Implicit Standard Materials Research Articles

A bipotential-based macroscopic fatigue criterion of porous materials with a pressure-sensitive and non-associated plastic solid matrix and comparison with numerical simulation

The formulation of macroscopic strength criteria under cyclic loading of ductile porous media containing a pressure sensitive and non-associated plastic matrix is still a pending issue. In this paper, a new approach is developed by using the bipotential theory. In this framework, non Generalized Standard Materials (GSM) are transformed into a class of Implicit Standard Materials (ISM), allowing the recovery of the flow rule normality in a weak form of an implicit relation. The classical shakedown theorems are extended to the homogenization of ISM. The solution of a thick wall tube under uniform pressure is established with the assumption of vanishing plastic strain increment over a single stabilized cycle. A macroscopic fatigue criterion is further delivered for the first time for porous materials with a non-associated Drucker–Prager type solid matrix. It is now possible to separate the effects of frictional and dilatancy angles. It is found that the shakedown limit load under hydrostatic loading is only related to the friction angle, but not the dilatancy one. The safety domain defined by the established criterion under general cyclic loads is comparatively reduced with the decrease of dilatancy angle. The new criterion is fully assessed by comparing the theoretical predictions to FEM-based step-by-step simulations for different values of porosity and frictional and dilatancy angles. Finally, the new criterion can also be applied to associated solid matrix as a special case and it significantly improves the accuracy of predictions provided by previous works based on the Melan’s static theorem.

A micro-mechanical constitutive model for heterogeneous rocks with non-associated plastic matrix as implicit standard materials

In this work, we shall propose a new micro-mechanical constitutive model for the estimation of effective elastic-plastic behaviors of heterogeneous rocks. A bi-potential based incremental variational (BIV) approach is developed in order to take into account non-uniform local strain fields of constituents. The studied materials are composed of a non-associated and pressure sensitive plastic matrix, elastic inclusions and/or voids. For clarity, the local behavior of matrix is first described by an elastic perfectly-plastic model. Based on the bi-potential theory to dealing with non-associated plastic flow, the solid matrix is considered as pertaining to implicit standard materials (ISMs). The effective incremental bi-potential and macroscopic stress tensor are then estimated through an extension of the incremental variational method initially established for generalized standard materials(GSMs). The accuracy of the BIV model is verified by comparing the model’s predictions with the reference results obtained from direct finite element simulations. Furthermore, by assuming that the general formulation obtained for the perfectly plastic matrix remains valid for each loading increment, the BIV model is extended to considering that the solid matrix exhibits an isotropic hardening by using an explicit algorithm. The accuracy of the extended BIV model is also validated by a series of comparisons with the reference solutions obtained by direct finite element simulations for both inclusion-reinforced composites and porous materials. Both local and macroscopic responses are compared. As an example of application, the extended BIV model is finally applied to estimating the mechanical responses of typical claystone and sandstone under different loading paths.

A computational strategy for the modeling of elasto-plastic materials under impact loadings

This paper presents a numerical strategy for the modeling of elasto-plastic materials under impact loadings. The bi-potential method is efficient in dealing with implicit standard materials constitutive laws, such as frictional contact. The Uzawa algorithm, based on the bi-potential method is applied to deal with the non-linearity of boundary conditions. For the non-linearity associated with materials, a Return Mapping Algorithm is applied to deal with the elasto-plastic constitutive laws. On the basis of the Updated Lagrangian formulation, we adopt a Rotationally Neutralized Objective hypothesis to describe the geometrically non-linear behavior. Equations of motion are integrated with a first order scheme. Three numerical examples are performed to verify the accuracy and to show the applicability of the proposed approach.

The Dirichlet–Neumann problem revisited after modelling a new class of non-smooth phenomena

Generalized Standard Materials are governed by maximal cyclically monotone operators and modelled by convex potentials. Gery de Saxce’s Implicit Standard Materials are modelled by biconvex bipotentials. Analyzing the intermediate class of n-monotone materials governed by maximal n-monotone operators and modelled by Fitzpatrick’s functions, we find out that n-monotonicity is a relevant criterion for the materials characterisation and classification. Additionally, the Fitzpatrick’s functions allow to describe the thermal or mechanical equilibrium equations of n-monotone materials by primal–dual two-fields variational principles. In doing so, we are led to Dirichlet–Neumann problems that we solve by Uzawa-type algorithms.

