This handbook is a collection of elasticity solutions. Many of the results presented here cannot be found in textbooks and are available in scientific articles only. Some of them were obtained in the closed form quite recently. The solutions have been thoroughly checked and reduced to a "user friendly" form. Every effort has been made to keep the book free of misprints. The theory of elasticity is a mature field and a large number of solutions are ava- able. We had to make choices in selecting material for this book. The emphasis is made on results relevant to general solid mechanics and materials science appli- tions. Solutions related to structural mechanics (beams, plates, shells, etc.) are left out. The content is limited to the linear elasticity. We are grateful to B. Nuller for several clarifications concerning the contact pr- lem and to V. Levin for suggestions on Eshelby's problem. We also appreciate a n- ber of remarks and comments made by L. Germanovich, I. Sevostianov, O. Zharii and R. Zimmerman. We are particularly indebted to E. Karapetian for a substantial help in putting the material together.

LIOn Delamination of Laminated Composites (a) Fiber-Reinforced Composites Considerable technological advances in the production of high-strength fibers (graphite, boron, etc.) have led to a wide use of light high-strength composite materials (graphite epoxy, boron-epoxy, etc.). It is expedient, to make thin walled composite rods, plates, and shells from such materials. Plates can be made by bonding a set of unidirectional thin fiber layers, Fig.l.l. Such plates are orthotropic, as a rule. A random short-fiber composite is shown in Fig. 1.2. Fiber-reinforced composites are widely used in thin-walled aircraft structures because of their specific high strength. For example, the graphite-epoxy composite is characterized by a unidirectional tensile strength of 1.4 GPa while the density is 1.6 Mg/rrt? . For comparison, we may take a steel (steel 4340) whose corresponding properties are identified by values like 1.2 GPa and 7.8 Mg/rrt? . 1. INTRODUCTION Figure 1.1 2 1.1. On Delamination of Laminated Composites Figure 1.2 3 1. INTRODUCTION It is characteristic for laminated plastic material to possess a fairly low bonding. Therefore, low-velocity impacts and defects in manufacturing lead to local delamination. (b) Linear Problems of Delamination Buckling Delamination can significantly reduce the compressive strength and stiffness of the laminate. Local delamination can be considered as a crack in the bond. Under buckling there appears a high interlaminate stress at the crack edge that leads to a spreading of the crack. Delamination growth can lead to structural instability."

Modern engineering materials subjected to unfavorable mechanical and environmental conditions decrease in strength due to the accumulation of microstructural changes. For example, considering damage in metals we can mention creep damage, ductile plastic damage, embrittlement of steels and fatigue damage. To properly estimate the value of damage when designing reliable structures it is necessary to formulate the damage phenomenon in terms of mechanics. Then it is possible to analyse various engineering problems using analytical and computational techniques. During the last two decades the basic principles of continuum damage mechanics were formulated and some special problems were solved. Many scientific papers were published and several conferences on damage mechanics took place. Now continuum damage mechanics is rapidly developing branch of fracture mechanics. This book is probably the first one on the subject; it contains a sys tematic description of the basic aspects of damage mechanics and some of its applications. In general, a theoretical description of damage can be rather compli cated. The experiments in this field are difficult (especially under multiax ial stress and non-proportional loading). Therefore, experimental data, as a rule, are scarce. Determination of functions and constants, which play a role in the complex variants of the theory, from available experimental data is often practically impossible. ix L.M. Kachanov The problems of damage mechanics are mainly engineering ones. Therefore, the author tries to avoid superfluous mathematical formalism. Some more details of the book's subject can be found in the list of con tents."

LIOn Delamination of Laminated Composites (a) Fiber-Reinforced Composites Considerable technological advances in the production of high-strength fibers (graphite, boron, etc.) have led to a wide use of light high-strength composite materials (graphite epoxy, boron-epoxy, etc.). It is expedient, to make thin walled composite rods, plates, and shells from such materials. Plates can be made by bonding a set of unidirectional thin fiber layers, Fig.l.l. Such plates are orthotropic, as a rule. A random short-fiber composite is shown in Fig. 1.2. Fiber-reinforced composites are widely used in thin-walled aircraft structures because of their specific high strength. For example, the graphite-epoxy composite is characterized by a unidirectional tensile strength of 1.4 GPa while the density is 1.6 Mg/rrt? . For comparison, we may take a steel (steel 4340) whose corresponding properties are identified by values like 1.2 GPa and 7.8 Mg/rrt? . 1. INTRODUCTION Figure 1.1 2 1.1. On Delamination of Laminated Composites Figure 1.2 3 1. INTRODUCTION It is characteristic for laminated plastic material to possess a fairly low bonding. Therefore, low-velocity impacts and defects in manufacturing lead to local delamination. (b) Linear Problems of Delamination Buckling Delamination can significantly reduce the compressive strength and stiffness of the laminate. Local delamination can be considered as a crack in the bond. Under buckling there appears a high interlaminate stress at the crack edge that leads to a spreading of the crack. Delamination growth can lead to structural instability."

This handbook is a collection of elasticity solutions. Many of the results presented here cannot be found in textbooks and are available in scientific articles only. Some of them were obtained in the closed form quite recently. The solutions have been thoroughly checked and reduced to a "user friendly" form. Every effort has been made to keep the book free of misprints. The theory of elasticity is a mature field and a large number of solutions are ava- able. We had to make choices in selecting material for this book. The emphasis is made on results relevant to general solid mechanics and materials science appli- tions. Solutions related to structural mechanics (beams, plates, shells, etc.) are left out. The content is limited to the linear elasticity. We are grateful to B. Nuller for several clarifications concerning the contact pr- lem and to V. Levin for suggestions on Eshelby's problem. We also appreciate a n- ber of remarks and comments made by L. Germanovich, I. Sevostianov, O. Zharii and R. Zimmerman. We are particularly indebted to E. Karapetian for a substantial help in putting the material together.

Modern engineering materials subjected to unfavorable mechanical and environmental conditions decrease in strength due to the accumulation of microstructural changes. For example, considering damage in metals we can mention creep damage, ductile plastic damage, embrittlement of steels and fatigue damage. To properly estimate the value of damage when designing reliable structures it is necessary to formulate the damage phenomenon in terms of mechanics. Then it is possible to analyse various engineering problems using analytical and computational techniques. During the last two decades the basic principles of continuum damage mechanics were formulated and some special problems were solved. Many scientific papers were published and several conferences on damage mechanics took place. Now continuum damage mechanics is rapidly developing branch of fracture mechanics. This book is probably the first one on the subject; it contains a sys tematic description of the basic aspects of damage mechanics and some of its applications. In general, a theoretical description of damage can be rather compli cated. The experiments in this field are difficult (especially under multiax ial stress and non-proportional loading). Therefore, experimental data, as a rule, are scarce. Determination of functions and constants, which play a role in the complex variants of the theory, from available experimental data is often practically impossible. ix L.M. Kachanov The problems of damage mechanics are mainly engineering ones. Therefore, the author tries to avoid superfluous mathematical formalism. Some more details of the book's subject can be found in the list of con tents."