework converting T-spline or NURBS models into Bézier meshes for isogeometric analysis. An a posteriori error estimator is established and used to guide the local refinement of T-spline models. Both standard T-splines with T-junctions and unstructured T-splines with extraordinary points are investigated in the examples. The obtained results are compared with existing solutions and those of ABAQUS. The numerical results confirm that the adaptive refinement strategy with T-splines could improve the convergence behaviors when compared with the uniform refinement strategy.

A hybrid strategy blending primal‐dual interior point and return mapping methods for a class of hypoelastic‐plastic models with memory surface

AbstractBy blending the primal‐dual interior point method (PDIPM) and the return mapping algorithm, we propose a hybrid strategy of implicit stress update for a class of hypoelastic‐plastic models with the hardening rule whose evolution is restricted by the memory surface. The inequality constraint relevant to the memory surface is replaced by the equality constraint by the introduction of a slack variable, and the duality gap is gradually reduced by using path‐following method. First, we formulate an optimization problem corresponding to the principle of maximum plastic dissipation for the standard von‐Mises plasticity with isotropic and kinematic hardening rules and its variant for the model with the memory surface. Next, after function forms employed for the elastic‐plastic model are specified, the PDIPM to realize an implicit stress update is briefly reviewed and then applied to the model involving the memory surface to replace the relevant inequality constraint by the equality constraint by the introduction of a slack variable. Then, we present a hybrid scheme that combines the standard return mapping algorithm with the PDIPM. The numerical accuracy of the proposed stress update algorithm for the conventional elastic‐plastic model is verified in comparison with the standard return mapping algorithm using iso‐error map. Also, targeting a notched round steel bar under cyclic loading with three different amplitudes, we demonstrate the performance for the stress update using the elastic‐plastic model with the memory surface. Finally, the capability of the proposed algorithm is proven through a typical real‐life example such that a steel bridge is subjected to earthquake for which the residual load carrying capacity must be estimated.

Regularized variational formulation for nonlinear dynamics of viscoplastic plates

In this work we present novel approach to Reissner–Mindlin plates, targeting the nonlinear dynamics applications and in particular robust scheme for improved performance in vibration and wave propagation computations, where the standard finite element isoparametric interpolations no longer provide the best approximation property. The first key ingredient of the present development is regularized variational formulation, recast in terms of stress resultants (bending moments and shear forces), which allows any convenient interpolations. The discrete approximation is here constructed by using the lowest order Raviart–Thomas vector space employed to discretize bending moment and shear force fields, providing the continuity of the stress resultants projections across the boundary between adjacent plate elements. The latter is of particular interest for dynamic analysis for it can capture smooth approximation for wave propagation phenomena. This particular space discrete approximation is further combined with the most appropriate time-integration schemes capable of enforcing either energy conservation or decay of high frequency modes, which delivers the schemes with higher stability and robustness in long-term computations. The developments are carried out for viscoplastic constitutive behavior of plate material, where the corresponding evolution equations are obtained by appealing to the penalty-like form of the principle of maximum plastic dissipation. The choice of viscoplasticity (rather than plasticity) is crucial in that allows development of second-order scheme by integrating simultaneously the momentum balance and viscoplastic strains evolution equations. Several numerical simulations are given to illustrate a very satisfying performance of the proposed hybrid-stress discrete approximation for thick plate element and of the energy conserving/decaying schemes.

A variationally consistent formulation of the thermo-mechanically coupled problem with non-associative viscoplasticity for glassy amorphous polymers

This study presents a variationally consistent formulation of the thermo-mechanically coupled problem with non-associative viscoplasticity for glassy amorphous polymers. First, the decomposition of the equivalent plastic strain is carried out to derive the variational consistent evolution law of the shear yield strength with reference to the analogous approach taken for formulating the Armstrong-Frederick model. Second, an alternative form of the dual viscoplastic dissipation potential is proposed to recast the principle of maximum plastic dissipation for viscoplasticity to be of the same form of rate-independent plasticity with the introduction of the extended yield function. Third, we address the optimization problem relevant to the thermo-mechanically coupled behavior of glassy amorphous polymers within the incremental variational framework with the help of the parameterization of flow rules. Thanks to the achieved variational consistency, the mathematical model derived by the proposed formulation can enjoy several benefits for ensuring the stability of the strongly coupled discretized equations in the monolithic method under the condition that the material behavior is stable. In addition, since the present formulation does not require the time derivative of the shear yield strength, the resulting evolution equation accommodates the time variations of temperature in terms of its material properties, implying that it is amenable to various thermal processes other than isothermal ones. Numerical examples are presented to demonstrate the capability of the proposed formulation in simulating actual compression tests conducted on a PMMA specimen that exhibits complex thermo-mechanically coupled phenomena.

