Mathematics and Mechanics of Solids recent issues

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.

On Saint-Venant's Principle for Elasto-Plastic Bodies

Knops, R.J., Villaggio, P. — Fri, 14 Aug 2009 02:29:27 PDT

This paper concerns Zanaboni’s version of Saint-Venant’s principle, which states that an elongated body in equilibrium subject to a self-equilibrated load on a small part of its smooth but otherwise arbitrary surface, possesses a stored energy that in regions of the body remote from the load surface decreases with increasing distance from the load surface. We here prove this formulation of Saint-Venant’s principle for elastic-plastic bodies. The present proof, which for linear elasticity considerably simplifies that developed by Zanaboni, depends crucially upon the principle of minimum strain energy to obtain a fundamental inequality that leads to the required result. Differential inequalities are not involved. The conclusion is not restricted to cylinders but is valid for plastic bodies of general geometries. Although no conditions are imposed on the plastic theories discussed here, counter-examples indicate that in certain circumstances the fundamental inequality, and hence the result, may be valid only for restricted data that includes the body’s minimum length.

Uniqueness and Growth of Solutions in Two-Temperature Generalized Thermoelastic Theories

Magana, A., Quintanilla, R. — Fri, 14 Aug 2009 02:29:27 PDT

In this work we study modifications of the non-classical models of thermoelasticity, the one proposed Green and Lindsay and the one stated by Lord and Shulman, to two-temperature setting. We prove uniqueness results for the solutions of the systems of equations that model both theories for isotropic material. We also provide growth estimates for the solutions.

Enhanced Stability of Coherently Strained Conical Quantum Dot Arrays Against Coarsening: Beyond the Small Slope Assumption

Gill, S.P.A. — Fri, 14 Aug 2009 02:29:27 PDT

An analytical model for the elastic energy of a system of conical heteroepitaxial quantum dots of finite slope is presented. An expression for the surface tractions at the dot—substrate interface is proposed. This includes a singularity in the stress field at the radial perimeter of the dot. The significance of this singularity increases as the slope of the dot increases. This dramatically increases the elastic interaction between dots. The stability of a hexagonal array of dots is found to be highly dependent on the strength of the stress singularity, with a system of highly sloped dots predicted to be metastable at much lower coverages than previously predicted. This could help explain the stability of bimodal island size distributions observed in some quantum dot systems.

The Eigenfrequencies of the Dirichlet and Neumann Problems for an Oscillating Finite Plate

Thomson, G.R., Constanda, C. — Fri, 14 Aug 2009 02:29:27 PDT

The Green’s tensors are constructed and the existence of discrete real spectra is proved for the displacement and traction boundary value problems in the case of a finite elastic plate with transverse shear deformation undergoing high-frequency vibrations.

An Abstract Framework for Elliptic Inverse Problems: Part 2. An Augmented Lagrangian Approach

Gockenbach, M. S., Khan, A. A. — Wed, 22 Jul 2009 02:12:15 PDT

The coefficient in a linear elliptic partial differential equation can be estimated from interior measurements of the solution. Posing the estimation problem as a constrained optimization problem with the PDE as the constraint allows the use of the augmented Lagrangian method, which is guaranteed to converge. Moreover, the convergence analysis encompasses discretization by finite element methods, so the proposed algorithm can be implemented and will produce a solution to the constrained minimization problem. All of these properties hold in an abstract framework that encompasses several interesting problems: the standard (scalar) elliptic BVP in divergence form, the system of isotropic elasticity, and others. Moreover, the analysis allows for the use of total variation regularization, so rapidly-varying or even discontinuous coefficients can be estimated.

Plastic Deformation of Bicrystals Within Continuum Dislocation Theory

Kochmann, D.M., Le, K.C. — Wed, 22 Jul 2009 02:12:15 PDT

Within continuum dislocation theory the plastic deformation of bicrystals under plane strain constrained shear is considered. An analytical solution is found in the symmetric case (for twins) which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. Similar features hold true also for the numerical solution in the general case.

Binary Mixtures of Elastic Solids with Microstructure

Iesan, D. — Wed, 22 Jul 2009 02:12:15 PDT

In this paper we establish a nonlinear theory of binary mixtures of elastic solids with microstructure. The independent constitutive variables are the displacement fields, displacement gradients, microdeformation tensors and their gradients. The basic equations are derived in Lagrangian description. The theory is linearized and a uniqueness theorem with no definiteness assumption on the constitutive coefficients is presented. The theory is used to study a special kind of microstructure in which the microdeformation tensor is isotropic. The problem of a concentrated body moment is investigated.

