Mean field modelling of the plastic behaviour of co-continuous dual-phase alloys with strong morphological anisotropy

This work addresses the plastic flow properties of a composite material in which the reinforcing phase is continuous and cannot be suitably represented by isolated ellipsoidal inclusions. The dual-phase metal under consideration is composed of a network of Inconel-601 fibres infiltrated by pure aluminium. Hence, both phases exhibit elastic–plastic behaviour and are continuous in the three dimensions of space. The fibre network presents a large morphological anisotropy that is reflected in the mechanical response of the composite. The modelling is based on Eshelby’s equivalent inclusion theory. Strain partitioning between the phases is computed incrementally based on tangent operators derived from the isotropic response of individual phases. Assessment of the model relies on extensive experimental data. Uniaxial tensile tests, involving measurement of the Lankford coefficient, have been performed at various temperatures on samples containing different volume fractions of fibres. Measurement of the phase stresses by neutron diffraction supplements the information provided by the macroscopic stress–strain curves. It is demonstrated that predictions are valid only when the micro–macro averaging scheme accounts for the co-continuous character of the constitutive phases.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

Effective pressure-sensitive elastoplastic behavior of particle-reinforced composites and porous media under isotropic loading

The composite under investigation consists of an elastoplastic matrix reinforced by elastic particles or weakened by pores. The material forming the matrix is pressure-sensitive. The Drucker–Prager yield criterion and a one-parameter non-associated flow rule are employed to formulate the yield behavior of the matrix. The objective of this work is to estimate the effective elastoplastic behavior of the composite under isotropic tensile and compressive loadings. To achieve this objective, the composite sphere assemblage model of Hashin [Z. Hashin, The elastic moduli of heterogeneous materials, ASME J. Appl. Mech. 29 (1962) 143–150] is used. Exact solutions are thus derived as estimations for the effective secant and tangent bulk moduli of the composite. The effects of the loading modes and phase properties on the effective elastoplastic behavior of the composite are analytically and numerically evaluated.     

A micromechanics-based elastoplastic damage model for granular materials at low confining pressure

This paper is devoted to the formulation of a micromechanics-based constitutive model for granular materials under relatively low confining pressure. The constitutive formulation is performed within the general framework of homogenization for granular materials. However, new rigorous stress localization laws are proposed. Some local constitutive relations are established under the consideration of irreversible thermodynamics. Macroscopic plastic deformation is obtained by considering local plastic sliding in a limit number of families of contact planes. The plastic sliding at each contact plane is described by a non-associated plastic flow rule, taking into account pressure sensitivity and normal dilatancy. Nonlinear elastic deformation related to progressive compaction of contacts is also taken into account. Material softening is described by involving damage process related to degradation of microstructure fabric. The proposed model is applied to some typical granular materials (sands). The numerical predictions are compared with experimental data.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

On the existence of indeterminate solutions to the equations of motion under non-associated flow

Under certain conditions, an indeterminate solution exists to the equations of motion for dynamic elastic–plastic deformation of materials using constitutive laws based on non-associated flow that suggests that an initially unbounded dynamic perturbation in the stress can develop from a quiescent state on the yield surface. The existence of this indeterminate solution has been alleged to discourage use of non-associated flow rules for both dynamic and quasi-static analysis theoretically. It is shown in this paper that the indeterminate solution that may solve the equations of motion is intrinsically dynamic, and it determinately goes to zero in the quasi-static limit regardless of other indeterminate parameters. Consequently, the existence of this unstable dynamic solution has no impact on stability and use of non-associated flow rules for analysis of the quasi-static problem. More importantly, for dynamic applications, it is also shown that the indeterminate solution solves the equations of motion only if critical restrictions are applied to the constitutive equations such that the effective modulus during loading is constant and the direction of the perturbation is unidirectional over a finite time interval. It is shown that common components of the constitutive laws used in metal forming and deformation analysis are inconsistent with these restrictions. So, these common models can be generalized to include non-associated flow for analysis of the dynamic problem without concern that the solution will become indeterminate.     

Review of Drucker’s postulate and the issue of plastic stability in metal forming

Drucker’s postulate defines a class of stable work hardening materials that are classified as non-energetic and is equivalent to the associated flow rule (AFR). The postulate has been shown to be a sufficient condition for plastic stability. However, experiments indicate that plastic deformation of aluminum and steel alloys does not adhere to the constraints of the AFR. Therefore, the requirement for accuracy suggests that the metal forming industry should also consider material models that are based on non-associated flow. But Drucker’s work raises the issue of stability when considering the use of non-associated flow in material models. While this concern is merited and many types of instability arises from certain types of non-associated flow, this has led to a widely accepted view that Drucker’s postulate is a necessary condition for stability. This perception is inhibiting the acceptance or consideration of more accurate material models that are suggested from the experimental observations about violations of the AFR. This paper proposes a specific class of material models based on non-associated flow and derives the constraints on this class of models to ensure stability. The existence of this class of non-AFR models proves that Drucker’s postulate is a sufficient but not necessary condition for stability. Furthermore, the class of models described in this paper is quite general and provides a framework for consideration of potentially more accurate material models while guaranteeing the same level of stability as typically associated with materials that satisfy Drucker’s postulate.     

