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E-raamat: Potential Method in Mathematical Theories of Multi-Porosity Media

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Potential Method in Mathematical Theories of Multi-Porosity MediaSuurem pilt
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This monograph explores the application of the potential method to three-dimensional problems of the mathematical theories of elasticity and thermoelasticity for multi-porosity materials.  These models offer several new possibilities for the study of important problems in engineering and mechanics involving multi-porosity materials, including geological materials (e.g., oil, gas, and geothermal reservoirs); manufactured porous materials (e.g., ceramics and pressed powders); and biomaterials (e.g., bone and the human brain).  

Proceeding from basic to more advanced material, the first part of the book begins with fundamental solutions in elasticity, followed by Galerkin-type solutions and Green’s formulae in elasticity and problems of steady vibrations, quasi-static, and pseudo-oscillations for multi-porosity materials.  The next part follows a similar format for thermoelasticity, concluding with a chapter on problems of heat conduction for rigid bodies. The final chapter then presents a number of open research problems to which the results presented here can be applied. All results discussed by the author have not been published previously and offer new insights into these models.

Potential Method in Mathematical Theories of Multi-Porosity Media will be a valuable resource for applied mathematicians, mechanical, civil, and aerospace engineers, and researchers studying continuum mechanics. Readers should be knowledgeable in classical theories of elasticity and thermoelasticity.

Arvustused

This monograph is a valuable contribution to mathematical physics. (Vladimir Mityushev, zbMATH 1481.74007, 2022)