An energy approach to Modified Cam-Clay plasticity and damage modeling of cohesive soils

An alternative approach to Modified Cam-Clay (MCC) critical state plasticity coupled with damage is proposed. The provided hyper-elastoplastic/damage framework is motivated by a desire to ensure thermodynamic consistency of model predictions, and is shown to satisfy the principle of maximum plastic dissipation, enabling enforcement of the plastic dissipation (and Clausius–Planck) inequality. A small strain Eshelby-like stress is derived as being energy-conjugate to the plastic strain rate, and relation to the damage-energy release rate is exploited to pose a coupled damage/yield MCC criteria (MCC-D). Relations between volumetric damage and porosity changes are examined, along with a method for estimating the damage state from bulk moduli measurements. The model is shown to simulate well existing high pressure compression measurements of Boulder clay, and connections between damage-related model parameters and behavior exemplifying different microstructural states is examined through numerical experiments.

A new class of plastic flow evolution equations for anisotropic multiplicative elastoplasticity based on the notion of a corrector elastic strain rate

We herein present a new continuum theory for both isotropic and anisotropic elastoplasticity at large strains. The new framework has the following properties: (1) It is valid for non-moderate large strains, (2) it is valid for both elastic and plastic anisotropy, (3) its description in rate form is parallel to that of the infinitesimal formulation, (4) it is compatible with the multiplicative decomposition, (5) results in a similar framework in any stress-strain work-conjugate pair, (6) it is consistent with the principle of maximum plastic dissipation and (7) does not impose any restriction on the plastic spin, which must be given as an independent constitutive equation. Furthermore, when formulated using logarithmic strain measures in the intermediate configuration: (8) it may be easily integrated using a classical backward-Euler rule resulting in an additive update. All these properties are obtained simply considering a plastic evolution in terms of a corrector rate of the proper elastic strain. This new continuum theory is a natural framework for elastoplasticity of both metals and soft materials and solves the (so-coined by Simo) rate issue.

On the performance of the Prange‐Hellinger‐Reissner finite element formulation for elasto‐plasticity at small strains

AbstractThis contribution deals with a stress‐displacement mixed finite element formulation for elasto‐plasticity within the framework of small deformations based on the Prange‐Hellinger‐Reissner (PHR) functional. The interpolations for the displacements are given by standard Lagrangian shape functions and a 5‐parameter discontinuous interpolation is introduced for the stresses which was published by [1]. Based on the principle of maximum plastic dissipation the flow rule and hardening law will be derived by regarding a von Mises yield criterion, see [2]. In contrast to [3], we apply a point‐wise enforcement of the flow rule, hardening law and loading/unloading conditions. This work is related to the physically nonlinear mixed finite element based on the Prange‐Hellinger‐Reissner formulations for elasto‐plasticity, [4]. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A novel Lagrangian approach for the stable numerical simulation of fault and fracture mechanics

The simulation of the mechanics of geological faults and fractures is of paramount importance in several applications, such as ensuring the safety of the underground storage of wastes and hydrocarbons or predicting the possible seismicity triggered by the production and injection of subsurface fluids. However, the stable numerical modeling of ground ruptures is still an open issue. The present work introduces a novel formulation based on the use of the Lagrange multipliers to prescribe the constraints on the contact surfaces. The variational formulation is modified in order to take into account the frictional work along the activated fault portion according to the principle of maximum plastic dissipation. The numerical model, developed in the framework of the Finite Element method, provides stable solutions with a fast convergence of the non-linear problem. The stabilizing properties of the proposed model are emphasized with the aid of a realistic numerical example dealing with the generation of ground fractures due to groundwater withdrawal in arid regions. A numerical model is developed for the simulation of fault and fracture mechanics.The model is implemented in the framework of the Finite Element method and with the aid of Lagrange multipliers.The proposed formulation introduces a new contribution due to the frictional work on the portion of activated fault.The resulting algorithm is highly non-linear as the portion of activated fault is itself unknown.The numerical solution is validated against analytical results and proves to be stable also in realistic applications.

