No abstract

A model to predict the forces on conical-nosed penetrators for normal impact into dry rock targets is developed. The target medium is described by a linear hydrostat, a linear shear failure-pressure relation, and the material density. A cylindrical cavity expansion approximation to the target response permits one-dimensional wave propagation calculations in the radial coordinate. The equations of motion are reduced, via a similarity transformation, to a nonlinear ordinary differential equation. This equation is solved numerically by a shooting technique which employs an asymptotic expansion to the solution near the wave front. Results include stress wave profiles in the target and curves for the stress on the penetrator nose as a function of its velocity for a wide range of realistic target parameters. Finally, results from the theory are compared with the deceleration history of a penetrator in a field test and reasonable correlation is observed.

No abstract

The linear version of the general theory of heat conduction is employed to study the problem of a half space subjected to step time inputs of temperature. The solution is obtained by the use of the Laplace transform on time and the sine transform on space. Exact solutions, wave-front, and long-time approximations are obtained. The long-time solutions satisfy the classical heat equation. It is shown that stable solutions may be obtained under more relaxed conditions than those prescribed by the general theory.

An arbitrary excitation of the plane x = 0 sends sound signals into the half-space x > 0 occupied by the viscous fluid. The governing third-order partial differential equation is solved exactly usi-•g the Laplace transform on time and the sine transform on space. New expressions for the most general solution are derived. The specific inputs considered in detail are the Dirac delta function, the Heaviside unit function, a decaying exponential, and a sinusoidal excitation. The final expressions are given in the form of real integrals and of exact power series. Short-time approximations are also given for a general input and for the four aforementioned specific inputs. Previously obtained approximations for the case of small attenuation coefficients are corrected and extended for the decaying exponential and the sinusoidal input. The results found indicate that viscosity tends to reduce the sharpness of the propagating disturbance and thus to smooth out any initially imposed discontinuity. The presence of the disturbance is felt immediately everywhere in the medium, and, in fact, the short-time approximations satisfy the parabolic heat equation.