Surface Transport in Continuum Mechanics

Fosdick, R., Huang Tang, — Wed, 22 Jul 2009 02:12:15 PDT

A moving surface is considered as a 3-dimensional submanifold in the 4-dimensional space-time setting. Elementary differential geometry is used to identify parameter-time and parameter independent normal-time derivatives, and their differences. A few elementary observations are made concerning surface transport theorems.

The Divergence Theorem for Divergence Measure Vectorfields on Sets with Fractal Boundaries

Silhavy, M. — Wed, 24 Jun 2009 07:28:00 PDT

A divergence measure vectorfield is an Rⁿ valued measure on an open subset U of Rⁿ whose weak divergence in U is a (signed) measure. The paper uses the product rule for the product of the divergence measure by a function from W ^1, (U) 3 established in Silhavy [Silhavy, M., submitted, 2007] to prove the divergence theorem for the divergence measure vectorfields on bounded open sets U. It is shown that the surface integral of the normal component of the vectorfield occurring in the classical divergence theorem has to be replaced by a continuous linear functional on the space of Lipschitz functions on the boundary1 the volume integral contains the duality pairing occurring in the product rule. The boundary of U is arbitrary, it can be even fractal in the sense that the normal to U cannot be defined.

A Theory of the Mechanics of Two Coupled Surfaces

Nadler, B. — Wed, 24 Jun 2009 07:28:00 PDT

In this work the mechanics of two coupled membrane-like surfaces are considered. It will be shown that under special restrictions the finite deformation of the two coupled surfaces can be described using only five field parameters. This is accomplished by introducing a pairing between the two surfaces. The surfaces are permitted to slip with respect to each other subjected to frictional slipping constitutive law, but restricted to maintain full contact at all time. Such a model can be used to model frictional slip in woven fabrics. The weak form of the equations are formulated to be used with the finite element method. This theory furnishes equations of motion and boundary conditions which have clear physical meaning.

On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework

Holzapfel, G. A., Ogden, R. W. — Wed, 24 Jun 2009 07:28:00 PDT

The mechanical testing of anisotropic nonlinearly elastic solids is a topic of considerable and increasing interest. The results of such testing are important, in particular, for the characterization of the material properties and the development of constitutive laws that can be used for predictive purposes. However, the literature on this topic in the context of soft tissue biomechanics, in particular, includes some papers that are misleading since they contain errors and false statements. Claims that planar biaxial testing can fully characterize the three-dimensional anisotropic elastic properties of soft tissues are incorrect. There is therefore a need to clarify the extent to which biaxial testing can be used for determining the elastic properties of these materials. In this paper this is explained on the basis of the equations of finite deformation transversely isotropic elasticity, and general planar anisotropic elasticity. It is shown that it is theoretically impossible to fully characterize the properties of anisotropic elastic materials using such tests unless some assumption is made that enables a suitable subclass of models to be preselected. Moreover, it is shown that certain assumptions underlying the analysis of planar biaxial tests are inconsistent with the classical linear theory of orthotropic elasticity. Possible sets of independent tests required for full material characterization are then enumerated.

Response of Anisotropic Nonlinearly Viscoelastic Solids

Rajagopal, K.R., Wineman, A.S. — Wed, 24 Jun 2009 07:28:00 PDT

Despite the technological relevance of anisotropic nonlinear viscoelastic solids, little effort has been expended in the development of specific constitutive theories. In this study we develop a constitutive model for describing the nonlinear response of anisotropic viscoelastic solids that might be well suited to describe the response of biological and geological solids. The model is an integral model that takes into account the history of deformation of the body. Using the model a few boundary value problems are studied, namely the time dependent extension and shearing of such bodies.

On the Coupling of Guided Waves Propagation in Piezoelectric Crystals Subject to Initial Fields

Simionescu-Panait, O., Ana, I. — Wed, 24 Jun 2009 07:28:00 PDT

This paper deals with the study of the coupling conditions for propagation of planar guided waves in a piezoelectric semi-infinite plane (called sagittal plane) subject to initial electro-mechanical fields. The piezoelectric material behaves linearly and without attenuation and the waveguide propagates in a normal mode. We suppose that the material is subject to initial electro-mechanical fields. If the sagittal plane is normal to a direct, resp. inverse dyad axis, we derive that the fundamental equations' system decomposes for particular choices of the initial electric field. In this way we obtain mechanical and piezoelectric waves generalizing the classical guided waves from the case without initial fields. Furthermore, we obtain a similar decomposition of mechanical and electrical boundary conditions, which enable us to characterize the obtained guided waves.