Morris Stern

A general boundary integral formulation for the numerical solution of plate bending problems

Abstract A direct approach is employed to obtain a general formulation of plate bending problems in terms of a pair of singular integral equations involving displacement, normal slope, bending moment and shear on the plate boundary. These equations are coupled with prescribed boundary conditions involving these same variables furnishing a convenient basis for numerical solution. A simple discretization scheme is described and two model problems treated to illustrate the nature and quality of some typical results.

The computation of stress intensity factors in dissimilar materials

A reciprocal work contour integral method for calculating stress intensity factors is extended to treat the problem of two bonded dissimilar materials containing a crack along the bond. The method is based on Bettis Reciprocal work theorem from which the singular stress intensities at the crack tip may be evaluated in terms of an integral involving tractions and displacements on a contour remote from the crack tip.

On the computation of stress intensities at fixed-free corners

Abstract A path independent integral formula is developed for the computation of the intensity of the stress singularity at a right corner where one edge is rigidly fixed and the other is free of traction. Numerical results are presented for the case of a strip compressed between rough rigid stamps and compared with previously published results for finite and semi-infinite strips and cylinders.

Wave reflection from a sediment layer with depth‐dependent properties

The reflection of plane acoustic waves at the water–sediment interface is analyzed. The sediments are modeled using Biot’s equations with depth‐dependent coefficients. Computations are made of the reflection coefficient as a function of the incident wave frequency and angle for representative sediment properties. The results are compared to those obtained by modeling the sediments as a homogeneous viscoelastic material and as a viscoelastic material with depth‐dependent properties. It is shown that both of these models yield predictions of the reflection coefficient that can differ substantially from the Biot model.

The numerical calculation of thermally induced stress intensity factors

A reciprocal work contour integral method for calculating the stress intensity factor at a crack on the interface between two dissimilar materials was given recently in [1]; other cases were treated in previous work [2, 3, 4]. All of the cases involved plane elastostatic problems in the absence of body force and thermal stresses. However, especially in the case of bonded dissimilar materials, thermally induced stresses are a common occurrence. In this note the contour integral method is extended to include the treatment of such problems. Let ll be a (two or three dimensional) region occupied by an isotropic linear elastic body with boundary 0II. Referred to an unstressed natural state at a uniform reference temperature, we denote by u the displacement, T the stress tensor, and 4~ the temperature increase resulting from a given loading and heating of the body. The equations governing the (uncoupled) static mechanical response of the body consist of the equilibrium equation

Boundary integral equations for bending of thin plates

There are a number of different ways to approach the formulation of boundary integral equations for plate bending. While in some sense these are equivalent (if correctly done) the differences becomes quite significant when the resulting formulations are implemented in the numerical solution of specific boundary value problems. Most early proposals for the direct numerical solution of plate bending boundary integral equations were based on so-called indirect formulations and generally were designed for specific classes of problems. One of the earliest significant examples is due to Jaswon and Maiti1 who propose a formulation for uniformly loaded clamped and simply supported plates based on the introduction of two source distribution densities on the plate boundary generating harmonic potentials which are then related to the plate displacement. A somewhat different formulation of the same type to treat uniformly loaded simply supported polygonal plates was proposed by Maiti and Chakrabarty2. Hansen3 derived two different boundary integral formulations designed mainly for plates containing holes with free edges.

An approximate shell theory for unrestricted elastic deformations

Abstract An approximate shell theory is formulated within the framework of the general theory of finite elasticity for unrestricted deformation. A deformation field is constructed throughout the shell based on the solution of the corresponding membrane problem. An additional deformation is then superposed on this “membrane state” and the resulting equations of motion for the final state linearized in the additional displacement. These equations are then “averaged” through the thickness of the shell to yield an approximate shell theory. Details are carried through for the case in which the additional deformation is itself linearized in the thickness variable so that normals to the middle surface remain straight and uniformly extended, but not necessarily normal. The resulting theory is then applied to the problem of a uniformly twisted, extended and inflated cylindrical shell.

An acoustic model of a laminar sand bed

It was postulated that a laminar sand bed may be modeled as an ensemble of randomly layered Biot media. The thickness of each layer was approximately half a grain diameter. The porosity variations in the vertical direction were matched to the mean and standard deviation of that of a structure of packed spherical grains. The effect of large-scale lateral variations in porosity was simulated by performing a coherent ensemble average of the acoustic output from several realizations of the randomly layered medium. The medium parameters were chosen to represent water-saturated sand. Specifically, the sand bed was modeled as bounded by a homogeneous water halfspace above, and a homogeneous poroelastic halfspace of equal average porosity below. Reflected and transmitted signals were computed. Coherent and random components of the reflected signal were calculated. The coherent parts were directly related to the reflected and the transmitted waves. Results showed significant differences between the modeled sand bed and an equivalent uniform Biot medium. In the modeled sand bed, the fast wave attenuation was found to be anisotropic, and a propagating slow wave was excited at most incident angles, except at normal incidence, which may explain the apparent failure to detect the slow wave of certain experiments.

Reflection and propagation loss over porous sediments

A full wave acoustic propagation model in a stratified viscoelastic medium [H. Schmidt and F. B. Jensen, J. Acoust. Soc. Am. 77, 813–825 (1985)] was combined with Biot’s theory for propagation and reflection in a porous medium [Stern et al., J. Acoust. Soc. Am. 77, 1781–1788 (1985)] to produce a model of acoustic propagation in a stratified porous medium. Sample results were compared with a normal mode model [T. Yamamoto, J. Acoust. Soc. Am. 73, 1587–1596 (1983)]. The model was extended to account for the effect of gas bubbles in the pore fluid [J. A. Hawkins and A. Bedford, J. Acoust. Soc. Am. Suppl. 1 88, S131 (1990)]. Significant differences between viscoelastic, poroelastic, and gassy poroelastic models were investigated. [Work supported by ONR.]

Coupling Boundary Integral and Finite Element Formulations for Nonlinear Halfspace Problems

In finite element applications involving static or quasistatic nonlinear inelastic response in unbounded regions, there is the question of how to truncate the region to be modeled and how to select appropriate boundary conditions to be imposed on the remote boundary. An example of an application in which this problem arises is the determination of the history of the deformation of an underground tunnel due to creep. Typically such problems are modeled by placing the exterior boundary “far enough away” so that the effect of fixing the boundary does not significantly alter the stress and deformation in areas of interest. There are two drawbacks to this approach. First, there is always some effect no matter how far away the boundary is placed, which makes it difficult for the analyst to judge precisely how much effect his choice of the location of the boundary is having on the solution. This is especially true in non- linear problems where intuition often fails. The second and perhaps more important drawback is that the boundary must be placed so far away from regions of interest that an unnecessarily large number of elements must be used, which increases the computational cost of the analysis.