Handbook of Elasticity Solutions

Frontmatter

Chapter 1. Basic equations of elasticity

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 2. Point forces and systems of point forces in the three-dimensional space and half-space

Abstract

This problem was solved by Lord Kelvin (W. Thomson) in 1848 (for reference, see Thomson, 1882). Point force F is applied at the origin of coordinates. Displacement vector u and stress tensor σ at the point x = x ₁ e ₁ + x ₂ e ₂ + x3e ₃ are

$$\begin{gathered} u = \frac{1}{{16\pi G\left( {1 - v} \right)}}\left[ {\left( {3 - 4v} \right)\frac{F}{R} + \frac{{F \cdot R}}{{{R^3}}}R} \right] \hfill \\ \sigma = \frac{1}{{8\pi G\left( {1 - v} \right)}}\frac{1}{{{R^3}}}\left[ {\left( {1 - 2v} \right)\left( {IF \cdot R - FR - RF} \right) - \frac{{3F \cdot R}}{{{R^2}}}RR} \right] \hfill \\ \end{gathered} $$

where R = x, \(R = \left| {\mathbf{R}} \right| = \sqrt {x_1^2 + x_2^2 + x_2^2} \).

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 3. Selected two-dimensional problems

Abstract

A body is said to be in the state of plane strain, parallel to x ₁ x ₂ plane, if component u ₃ of the displacement vector is zero or constant and components u ₁ and u ₂ are functions of x ₁ and x ₂, but not of x ₃. Then ε ₁₃ = ε ₂₃ = ε ₃₃ = 0.

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 4. Three-dimensional crack problems for the isotropic or transversely isotropic infinite solid

Abstract

This chapter gives displacements, stresses, stress intensity factors (SIFs) and displacement discontinuities (crack opening displacements, CODs) in an infinite solid containing one crack. The solid is assumed to be either isotropic or transversely isotropic; in the latter case, the crack is parallel to the isotropy plane.

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 5. A crack in an infinite isotropic two-dimensional solid

Abstract

The stress field in an infinite linear elastic solid with uniform stresses σ _ij ^∞ at infinity and containing a crack with a unit normal n, can be represented as a superposition of (1) the homogeneous state σ _ij ^∞ and (2) the stress state in a solid with stresses vanishing at infinity and the crack faces loaded by tractions niσ _ij ^∞ . This latter problem is considered.

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 6. A crack in an infinite anisotropic two-dimensional solid

Abstract

Elastic stiffnesses and compliances are denoted by C _ijkl and S _ijkl , respectively, so that Hooke’s law takes the form

$${\sigma _{ij}} = {C_{ijkl}}{\varepsilon _{kl}}\quad or\quad {\varepsilon _{ij}} = {S_{ijkl}}{\sigma _{kl}}$$

.

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 7. Thermoelasticity

Abstract

In this section, basic governing equations of thermoelasticity are summarized. The solid is assumed isotropic, from the point of view of both the elastic properties and the thermal conductivity.

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 8. Contact problems

Abstract

The following quantities are of interest in contact problems:

• Distribution of stresses in the contact zone;
• Area of the contact zone;
• Rotation angle of the punch
• Settlement of the punch (its vertical displacement as a function of applied loads).

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 9. Eshelby’s problem and related results

Abstract

This problem, formulated by Eshelby (1957, 1959, 1961), constitutes one of the major advances in the theory of elasticity of the 20th century. It is of central importance for material science applications involving various inhomogeneities (pores, cracks, particles undergoing phase transformations). One of the important applications is the problem of effective elastic properties of materials with multiple inhomogeneities.

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 10. Elastic space containing a rigid ellipsoidal inclusion subjected to translation and rotation

Abstract

A rigid ellipsoidal inclusion is embedded into an infinite elastic space. It is given a small displacement and a small rotation. This chapter gives the resulting elastic fields (solution was given by Lur’e (1970) in a somewhat incomplete form and with errors; for a corrected solution in the complete form, see Kachanov et al. (2001).

Mark Kachanov, Boris Shafiro, Igor Tsukrov

Chapter 11. Basic stress intensity factors (SIFs) and stress concentrations (2-D configurations)

Abstract

SIFs selected for this section are those relevant for the basic fracture mechanics. Much larger collections, with applications to various structural mechanics problems, can be found in several handbooks of SIFs (Sih, 1973, Tada et al., 1973, Rooke & Cartwright, 1976, Mukarami, 1987).

Mark Kachanov, Boris Shafiro, Igor Tsukrov

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