Steady planar ideal flow of anisotropic materials

This paper extends the ideal flow theory, which is well known for isotropic rigid perfectly plastic materials, to quite general orthotropic materials which comply with the principle of maximum plastic dissipation. The new theory is restricted to steady planar flow. The original ideal flow theory is widely used as the basis for inverse methods for the preliminary design of metal forming processes driven by minimum plastic work. The new theory extends this area of application to orthotropic materials. Moreover, another design criterion based on the Cockroft–Latham ductile fracture criterion is incorporated in the theory. To this end, the extended Bernoulli’s theorem relating pressure and velocity along any streamline during the steady planar flow of rigid perfectly plastic solids when the streamline is coincident everywhere with a principal stress trajectory is used. In particular, this theorem and the concept of ideal flow combine to evaluate the integral involved in the ductile fracture criterion. The final result is a simple relation between process parameters and the constitutive parameter involved in the ductile fracture criterion. The simplicity of this relation makes it suitable for quick design of metal forming processes.

Flow rules for rigid-plastic strain kinematic hardening solids

Abstract The associated flow rule is commonly supposed to be one of the cornerstones of classical plasticity theory for metals though some experimental results do not accord with it. This paper investigates the flow rule of plastic deformation rate for von Mises materials exhibiting kinematic hardening. It is found that the associated flow rule is not valid in rigid-plastic deformations. The associated flow rule for von Mises materials exhibiting kinematic hardening is modified. The modified associated flow rule implies that the vector of plastic deformation rate need not be perpendicular to the yield surface in nine-dimensional stress space. Finally, the principle of maximum plastic dissipation is modified as well.

Eulerian Framework for Inelasticity Based on the Jaumann Rate and a Hyperelastic Constitutive Relation—Part II: Finite Strain Elastoplasticity

An Eulerian rate formulation of finite strain elastoplasticity is developed based on a fully integrable rate form of hyperelasticity proposed in Part I of this work. A flow rule is proposed in the Eulerian framework, based on the principle of maximum plastic dissipation in six-dimensional stress space for the case of J2 isotropic plasticity. The proposed flow rule bypasses the need for additional evolution laws and/or simplifying assumptions for the skew-symmetric part of the plastic velocity gradient, known as the material plastic spin. Kinematic hardening is modeled with an evolution equation for the backstress tensor considering Prager’s yielding-stationarity criterion. Nonlinear evolution equations for the backstress and flow stress are proposed for an extension of the model to mixed nonlinear hardening. Furthermore, exact deviatoric/volumetric decoupled forms for kinematic and kinetic variables are obtained. The proposed model is implemented with the Zaremba–Jaumann rate and is used to solve the problem of rectilinear shear for a perfectly plastic and for a linear kinematic hardening material. Neither solution produces oscillatory stress or backstress components. The model is then used to predict the nonlinear hardening behavior of SUS 304 stainless steel under fixed-end finite torsion. Results obtained are in good agreement with reported experimental data. The Swift effect under finite torsion is well predicted by the proposed model.

A variational principle of elastoplasticity and its application to the modeling of frictional materials

Starting from a thermomechanical description of elastoplasticity, a stress-based variational principle is derived. The principle, which generalizes von Mises’s principle of maximum plastic dissipation, reproduces the conventional elastic/hardening-plastic framework applicable to metals as a special case and further proves to be suitable for developing constitutive models for frictional materials. Application of the principle to the isotropic and triaxal compression behaviour of sands is considered by means of a non-conventional extension of the modified Cam clay model. The new model allows for the specification of arbitrary stress-dilatancy relations without altering the yield potential or introducing a separate flow potential. Moreover, the elastoplastic tangent modulus is always symmetric, regardless of the degree of apparent nonassociativity.

On stress resultant plasticity and viscoplasticity for metal plates

In this work we derive elastoplastic and elastoviscoplastic finite element formulations for stress resultant bending analysis of thin metal plates. The principle of maximum plastic dissipation is used to obtain the ingredients of the small strain stress resultant plate elastoplasticy with state variables describing general isotropic and linear kinematic hardening. The ingredients of the plate stress resultant elastoviscoplasticy are further obtained by using the penalty-like form of the principle of maximum plastic dissipation. Such an approach enables single framework for numerical implementation of both considered inelastic stress resultant plate material models. The implementation is based on the spectral decomposition algorithm. For spatial discretization we use simple and robust quadrilateral finite element. A set of numerical examples is presented to illustrate the approach and to discuss the accuracy of the stress resultant inelastic plate formulations.

