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Abstract This paper deals with a new numerical boundary variation method for solving the generalized Cheeger problem. This problem (also called weighted Cheeger problem) – modeling landslides, snow avalanches and other geophysical flows – aims to find the safety factor and the collapse domain (onset flow region). It is formulated in the space of bounded variations functions and rewritten in terms of a shape optimization problem, involving only boundary valued functions. We propose here a numerical scheme, based on a boundary variation method (without changing the topology). For that, we have computed the shape derivative of the Cheeger functional and derived a surface divergence formula for surface defined functions. For the spatial discretization we only use a shape boundary discretization. Even if sometimes the choice of the initial shape of the algorithm had to be done using a global optimization method, the boundary variation method is very attractive. Finally, we illustrate the proposed method with numerical computations of the onset velocity regions for certain physical sound problems (in two and three dimensions). [ABSTRACT FROM AUTHOR]

In this paper we address the following design problem: what is the shape of a plate and the associated pre-stress that relaxes toward a given three-dimensional shell ? As isometric transformations conserve the gaussian curvature, three-dimensional non-developable shells cannot be obtained from the relaxation of pre-strained plates by using isometric transformations only. Overcoming this geometric restriction, including small-strains and large rotations, solves the problem for small areas only. This paper dispenses with the small-area restriction to cover three-dimensional shells fully by using shell-strips. Since shell-strips have an additional geometric parameter, we show that under suitable assumptions that relate the width of the strip to the curvature of the shell, we are able to design arbitrary shell surfaces by covering them with shell-strips. As an illustration, we provide optimized covers of the sphere in a variety of different surface-strips relaxed from plate-strips with homogeneous and isotropic pre-stress. Moreover, we propose the design of the torus, of the helicoid and of the non-developable Möbius band, which requires inhomogeneous and anisotropic pre-stress. • Obtain 3D shells using pre-stress relaxation starting from planar pre-stressed ribbons. • Design of both planar shape and pre-strain needed to cover non-developable shells. • Solution to design problems for to the sphere, the torus and the Mobius band [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a macroscopic anisotropic strain rate potential, which can describe both the anisotropy and tension-compression asymmetry of the plastic response of textured metals is derived. This strain rate potential is the exact work-conjugate of the anisotropic stress potential CPB06 of . Application of the developed strain rate potential to HCP high-purity alpha-titanium is presented. [Copyright &y& Elsevier]

Abstract: We present a new limit analysis method, originated in and called discontinuous velocity domain splitting method (DVDS). DVDS is a mesh free method which focuses on the strain localization and completely neglect the bulk deformations. It considers the kinematic variational principle on a special class of virtual velocity fields to get an upper bound of the limit load. To construct this class of virtual velocity fields, the rigid-plastic body is splinted into simple connected sub-domains and on each such sub-domain a rigid motion is associated. The discontinuous collapse flow velocity field results in localized deformations only, located at the boundary of the sub-domains. In the numerical applications of the DVDS method, we introduce a numerical technique based on a level set description of the partition of the rigid-plastic body and on genetic minimization algorithms. For the anti-plane flow of a von Mises material, DVDS is exact in solving the limit analysis problem: the collapse flow is a rigid motion of a sub-domain. The associated deformation rate is localized on the smooth boundary of the moving sub-domain representing the fracture surface. In the case of in-plane deformation of pressure insensitive materials, the internal boundaries of the sub-domains are parts of circles or straight lines, tangent to the collapse velocity jumps. In this case, DVDS reduces to the block decomposition method, which was intensively used to get analytical upper bounds of the limit loads. When applied to the two notched tensile problem of a von Mises material, DVDS gives excellent results with a low computational cost. Furthermore, DVDS was applied to model collapse in pressure sensitive plastic materials. Illustrative examples for homogenous and heterogeneous Coulomb and Cam-Clay materials shows that DVDS gives excellent prediction of limit loads and on the collapse flow. [Copyright &y& Elsevier]

