Koffi Enakoutsa

A model for elastic flexoelectric materials including strain gradient effects

A constitutive model for elastic flexoelectric materials under small deformation based on second gradient continuum theory is developed, using a Toupin-like variational formulation to simultaneously obtain constitutive relations, balance equations and boundary conditions. The model includes three different electromechanical “stresses”: a higher-order stress, an extended local electric force and a generalized Cauchy stress tensor. The constitutive equations of the model are obtained by postulating an internal energy density function which depends on both the strain and its gradient as well as the polarization. Finally, as an application of the model, we derive the explicit analytical expressions of the polarization and displacement vector fields for the problem of the polarization induced over a thin spherical shell subjected to hydrostatic loading conditions.

Some new applications of the GLPD micromorphic model of ductile fracture

This paper presents some new applications of a model of ductile rupture of porous ductile materials proposed by Gologanu, Leblond, Perrin and Devaux (GLPD). The first application is concerned with the relationship between this model and the class of generalized standard materials. We show that the GLPD model fits in this class, provided that the porosity is not allowed to change and the hypothesis of linearized theory is adopted. The advantage of this property is that it automatically warrants the unicity of the solution for the ‘projection’ problem of the (supposedly) elastic stress predictor onto the GLPD’s yield locus. The second application leads to some exact analytical solutions of the GLPD model constitutive equations for the problems of an elastic hollow sphere in the framework of linearized theory, and viscous in large deformations. Comparisons between the numerical predictions of the GLPD model and the analytical solutions confirm the robustness of the numerical scheme used to implement this model into SYSTUS© finite element (FE) code. Thus, these exact analytical solutions can be used to validate the implementation of the GLPD model in another finite element code. In the third application, comparisons between experimental and numerical load vs. displacement curves for an axisymmetric pre-cracked specimen made of a typical stainless steel are found to yield satisfactory results.

An analytic benchmark solution to the problem of a generalized plane strain hollow cylinder made of micromorphic plastic porous metal and subjected to axisymmetric loading conditions

We provide a full analytical solution for the problem of a generalized plane strain circular hollow cylinder subjected to axisymmetric loading conditions. The matrix of the cylinder obeys a micromorphic plasticity theory as proposed by Gologanu, Leblond, Perrin and Devaux. The solution gives explicit expressions for the displacement, the strain and its gradient, as well as the ordinary and generalized stress fields. The newly derived solution satisfies the equilibrium equations and is shown to be an extension of the solution of the same model problem using (von Mises) classical plasticity theory.

Damage smoothing effects in a delocalized rate sensitivity model for metals

It has been long time established that application of damage delocalization method to softening constitutive models yields numerical results that are independent of the size of the finite element. However, the prediction of real-world large and small scale problems using the delocalization method remains in its infancy. One of the drawbacks encountered is that the predicted load versus displacement curve suddenly drops, as a result of excessive smoothing of the damage. The present paper studies this unwanted effect for a delocalized plasticity/damage model for metallic materials. We use some theoretical arguments to explain the failure of the delocalized model considered, following which a simple remedy is proposed to deal with it. Future works involve the numerical implementation of the new version of the delocalized model in order to assess its ability to reproduce real-world problems.

Combined polarization field gradient and strain field gradient effects in elastic flexoelectric materials

We consider a constitutive model for the behavior of elastic flexoelectric materials including strain gradient fields and polarization gradient fields. This model is based on a stored elastic energy density function which depends on four independent variables: the polarization field and the polarization field gradient as well as the strain field and the strain field gradient. A generalized Toupin variational approach is utilized to find the governing equations (constitutive relations, equilibrium equations and boundary conditions) of the material. The present model is then applied to the problem of a thick walled cylindrical tube of elastic isotropic flexoelectric material, subjected to axisymmetric loading. The resulting radial displacement field noticeably differs from the elastic and strain gradient elastic cases.

A generalized constitutive elasticity law for GLPD micromorphic materials, with application to the problem of a spherical shell subjected to axisymmetric loading conditions

In this work we propose to replace the GLPD hypo-elasticity law by a more rigorous generalized Hookes law based on classical material symmetry characterization assumptions. This law introduces in addition to the two well-known Lames moduli, five constitutive constants. An analytical solution is derived for the problem of a spherical shell subjected to axisymmetric loading conditions to illustrate the potential of the proposed generalized Hookes law.