Bipotential versus return mapping algorithms: Implementation of non associated flow rules

Numerical implementation of constitutive laws involves specific incremental methods. The “return mapping” (Simo and Hughes, 1998) and the “bipotential” (de Saxcé, 1992) are one of those, associated respectively to two different classes of materials: the General Standard Materials (GSM) for the return mapping and the Implicit Standard Materials (ISM) for the bipotential.The objective of this paper is then to compare the implementation of those both methods in the case of non associated flow rules in plasticity.In the first section, the properties of the different previous material classes will be recalled and the methods of “return mapping” and “bipotential” will be detailed. The comparison of both methods is realised on the non linear kinematic hardening rule of Armstrong–Frederick (Armstrong and Frederick, 1966) in a second section and the details are given in a third part. The numerical implementation is realised in Abaqus/Standard 6.11 by the means of a UMat subroutine and the practical simple case of tension–compression is analysed in a last section.

Modelling of implicit standard materials. Application to linear coaxial non-associated constitutive laws

We analyze the relation between Géry de Saxcé's bipotentials representing non-associated constitutive laws and Fitzpatrick's functions representing maximal monotone multifunctions. Revisiting the model of elastic materials initiated by Robert Hooke, we illustrate that Fitzpatrick's representation of monotone operators coming from convex analysis provides a constructive method to discover the best bipotential for modelling an Implicit Standard Material.

A Class of Non-associated Materials: n-Monotone Materials—Hooke’s Law of Elasticity Revisited

Generalized Standard Materials are governed by maximal cyclically monotone operators and modeled by convex potentials. Gery de Saxce’s Implicit Standard Materials are modeled by biconvex bipotentials. We analyze the intermediate class of n-monotone materials governed by maximal n-monotone operators and modeled by Fitzpatrick’s functions. Revisiting the model of elastic materials initiated by Robert Hooke, and insisting on the linearity, coaxiality and monotonicity properties of the constitutive law, we illustrate that Fitzpatrick’s representation of n-monotone operators coming from convex analysis provides a constructive method to discover the best bipotential modeling a n-monotone material. Giving up the symmetry of the linear constitutive laws, we find out that n-monotonicity is a relevant criterion for the materials characterization and classification.

A variational formulation for constitutive laws described by bipotentials

We propose a rigorous variational formulation and an algorithm for solving the discretisation in time of the evolution problem for an implicit standard material, in terms of bipotentials.

Plastic limit analysis with non linear kinematic strain hardening for metalworking processes applications

Limit analysis approaches are widely used to deal with metalworking processes analysis; however, they are applied only for perfectly plastic materials and recently for isotropic hardening ones excluding any kind of kinematic hardening. In the present work, using Implicit Standard Materials concept, sequential limit analysis approach and the finite element method, our objective consists in extending the limit analysis application for including linear and non linear kinematic strain hardenings. Because this plastic flow rule is non associative, the Implicit Standard Materials concept is adopted as a framework of non standard plasticity modeling. The sequential limit analysis procedure which considers the plastic behavior with non linear kinematic strain hardening as a succession of perfectly plastic behavior with yielding surfaces updated after each sequence of limit analysis and geometry updating is applied. Standard kinematic finite element method together with a regularization approach is used for performing two large compression cases (cold forging) in plane strain and axisymmetric conditions.

