Crack branching at low tip speeds: spilling the T
Using the criterion that a crack will extend along the direction of maximum circumferential stress, this paper demonstrates the influence of the coupling between the crack-parallel T-stress and the tip speed on the directional (in)stability of dynamics cracks in brittle materials, i.e., branching, turning, and limiting velocities. The proposed (in)stability criterion evolves within the theory of dynamic fracture: we build on the work of Ramulu and Kobayashi (1983) by introducing a reference distance ahead of the crack-tip to incorporate the contribution of the higher-order terms in the asymptotic solution of the elastic crack-tip fields. The theoretical aspect is first explored, a methodology to numerically (and experimentally) advocate the instability—as a co-action of T-stress and a fast-running crack—is then proposed and validated on Borden et al. (2012)’s branching benchmark. An experimental setup combining Ultra-High-Speed High-Resolution imaging with advanced Digital Image Correlation algorithms and a novel crack-branching inertial impact test enables for never-seen-before quantification of the rich dynamical behaviour of the fracture. This permits the experimental validation of the developed crack (in)stability criterion.
17 août 2023
The average conformation tensor of inter-atomic bonds as an alternative state variable to the strain tensor: definition and first application — the case of nanoelasticity
Most of the mechanical models for solid state materials are in a methodological framework where a strain tensor, whatever it is, is considered as a thermodynamic state variable. As a consequence, the Cauchy stress tensor is expressed as a function of a strain tensor—and, in many cases, of one or more other state variables, such as the temperature. Such a choice for the kinematic state variable is clearly relevant in the case of infinitesimal or finite elasticity. However, one can ask whether an alternative state variable could not be considered. In the case of finite elastoplasticity, the choice of a strain tensor as the basic, kinematic state variable is not totally without issue, in particular in relation to the physical meaning of the internal state variable describing the permanent strains. In any case, this paper proposes an alternative to the strain tensor as a state variable, which is not based on the deformation (Lagrangian) gradient: the average conformation tensor of inter-atomic bonds. The purpose, however, is restricted to (1) a particular type of materials, namely the pure substances (copper or aluminum, for instance), (2) the nanoscale, and (3) the case of elasticity. The very simple case of two atoms of a pure substance in the solid state is first considered. It is shown that the kinematics of the inter-atomic bond can be characterized by a so called ``conformation'' tensor, and that the tensorial internal force acting on it can be immediately deduced from a single scalar function, depending only on the conformation tensor: the state potential of free energy (or interaction potential). Using an averaging procedure, these notions are then extended to a finite set of atoms, namely an atom and its first neighbours, which can be seen as the ``unit cell'' of a pure substance in the solid state considered as a discrete medium. They are also transposed to the Continuum case, where an expression of the Cauchy stress tensor is proposed as the first derivative of a state potential of density (per unit mass) of average free energy of inter-atomic bonds, which is an explicit function of the average conformation tensor of inter-atomic bonds. By applying a standard procedure in Continuum Thermodynamics, it is then shown that the objective part of the material derivative of this new state variable, at least in the case when the pure substance can be considered as an elastic medium, is equal to the symmetric part of the Eulerian velocity gradient, that is the rate of deformation tensor. In the case of uniaxial tension, a simple relationship is eventually set out between the average conformation tensor and a strain tensor, which is correctly approximated by the usual infinitesimal strain tensor as long as the conformation variations (from an initial state of conformation) are ``small''. From this latter result, and assuming an elastic behavior, a simple expression for the state potential of density of average free energy is inferred, showing great similarities with—but not equivalent to—the classical model of isotropic, linear elasticity (Hooke's law).
05 juin 2023