1 Introduction
1(24)
1.1 Brief Review of the Theories of Multi-Porosity Materials
1(7)
1.1.1 Single Porosity Materials
1(2)
1.1.2 Double Porosity Materials
3(3)
1.1.3 Triple Porosity Materials
6(1)
1.1.4 Multi-Porosity Materials
7(1)
1.2 Potential Method: An Overview
8(2)
1.3 Notation
10(2)
1.4 Equations of Thermoelasticity
12(5)
1.5 Equations of Elasticity
17(1)
1.6 Matrix Representation of the Basic Equations
18(3)
1.7 Stress Operators
21(4)
2 Fundamental Solutions in Elasticity
25(32)
2.1 Fundamental Solutions: An Overview
25(2)
2.2 Equations of Steady Vibrations and Pseudo-Oscillations
27(7)
2.3 Equations of Quasi-Static
34(3)
2.4 Equations of Equilibrium
37(3)
2.5 Equations of Steady Vibrations of Rigid Body
40(1)
2.6 Equations of Equilibrium of Rigid Body
41(2)
2.7 Basic Properties of Fundamental Solutions
43(7)
2.8 Singular Solutions in Elasticity
50(7)
3 Galerkin-Type Solutions and Green's Formulas in Elasticity
57(26)
3.1 On the Representation of General Solution
57(1)
3.2 Galerkin-Type Solutions
58(4)
3.2.1 Equations of Steady Vibrations
58(1)
3.2.2 Equations in the Laplace Transform Space
59(1)
3.2.3 Equations of Quasi-Static
60(1)
3.2.4 Equations of Equilibrium
61(1)
3.2.5 Equations of Steady Vibrations for Rigid Body with Quadruple Porosity
62(1)
3.3 Green's Formulas in Mathematical Physics
62(1)
3.4 Green's Formulas for Equations of Steady Vibrations
63(8)
3.4.1 Green's First Identity
64(3)
3.4.2 Green's Second Identity
67(2)
3.4.3 Green's Third Identity
69(2)
3.5 Green's Formulas for Equations in the Laplace Transform Space
71(4)
3.5.1 Green's First Identity
72(1)
3.5.2 Green's Second Identity
73(1)
3.5.3 Green's Third Identity
74(1)
3.6 Green's Formulas for Quasi-Static Equations
75(2)
3.7 Green's Formulas for Equilibrium Equations
77(2)
3.8 Green's Formulas for Steady Vibrations Equations of Rigid Body
79(2)
3.9 Green's Formulas for Equilibrium Equations of Rigid Body
81(2)
4 Problems of Steady Vibrations of Rigid Body
83(14)
4.1 Basic Boundary Value Problems
83(1)
4.2 Uniqueness Theorems
84(2)
4.3 Basic Properties of Potentials
86(6)
4.3.1 Single-Layer Potential
87(2)
4.3.2 Double-Layer Potential
89(1)
4.3.3 Volume Potential
90(2)
4.4 Fredholm Operators
92(1)
4.5 Existence Theorems
93(4)
5 Problems of Equilibrium of Rigid Body
97(16)
5.1 Basic Boundary Value Problems
97(1)
5.2 Uniqueness Theorems
98(1)
5.3 Basic Properties of Potentials
99(5)
5.3.1 Single-Layer Potential
100(2)
5.3.2 Double-Layer Potential
102(1)
5.3.3 Volume Potential
103(1)
5.4 Existence Theorems
104(9)
6 Problems of Steady Vibrations in Elasticity
113(18)
6.1 Basic Boundary Value Problems
113(1)
6.2 Uniqueness Theorems
114(5)
6.3 Basic Properties of Potentials
119(3)
6.4 Singular Integral Operators
122(1)
6.5 Existence Theorems
123(8)
7 Problems of Quasi-Static in Elasticity
131(26)
7.1 Basic Boundary Value Problems
131(1)
7.2 Uniqueness Theorems
132(2)
7.3 Basic Properties of Potentials
134(3)
7.4 Singular Integral Operators
137(1)
7.5 Existence Theorems
138(6)
7.6 Solution of the First BVPs by Fredholm's Integral Equations
144(11)
7.6.1 Auxiliary Boundary Value Problems
144(1)
7.6.2 Uniqueness Theorems for the Auxiliary BVPs
144(6)
7.6.3 Basic Properties of Potential
150(2)
7.6.4 Existence Theorems
152(3)
7.7 On the Problems of Equilibrium in Elasticity
155(2)
8 Problems of Pseudo-Oscillations in Elasticity
157(10)
8.1 Basic Boundary Value Problems
157(1)
8.2 Uniqueness Theorems
158(2)
8.3 Basic Properties of Potentials
160(2)
8.4 Singular Integral Operators
162(2)
8.5 Existence Theorems
164(3)
9 Problems of Steady Vibrations in Thermoelasticity
167(32)
9.1 Fundamental Solution
168(8)
9.2 Galerkin-Type Solution
176(2)
9.3 Green's Formulas
178(6)
9.3.1 Green's First Identity
179(2)
9.3.2 Green's Second Identity
181(1)
9.3.3 Green's Third Identity
182(2)
9.4 Basic Boundary Value Problems
184(1)
9.5 Uniqueness Theorems
184(5)
9.6 Basic Properties of Potentials
189(2)
9.7 Singular Integral Operators
191(1)
9.8 Existence Theorems
192(7)
10 Problems of Pseudo-Oscillations in Thermoelasticity
199(10)
10.1 Fundamental Solution
199(3)
10.2 Galerkin-Type Solution
202(1)
10.3 Green's Formulas
203(5)
10.3.1 Green's First Identity
203(2)
10.3.2 Green's Second Identity
205(1)
10.3.3 Green's Third Identity
206(2)
10.4 Basic Boundary Value Problems
208(1)
11 Problems of Quasi-Static in Thermoelasticity
209(38)
11.1 Fundamental Solution
210(6)
11.2 Galerkin-Type Solution
216(1)
11.3 Green's Formulas
217(4)
11.3.1 Green's First Identity
217(1)
11.3.2 Green's Second Identity
218(2)
11.3.3 Green's Third Identity
220(1)
11.4 Basic Boundary Value Problems
221(1)
11.5 Uniqueness Theorems
222(3)
11.6 Basic Properties of Potentials
225(2)
11.7 Singular Integral Operators
227(1)
11.8 Existence Theorems
228(5)
11.9 Solution of the First BVPs by Fredholm's Integral Equations
233(14)
11.9.1 Basic Properties of the Pseudostress Operator in Thermoelasticity
233(2)
11.9.2 Auxiliary Boundary Value Problems
235(1)
11.9.3 Uniqueness Theorems for the Auxiliary BVPs
236(4)
11.9.4 Basic Properties of Potential
240(2)
11.9.5 Existence Theorems
242(5)
12 Problems of Heat Conduction for Rigid Body
247(26)
12.1 Fundamental Solution
248(4)
12.2 Galerkin-Type Solution
252(2)
12.3 Green's Formulas
254(4)
12.3.1 Green's First Identity
254(2)
12.3.2 Green's Second Identity
256(1)
12.3.3 Green's Third Identity
257(1)
12.4 Basic Boundary Value Problems
258(1)
12.5 Uniqueness Theorems
259(2)
12.6 Basic Properties of Potentials
261(6)
12.6.1 Single-Layer Potential
262(2)
12.6.2 Double-Layer Potential
264(1)
12.6.3 Volume Potential
265(2)
12.7 Existence Theorems
267(3)
12.8 On the Problems of Equilibrium in Thermoelasticity
270(3)
13 Future Research Perspectives
273(10)
References 283(16)
Index 299
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