Mathematical and numerical modeling of the non-associated plasticity of soils—Part 2: Finite element analysis

The method of the implicit standard material has allowed the formulation of a consistent mathematical model of the boundary value problem for the non-associated plasticity of soil. The mean accomplished steps are the achievement of the bipotential function, the recovering of the stress–strain relationship under a normality rule, introduction of the bifunctional and the proof of the solution existence. Here the mathematical model is discretized by the finite element method. First, the stress update scheme was formulated, the tangent matrix is explicitly derived and then the non-linear system is solved by the Newton–Raphson method where a new algorithm using a symmetrical tangent matrix is improved. This is in opposition to conventional non-associated plasticity, which uses a non-symmetric tangent matrix. Through the numerical examples we show the feasibility and the efficiency of the algorithm. It is also seen that we perform some studies of the numerical solutions, particularly the comparison between associated and non-associated limit load.

Mathematical and numerical modeling of the non-associated plasticity of soils—Part 1: The boundary value problem

The soil is characterized by the influence of the hydrostatic stress, which leads to a yield surface with a shape of a pyramid for Mohr–Coulomb criteria and a shape of a cone for Drucker–Prager one. These materials are also characterized by a non-associated plasticity where the plastic yielding rule does not follow the normality rule. The usual mechanical models use two independent functions to describe this particular collapse. Unfortunately, this manner broke the model formulation. The purpose of this work is to present a consistent formulation of the non-associated plasticity of soil. The frame of the mathematical analysis is the concept of the implicit standard material. The cornerstone of this new idea is the construction of a single function called the bipotential playing in the same time the roles of the yield surface and the plastic potential. The bipotential concept is then intended to involve the constitutive law, cover the normality rule even for the non-associated soil and the proof of the solution existence. The formulation was initially performed for the case of a regular point out of the cone apex and in present, it is extended to the irregular point located at the apex. The paper presents firstly the implicit standard material method. Then, the methodology to build a full model for the boundary value problem is detailed. Particular expressions and relations are sufficiently explained and discussed. Attention is made to the evolution problem and the variational principles related to the elastic–plastic behavior.

Shakedown analysis: Comparison between models with the linear unlimited, linear limited and non-linear kinematic hardening

In this paper, shakedown analysis for plasticity models with hardening is considered by using the framework of implicit standard materials. It is shown that the concept of bipotential allows not only to retrieve some classical results in associated plasticity but also to consider non-associated laws e.g. the non-linear kinematic hardening rule. Furthermore, the comparison of the shakedown load for the three kinds of hardening on an example shows substantial differences and thus the importance of considering the non-linear kinematic hardening in shakedown analysis.

On Hill's Bipotential

AbstractA new formulation of the constitutive law of materials based on the concept of bipotential was introduced by de Saxcé. This concept is able to describe non‐associated laws and to suggest robust numerical algorithms. Materials that admit a bipotential to model their behaviour constitute the class of Implicit Standard Materials (ISM). This class contains, as a sub‐class, the Generalized Standard Materials (GSM). In this paper, we present a bipotential expressing the coaxiality of two tensors with preservation of the order of the eigenvalues. We call it Hill's bipotential. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A comparison between analytical calculations of the shakedown load by the bipotential approach and step-by-step computations for elastoplastic materials with nonlinear kinematic hardening

The class of generalized standard materials is not relevant to model the nonassociative constitutive equations. The bipotential approach, based on a possible generalization of Fenchel’s inequality, allows the recovery of the flow rule normality in a weak form of an implicit relation. This defines the class of implicit standard materials. For such behaviours, this leads to a weak extension of the classical bound theorems of the shakedown analysis. In the present paper, we recall the relevant features of this theory. Considering an elastoplastic material with nonlinear kinematic hardening rule, we apply it to the problem of a sample in plane strain conditions under constant traction and alternating torsion in order to determine analytically the interaction curve bounding the shakedown domain. The aim of the paper is to prove the exactness of the solution for this example by comparing it to step-by-step computations of the elastoplastic response of the body under repeated cyclic loads of increasing level. A reliable criterion to stop the computations is proposed. The analytical and numerical solutions are compared and found to be closed one of each other. Moreover, the method allows uncovering an additional ‘2 cycle shakedown curve’ that could be useful for the shakedown design of structure.