Normality Structures With Thermodynamic Equilibrium Points

AbstractEnriched by the nonlinear Onsager reciprocal relations and thermodynamic equilibrium points (Onsager, Phys. Rev., 37, pp. 405–406; 38, pp. 2265–2279), an extended normality structure by Rice (1971, J. Mech. Phys. Solids, 19, pp. 433–455) is established in this paper as a unified nonlinear thermodynamic theory of solids. It is revealed that the normality structure stems from the microscale irrotational thermodynamic fluxes. Within the extended normality structure, this paper focuses on the microscale thermodynamic mechanisms and significance of the convexity of flow potentials and yield surfaces. It is shown that the flow potential is convex if the conjugate force increment cannot not oppose the increment of the rates of local internal variables. For the Rice fluxes, the convexity condition reduces to the local rates being monotonic increasing functions with respect to their conjugate forces. The convexity of the flow potential provides the thermodynamic system a capability against the disturbance of the thermodynamic equilibrium point. It is proposed for time-independent behavior that the set of plastically admissible stresses determined by yield conditions corresponds to the set of thermodynamic equilibrium points. Based on that viewpoint, the intrinsic dissipation inequality is just the thermodynamic counterpart of the principle of maximum plastic dissipation and requires the convexity of the yield surfaces.

Cam-Clay plasticity. Part V: A mathematical framework for three-phase deformation and strain localization analyses of partially saturated porous media

We present a mathematical framework for deformation and strain localization analyses of partially saturated granular media using three-phase continuum mixture theory. First, we develop conservation laws governing a three-phase mixture to identify energy-conjugate expressions for constitutive modeling. Energy conjugate expressions identified relate a certain measure of effective stress to the deformation of the solid matrix, the degree of saturation to the matrix suction, the pressure in each phase to the corresponding intrinsic volume change of this phase, and the seepage forces to the corresponding pressure gradients. From the second of law of thermodynamics we obtain the dissipation inequality; from the principle of maximum plastic dissipation we derive a condition for the convexity of the yield function. Then, we formulate expressions describing conditions for the onset of tabular deformation bands under locally drained and locally undrained conditions. Finally, we cast a specific constitutive model for partially saturated soils within the proposed mathematical framework, and implement it in the context of return mapping algorithm of computational plasticity. The proposed constitutive model degenerates to the classical modified Cam-Clay model of soil mechanics in the limit of full saturation. Numerical examples are presented to demonstrate the performance of the return mapping algorithm as well as illustrate the localization properties of the model as functions of imposed deformation and matrix suction histories.

Numerical concepts for rate-independent single crystal plasticity

We provide a new time integration algorithm for rate-independent single crystal plasticity. The algorithm is based on an augmented Lagrangian formulation of the principle of maximum plastic dissipation. It is a synthesis of Lagrangian multiplier and penalty formulations avoiding their drawbacks (active set search, pseudo-inverse and approximate fulfillment of the yield condition respectively). The algorithm is physically motivated and especially provides a clear concept how to determine the set of active slip systems. The method is stable and efficient, several representative numerical examples are shown.

Viscoplasticity model at finite deformations with combined isotropic and kinematic hardening

In this work, we construct a phenomenological constitutive model of viscoplasticity at finite strains, which generalizes the classical Perzyna or Duvaut–Lions models to finite strains. The latter is accomplished with a minimum number of hypothesis, including the multiplicative decomposition of deformation gradient, a definition of the elastic domain and finally a penalty-like, viscoplastic regularization of the principle of maximum plastic dissipation. The model is extended to include the isotropic and kinematic hardening of Prager–Ziegler type. Numerical computation of the viscoplastic flow at finite strain is also discussed, along with the corresponding simplification resulting from a convenient choice of the logarithmic strain measure. Several illustrative numerical examples are presented in order to demonstrate the ability of the proposed model to remove the deficiencies of some currently used models.

On the Calculation of Plastic Spin in a Model of Crystal Plasticity Based on Symmetric Tensors

AbstractIn a model of anisotropic plasticity based on symmetric tensors the plastic spin is not determined by the principle of maximum plastic dissipation. The spin can be derived by introducing additional kinematic relations and applying a generalized inverse based on a minimum principle. A procedure specialized for the case of crystal plasticity is shown.

Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditions

In this paper we present the formulation and numerical analysis of a finite deformation model of coupled poro-plasticity. Fully saturated conditions are assumed. The model is based on a multiplicative decomposition of the deformation gradient in elastic and plastic parts, together with an additive elastoplastic decomposition of the fluid content. The final relations describing the evolution of the different fields are then obtained in the thermodynamic framework furnished by the principle of maximum plastic dissipation. The particular case of an associated Drucker-Prager model in effective stresses is considered as a model example. A complete characterization of the dissipative structure of the problem of evolution is described through the derivation of an a priori stability estimate. We describe in detail the finite element implementation of the model in the context of staggered algorithms for the solution of the physically coupled problem. Finally, we present representative numerical simulations to illustrate the features of the proposed models. In particular, the case of strain localization in globally undrained conditions is studied in the proposed framework.