Features of the stress field near tunnel inclusions in an inhomogeneous anisotropic space

This paper proposes a method to solve problems for interface tunnel defects in a piecewise-homogeneous elastic material that is under generalized plane strain and has no planes of elastic symmetry. The method is based on integral relations between the discontinuities and sums of the components of the displacement vector and stress tensor at the interface. Closed-form solutions are obtained for a system of interface tunnel inclusions with mixed contact conditions between the space and the inclusions. The dependences of the indices of singularity of the solutions on orthogonal coordinate transformation are established for different combinations of materials of monoclinic and orthorhombic systems. The effect of the antiplane component on the behavior of the solutions is revealed __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 36–45, June 2008.     

The Galois unsolvability of the sextic equation of anisotropic elasticity

By an approximate numerical application of Galois theory it is proved that the sextic equation of anisotropic elasticity for cubic symmetry is in general unsolvable in radicals, elementary transcendental functions, or elliptic modular functions and that its group is the full symmetric group. This implies the same unsolvability for tetragonal, orthorhombic, monoclinic, and triclinic symmetry. A separate investigation proves the same unsolvability for trigonal symmetry. Special cases of cubic symmetry which might have solvable equations are examined. Directions restricted to {111} or {112} planes give unsolvable equations, in contrast to {100} and {110} planes. Three additional classes of elastic constants which give solvable equations are found but only two limiting cases are physically possible. An extensive survey suggests that any further special elastic constants are rather unlikely.     

Analytical solutions for anisotropic bimaterials under several boundary conditions on the interface

Summary Analytical solutions are proposed for the stress and displacement fields induced by in-plane loading of a bimaterial under several boundary conditions. The two joined orthotropic layers forming the bimaterial are allowed to possess different types of anisotropy with their planes of elastic symmetry arbitrarily inclined with respect to the horizontal.     

Optimization of the strain energy density in linear anisotropic elasticity

The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.     

Identification of symmetry type of linear elastic stiffness tensor in an arbitrarily orientated coordinate system

We develop a method through the mirror plane (MP) to identify the symmetry type of linear elastic stiffness tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the irreducible decomposition of high-order tensor into a set of deviators and the multipole representation of a deviator into a scalar and a unit-vector set. Since a unit-vector depends on two Euler angles, we can illustrate the MP normals of the elastic tensor as zeros of a characteristic function on a unit disk and identify its symmetry immediately, which is clearer and simpler than the methods proposed before. Furthermore, by finding the common MPs of three unit-vector sets using Fortran recipes, we can also analytically recognize the symmetry type first and then recover the natural coordinate system associated with the linear elastic tensor. The structures of linear elastic stiffness tensors of real materials with all possible anisotropies are investigated in detail.     

Circles on the slowness surface of a cubic elastic material

It is well-known that either the outer or the medial sheet of the slowness surface of an elastic material with cubic symmetry intersects the cube faces in circles. In is shown here that there exist on the next sheet (medial or outer) three pairs of circles centred on the symmetry axes and situated in planes parallel to the cube faces.     

Identifying Symmetry Classes of Elasticity Tensors Using Monoclinic Distance Function

We present a method to identify the symmetry class of an elasticity tensor whose components are given with respect to an arbitrarily oriented coordinate system. The method is based on the concept of distance in the space of tensors, and relies on the monoclinic or transversely isotropic distance function. Since the orientation of a monoclinic or transversely isotropic tensor depends on two Euler angles only, we can plot the corresponding distance functions on the unit sphere in ℝ³ and observe the symmetry pattern of the plot. In particular, the monoclinic distance function vanishes in the directions of the normals of the mirror planes, so the number and location of the zeros allows us to identify the symmetry class and the orientation of the natural coordinate system. Observing the approximate locations of the zeros on the plot, we can constrain a numerical algorithm for finding the exact orientation of the natural coordinate system.     

