Mathematics and Mechanics of Solids current issue

Finite Deformations and Motions of a Pre-stressed Incompressible Elastic Tube

Carroll, M.M. — Fri, 16 Oct 2009 01:46:28 PDT

Rivlin’s exact solution for finite bending of a rectangular block of incompressible isotropic elastic material into a circular cylindrical sector is specialized to the case of complete bending, in which two ends of the block are brought together. These ends may be glued together to form a circular cylindrical tube without introducing any stress discontinuity. Several boundary value problems that admit exact solutions, radial inflation or compaction, eversion, steady rotation, radial oscillation, torsion, azimuthal shearing and telescopic shearing, for a natural (unstressed) tube admit similar exact solutions for the pre-stressed tube. Except for the last two shearing deformations, these solutions for both the natural tube and the pre-stressed tube are independent of the strain energy. The availability of similar exact solutions for a naturally circular tube and for a pre-stressed tube that was originally a rectangular block should provide useful information about the effects of pre-stress on non-linearly elastic response.

The Numerical Computation of the Critical Boundary Displacement for Radial Cavitation

Negron-Marrero, P. V., Sivaloganathan, J. — Fri, 16 Oct 2009 01:46:28 PDT

We study radial solutions of the equations of isotropic elasticity in two dimensions (for a disc) and three dimensions (for a sphere). We describe a numerical scheme for computing the critical boundary displacement for cavitation based on the solution of a sequence of initial value problems for punctured domains. We give examples for specific materials and compare our numerical computations with some previous analytical results. A key observation in the formulation of the method is that the strong—ellipticity condition implies that the specification of the normal component of the Cauchy stress on an inner pre—existing but small cavity, leads to a relation for the radial strain as a function of the circumferential strain. To establish the convergence of the numerical scheme we prove a monotonicity property for the inner deformed radius for punctured balls.

An Alternative Form of Propagation Criterion for Two Collinear Cracks under Compression

Zhu, Z. — Fri, 16 Oct 2009 01:46:28 PDT

Under compression, cracks extend, branch and coalesce. These fracturing processes have received much attention recently. In this paper, an attempt is made to find the analytical solution of stress intensity factors for the special case of cracks situated along a straight line, and to set up a fracture criterion. Under compression, cracks close and the crack surface friction can resist crack surface sliding. Considering crack surface friction, a set of complex stress functions is proposed for the special case of cracks situated along a straight line. The analytical solution is formulated, and for the case of only two collinear cracks inside an infinite plate, the exact analytical solution of stress intensity factor is presented. Finally, an alternative form of crack propagation criterion for two collinear cracks under compression is developed, which is expressed in terms of principal stresses. For the case of materials without pre-existing macrocracks, this new propagation criterion becomes the well known Coulomb—Mohr criterion.

Analytical Solution for a Pressurized Thick-Walled Spherical Shell Based on a Simplified Strain Gradient Elasticity Theory

Gao, X.-L., Park, S.K., Ma, H.M. — Fri, 16 Oct 2009 01:46:28 PDT

The problem of a pressurized thick-walled spherical shell is analytically solved using a simplified strain gradient elasticity theory. The closed-form solution derived contains a material length scale parameter and can account for microstructural effects, which qualitatively differs from Lamé’s solution in classical elasticity. When the strain gradient effect (a measure of the underlying material microstructure) is not considered, the newly derived strain gradient elasticity solution reduces to Lamé’s classical elasticity solution. To illustrate the new solution, a sample problem with specified geometrical parameters, pressure values and material properties is solved. The numerical results reveal that the magnitudes of both the radial and tangential stress components in the shell wall given by the current strain gradient solution are smaller than those given by Lamé’s solution. Also, it is quantitatively shown that microstructural effects can be large and Lamé’s solution may not be accurate for materials exhibiting significant microstructure dependence.