G. Herrmann

Conservation laws and the material momentum tensor for the elastic dielectric

Abstract It is shown that a Langrangian formulation of continuum mechanics can provide not only the equations of motion, but the conservation laws related to the material symmetries in a perfect continuum interacting with an external electric field. These conservation laws in the presence of defects lead to the path-independent integrals widely used in fracture mechanics. They are basically related to the “material force” on a defect in a continuum. The quantity playing the role of the physical stress tensor in this formulation is the material momentum tensor. A material force in the form of a path-independent integral for the elastic dielectric is derived employing Toupins [1] formulations.

Weakening of an elastic solid by a rectangular array of cracks

An infinite elastic solid containing a double-periodic rectangular array of slit-like cracks is considered. The solid is subjected to a uniform stress resulting in a state of plane strain. The cracks are represented as suitable distributions of dislocations which are determined from a singular integral equation. This equation is solved numerically in an efficient manner using an expansion of the non-singular part of the kernel in a series of Chebyshev polynomials. Values of the stress intensity factors are presented, as well as the change in strain energy due to the presence of the cracks. Also, the effective elastic constants of a sheet having a rectangular array of cracks are given as functions of the crack spacing.

Floquet waves in anisotropic periodically layered composites

The field equations governing the plane harmonic elastic motions of anisotropic stratified media are written in the form of a matrix system of differential equations, where the dependent variables are the displacements and tractions acting across planes normal to the direction of stratification. In the case of periodically layered media, Floquet’s theorem and the propagator matrix method can be applied to solve the governing sextic matrix equation. In the absence of sources or body forces, the general motion of the layered medium is described by a combination of six partial waves (Floquet waves). A closed‐form algebraic solution for the dispersion equation of such waves is derived. Numerical results describing the dispersion spectrum of a cross‐ply periodic laminated are discussed in detail.

Conservation laws in nonhomogeneous plane elastostatics

Abstract Conservation laws in nonhomogeneous elastostatics are studied by means of a special version of Noethers theorem on invariant variational problems. The investigation is restricted to isotropic, linearly elastic bodies with smoothly varying elastic moduli and subjected to plane strain deformation fields. By applying Lies infinitesimal criterion on the invariance of the action integral under a continuous transformation group, it is found that the materials admitting conservation laws are those for which the elastic parameters satisfy a first-order linear partial differential equation. While the general solution to this equation is readily available, a special emphasis is placed on extending Rices J -integral to certain classes of materials with varying Youngs modulus in the direction of a crack line. The extended integrals are related to stress intensity factors at the crack tip, and their use is illustrated by applying them to two simple examples. Also, the circumstances under which these integrals can be generalized to the three-dimensional case are briefly discussed.

An elementary theory of defective beams

SummaryMaterial conservation and balance laws of elementary beam theory have been derived. The application to beams with discontinuities in the stiffness results in a surprisingly simple formula to calculate stress intensity factors of cracked beams.

Wave Propagation in Piezoelectric Layered Media with Some Applications

In this article we introduce a systematic methodology to investigate wave propagation in piezoelectric layered media. It is based on a matrix formalism and the con cept of the surface impedance tensor which relates the components of particle displace ment and the normal component of the electric displacement along a surface to the electric potential and the components of traction acting along the same surface. Once the surface impedance tensor for a single layer is calculated, a simple recursive algorithm allows the evaluation of the surface impedance tensor for any number of layers. As an example, the surface impedance tensor is used to find the dispersion curves for a bilaminated piezoelec tric plate. Also the dispersion curves for the subsonic interfacial waves when the plate is in contact with a nonconducting acoustic fluid is investigated. Floquet theory is also ap plied to study wave propagation when the layered medium is periodic.

Acoustic measurements of stress fields and microstructure

A very precise system for measuring two-dimensional velocity fields in solid samples has been used for nondestructive measurements of both externally applied and residual inhomogeneous stresses in solids,J integrals, stress intensity factors of cracks, and hardness of quenched steel. The longitudinal velocity measurement is based on precise determination of the propagation transit time through the stressed solid specimen using a small diameter, water-coupled acoutic transducer, which is scanned mechanically over the sample. Changes in velocity are then related to changes of stress in the sample by the theory of acoustoelasticity. Similar measurements show a high degree of correlation between longitudinal velocity changes and changes in microstructure in steel samples. Applications to problems of solid mechanics and material science illustrate the utility of this nondestructive measuring technique.

On estimates of stress intensity factors for cracked beams and pipes

Abstract This paper presents a discussion and an extension of a method, advanced by Kienzler and Herrmann [ Acta Mech . 62 , 37–46 (1986)], of evaluating the stress intensity factors of cracked beams based on an elementary beam theory estimation of strain energy release rate as the crack is widened into a fracture band. Recently, Bazant [ Engng Fracture Mech . 36 , 523–525 (1990)] justified the approach taken by Kienzler and Herrmann as a good approximation by showing that the strain energy release rate for crack extension is closely related to, but usually not equal to, the energy release for crack widening into a band. This indicates that an additional factor needs to be introduced to improve the approximations. We show here that this factor can be obtained through asymptotic matching with standard limiting crack solutions. As an extension, we use the beam theory estimation to compute stress intensity factors for a circumferentially cracked cylindrical pipe in bending and tension and find good agreement with the exact solutions obtained from complete shell analysis. Applications of the beam theory analysis are under progress for numerous other crack problems of practical interest.

An integral equation method and its application to defect mechanics

Combining the body force method with the complex stress function theory, a complex integral equation method is presented to study interaction problems between holes and other defects in an infinite or semi-infinite elastic plane. A closed form solution of the integral equation is obtained for the generalized Kirsch problem in terms of the Laurent series expansion technique. This solution leads to a very simple formula to calculate the distribution of hoop stresses at the boundary of a circular hole in a plane subjected to arbitrary loads. To show the application of the integral equation method, three specific problems are treated analytically and/or numerically. Investigation of the asymptotic behaviour of stress distribution in neighbouring defects is of special interest. It was found that the hole to hole and hole to boundary interaction are governed by a 1√ϵ singularity, where ϵ stands for a non-dimensional distance between the holes and the hole or the boundary, respectively.

Further aspects of the elastic field for two circular inclusions in antiplane elastostatics

The heterogenization technique is applied to the problem of two circular inclusions of arbitrary radii and of different shear moduli and perfectly bonded to a matrix, of infinte extent, subjected to arbitrary loading