The Remarkable Nature of Radially Symmetric Deformation of Spherically Uniform Linear Anisotropic Elastic Solids

Antman and Negron-Marrero [1] have shown the remarkable nature of a sphere of nonlinear elastic material subjected to a uniform pressure at the surface of the sphere. When the applied pressure exceeds a critical value the stress at the center r=0 of the sphere is infinite. Instead of nonlinear elastic material, we consider in this paper a spherically uniform linear anisotropic elastic material. It means that the stress-strain law referred to a spherical coordinate system is the same for any material point. We show that the same remarkable nature appears here. What distinguishes the present case from that considered in [1] is that the existence of the infinite stress at r=0 is independent of the magnitude of the applied traction σ0 at the surface of the sphere. It depends only on one nondimensional material parameter κ. For a certain range of κ a cavitation (if σ0>0) or a blackhole (if σ0<0) occurs at the center of the sphere. What is more remarkable is that, even though the deformation is radially symmetric, the material at any point need not be transversely isotropic with the radial direction being the axis of symmetry as assumed in [1]. We show that the material can be triclinic, i.e., it need not possess a plane of material symmetry. Triclinic materials that have as few as two independent elastic constants are presented. Also presented are conditions for the materials that are capable of a radially symmetric deformation to possess one or more symmetry planes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Elastic bodies reinforced with inextensible surfaces

The constraint of in-plane rigidity is examined within the general framework of the theory of internally constrained materials. It is shown that, for in-plane rigid materials, local strain and active stress are both defined by vectorial quantities. Representation formulae for the elastic response mapping are established in the cases of transverse isotropy and maximal symmetry, compatible with the constraint manifold. The equilibrium problem for an elastic body reinforced with parallel inextensible planes is also considered. In particular, universal solutions for bodies with maximal material symmetry are determined within the class of deformations which leave rigid every reinforcing plane.     

Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics

Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. This paper describes procedures for the computer implementation of the absolute nodal coordinate formulation' for flexible multibody applications. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The configuration of the finite element is defined using global displacement coordinates and slopes. By using this mixed set of coordinates, beam and plate elements can be treated as isoparametric elements. As a consequence, the dynamic formulation of these widely used elements using the absolute nodal coordinate formulation leads to a constant mass matrix. It is the objective of this study to develop computational procedures that exploit this feature. In one of these procedures, an optimum sparse matrix structure is obtained for the deformable bodies using the QR decomposition. Using the fact that the element mass matrix is constant, a QR decomposition of a modified constant connectivity Jacobian matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. An alternate computational procedure based on Cholesky decomposition is also presented in this paper. This alternate procedure, which has the same computational advantages as the one based on the QR decomposition, leads to a square velocity transformation matrix. The computational procedures proposed in this investigation can be used for the treatment of large deformation problems in flexible multibody systems. They have also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions.     

Third-order elastic,piezoelectric, and dielectric constants

The definitions of the third-order elastic, piezoelectric, and dielectric constants and the properties of the associated tensors are discussed. Based on the energy conservation and coordinate transformation, the relations among the third-order constants are obtained. Furthermore, the relations among the third-order elastic, piezoelectric, and dielectric constants of the seven crystal systems and isotropic materials are listed in detail.These third-order constants relations play an important role in solving nonlinear problems of elastic and piezoelectric materials. It is further found that all third-order piezoelectric constants are 0 for 15 kinds of point groups, while all third-order dielectric constants are0 for 16 kinds of point groups as well as isotropic material. The reason is that some of the point groups are centrally symmetric, and the other point groups are high symmetry.These results provide the foundation to measure these constants, to choose material, and to research nonlinear problems. Moreover, these results are helpful not only for the study of nonlinear elastic and piezoelectric problems, but also for the research on flexoelectric effects and size effects.     

A note on bi-orthogonality relations for elastic cylinders of general cross section

The bi-orthogonality relation satisfied by the elastodynamic (or elastostatic) eigenfunctions of a cylindrical rod of general cross section is obtained by a simple argument. The relation is shown to depend only upon (i) the elastic reciprocal theorem, and (ii) the elastic symmetry of the cylinder in planes perpendicular to its generators.This work was supported in part by N.R.C. grant No. A9117, while the author was visiting the University of British Columbia, Vancouver B.C. Canada.     

