Anisotropic Elasticity with Matlab

To study the behavior of an elastic continuous medium, the theory of elasticity is a generally accepted model. A simple idealized linear stress-strain relationship gives a good description of the mechanical properties of many elastic materials around us. By this relation, we need 21 elastic constants to describe a linear anisotropic elastic material if the materials do not possess any symmetry properties. Consideration of the material symmetry may reduce the number of elastic constants. If the two-dimensional deformation is considered, the number of elastic constants used in the theory of elasticity can be further reduced. If the materials are under thermal environment, additional thermal properties are needed to express the temperature effects on the stress-strain relation. If the materials exhibit the piezoelectric effects, the stress-strain relation should be further expanded to include the electric displacements and the electric fields. If not only the inplane deformation but also the out-of-plane deflection are considered for the laminates made by laying up various unidirectional fiber-reinforced composites, the elastic constants will generally be reorganized into the extensional, coupling and bending stiffnesses to suit for the classical lamination theory. Since the computer program developed in this book covers all these kinds of materials, their constitutive relations are now described in this Chapter. Further extensions to magneto-electro-elastic and viscoelastic materials will then be described in Chaps. 11 and 12.

In engineering applications if a thin plate is considered, most of the stress analyses will concentrate on the plane variation instead of the distribution in thickness direction, and hence the complex variable formalism which combines two variables into one becomes a useful and powerful tool. When a thin plate is subjected to inplane loads, uniform distribution across the plate thickness is usually assumed for the deformations and stresses, and hence the two-dimensional analysis can be employed. When a thin plate is under transverse loads or bending moments, linear distribution across the plate thickness is usually assumed and the analysis is generally called plate bending analysis. By taking the stress resultants and the deformations of the middle surface as the basic functions, the plate bending analysis can also be treated with two plane coordinate variables. When the thin plates are made by laying up various layers, the inplane and plate bending deformations may uncouple or couple depending on the symmetry or unsymmetry with respect to the middle surface of the plate. In this case, the coupled stretching-bending analysis should be employed. Since all these kinds of problems have been well treated by using the theory of complex variable, summarizing the formalisms introduced in the book (Hwu Anisotropic Elastic Plates, Springer, New York, 2010) several complex variable formalisms are presented in this chapter such as Lekhnitskii formalism and Stroh formalism for two-dimensional analysis and plate bending analysis, and Stroh-like formalism for the coupled stretching-bending analysis. In addition, the extended versions of Stroh formalism for thermoelastic problems and piezoelectric materials are also presented in the related sections of this chapter. Further extensions to magneto-electro-elastic and viscoelastic materials will be described in Chaps. 11 and 12. Moreover, through the use of Radon transform, the Stroh formalism can also be applied to the three-dimensional analysis, whose detailed description will be presented in Sect. 15.​9.​1.

This book is concerned with the computer program using matlab language to code the analytical solutions and boundary element methods formulated by Stroh formalism. To achieve this goal, the program structures detailing the computational procedure as well as the nomenclature of control parameters, global variables, input, and output are described in Sect. 3.1. The entire computer program named as AEPH (Anisotropic Elastic Plates—Hwu) is performed by the main program together with several functions to carry out each different task. Section 3.2 briefly describes the main program and the tasks of some functions. The functions for the input of material properties, the calculation of material eigenvalues and eigenvectors, the calculation of analytical solutions, the double check, and the output are described in Sects. 3.3 to 3.7. The details of the computer codes of main program and all functions created in AEPH are collected and presented in Appendix F, which are listed according to the alphabetic order of function name. Examples for the preparation of input files of the problems with elastic, thermal, and piezoelastic properties are shown in Sect. 3.8.

The general solution (2.​6) of anisotropic elasticity solved by Stroh formalism shows that to calculate the physical responses such as displacements, strains, and stresses, one needs to know the material eigenvalues \(\mu_{\alpha }\), the material eigenvector matrices A and B, and the complex function vector f(z).

The infinite space, half-space or bi-material considered in the previous chapter has at most one straight line boundary. More straight line boundaries are encountered for a wedge with two sides or a multi-material wedge with multiple interfaces and two outer sides. Several different boundary conditions of wedges have been considered and solved in Hwu (2010). Since most of the wedge problems are considered asymptotically around the wedge apex not for the entire region, instead of the complete solutions only certain particular solutions or homogeneous solutions (or called eigenfunctions) are obtained analytically. Sections 5.1 and 5.2 provide particular solutions for the wedges subjected to forces on the edges and apex, and Sect. 5.3 provides the orders of stress singularity as well as eigenfunctions for multi-material wedge spaces and multi-material wedges with four different boundary conditions. The near tip solutions, the unified definition of stress intensity factors, and the path-independent H-integral for interface corners are discussed and coded in Sect. 5.4. Common functions and examples are then presented in the last two sections.

