Most soft materials resist volumetric changes much more than shape distortions. This experimental observation led to the introduction of the incompressibility constraint in the constitutive description of soft materials. The incompressibility constraint provides analytical solutions for problems which, otherwise, could be solved numerically only. However, in the present work, we show that the enforcement of the incompressibility constraint in the analysis of the failure of soft materials can lead to somewhat nonphysical results. We use hyperelasticity with energy limiters to describe the material failure, which starts via the violation of the condition of strong ellipticity. This mathematical condition physically means inability of the material to propagate superimposed waves because cracks nucleate perpendicular to the direction of a possible wave propagation. By enforcing the incompressibility constraint, we sort out longitudinal waves, and consequently, we can miss cracks perpendicular to longitudinal waves. In the present work, we show that such scenario, indeed, occurs in the problems of uniaxial tension and pure shear of natural rubber. We also find that the suppression of longitudinal waves via the incompressibility constraint does not affect the consideration of the material failure in equibiaxial tension and the practically relevant problem of the failure of rubber bearings under combined shear and compression.

Transverse impact response of a linear elastic Kevlar ® KM2 fiber yarn was determined at various striking speeds from Hopkinson bar and gas gun experiments incorporated with high-speed photography techniques. Upon transverse impact, a triangle shape was formed in the fiber yarn. Both longitudinal and transverse waves were produced and propagated outwards the fiber yarn. Both the angle of the triangle and Euler transverse wave speed vary with striking speeds. The relationship between the Euler transverse wave speed and the striking speed was determined. The transverse impact response of the fiber yarn was also analyzed with a model, which agrees well with the experimental results. The model shows that the longitudinal wave speed is critical in the ballistic performance of the fiber yarn. At a certain striking speed, a higher longitudinal wave speed produces a higher Euler transverse wave speed, enabling us to spread the load and dissipate the impact energy faster, such that the ballistic performance of the fiber yarn is improved.

This study is concerned with longitudinal displacement waves propagating in an elastic cylindrical rod submerged in a viscous fluid. Provided that the wave propagation velocity in the rod is small compared with the velocity of sound in the surrounding fluid and the wavelength is large compared with the thickness of the boundary layer around the rod, an analytical relation is obtained between the wave number and the frequency. The presence of the fluid makes the waves disperse—the short waves become faster than the long ones.

Lagrangian equations of motion for finite amplitude azimuthal shear wave propagation in a compressible isotropic hyperelastic solid are obtained in conservation form with a source term. A Godunov-type finite difference procedure is used along with these equations to obtain numerical solutions for wave propagation emanating from a cylindrical cavity, of fixed radius, whose surface is subjected to the sudden application of a spatially uniform azimuthal shearing stress. Results are presented for waves propagating radially outwards; however, the numerical procedure can also be used to obtain solutions if waves are reflected radially inwards from a cylindrical outer surface of the medium. A class of strain energy functions is considered, which is a compressible generalization of the Mooney-Rivlin strain energy function, and it is shown that, for this class, an azimuthal shear wave can not propagate without a coupled longitudinal wave. This is in contrast to the problem of finite amplitude plane shear wave propagation with the neo-Hookean generalization, for which a shear wave can propagate without a coupled longitudinal wave. The plane problem is discussed briefly for comparison with the azimuthal problem.

An exact solution for scattering of ultrasound from a spherically orthotropic shell is presented. The shell is assumed to be embedded in an isotropic elastic medium, and the core surrounded by the shell is also assumed to be isotropic. The shell itself is assumed to be “spherically orthotropic,” with five independent elastic constants (the spherical analog of a transversely isotropic material in Cartesian coordinates). Field equations for this material are presented, and these equations are shown to be separable. Working with the displacement vector, we find that the radius dependent part of the solution satisfies coupled second-order ordinary differential equations. This system of equations is solved using the method of Frobenius, and results in four independent series determined by material properties to within a multiplicative constant. Use of boundary conditions expressed in terms of stresses and displacements at the inner and outer shell radii completes the solution. Numerical results for a range of shell elastic constants show that this solution matches known analytic results in the special case of isotropy and matches previously developed finite difference results for anisotropic elastic constants. The effect of shell anisotropy on far-field scattering amplitude is explored for an incident plane longitudinal wave.

The scattering of elastic waves by a circular crack situated in a transversely isotropic solid is studied here. The axis of material symmetry and the axis of the crack coincides. The incident wave is taken as a plane longitudinal wave propagating perpendicular to the crack surface. A Hankel transform representation of the scattered field is used, and after some manipulations using the boundary conditions this leads to an integral equation over the crack for the displacement jump across the crack. This jump is expanded in a series of Legendre polynomials which fulfill the correct edge condition and the integral equation is projected on the same set of Legendre polynomials. The far field is computed by the stationary phase method. A few numerical computations are carried out for both isotropic and anisotropic solids. Results for the isotropic solid compare favorably with those available in the literature.