On the role of symmetry in Saint-Venant’s problem

In the relaxed Saint-Venant’s elastic problem, in virtue of Saint-Venant’s Postulate, the pointwise assignments of tractions at cylinder plane ends are replaced by the assignments of the corresponding resultant forces and moments. The solution indeterminacy so introduced is traditionally overcome by postulating that some specific features characterize the elastic state. In this work a relaxed incremental equilibrium problem is posed for a heterogenous anisotropic cylinder, whose tangent elasticity tensor field possesses the usual major and minor symmetries, is positive definite, independent from the axial coordinate and endowed with a plane of elastic symmetry orthogonal to the cylinder axis. Symmetry has been consistently employed to formulate the basic problems of extension, bending, torsion and flexure as symmetric and antisymmetric problems respectively. It is shown that Saint-Venant’s postulate, momentum balance and symmetry are sufficient, without resorting to any a priori assumption, to determine the general form of the displacement field and to remove the solution indeterminacy.     

Saint-Venant's problem with Voigt's hypotheses for anisotropic solids

We seek for a solution of Saint-Venant's problem for inhomogeneous and anisotropic materials under the assumptions, introduced by Voigt, that the stress is either constant along the axis of the cylinder or depends linearly on the axial coordinate. We first prove the uniqueness of the solution in terms of resultants, then we exhibit an explicit formula for such a solution; we show finally how Clebsch's hypothesis, that the stress vector on axial planes is parallel to the axis, is compatible with Voigt's hypotheses provided that the symmetry group of the material comprising the cylinder contains the reflections on the cross-section.     

A general metrology of stress on crystalline silicon with random crystal plane by using micro-Raman spectroscopy

The requirement of stress analysis and measurement is increasing with the great development of heterogeneous structures and strain engineering in the field of semiconductors. Micro-Raman spectroscopy is an effective method for the measurement of intrinsic stress in semiconductor structures. However, most existing applications of Raman-stress measurement use the classical model established on the (001) crystal plane. A non-negligible error may be introduced when the Raman data are detected on surfaces/cross-sections of different crystal planes. Owing to crystal symmetry, the mechanical, physical and optical parameters of different crystal planes show obvious anisotropy, leading to the Raman-mechanical relationship dissimilarity on the different crystal planes. In this work, a general model of stress measurement on crystalline silicon with an arbitrary crystal plane was presented based on the elastic mechanics, the lattice dynamics and the Raman selection rule. The wavenumber-stress factor that is determined by the proposed method is suitable for the measured crystal plane. Detailed examples for some specific crystal planes were provided and the theoretical results were verified by experiments.     

Anholonomic configuration spaces and metric tensors in finite elastoplasticity

Deformation mappings are considered that correspond to the motions of lattice defects, elastic stretch and rotation of the lattice, and initial defect distributions. Intermediate (i.e., relaxed) configuration spaces associated with these deformation maps are identified and then classified from the differential-geometric point of view. A fundamental issue is the proper selection of coordinate systems and metric tensors in these configurations when such configurations are classified as anholonomic. The particular choice of a global, external Cartesian coordinate system and corresponding covariant identity tensor as a metric on an intermediate configuration space is shown to be a constitutive assumption often made regardless of the existence of geometrically necessary crystal defects associated with the anholonomicity (i.e., the non-Euclidean nature) of the space. Since the metric tensor on the anholonomic configuration emerges necessarily in the definitions of scalar products, certain transpose maps, tensorial symmetry operations, and Jacobian invariants, its selection should not be trivialized. Several alternative (i.e., non-Euclidean) representations proposed in the literature for the metric tensor on anholonomic spaces are critically examined.     

Reduced equilibrium equations for perfectly elastic materials

Sufficient conditions are given on the coordinate systems which enable reduced equilibrium equations to be derived for perfectly elastic materials involving deformations which depend in an essential way only on two of the three coordinates. Reduced equilibrium equations given previously for plane and axially symmetric deformations are special cases of the equations given here. These equations considerably reduce the calculations involved in investigating possible solutions of finite elasticity, either exact semi-inverse solutions or approximate perturbation solutions. Moreover a formula for the pressure function appearing in the reduced equilibrium equations is given which relates to the corresponding pressure function associated with the inverse deformation. This formula is similar to one given previously for fully three dimensional deformations.     

Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors

We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a priory knowledge of the symmetry axes.