The solutions to the problems of two-dimensional anisotropic elastic solids containing elliptical holes or polygon-like holes have been presented in Hwu (Anisotropic elastic plates. Springer, New York, 2010) for various kinds of loading conditions. Exact closed-form solutions are obtained for the elliptical holes, which are also applicable for the circular holes and cracks. Whereas the solutions to the polygon-like holes are generally approximate. The ellipse, triangle, oval, square, pentagon, and so on are all included as special cases of polygon-like holes. Most of the solutions presented in this chapter are expressed in terms of the variables mapped from an ellipse or a polygon to a unit circle. Since the transformation is in general not one-to-one, special attention shall be made on the proper selection of the mapped variables. With this concern, some remarks are added in the related sections to assist the presentation of the associated computer codes. Section 6.1 covers the problems of elliptical holes under uniform load, in-plane bending, arbitrary load along the hole boundary, point force and dislocation. Section 6.2 includes the problems of polygon-like holes under uniform load and in-plane bending. Functions for common use and examples for hole problems are then presented in the last two sections.

Linear elastic fracture mechanics is established based upon the knowledge of stress field in the vicinity of a crack. Since cracks are special cases of wedges, the near tip solutions for the general cases of cracks can be derived from the eigenfunctions presented in Chap. 5 for the multi-material wedges. A straight crack is also a special case of an elliptical hole with zero minor axis, and hence the full field solutions of crack problems can be obtained from their associated hole problems. Therefore, in the first two sections of this chapter, most of the computer codes are designed based upon the related functions of the previous two chapters. Sections 7.3 and 7.4 provide the solutions and computer codes for the problems of collinear cracks and collinear interface cracks. Most of the codes include the calculation of fracture parameters such as stress intensity factors, crack opening displacements and energy release rates. Examples using the designed computer codes for crack problems are then presented at the last section of this chapter.

Inclusion is a foreign solid enclosed in the matrix. Due to the material discontinuity across the interface, stress distributions around the inclusion have a significant change. Some analytical solutions were presented in Hwu (Anisotropic elastic plates. Springer, New York, 2010) for certain specific situations in which the inclusions can be elastic or rigid whose shapes may be elliptical, line or polygon-like. Based upon these solutions, some matlab functions are designed. Sections 8.1 and 8.2 shows the functions for the problems with elastic or rigid inclusions subjected to a uniform load at infinity or a concentrated force at the matrix. The interactions between inclusion and dislocations are considered in Sect. 8.3, in which the dislocations may be located outside, inside or on the interface. By treating the crack as a distribution of dislocations, the inclusions with cracks outside, inside, penetrating, or along the interface are then considered in Sect. 8.4. Functions for common use and examples for inclusion problems are presented in the last two sections.

Several different contact problems presented in Hwu (Anisotropic elastic plates. Springer, New York, 2010) are coded with matlab in this chapter. If the punches are assumed to be rigid, according to the geometry of the boundary three different contact problems are considered in the first three sections. The first is a rigid punch on a half-plane, the second is a rigid stamp indentation on a curvilinear hole boundary, and the third is a rigid punch on a perturbed surface. No matter which kind of boundary geometries and punch shapes, in these three sections the punches are assumed to be perfect bonded with their contact surfaces. The problems of sliding punches with or without friction are then considered in Sect. 9.4. Different from the first four sections whose indenters are assumed to be rigid, the contact between two elastic bodies is considered in Sect. 9.5. Functions for common use and examples for contact problems are presented in the last two sections.

The Stroh formalism for two-dimensional linear anisotropic elasticity is extended in this chapter to the uncoupled steady state thermoelastic problems. Based upon the extended Stroh formalism, several different kinds of thermoelastic problems have been solved and presented in the literature such as holes, cracks, rigid inclusions, interface cracks, and multi-materials wedges. Since some of the solutions presented in Hwu (2010) are solved only in principle without further showing their explicit form, to assist in programming the computer codes, in this chapter some solutions are rewritten in explicit expressions and necessary remarks are provided.