Trapped modes in bent elastic rods

We investigate the existence of trapped modes in elastic rods of constant circular cross-section that possess bends of arbitrary curvature and straighten out at infinity; such trapped modes consist of finite energy localized in regions of maximal curvature. An asymptotic model assuming smallness of dimensionless curvature is developed to describe the trapping. Existence conditions depending on Poisson’s ratio are offered, and the equations from which they derive are numerically validated. A physical explanation of why trapped modes should be expected is also given.     

Evaluation of advanced anisotropic models with mixed hardening for general associated and non-associated flow metal plasticity

The main objective of this paper is to develop a generalized finite element formulation of stress integration method for non-quadratic yield functions and potentials with mixed nonlinear hardening under non-associated flow rule. Different approaches to analyze the anisotropic behavior of sheet materials were compared in this paper. The first model was based on a non-associated formulation with both quadratic yield and potential functions in the form of Hill’s (1948). The anisotropy coefficients in the yield and potential functions were determined from the yield stresses and r-values in different orientations, respectively. The second model was an associated non-quadratic model (Yld2000-2d) proposed by Barlat et al. (2003). The anisotropy in this model was introduced by using two linear transformations on the stress tensor. The third model was a non-quadratic non-associated model in which the yield function was defined based on Yld91 proposed by Barlat et al. (1991) and the potential function was defined based on Yld89 proposed by Barlat and Lian (1989). Anisotropy coefficients of Yld91 and Yld89 functions were determined by yield stresses and r-values, respectively. The formulations for the three models were derived for the mixed isotropic-nonlinear kinematic hardening framework that is more suitable for cyclic loadings (though it can easily be derived for pure isotropic hardening). After developing a general non-associated mixed hardening numerical stress integration algorithm based on backward-Euler method, all models were implemented in the commercial finite element code ABAQUS as user-defined material subroutines. Different sheet metal forming simulations were performed with these anisotropic models: cup drawing processes and springback of channel draw processes with different drawbead penetrations. The earing profiles and the springback results obtained from simulations with the three different models were compared with experimental results, while the computational costs were compared. Also, in-plane cyclic tension–compression tests for the extraction of the mixed hardening parameters used in the springback simulations were performed for two sheet materials.     

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.     

A cyclic plasticity model for single crystals

The Armstrong–Frederick type kinematic hardening rule was invoked to capture the Bauschinger effect of the cyclic plastic deformation of a single crystal. The yield criterion and flow rule were built on individual slip systems. Material memory was introduced to describe strain range dependent cyclic hardening. The experimental results of copper single crystals were used to evaluate the cyclic plasticity model. It was found that the model was able to accurately describe the cyclic plastic deformation and properly reflect the dislocation substructure evolution. The well-known three distinctive regimes in the cyclic stress–strain curve of the copper single crystals oriented for single slip can be reproduced by using the model. The model can predict the enhanced hardening for crystals oriented for multislip, showing the model's ability to describe anisotropic cyclic plasticity. For a given loading history, the model was able to capture not only the saturated stress–strain response but also the detailed transient stress–strain evolution. The model was used to predict the cyclic plasticity under a high–low loading sequence. Both the stress–strain responses and the microstructural evolution can be appropriately described through the slip system activation.     

Very Large Poisson’s Ratio with a Bounded Transverse Strain in Anisotropic Elastic Materials

In a recent paper by Ting and Chen [18] it was shown by examples that Poisson’s ratio can have no bounds for all anisotropic elastic materials. With the exception of cubic materials, the examples presented involve a very large transverse strain. We show here that a very large Poisson’s ratio with a bounded transverse strain exists for all anisotropic elastic materials. The large Poisson’s ratio with a bounded transverse strain occurs when the axial strain is in the direction very near or at the direction along which Young’s modulus is very large. In fact the transverse strain has to be very small for the material to be stable. If the non-dimensionalized Young’s modulus is of the order δ⁻¹, where δ is very small, the axial strain, the transverse strain and Poisson’s ratio are of the order δ, δ^1/2 and δ^−1/2, respectively. Mathematics Subject Classifications (2000) 74B05, 74E10.T.C.T. Ting: Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

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.     