A technique for the complete nondestructive evaluation of plane states of residual stress is presented. This technique is based on the acoustoelastic effect in which the presence of the residual stress causes a shift in the speed at which a wave propagates through the material. The particular acoustoelastic technique considered here employs longitudinal waves propagating normal to the plane of the stress. Such waves experience a shift in propagation speed which, for an isotropic material, is proportional to the sum of the principal stresses. A Poisson’s equation for the in-plane shear stress is obtained from the two-dimensional equilibrium equations in which the forcing function is obtained directly from the measured velocity variations. Once this equation is integrated for the shear stress, the normal stresses may be evaluated directly from the equilibrium equations. In this paper, the basic equations are derived for the case of an anisotropic material. The experimental and numerical procedures are reviewed, and results of residual stresses in an aluminum ring are presented.

Axisymmetric end problems of longitudinal wave propagation are studied in a semi-infinite isotropic solid circular cylinder which is free of traction on its cylindrical surface. An accurate and computationally efficient method of solution is presented which can exploit the asymptotic behavior for high harmonics in the radial direction. The stresses and displacements are expanded in terms of the eigenfunctions of the case of a lubricated-rigid cylindrical surface condition. The expansions are used to construct a stiffness matrix relating the harmonics of stress and displacement for the traction-free case, which is shown to approach asymptotically that of the case of the mixed condition. Unlike other approaches such as finite element or boundary integral methods, which typically require the solution of large systems of equations for rapidly varying end conditions, the present formulation can lead to a coupled system of equations for lower spatial harmonics and a weakly coupled system for higher spatial harmonics. Due to the small number of equations in the coupled system, the present approach is very effective in handling general boundary conditions, and is particularly efficient for end conditions with rapid spatial variation.

This article describes the measurement and analysis of plate and soil response under low velocity impact. A free-drop impact system was developed to generate the dynamic loading on the plate free surface. The radial strain of the target plate, the longitudinal wave speed and the acceleration of the sand were measured. The measured wave speed data were then used to evaluate the elastic constants of the sand. An analysis based on linear elastodynamics was developed for transient waves on a thin plate resting on an elastic half space. The contact stresses and the normal displacements of the plate were taken as unknown functions. The contact between the plate and the half space were assumed frictionless. The experimental results of the radial strain at the bottom of the target plate and the acceleration of the sand beneath the center of the target plate were compared with the analytical solution. The arrival time, the duration, and the magnitude have good correlation between the analysis and experiment. The overall results appear good and provide an understanding of the transmission of impact load through the plate, the interaction between the plate and the sand, and the propagation of the load into the sand.

The equations governing three-dimensional elastodynamic scattering from planar cracks are formulated and solved in the frequency domain by Boundary Integral Equation (BIE) methods. The formulation requires a regularization of all nonintegrable kernels in the representation integral for the scattered stress field. The regularization procedure is novel in that it requires an initial discretization of the crack. The resulting discretized system of integral equations can be solved explicitly for the unknown crack-opening displacements. The crack-opening displacements, in conjunction with the appropriate representation integral, have been used to calculate far-field quantities of physical interest. Numerical results are compared with those from earlier papers dealing with a penny-shaped crack under normal incidence of a longitudinal wave field. The code has also been applied to elliptic crack geometries of various aspect ratios under normal longitudinal wave incidence. Numerical results are given for crack-opening displacements, scattered far fields, and scattering cross sections. The numerical results indicate that at sufficiently high frequencies, the scattering process is significantly affected by the extra length parameter of the elliptic crack.

Propagation of longitudinal waves in isotropic homogeneous elastic plates is studied in the context of the linear theory of nonlocal continuum mechanics. To determine the nonlocal moduli, the dispersion equation obtained for the plane longitudinal waves in an infinite medium is matched with the parallel equation derived in the theory of atomic lattice dynamics. Using the integroalgebraic representation of the stress tensor and the Fourier transform, the system of two coupled differential field equations is solved in the standard manner giving the frequency equations for the symmetric and antisymmetric wave modes. It is found that the short wave speed in the Poisson medium differs by about 13 percent from the speed established in the classical theory. A numerical example is given.

A general formulation is presented for the analysis of stress wave propagation through the junction of rectangular bars. The analysis is applied to the case of two bars meeting at right angles and is used to theoretically predict the passage of longitudinal waves through the junction. An experimental investigation of the phenomenon, using dynamic photoelasticity is conducted with a high-speed multiple spark gap camera of the Cranz-Schardin type. Three different geometries are tested to represent the most common types of junctions encountered in practice. In each of the cases, experimentally obtained results are observed to be very consistent with the theoretical predictions.