Modeling the Dynamic Failure of Railroad Tank Cars Using a Physically Motivated Internal State Variable Plasticity/Damage Nonlocal Model

We used a physically motivated internal state variable plasticity/damage model containing a mathematical length scale to idealize the material response in finite element simulations of a large-scale boundary value problem. The problem consists of a moving striker colliding against a stationary hazmat tank car. The motivations are (1) to reproduce with high fidelity finite deformation and temperature histories, damage, and high rate phenomena that may arise during the impact accident and (2) to address the material postbifurcation regime pathological mesh size issues. We introduce the mathematical length scale in the model by adopting a nonlocal evolution equation for the damage, as suggested by Pijaudier-Cabot and Bazant in the context of concrete. We implement this evolution equation into existing finite element subroutines of the plasticity/failure model. The results of the simulations, carried out with the aid of Abaqus/Explicit finite element code, show that the material model, accounting for temperature histories and nonlocal damage effects, satisfactorily predicts the damage progression during the tank car impact accident and significantly reduces the pathological mesh size effects.

Modeling ductile fracture in metals involving two populations of voids – influence of continuous nucleation of secondary voids upon growth and coalescence of primary voids

This paper is devoted to the study of the influence of the nucleation of small, secondary voids upon the growth and coalescence of large, primary cavities in porous ductile metals. For this purpose, a simple model is considered: a hollow sphere subjected to hydrostatic loading whereby the central hole simulates a large, primary void, and the presence of secondary voids in the surrounding matrix is accounted for by the Gurson–Tvergaard–Needleman homogenized model for plastic porous materials. Continuous nucleation of small voids in the matrix is described by the straightforward phenomenological formula proposed by Pineau and Joly. Initially, the problem is solved analytically, then numerically when a non-uniform distribution of secondary voids develops in the matrix. These elements are used to define a simplified model in which only a small number of internal parameters describe the growth of primary voids coupled with the nucleation, growth and coalescence of secondary voids. Comparisons between the results derived from this simplified model and those obtained through the numerical computations are used to demonstrate the accuracy of the proposed model.

The method of virtual power in a micromorphic theory of ductile fracture in metals

We use the method of virtual power to rigorously establish the balance equations and boundary conditions in the context of a micromorphic theory developed by Gologanu, Leblond, Perrin and Devaux (GLPD) to solve the pathological mesh size effects in numerical simulations of problems involving ductile rupture. As an example, we derive these equations for the problem of circular bending of a beam deformed in plane strain. Also, we provide links between the outcome of the method and the micromorphic theory of Germain. In particular, we show that, with a minor modification, the modified GLPD theory, which can easily fit into a finite element subroutine, is equivalent to Germain micromorphic theory. The paper ends with some comparisons with the general second gradient theory.

Localization effects in Bammann-Chiesa-Johnson metals with damage delocalization

The presence of softening in the Bammann-Chiesa-Johnson (BCJ) material model presents a major physical drawback: the unlimited localization of strain which results in spurious zero dissipated energy at failure. This difficulty resolves when the BCJ model is modified to incorporate a nonlocal evolution equation for the damage, as proposed by Pijaudier-Cabot and Bazant (1987). The objective of this work is to theoretically assess the ability of such a modified BCJ model to prevent unlimited localization of the strain. To that end, we investigate two localization problems in nonlocal BCJ metals: appearance of a spatial discontinuity of the velocity gradient in a finite, inhomogeneous body, and localization into finite bands. We show that in spite of the softening arising from the damage, no spatial discontinuity occurs in the velocity gradient. Also, we find that the dissipation energy is continuously distributed in nonlocal BCJ metals and therefore can not localize into zones of vanishing volume. As a result, the appearance of any vanishing width adiabatic shear band is impossible in a nonlocal BCJ metal. Finally, we study the finite element (FE) solution of shear banding in a rectangular mesh, deformed in plane strain tension and containing an imperfection, thereby illustrating the effects of imperfections on the localization of the strain.Copyright