Asymptotic analysis of wave propagation in a finite viscoplastic bar

Summary Axially symmetric wave propagation in a finite extent cylindrical sample, whose viscoplastic behaviour is described by the Bodner-Partom model, involves variables which can be scaled so as to bring out a small parameter characteristic of the problem at hand. Asymptotic solutions can be derived from power series expansions of the variables with respect to the small parameter.In the one-dimensional case, a finite difference solution is shown to agree with the zero order term of the asymptotic expansion which is closely related to the Hopkinson pressure bar simplified theory.Then it is shown in a special case that retaining the sole first term of each asymptotic expansion amounts to eliminating the needlessly intricate fluctuations of the exact solution due to successive wave reflections at the specimen's both ends, while keeping the essentials of the response and simplifying the numerical work.Afterwards, the relative orders of magnitude of the various stress components with respect to the small parameter are estimated in the more general framework of the two-dimensional problem, with a view to re-deriving the so-called inertia correction; the latter relies on an appropriate definition of the mean stress experienced by the sample and uses the first two terms of the asymptotic expansions.This study suggests the applicability of the method in a more general scope, taking into account in a consistent way other phenomena (friction and lateral motion of the adjacent bars, for instance) or more refined constitutive equations.Notation a=h/R aspect ratio of specimen - A specimen cross-section area - A ₁ bar cross-section area - c elastic wave speed in specimen - c ₁ elastic wave speed in bars - E specimen Young's modulus - E ₁ bar Young's modulus - h specimen length - H Z/E - r radial coordinate - R specimen radius - s _ij stress deviator - t time - T incident signal duration - u _i physical displacement in specimen - U _i dimensionless displacement in specimen - V ₀,V ₁ dimensionless velocities on end faces - w _p plastic work - W _p dimensionless plastic work - x axial coordinate - Z Bodner-Partom hardening parameter - small parameter - zero order longitudinal extension rate - strain rate tensor - dimensionless axial coordinate - dimensionless axial coordinate - polar angle - Poisson's ratio - specimen density - _ij stress tensor - _ij dimensionless stress tensor - equivalent stress - dimensionless time - dimensionless viscoplastic strain rate in axial direction     

Steady-State Crack Velocity in a Viscoelastic Composite Material

The velocity of a semi-infinite crack slowly propagating in aninfinitely long strip made of a viscoelastic composite material isdetermined according to Christensen's fracture criterion. The edges ofthe strip are subjected to uniform opposite displacements normal to thecrack plane. The crack velocity is obtained from an energy balanceequation involving the energy dissipated in the whole strip; the latteris evaluated using an approximate, but sufficiently accurate, expressionof the stress field in the structure obtained by taking into account thestrong anisotropy of the long fibre composite material of interest. Thisnew version of Christensen's criterion compares favourably with theoriginal one and gives crack velocity predictions very close to thoseprovided by Schapery's criterion.     

Application of viscoelastic fracture criteria to progressive crack propagation in polymer matrix composites

With the view of comparing local and global viscoelastic fracture criteria, an extension of Christensen's criterion to composite materials with stiff elastic fibers is proposed. Several versions of this criterion are compared with Schapery's local approach of the same phenomenon. The asymptotic version of Christensen's criterion for rapid crack propagation is found suitable for the material investigated, at room temperature. Dissipation in the specimen has two main sources: the undamaged material on one hand, and the damaged material inside the `failure zone' close to the crack tip on the other. The respective roles of these two kinds of dissipation are assessed.     

Free end effect on the stress and displacement distributions in a cracked composite strip

Taking into account the marked anisotropic character of carbon fibre composite materials, a small parameter related to material properties is introduced. The stress field in a semi-infinite strip with a semi-infinite crack whose tip is close to the strip free end is investigated through a singular perturbation method. For slow crack propagation, the quasi-static character of the stress field is established. An accurate asymptotic solution is derived, which allows the interaction between the strip free end and the stress field around the crack tip to be studied. For the carbon–epoxy material investigated, this interaction becomes negligible when the distance between the crack tip and the strip free end is greater than 1.5h, where h is the strip half-height.