Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

This paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

Modeling Intersonic Crack Propagation in Fiber Reinforced Composites With Contact/Cohesive Laws

Dynamic crack propagation in an unidirectional Carbon/Epoxy composite is studied through finite element analyses in total Lagrangian co-ordinates. A finite deformation anisotropic visco-plastic model is used to describe the constitutive response of the composite. Crack initiation and propagation is simulated by embedding zero thickness interface element along the possible crack path. An irreversible cohesive law is used to describe the evolution of normal and shear tractions as a function of displacement jumps. The compressive response prior to interface failure is analyzed using contact impenetrability conditions. The failure of the first interface element at the pre-notch tip models crack initiation. Crack propagation is modeled through consecutive failure of interface elements. Dynamic crack propagation phenomena are studied in terms of crack initiation time, crack speed, mode I and mode II displacement jumps and tractions associated with the failure of interface elements, effective plastic strain at the crack tip and path independent integral J′. Analyses are first carried out for the dynamic crack propagation along bi-material interfaces. The results obtained from present analyses agree well with literature data. Detailed analyses are carried out for a pre-notched unidirectional Carbon/Epoxy composite material. The impact velocity in the analyses is an imposed velocity over an assumed impact region and remains constant throughout the analysis. Analyses are carried out at impact velocities of 5, 10, 20, 30 and 40 m/s, assuming the crack wake is frictionless. Moreover, analyses at impact velocities of 30 and 40 m/s are also carried out with a friction coefficient of 0.5 along the crack surfaces. The analyses established intersonic crack speed in the fiber reinforced composite material. Intersonic crack propagation for the impact velocities of 40 m/s is 400% of the shear wave speed and 87% of the longitudinal wave speed. Detailed discussion is given on the features of sub-sonic and intersonic crack propagation in Carbon/Epoxy composite materials. It is shown that the friction coefficient along the crack surface plays an important role by smearing the discontinuous field that develops behind the crack tip and by reducing crack speed in the intersonic regime. The analyses show that the contour integral J′ computed at near field contours are path independent and can serve as a parameter for characterizing intersonic crack propagation.

Intersonic Crack Propagation—Part II: Suddenly Stopping Crack

In Part I of this series, we have obtained the fundamental solution for a mode II intersonic crack which involves a crack moving uniformly at a velocity between the shear and longitudinal wave speeds while subjected to a pair of concentrated forces suddenly appearing at the crack tip and subsequently acting on the crack faces. The fundamental solution can be used to generate solutions for intersonic crack propagation under arbitrary initial equilibrium fields. In this paper, Part II of this series, we study a mode II crack suddenly stopping after propagating intersonically for a short time. The solution is obtained by superposing the fundamental solution and the auxiliary problem of a static crack emitting dynamic dislocations such that the relative crack face displacement in the fundamental solution is negated ahead of where the crack tip has stopped. We find that, after the crack stops moving, the stress intensity factor rapidly rises to a finite value and then starts to change gradually toward the equilibrium value for a static crack. A most interesting feature is that the static value of stress intensity is reached neither instantaneously like a suddenly stopping subsonic crack nor asymptotically like a suddenly stopping edge dislocation. Rather, the dynamic stress intensity factor changes continuously as the shear and Rayleigh waves catch up with the stopped crack tip from behind, approaches negative infinity when the Rayleigh wave arrives, and then suddenly assumes the equilibrium static value when all the waves have passed by. This study is an important step toward the study of intersonic crack propagation with arbitrary, nonuniform velocities.