Rate Sensitive Solids Research Articles

A generalized forward gradient procedure for rate sensitive constitutive equations

AbstractMany forward gradient schemes have been proposed for the time‐integration of the stiff constitutive equations of rate sensitive solids. It is shown here that one of these methods can be interpreted as a hybrid ordinary differential equation integrator which combines explicit and semi‐implicit Runge‐Kutta methods. This observation permits development of higher order schemes, illustrated here by one of second order. An embedded first order estimate provides a reliable step‐size control. The method is applied to an overstress model and to an internal variable model, and is used in a finite element analysis of hydrostatic bulging of sheet metal.

High speed deep drawing of hardening and rate sensitive solids with small interfacial friction

An asymptotic expansion solution for a hardening visco-plastic material undergoing a high rate deep drawing process is derived and utilized. The solution incorporates a wide range of variables, some of them not previously studied in this context, as, for example, the dynamic load induced by the high speed of the operation, the material rate sensitivity index and the frictional constraints along the blank/die interfaces. The behaviour of the suggested solution reconstitutes some observed phenomena, like the loading path of the moving punch, the limiting drawing ratio before rupture intercepts and the onset of blank thinning. Comparisons to available experiments and numerical solutions are made.

Dynamic crack-tip fields in rate-sensitive solids

T he asymptotic stress and strain fields near the tip of a crack which propagates dynamically in a rate-sensitive solid are obtained under anti-plane shear and plane strain conditions. The problem is formulated within the context of a small-strain theory for a solid whose mechanical behavior under high strain rates is described by an elastic-viscoplastic constitutive relation. It is shown that, if the stresses are singular at the crack-tip, the viscoplastic relation is equivalent asymptotically to an elastic-non-linear viscous relation. Furthermore, for a certain range of the material parameter which characterizes the rate-sensitivity of the material, the elastic strain-rates near the propagating crack tip are shown to have the same asymptotic radial dependence near the propagating crack-tip as the inelastic strain-rates. This determines the order of the stress singularity uniquely. The governing equations for anti-plane shear and plane strain are then derived. The numerical results for the stress and strain fields are presented for anti-plane shear and plane strain. For the present model, the results suggest that under small-scale yielding conditions, there exists a minimum velocity for stable steady crack propagation. The implication that a terminal velocity for a running crack may exist is also discussed.

Fatigue in rate sensitive solids : Wnuk, MP South Dakota State Uinv. Brookings, USA 19R Int. J. Fracture, V10, N2, June, 1974, P223–226

Fatigue in rate sensitive solids : Wnuk, MP South Dakota State Uinv. Brookings, USA 19R Int. J. Fracture, V10, N2, June, 1974, P223–226

Fatigue in rate sensitive solids

The fatigue process, viewed as a sequence of slow growth periods, is described by a non-linear differential equation of the first order which includes the “creep component” of crack growth in a visco-elastic solid. It is shown that fracture in a rate sensitive medium may extend even under sustained or decreasing loads. The total rate of growth consists therefore of two terms where the first term is a familiar power law, valid for a high-cycle range and limited plasticity effects, while the second one accounts for the time-dependent contribution. Here f denotes frequency, Y is the yield stress, Kcdenotes fracture toughness, ΔK stands for the stress intensity range while the crack closure correction α, the rate sensitivity C and the constant R * are defined in the text.