Gilles Perrin

Self-healing slip pulse on a frictional surface

Guided by seismic observations of short-duration radiated pulses in earthquake ruptures, Heaton (1990) has postulated a mechanism for the frictional sliding of two identical elastic solids that consists in the subsonic propagation of a self-healing slip velocity pulse of finite duration along the interface. The same type of pulse may be conjectured for inhomogeneous slip along sufficiently large, and compliant, technological surfaces. We analyze such pulses, first as steady traveling waves which move at constant speed, and without alteration of shape, on the interface between joined elastic half-spaces, and later as transient disturbances along such an interface, arising as slip rupture propagates spontaneously from an over-stressed nucleation site. The study is conducted in the framework of antiplane elastodynamics; normal stress is uniform and alteration of it is not considered. We show that not all constitutive models allow for steady traveling wave pulses: the static friction threshold subsequent to the relocking of the fault must increase with time. That is, such solutions do not exist for pure velocity-dependent constitutive models, in which the stress-resisting slip on the ruptured surface is a continuously decreasing function of the instantaneous sliding rate (but not of its previous history or of other measures of the evolving state of the surface). Further, even for constitutive models that include both the rate- and state-dependence of friction, such as the laboratory-based constitutive models for friction as developed by Dieterich (1979, 1981) and Ruina (1983), steady pulse solutions do not exist for versions, like one discussed by Ruina (1983), which do not allow (rapid) restrengthening in truly stationary contact. For a particular class of rate- and state-dependent laws which includes such restrengthening, we establish parameter ranges for which steady pulse solutions exist, and use a numerical method stabilized by a Tikhonov-style regularization to construct the solutions. The numerical method used for the transient analysis adopts Fourier series representations for the spatial dependence of stress and slip along the interface, with the (time-dependent) coefficients in those Fourier series being related to one another in a way which obtains from exact solution to the equations of elastodynamics. This allows an efficient numerical method, based on use of the Fast Fourier Transform in each time step, with the frictional constitutive law enforced at the FFT sample points along the interface. Solutions based on a law that includes restrengthening in stationary contact show that spontaneous rupture propagation will occur either in the self-healing slip pulse mode (but not generally as a steady pulse) or in the classical enlarging-crack mode, depending on the values of parameters which enter the constitutive law. This analysis suggests that the strictly steady, traveling wave pulse solutions may either be unstable or have a limited basin of attraction.

Accelerated void growth in porous ductile solids containing two populations of cavities

Abstract This paper presents an analytical and numerical study of accelerated void growth in porous ductile solids arising from the presence of two populations of cavities very different in size. It is based on the model problem of some hollow sphere made of porous plastic material and subjected to hydrostatic tension. The central hole plays the role of a typical big cavity of the first population while those dispersed in the matrix stand for the small cavities of the second one. The behavior of the matrix is supposed to obey Gursons famous “homogenized” model for porous ductile solids (Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: part I — yield criteria and flow rules for porous ductile media. ASME J Engng Materials Technol 99, 2–15). The analytic solution of this model problem shows that the small voids located near the big one grow twice as fast as the latter void. This suggests that in a subsequent step, these small cavities may reach coalescence prior to the big ones, thus creating spherical shells of ruined matter around the cavities of the first population and leading to accelerated growth of the latter cavities; this scenario is in agreement with experimental evidence. Since this subsequent step is not amenable to a complete analytic solution, it is studied numerically. Finally, a simplified model reproducing the two steps of void growth (prior to coalescence of the small voids and after it has started) is developed on the basis of the analytical solution for the first step and some elements of a similar solution for the second one. The results derived from this simplified model are in good quantitative agreement with those obtained through the complete numerical simulations.

Disordering of a dynamic planar crack front in a model elastic medium of randomly variable toughness

Abstract Rice et al . (1994, J. Mech. Phys. Solids 42 , 813–843) analyse the propagation of a planar crack with a nominally straight front in a model elastic solid with a single displacement component. Using the form of their results for a strictly linearized perturbation from a straight crack front which moves at uniform speed, we give the corresponding first-order expression for the deviation of a crack front from straightness as a direct integral expression in the deviation of the material toughness from uniformity in the crack plane. We then use this expression to analyse the autocorrelation of the crack front position when the toughness deviations are random. We find that the root mean square deviation in position diverges logarithmically with travel distance across the random toughness region, as do the variances of the propagation velocity and slope of the crack front. That is, according to strictly linearized analysis, perturbed about the solution for a uniformly moving crack front, the perturbations from straightness and from uniform propagation speed should grow without bound in the presence of random deviations in toughness. What is remarkable about this result is that, according to the same strictly linearized analysis, if the toughness is completely uniform over the remaining part of the fracture plane, after encounter with a region of nonuniform toughness, the moving crack front becomes asymptotically straight with increase of time. Nonlinearities, not considered here, must control how statistically disordered the crack front can ultimately become as it propagates through a region of random toughness variation. Also, because of the logarithmic nature of the growth, significant disorder can occur in response to small perturbations only when the crack moves over a great distance compared to the correlation length scale in the fracture toughness.

Functional methods and effective potentials for non-linear composites

Abstract A formulation of variational principles in terms of functional integrals is proposed for any type of local plastic potentials. The minimization problem is reduced to the computation of a path integral. This integral can be used as a starting point for different approximations. As a first application, it is shown how to compute to second order the weak-disorder perturbative expansion of the effective potentials in random composite. The three-dimensional results of Suquet and Ponte-Castaneda (Suquet, P., Ponte-Castaneda, P., 1993. Small-contrast perturbation expansions for the effective properties of nonlinear composites. C. R. Acad. Sci. (Paris) Ser. II 317, 1515–1522) for the plastic dissipation potential with uniform applied tractions are retrieved and extended to any space dimension, taking correlations into account. In addition, the viscoplastic potential is also computed for uniform strain rates.

Numerical Assessment of a Micromorphic Model of Ductile Rupture

All constitutive models involving softening predict unlimited strain localization, and the famous Gurson [1] model of ductile rupture is no exception. An improved variant of this model aimed at solving this problem was derived by Gologanu et al. [2] from some refinement of Gurson’s original homogenization procedure. They obtained a new model of “micromorphic” nature, involving the second gradient of the macroscopic velocity and generalized macroscopic stresses of “moment” type.

Unified Constitutive Equations to Describe Elastoplastic and Damage Behavior of X100 Pipeline Steel

The deformation and fracture of the high strength steel which is used to fabricate grade X100 steel pipe were investigated at room temperature, on plate and pipe. Anisotropic behaviour was characterized by using tensile tests conducted along different directions. The fracture toughness of plate and pipe materials are compared using compact tension (CT) and Charpy V-notched tests. In both cases, these materials give rise to good fracture properties. The aim of the present study is to establish unified constitutive equations able to describe the elastoplastic and damage behavior of one X100 linepipe steel. The model is then used to simulate three point bending and compact tension tests.