Ali R. Hadjesfandiari

Theory of Boundary Eigensolutions in Engineering Mechanics

A theory of boundary eigensolutions is presented for boundary value problems in engineering mechanics. While the theory is quite general, the presentation here is restricted to potential problems. Contrary to the traditional approach, the eigenproblem is formed by inserting the eigenparameter, along with a positive weight function, into the boundary condition. The resulting spectra are real and the eigenfunctions are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of nonsmooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the convergence characteristics are also introduced. Furthermore, the connection is made to integral equation methods and variational methods. This paves the way toward the development of new computational formulations for finite element and boundary element methods. Two numerical examples are included to illustrate the applicability.

Size-dependent thermoelasticity

In this paper a consistent theory is developed for size-dependent thermoelasticity in heterogeneous anisotropic solids. This theory shows that the temperature change can create not only thermal strains, but also thermal mean curvatures in the solids. This formulation is based on the consistent size-dependent continuum mechanics in which the couple-stress tensor is skew-symmetric. Here by including scale-dependent measures in the energy and entropy equations, the general expressions for force- and couple-stresses, as well as entropy density, are obtained. Next, for the linear material the constitutive relations and governing coupled size-dependent thermoelasticity equations are developed. For linear material, one can see that the thermal properties are characterized by the classical symmetric thermal expansion tensor and the new size-dependent skew-symmetric thermal flexion tensor. Thus, for the most general anisotropic case, there are nine independent thermoelastic constants. Interestingly, for isotropic and cubic materials the thermal flexion tensor vanishes, which shows there is no thermal mean curvature

Boundary eigensolutions in elasticity II. Application to computational mechanics

The theory of fundamental boundary eigensolutions for elastostatic problems, developed in Part I, is applied to formulate methods for computational mechanics. This theory shows that every elastic solution can be written as a linear combination of some fundamental boundary orthogonal deformations, thus providing a generalized Fourier expansion. One finds that traditional boundary element and finite element methods are largely consistent with this theory, but do not harness its full power. This theory shows that these computational methods are indirectly a generalized discrete Fourier analysis. Furthermore, by utilizing suitable boundary weight functions, boundary element and finite element formulations may be written exclusively in terms of bounded quantities, even for non-smooth problems involving notches, cracks, mixed boundary conditions and bi-material interfaces. The close relationship between the resulting boundary element and finite element methods also becomes evident. Both use displacement and surface traction as primary variables. A new degree-of-freedom concept is introduced, along with a stiffness tensor that enables one to visualize a finite element method via a boundary discretization process, just as in a boundary element approach. Global convergence characteristics of the traction-oriented finite element method are also developed. Comparisons with closed-form fundamental boundary eigensolutions for a circular elastic disc are presented in order to provide a means for assessing the numerical methods. Several other numerical examples are solved efficiently by using the concept of boundary eigensolutions in an indirect fashion. The results indicate that the algorithms follow the underlying theory and that solutions to non-smooth problems can be obtained in a systematic manner. Beyond this, the concept of boundary eigensolutions provides an alternative view of computational continuum mechanics that may lead to the development of other non-traditional approaches.

Boundary eigensolutions in elasticity. I. Theoretical development

The theory of fundamental boundary eigensolutions for elastostatic boundary value problems is developed. The underlying fundamental eigenproblem is formed by inserting the eigenparameter and a tensor weight function into the boundary condition, rather than into the governing differential equation as is often done for vibration problems. The resulting spectra are real and the eigenfunctions (eigendeformations) are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of non-smooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the behavior of eigensolutions in the domain, integral equation methods, variational methods, convergence characteristics, flexibility and stiffness kernels, and solutions to problems with body forces are also introduced. Of particular note are the integral equation and variational formulations that lead to the development of new computational formulations for boundary element and finite element methods, respectively. An example with closed form and numerical results is included to illustrate some aspects of the theory.

On the symmetric character of the thermal conductivity tensor

In this paper, the symmetric character of the conductivity tensor for linear heterogeneous anisotropic material is established as the result of arguments from tensor analysis and linear algebra for Fourier’s heat conduction. The non-singular nature of the conductivity tensor plays the fundamental role in establishing this statement.

Boundary Element Analysis of Thermoelastic Effects in Size-Dependent Mechanics

A new boundary element formulation is developed to analyze two-dimensional size-dependent thermoelastic response in linear isotropic couple stress materials. The model is based on the recently developed consistent couple stress theory, in which the couple-stress tensor is skew-symmetric. The size-dependency effect is specified by one characteristic parameter length scale l, while the thermal effect is quantified by the classical thermal expansion coefficient α and conductivity k. We discuss the boundary integral formulation and numerical implementation of this size-dependent thermoelasticity boundary element method (BEM). Then, we apply the resulting BEM formulation to a computational example to validate the numerical implementation and to explore thermoelastic couplings as the non-dimensional characteristic scale of the problem is varied. Interestingly, for a cantilever beam with a transverse temperature gradient, we find significantly reduced non-dimensional tip deflections as the beam depth h approaches the material characteristic length scale l. On the other hand, when l/h < 0.01, the classical size-independent deflections are recovered.Copyright

Boundary Element Analysis of Interface Cracks

A new boundary element formulation is developed to determine the complex stress intensity factors associated with cracks on the interface between dissimilar materials. This represents an extension of the methodology developed previously by the authors for determination of free-edge generalized stress intensity factors on bimaterial interfaces, which employs displacements and weighted tractions as primary variables. However, in the present work, the characteristic oscillating stress singularity is addressed through the introduction of modified weighting functions and corresponding numerical quadrature formulas. As a result, this boundary-only approach provides extremely accurate mesh-independent solutions for a range of interface crack problems. A number of computational examples are considered to assess the performance of the method in comparison with analytical solutions and previous work on the subject.Copyright