It is well known that piezoelectric and piezomagnetic materials have the ability of converting energy from one form (between electric/magnetic and mechanical energies) to the other. In other words, these materials can produce an electric or a magnetic field when deformed and undergo deformation when subjected to an electric or a magnetic field. If a multilayered composite is made up of different layers such as a fiber-reinforced composite layer and a composite layer consisting of the piezoelectric materials and/or piezomagnetic materials, it may exhibit magnetoelectric effects that are more complicated than those of single-phase piezoelectric or piezomagnetic materials. Because of this intrinsic coupling phenomenon, piezoelectric, piezomagnetic and magneto-electro-elastic (MEE) materials are widely used as sensors and actuators in intelligent advanced structure design. To study the electromechanical behaviors of piezoelectric materials, magnetomechanical behaviors of piezomagnetic materials, and the magneto-electro-mechanical behaviors of MEE materials, suitable mathematical modeling becomes important. As stated in Chap. 2 the expanded Stroh formalism for piezoelectric materials preserves most essential features of Stroh formalism, it becomes a popular tool for the study of piezoelectric anisotropic elasticity. By proper replacement of piezoelectric properties with piezomagnetic properties, the expanded Stroh formalism for piezoelectric materials can be applied to the cases with piezomagnetic materials. Moreover, further expansion can also be applied to the problems with MEE materials. Most of the matlab functions designed previously for the anisotropic elastic materials can be applied directly to the piezoelectric, piezomagnetic, and MEE materials. Actually, this function was originally designed for anisotropic elastic materials, due to the equivalent mathematical form it can now be applied to piezoelectric materials. Same situation is applicable for the piezomagnetic/MEE materials and other functions coded in AEPH. Since all the formulations for the piezomagnetic materials are exactly the same as those of piezoelectric materials, no further discussions will be provided in this chapter for the piezomagnetic materials. Because the related formulations for the piezoelectric materials have been stated in Chaps. 1 and 2, the first two sections of this chapter will focus on the constitutive laws and expanded Stroh formalism of the MEE materials. The functions presented in the following sections of this chapter are just collections of some particular problems such as holes, multi-material wedges, and cracks. For the problems that are not collected in this chapter such as half-plane, bi-material, inclusion, contact, and thermoelastic problems, functions of AEPH can still be implemented by simply choosing the associated Ltype with Ptype \(\ne\) 0.

Viscoelastic materials exhibit a time and rate dependence that is completely absent in the elastic materials. To understand the mechanical behavior of viscoelastic materials, solutions for the deformations and stresses are generally required. Through the use of correspondence principle between linear elasticity and linear viscoelasticity, the Stroh formalism for linear anisotropic viscoelasticity will be introduced in this chapter. With this formalism, the viscoelastic solids can be treated effectively in Laplace domain and the solutions in time domain can be obtained by numerical inversion of the Laplace transform. Solutions for the related viscoelastic problems of holes, cracks, inclusions, interface corners, and contact (only for complete indentation) discussed previously for anisotropic elastic materials, will then be presented in this chapter. Although the elastic-viscoelastic correspondence principle is simple and directly related to its corresponding elastic system, it can only be applied to the cases with time-invariant boundaries. To solve the general problems of anisotropic viscoelasticity, an alternative approach called time-stepping method will be introduced later in Sect. 15.8.3. By that approach, the problems with time-dependent boundaries can also be solved through their associated solutions of elasticity.

In Chaps. 4–12 the deformation of the anisotropic elastic plates is considered to be two-dimensional. If the plate will bend after the action of external force, the bending theory of anisotropic elastic plates described in Sect. 2.​2 should be considered. In this theory, the plate is composed of the anisotropic elastic materials having one plane of elastic symmetry parallel to the middle plane and the plate thickness is small relative to the other dimensions. Based upon the theory of plate bending, the Stroh-like bending formalism was developed and introduced in Sect. 2.​2.​2. With the Stroh-like bending formalism, some analytical solutions and their associated computer codes for the anisotropic elastic plates with holes/cracks/inclusions subjected to out-of-plane bending moments are presented in this chapter.

If the laminates are unsymmetric they will be stretched as well as bent even under pure in-plane forces or pure bending moments. To study the mechanical behavior of thin laminated plates, the coupled stretching-bending theory of laminates was developed. Since this theory considers linear variation of displacements across the thickness direction, by separating the thickness dependence it has been shown that the general solutions can be obtained through the complex variable formalism. In this chapter, the counterpart of Stroh formalism generally called Stroh-like formalism introduced in Sect. 2.​3 is employed to deal with the problems of coupled stretching-bending analysis such as holes/cracks/inclusions in laminates under uniform stretching and bending moments, under uniform heat flow, and their associated Green’s functions.