Ahmed Benallal

Strain localization and bifurcation in a nonlocal continuum

Abstract The conditions for localization and wave propagation in a strain softening material described by a nonlocal damage-based constitutive relation are derived in closed form. Localization is understood as a bifurcation into a harmonic mode. The criterion for bifurcation is reduced to the classical form of singularity of a pseudo “acoustic tensor”; this tensor is not a material property as it involves the wavelength of the bifurcation mode through the Fourier transform of the weight function used in the definition of the nonlocal damage. A geometrical solution is provided to analyse localization. The conditions for the onset of bifurcation are found to coincide in the nonlocal and in the corresponding local cases. In the nonlocal continuum, the wavelength of the localization mode is constrained to remain below a threshold which is proportional to the characteristic length of the continuum. The analysis in dynamics exhibits the well-known property of wave dispersion. In some instances i.e. for large wavelength modes, wave celerities become imaginary, but waves with a sufficiently short wavelength are found to propagate during softening in all the situations.

Theoretical and computational aspects of a thermodynamically consistent framework for geometrically linear gradient damage

This paper presents the theory and the numerics of an isotropic gradient damage formulation within a thermodynamical background. The main motivation is provided by localization computations whereby classical local continuum formulations fail to produce physically meaningful and numerically converging results. We propose a formulation in terms of the Helmholtz free energy incorporating the gradient of the damage field, a dissipation potential and the postulate of maximum dissipation. As a result, the driving force conjugated to damage evolution is identified as the quasi-nonlocal energy release rate, which essentially incorporates the divergence of a vectorial damage flux besides the strictly local energy release rate. On the numerical side, besides balance of linear momentum, the algorithmic consistency condition must be solved in weak form. Thereby, the crucial issue is the selection of active constraints which is solved by an active set search algorithm borrowed from convex nonlinear programming. In the examples, we compare the behavior in local damage with the performance of the gradient formulation.

An experimental investigation of cyclic hardening of 316 stainless steel and of 2024 aluminium alloy under multiaxial loadings

Abstract This paper is concerned with the experimental behavior of a 316 stainless steel and a 2024 aluminium alloy at room temperature and under complex nonproportional strainings in tension-torsion. The basic features of this behavior are underlined and their interactions emphasized. It is observed that the response of these materials under general loading paths is a balance between hardening and softening occuring respectively when the nonproportionality of the straining path is increased or decreased.

Continuum damage mechanics and local approach to fracture: Numerical procedures

Abstract In this paper, continuum damage mechanics is applied to the prediction of the failure of structures. The numerical implementation of this theory within the framework of the finite element method is described in details for both initiation and propagation problems. Practical examples are given to demonstrate the usefulness of this so-called ‘local approach to fracture’ in the case of creep and ductile damages.

Bifurcation and stability issues in gradient theories with softening

A bifurcation and stability analysis is carried out here for a bar made of a material obeying a gradient damage model with softening. We show that the associated initial boundary-value problem is ill posed and one should expect mesh sensitivity in numerical solutions. However, in contrast to what happens for the underlying local damage model, the damage localization zone has a finite thickness and stability arguments can help in the selection of solutions.

Nonlocal continuum effects on bifurcation in the plane strain tension-compression test

Abstract The paper examines nonlocal effects on bifurcation phenomena. A gradient plasticity model is used where a characteristic length is introduced in the yield criterion. Hills well known framework of bifurcation theory is shown to hold in the presence of normality and a sufficient condition for uniqueness is given. Further, the regularizing effects of nonlocality are underlined. It is also shown that the underlying local continuum, obtained when the length scale goes to zero, always provides a lower bound for bifurcation stresses for the nonlocal continuum. Detailed analysis of bifurcation phenomena in the plane strain tension-compression test is carried out and compared to the results of Hill and Hutchinson for the local continuum. The results are qualitatively the same in the long wavelength domain while they differ markedly in the short wavelength domain. In this last case and in the elliptic regime, bifurcation modes disappear in tension while the corresponding stresses are significantly increased in the compressive regime.