The problem of the diffraction of normally incident longitudinal waves on a Griffith crack located in an infinite soft ferromagnetic elastic solid is considered. It is assumed that the solid is a homogeneous and isotropic one and is permeated by a uniform magnetostatic field normal to the crack surfaces. Fourier transforms are used to reduce the problem to two simultaneous dual integral equations. The solution to the integral equations is expressed in terms of a Fredholm integral equation of the second kind having the kernel that is a finite integral. The dynamic singular stress field near the crack tip is obtained and the influence of the magnetic field on the dynamic stress intensity factor is shown graphically in detail. Approximate analytical expressions valid at low frequencies are also obtained and the range of validity of these expressions is examined.

The interaction of a longitudinal wave with a narrow cavity or slot that contains an inclusion is considered. A singular integral equation is derived for the scattered field and the integral equation is solved numerically by a Gauss-Chebyshev technique. The stress intensity at the ends of the slot is obtained as a function of frequency and inclusion stiffness for both a continuous and discrete element inclusion.

The shadow spots which are obtained in using the optical method of caustics to experimentally determine dynamic stress-intensity factors are usually interpreted on the basis of a static elastic crack model. In this paper, an attempt is made to include both crack-tip plasticity and inertial effects in the analysis underlying the use of the method in reflection. For dynamic crack propagation in the two-dimensional tensile mode which is accompanied by a Dugdale-Barenblatt line plastic zone, the predicted caustic curves and corresponding initial curves are studied within the framework of plane stress and small scale yielding conditions. These curves are found to have geometrical features which are quite different from those for purely elastic crack growth. Estimates are made of the range of system parameters for which plasticity and inertia effects should be included in data analysis when using the method of caustics. For example, it is found that the error introduced through the neglect of plasticity effects in the analysis of data will be small as long as the distance from the crack tip to the initial curve ahead of the tip is more than about twice the plastic zone size. Also, it is found that the error introduced through the neglect of inertial effects will be small as long as the crack speed is less than about 20 percent of the longitudinal wave speed.

A method for studying the behavior of transient longitudinal waves in fiber-reinforced composites, both in high and low frequency ranges, is presented. The results agree well with the solutions obtained by means of the transform method. The dispersive effect of waves to the interpretation of experimental results is also demonstrated by comparing the results obtained from the ultrasonic pulse technique.

Elastodynamic stress-intensity factors accompanying the three-dimensional steady-state elastodynamic response of an unbounded solid containing a semi-infinite crack, are investigated in this paper. Analytic solutions are obtained by application of the Fourier transform technique in conjunction with the Wiener-Hopf method. Two specific examples are worked out. For the diffraction of a plane longitudinal wave, which is incident under an arbitrary angle with the edge of the crack, explicit expressions are presented for the Modes I, II, and III stress-intensity factors. The variation of these quantities with the direction of the incident wave was worked out in detail, and the results are displayed in several figures. The second example deals with the elastodynamic field generated by the application of normal point loads of equal magnitude but opposite sense to the surfaces of the crack. For this case relatively simple approximate expressions are derived for the Mode I stress-intensity factor.

A rigid rectangular foundation embedded in an elastic half space moves in a direction perpendicular to the surface of the half space, Fig. 1. The model under consideration is of the plane-strain type. By application of the Laplace, Fourier, and Kontorovich-Lebedev (K-L) transforms, the equation of motion for the foundation is derived. The transient response of the foundation is exact during the period of time required for a longitudinal wave to traverse the base of the foundation twice. Thus the process of multiple diffractions at the corners of the foundation is taken into account.

A precursor boundary concept is introduced which decouples the stress-free boundary conditions at the parallel surfaces of the plate. The resulting orthogonal eigenfunctions are longitudinal and transverse waves. This property is exploited so as to satisfy stress-free conditions at the edge. The theoretical analysis reveals that the reflected waves are reordered so that the longitudinal and transverse modes represent the gradients of the incident transverse and longitudinal waves.

A numerical study of plane longitudinal waves in a nonlinear viscoelastic material is presented. The constitutive relationship and the conservation equations, in Lagrangian form, are formulated in an explicit first-order finite-difference manner. The mechanical behavior of the material is described by means of state and orientation variables and the associated differential equations. With the use of the numerical procedure we model wave-propagation experiments in polymethyl methacrylate and derive a constitutive relationship for that material. We then use this constitutive equation in a numerical study of the evolution of steady-state waves and we show that the time for the formation of these waves is inversely proportional to magnitude of the imposed velocity.