Abstract: A wedged configuration with Coulomb friction is a nontrivial equilibrium state of a linear elastic body in a frictional unilateral contact with a rigid body under vanishing external loads. We analyze here the relation between the geometry of the elastic body and the friction coefficient for which wedged configurations exist in a 3-D context. The critical friction coefficient, μ w , is defined as the infimum of a supremal functional defined on the set of admissible normal displacement and tangential stresses. For friction coefficients μ with μ > μ w the wedged problem has at least a solution and for μ < μ w there exits no wedged configurations. For the in-plane problem we discuss the link between the critical friction and the smallest real eigenvalue μ s which is related to the loss of uniqueness. The wedged problem is stated in a discrete framework using a mixed finite element approach and the (discrete) critical friction coefficient is introduced as the solution of a global minimization problem involving a non differentiable and non-convex functional. The existence of , the displacement field of a critical wedged state, is proved and a specific numerical method, based on a genetic algorithm, was developed to compute the critical wedged configurations. Some techniques to handle the discontinuities of the normal vector on the contact surface are presented and the analysis is illustrated with three numerical simulations. [Copyright &y& Elsevier]

The dynamic evolution with frictional contact of an elastic body is considered. In modeling the contact the Tresca model and a slip-dependent friction law are used. The existence of a solution is proved in the two-dimensional case. The uniqueness is proved for the one-dimensional shearing problem. The convergence, for a vanishing viscosity, of the unique solution of the viscoelastic problem to a solution of the elastic problem is obtained. [Copyright &y& Elsevier]

• The first use of a "minimal" 2D Eulerian plasticity approach in modeling micro-pillars compression. • The model reproduces ductile fracture (kink band localization) and its orientation dependence. • Engineered inhomogeneities could give a more uniform deformation, as in experiments. The mechanical response of almost pure single-crystal micro-pillars under compression exhibits a highly localized behavior that can endanger the structural stability of a sample. Recent experiments revealed that the mechanical response of a crystal is very sensitive to both the presence of a quenched disorder in the sample, and the orientation of the crystal. In this work, we study the influence of disorder and crystal orientation on the large 2-D plane strain response of a FCC crystal with three active glide planes using a very simple Eulerian plasticity model in the FE framework. Our numerical and theoretical results on clean crystal pillars suggest that a single plane or many gliding planes can be activated depending on the crystal orientation. While in the former case, the deformation is localized, leading to ductile rupture, in the latter, a complex interplay between active planes takes place, resulting in a more uniform deformation. The strain-localization can be avoided when inhomogeneities are engineered inside the crystal, or the crystal orientation is altered because of the activation of multiple slip systems, resulting in a "patchwork" of the distribution of the slip systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the discrete contact problem governed by Coulomb’s friction law. We propose and study a new technique using mixed finite elements with two multipliers in order to determine numerically critical friction coefficients for which multiple solutions to the friction problem exist. The framework is based on eigenvalue problems and it allows to exhibit non-uniqueness cases involving an infinity of solutions located on a continuous branch. The theory is illustrated with several computations which clearly show the accuracy of the proposed method. [Copyright &y& Elsevier]

• We study dynamic/quasi-static stability of strain-driven damage evolution in solids. • We deduce simple stability criteria for micromechanics-based models of damage. • We apply the stability criteria to non-interacting cracks with self-similar growth. • We find a critical crack density parameter differentiating stable/unstable behaviour. • The critical parameter depends only on the chosen configuration and Poisson's ratio. We investigate the stability of dynamic and quasi-static strain-driven processes for a large class of micromechanics-based models of damage in brittle solid materials. A special attention is paid to the compatible models having the same equilibrium states where the dynamic damage law is constructed from a quasi-static one. We deduce a simple stability criterion for non-associated damage models. The unstable equilibria of quasi-static models are related to the solution multiplicity, shocks and non-smooth stress-strain curves. Also, we deduce the stability of the dynamic models from the linear stability of the associated damage differential system. For compatible models the dynamic and quasi-static stability criteria coincide. We use the stability criteria to analyse mechanically non-interacting cracks with self-similar growth, and find a critical crack density parameter that distinguishes between stable and unstable behaviours. It depends on the chosen configuration and Poisson's ratio, but it is independent of the other micromechanical properties. The configurations with the crack densities smaller than the critical value are mechanically unstable but cracks are activated at high stress levels, generating a sharp stress drop in the stress-strain diagram. For crack densities above the critical value, the model is stable and the stress varies smoothly. For dynamic loadings of unstable configurations a high strain rate is associated with a smooth stress-strain curve, while a low strain rate induces an abrupt stress drop in the stress-strain relation. This kind of instability is qualitatively the same for porous brittle plates under uni-axial compression and for the self-consistent effective elasticity and the wing crack models. [ABSTRACT FROM AUTHOR]