A kinematic elastic nonshakedown theorem for implicit standard materials

Criteria for a priori recognition of the type of steady-state response induced by cyclic loads and prediction whether a structure will shakedown elastically or not, without the necessity of performing a step-by-step full analysis, have considerable importance. Melan and Koiter theorems provide criteria that guarantee whether elastic shakedown occurs or not under cyclic loads in case of perfect plasticity. However, there remain some aspects of the shakedown theory which deserve further study. One of these, concerned with more realistic nonassociative elastic–plastic constitutive material models, allowing for nonlinear kinematic and isotropic hardening suitable to describe the cyclic plastic behaviour of metallic materials, has strong motivation. Koiter's elastic nonshakedown theorem is reconsidered here, with the objective of extending it to the de Saxce's implicit standard material class, which contains a wide class of nonassociative elastic–plastic material behaviours. Shakedown analysis is formulated by a kinematic approach based on the plastic accumulation mechanism concept due to Polizzotto. A sufficient condition for elastic nonshakedown and a distinct necessary condition are established. Then, an upper bound to the shakedown multiplier is evaluated.

A new approach to shakedown analysis for non-standard elastoplastic material by the bipotential

On the basis of the implicit standard materials that introduces a function, called bipotential, depending on both the stress and plastic strain rate, this paper is devoted to present a new approach of shakedown analysis for non standard elastoplastic materials. The bipotential theory was successfully applied to geomaterials with non-associated flow rule and Coulomb's dry friction law. The present analysis is different to the Melan's potential and it is based on a corner stone inequality satisfied by the bipotential and the existence of time-independent residual stress field. The deduction of bound theorems, static and kinematic, is detailed in the present article.

The implicit standard material theory for modelling the nonassociative behaviour of metals

Modelling the elastoplastic or elastoviscoplastic behaviour of metallic materials exhibiting strain hardening and damage leads to complex nonassociative constitutive equations, sources of many theoretical and numerical troubles. The usual modelling of a nonassociative constitutive equation leads to the loss of the interesting and very useful properties of generalised standard materials deriving from the key concepts of convexity and normality. The argument that will be developed is that the bipotential concept is an appropriate answer. In the first part, after introducing the state variables generally used to describe the behaviour of metallic materials, the constitutive equations subjected to the principles of thermodynamics are derived from two potentials. The state potential gives the state laws, and the bipotential of dissipation delivers the evolution laws for state variables, through the implicit normality assumption. The second part is devoted to several particular applications to metal elastoplasticity and elastoviscoplasticity models.

Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach

The class of generalised standard materials is not relevant to model the non-associative constitutive equations. The possible generalisation of Fenchel's inequality proposed by de Saxcé allows the recovery of flow rule normality for non-associative behaviours. The normality rule is written in the weak form of an implicit relation. This leads to the introduction of the class of implicit standard materials. This formulation is applied to constitutive equations involving non-linear kinematic hardening, indispensable to describe accurately and realistically the cyclic plasticity of metallic materials. For these plastic flow rules shakedown bound theorems can be extended; an analytical example of the shakedown of a thin-walled tube under constant traction and alternate cyclic torsion is considered and the obtained solution is proved to be exact.

Matériaux viscoplastiques et lois de normalité implicites

Some materials interesting for engineering applications do not come with scope of generalized standard material (GSM) class for which efficient and robust algorithms are available. Typical examples are constitutive laws of geomaterials. Nevertheless, they are not the only ones. For instance, laws describing the behaviour of metals under cyclic loading which are accurately approached by non standard (visco-)plastic laws. The good properties of GSM result from the generalized normality rule and the convexity of the associated pseudo-potential. The concept of bi-potential, introduced by de Saxcé, extends the notion of pseudo-potential and enables to write the evolution laws as implicit normality rules. These materials belong to the implicit standard material (ISM) class, of which the GSM class is a subset. The aim of this paper is to show that the viscoplastic model with non-linear kinematic hardening proposed by Chaboche et al. belongs to this class of materials.

Sur l'endommagement et les matériaux standard implicites

The class of implicit standard materials, based on the introduction of a bipotential, has been proposed by de Saxcé. The kinetic isotropic damage law of Lemaitre is presented here as a new example of application.