A bipotential-based limit analysis and homogenization of ductile porous materials with non-associated Drucker–Prager matrix

In Gurson's footsteps, different authors have proposed macroscopic plastic models for porous solid with pressure-sensitive dilatant matrix obeying the normality law (associated materials). The main objective of the present paper is to extend this class of models to porous materials in the context of non-associated plasticity. This is the case of Drucker–Prager matrix for which the dilatancy angle is different from the friction one, and classical limit analysis theory cannot be applied. For such materials, the second last author has proposed a relevant modeling approach based on the concept of bipotential, a function of both dual variables, the plastic strain rate and stress tensors. On this ground, after recalling the basic elements of the Drucker–Prager model, we present the corresponding variational principles and the extended limit analysis theorems. Then, we formulate a new variational approach for the homogenization of porous materials with a non-associated matrix. This is implemented by considering the hollow sphere model with a non-associated Drucker–Prager matrix. The proposed procedure delivers a closed-form expression of the macroscopic bifunctional from which the criterion and a non-associated flow rule are readily obtained for the porous material. It is shown that these general results recover several available models as particular cases. Finally, the established results are assessed and validated by comparing their predictions to those obtained from finite element computations carried out on a cell representing the considered class of materials.     

A finite strain theory for elastic-plastic deformation

An elastic-plastic theory that is applicable when the elastic part of the strain is finite is proposed. A flow rule for an incompressible solid is obtained from Drucker's postulate [1]. Isothermal simple shear of a material which is neo-Hookean both before yielding and during elastic unloading after yielding is considered as an application of the theory. The problem is solved for two yield conditions and associated flow rules.     

Consequences of the dissipative restrictions in finite anisotropic elasto-plasticity

The paper is devoted to the dissipation postulate in anisotropic finite elastoplasticity, properly formulated in terms of the total strain histories, on small cycles only. The equivalence between the dissipation postulate and the existence of the stress potential together with the dissipation inequality is proved. The modified flow rules compatible with the dissipation postulate follow as a necessary condition. The convexity and normality properties can be treated as an equivalent issue of the dissipation postulate only within the framework of Σ models. We identify such classes of Σ-models based on the pre-image theorem. The difficulties arise from the non-injectivity of the Mandel's stress measure, as dependent on the elastic strain. We define the yield stress function and the admissible elastic stress range in Σ-space. The equivalence is achieved only if it possible to construct the elastic range in strain space, having just the topological properties originally assumed as a basis of the dissipation postulate. The normality to the admissible elastic stress range does not mean an associative flow rule. The results are exemplified for transversely isotropic elasto–plastic materials as well as for models with small elastic strains.     

Multi-scale defect interactions in high-rate brittle material failure. Part I: Model formulation and application to ALON

Within this two part series we develop a new material model for ceramic protection materials to provide an interface between microstructural parameters and bulk continuum behavior to provide guidance for materials design activities. Part I of this series focuses on the model formulation that captures the strength variability and strain rate sensitivity of brittle materials and presents a statistical approach to assigning the local flaw distribution within a specimen. The material model incorporates a Mie–Grüneisen equation of state, micromechanics based damage growth, granular flow and dilatation of the highly damaged material, and pore compaction for the porosity introduced by granular flow. To provide initial qualitative validation and illustrate the usefulness of the model, we use the model to investigate Edge on Impact experiments (Strassburger, 2004) on Aluminum Oxynitride (AlON), and discuss the interactions of multiple mechanisms during such an impact event. Part II of this series is focused on additional qualitative validation and using the model to suggest material design directions for boron carbide.     

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.     

Inversion of Ramberg–Osgood equation and description of hysteresis loops

The Ramberg–Osgood equation has been approximately inverted. Four orders of approximations providing progressively more accurate inversions are considered. The second order inversion is used to develop closed-form relations for stress in terms of strain and for cyclic stress in terms of cyclic strain. Explicit relations between the cyclic-strength coefficient and the cyclic strain-hardening exponent and between the cyclic strain-hardening exponent and the elastic modulus are developed. Application to nine different engineering metals shows that the proposed approximate inversion provides a powerful tool for describing stress–strain relations. Examples of hysteresis loops for a number of strain time-histories are presented. Also, it is shown that augmenting the strain time-history with fictitious strain segments to close all the hysteresis loops yields the same stress–strain relation as that obtained through the rain-flow counting method.     

Exact solution of rotating FGM shaft problem in the elastoplastic state of stress

Plane strain analytical solutions to estimate purely elastic, partially plastic and fully plastic deformation behavior of rotating functionally graded (FGM) hollow shafts are presented. The modulus of elasticity of the shaft material is assumed to vary nonlinearly in the radial direction. Tresca’s yield criterion and its associated flow rule are used to formulate three different plastic regions for an ideal plastic material. By considering different material compositions as well as a wide range of bore radii, it is demonstrated in this article that both the elastic and the elastoplastic responses of a rotating FGM hollow shaft are affected significantly by the material nonhomogeneity.