Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics

Complete dynamical PDE systems of one-dimensional nonlinear elasticity satisfying the principle of material frame indifference are derived in Eulerian and Lagrangian formulations. These systems are considered within the framework of equivalent nonlocally related PDE systems. Consequently, a direct relation between the Euler and Lagrange systems is obtained. Moreover, other equivalent PDE systems nonlocally related to both of these familiar systems are obtained. Point symmetries of three of these nonlocally related PDE systems of nonlinear elasticity are classified with respect to constitutive and loading functions. Consequently, new symmetries are computed that are: nonlocal for the Euler system and local for the Lagrange system; local for the Euler system and nonlocal for the Lagrange system; nonlocal for both the Euler and Lagrange systems. For realistic constitutive functions and boundary conditions, new dynamical solutions are constructed for the Euler system that only arise as symmetry reductions from invariance under nonlocal symmetries.     

Micromechanical analysis of the finite element calculation of a diffusional transformation

A micromechanical model of a diffusional transformation has been developed, which describes the progress of the transformation within a three-dimensional “unit cell” submitted to an external stress state. The example chosen is that of an isothermal pearlitic transformation of a steel. The transformation plastic strain is due to interactions occurring between the local stresses and the transformation process, resulting in an oriented plastic flow in or in the vicinity of the material layers swept by the transformation front. The analysis of the local mechanical states of the simulation provides a good interpretation of the evolution of transformation plastic strain, when considering the effect of applied stress level and the way mechanical properties are imposed on the newly formed phase. In particular, the normality law for transformation plasticity is related to the shape of the local plastic zones. The discrepancy observed between simulation and experience is then discussed, following two main points: the influence of the behaviour law of the phases and the way interactions between neighbouring cells are prescribed. The difficulty and importance of obtaining realistic mechanical properties of the forming pearlite is pointed out. This revised version was published online in November 2006 with corrections to the Cover Date.     

A mesoscopic wave model for textile materials in large deformations

A constitutive law for large transformations of soft structures is given, starting from a micropolar constitutive law, where the microrotation takes into account the microstructure effect. In fabrics a wave defines the mesoscopic scale. The hyperelastic law takes into account the extension in the fabric. The mesoscopic wave model is applied in the present paper to predict the uniaxial extension of plain weave fabrics.     

Modelling of curved interfaces in composite shells

Contact laws for thin curved interfaces in elastic multistructures have been established, within the framework of thin shell mechanics. Limit contact laws for the interface are obtained in a deductive manner, using an asymptotic expansion of the solution vs. the interface thickness. The limit solution is shown to depend on the ratio between the thickness of the shell and the radius of curvature of the mean plane of the shell. Different behaviours are evidenced, according to the scaling of the mechanical properties of the intermediate shell and the relative amount of curvature vs. thickness. The cases of very weakly curved shells, weakly curved shells, and strongly curved shells are highlighted. The effect of curvature on stress generation is evidenced, and the stress distribution is represented in some specific cases.     

Composites with auxetic inclusions showing both an auxetic behavior and enhancement of their mechanical properties

Composite materials made of auxetic inclusions and giving rise overall to negative Poisson’s ratio are considered, adopting a two-steps micromechanical approach for the calculation of their effective mechanical properties. The inclusions consist of periodic beam lattices, whose equivalent mechanical properties are calculated by a discrete homogenization scheme in a first step. The hexachiral and hexagonal reentrant lattices are considered as representative of the two main deformation mechanisms responsible for auxeticity. In a second step, the equivalent properties of the composite are calculated from numerical homogenization using the finite element method. It is shown that both an auxetic behavior and enhanced moduli can be obtained for not too slender micro-beams.     

Calculation of Thermal Stresses in Glass-Ceramic Composites

Opto-electronics make intensive use of composite materialsbased on amorphous materials, which can be considered as smart materialssince they are capable of high performances in their final state.Particularly, glass-ceramic composites involved in welding operationsfor microelectronics applications are subjected to important thermalstresses during their production, which can deteriorate their propertiesat room temperature, until the failure stage is reached. It is thenessential to be able to predict the evolution of the internal stressesgenerated during the cooling. We have performed finite-elementsimulations in order to quantify the stress evolutions for differentcomposite geometries: a ceramic fiber embedded within a glass matrix, aspherical particle located at the center of a spherical glass matrix,and a dispersion of spherical ceramic particles, this last case beingthe most representative of reality. The thermomechanical modeling of theglassy matrix takes into account its viscoelastic behavior, and theglass transition is described by the decrease during cooling of the freevolume as a function of the temperature history. The combined effect ofthe differential thermal strain during the transition and mechanicalrelaxation of glass on stress evolutions is evidenced. It is shown thatthe consideration of a periodical or random distribution of sphericalceramic particles leads to similar stress profiles.     

Analysis of Contact and Rupture Behaviour Under Non-Constant Contact Surface Conditions

The present paper addresses the problem of contact between a rigid hemisphere and a thin elastic layer strongly bonded on a rigid plane support, which can be thought of as an adhesive obeying a geometrically non-linear behaviour due to the change of contact area. Using the asymptotic expansion method from a three-dimensional analysis of the layer, a two-dimensional model is derived, under the assumptions of large displacements and small strains. The leading term of the solution of the asymptotic development is such that the displacement field varies linearly through the layer thickness and the stress tensor is constant. A quasi-linear relation is obtained between the area of contact and the penetration of the hemisphere within the layer, and the variation with penetration of the compressive load exerted by the hemisphere is seen to give satisfactory agreement with experiments. In the last part, we present theoretical results concerning the rupture behaviour; the effect of adhesion energy between the hemisphere and the layer on the radius of curvature at the rupture point between both solids is assessed. Further, the thickness of an hypothetical interphase through which failure propagates is determined theoretically.     

Adhesion of a Rigid Punch to a Thin Elastic Layer

Adhesion between a rigid flat cylindrical punch and an elastic layer has been investigated. FE analysis was employed to determine the layer stiffness. Linear elastic fracture mechanics was then used to determine the energy release rate, G_a, per unit of bonded area for a circular debond propagating inwards from the edge of the punch. The calculations showed a strong effect of Poisson's ratio for thin layers, small departures from complete incompressibility causing large reductions in stiffness and hence in detachment force. Experiments were performed with an aluminum punch adhered to a rubber layer using a rubber-based adhesive. The ratio of punch radius to layer thickness was varied over the range 0.07 to 3.3. Detachment forces were measured and compared with calculated values. Reasonable agreement was obtained for thick layers but not for thin ones, possibly because of a change in the mode of failure.