Effects of temperature and thermo-mechanical couplings on material instabilities and strain localization of inelastic materials

A general framework for rate-independent, small-strain, thermoinelastic material behaviour is presented, which includes thermo-plasticity and -damage as particular cases. In this context, strain localization and the development of material instabilities are investigated to highlight the roles of thermal effects and thermomechanical couplings. During a loading process, it is shown that two conditions play the essential roles and correspond to the singularity of the isothermal and the adiabatic acoustic tensors. Under quasi-static conditions, strain localization (in a classical sense) may occur when either of these two conditions is met. It involves a jump in temperature rate in the latter case, whereas temperature rate remains continuous in the former, but a discontinuity in the spatial derivatives of the heat flux must occur. This is consistent with the condition of stationarity of acceleration waves, which are shown to be homothermal and propagate with a velocity related to the eigenvalues of the isothermal acoustic tensor. A linear perturbation analysis further clarifies the above findings. In particular, for a quasi-static path of an infinite medium, failure of positive definiteness of either of the acoustic tensors corresponds to bifurcations in wave-like modes of arbitrary wave-length and infinite rate of growth. Under dynamic conditions, unbounded growth of perturbations is associated only to the short wavelength regime and corresponds to divergence growth or flutter phenomena relative to the isothermal acoustic tensor.

Localization analysis via a geometrical method

A geometrical technique is proposed in order to solve explicitly the critical conditions at localization for a quite general constitutive behaviour with isotropic elastic properties. These critical conditions are shown to be closely related to the spectral properties (eigenvalues and eigenvectors) of the sum and difference of two tensors describing the inelastic effects. When these two tensors are coaxial, it is shown that the normal to a potential localization plane always lies in one of their principal planes. It is also demonstrated that, depending on the constitutive behaviour and the loading conditions, several expressions for the critical hardening modulus at localization are available and their respective domain of validity well defined. The roles and interactions of both deviatoric and hydrostatic non-associativities in the critical conditions of localization are emphasized.

Bifurcation and Localization in Rate-Independent Materials. Some General Considerations

This work deals with some aspects of bifurcation and localization phenomena for solids made of rate-independent materials. Only the theoretical developments are presented. Physical non-linearities (plasticity, damage, ...) and geometrical non-linearities are taken into account. The analysis is limited to quasi-static loadings. A full and complete analysis of the rate problem for incrementally linear solids is carried out. The first order rate problem is formulated and analysed in the framework of modern theory of linear elliptic boundary value problems. Three conditions are necessary and in the same time sufficient for this problem to be well-posed. These conditions are local in nature and are used to describe localization phenomena.

A numerical study on the influence of the Portevin–Le Chatelier effect on necking in an aluminium alloy

The constitutive relation proposed by McCormick (1988 Acta Metall. 36 3061–7) for materials exhibiting negative steady-state strain-rate sensitivity and the Portevin–Le Chatelier (PLC) effect is incorporated into an elastic–viscoplastic model for metals with plastic anisotropy. The constitutive model is implemented in LS-DYNA for corotational shell elements. Plastic anisotropy is taken into account by use of the yield criterion Yld2000/Yld2003 proposed by Barlat et al (2003 J. Plast. 19 1297–319) and Aretz (2004 Modelling Simul. Mater. Sci. Eng. 12 491–509). The parameters of the constitutive equations are determined for a rolled aluminium alloy (AA5083-H116) exhibiting negative steady-state strain-rate sensitivity and serrated yielding. The parameter identification is based on existing experimental data. A numerical investigation is conducted to determine the influence of the PLC effect on the onset of necking in uniaxial and biaxial tension for different overall strain rates. The numerical simulations show that the PLC effect leads to significant reductions in the strain to necking for both uniaxial and biaxial stress states. Increased surface roughness with plastic deformation is predicted for strain rates giving serrated yielding in uniaxial tension. It is likely that this is an important reason for the reduced critical strains. The characteristics of the deformation bands (orientation, width, velocity and strain rate) are also studied.