  # Linear and Non-Linear Deformations of Elastic Solids

## 1st Edition

CRC Press

536 pages | 87 B/W Illus.

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### Description

Linear and Non-Linear Deformations of Elastic Solids aims to compile the advances in the field of linear and non-linear elasticity through discussion of advanced topics. Broadly classified into two parts, it includes crack, contact, scattering and wave propagation in linear elastic solids and bending vibration, stability in non-linear elastic solids supported by MATLAB examples. This book is aimed at graduate students and researchers in applied mathematics, solid mechanics, applied mechanics, structural mechanics and includes comprehensive discussion of related analytical/numerical methods.

PART ONE

1. Basic fundamentals and an overview
1. Introduction
2. Basic stress system
3. Equation of motion and various potentials
4. Various transforms used
5. General form of the elastic wave equation
6. Reciprocity principle and representation theorem
7. General solution of the equation of motion for an arbitrary force system
8. Green’s function in an infinite medium
9. Principle of fracture mechanics

2. One or two-dimensional singular integral equation in contact and crack and method of solution
1. Introduction
2. Crack boundary condition
3. Boundary condition for punch or indentation problem
4. Basic form of singular integral equation for crack and punch problems
5. Method of solution of one-dimensional singular integral equation
6. Basic integral equation in crack and punch problem in planar surface
7. Direct method of solution for two-dimensional singular integral solution in　elliptic region
8. Potential method for two-dimensional singular integral solution
9. Derivation in terms of Jacobi’s polynomial
10. Applications

3. Two-dimensional contact and crack problem in isotropic elastic media:Complex variable technique
1. Introduction
2. Complex representation of the plane elasticity problem
3. Complex potentials in semi-infinite medium
4. First fundamental problem for the semi-infinite medium
5. Green’s function in infinite and semi-infinite media
6. Contact problem for the half plane
7. Flat punch
8. Hertz indentation
9. Stress in the medium for Hertz’ indentation
10. Formulation of the crack problem
11. Line crack at the interface of two elastic media
12. Stress intensity factor in interface medium
13. Stress intensity factor
14. Crack tip singularity: stress intensity factor determination in wedge
15. General Observation

4. Two-dimensional contact and crack problem in anisotropic media
1. Introduction
2. Green’s function in an anisotropic medium
3. Line source and dislocation in an infinite medium
4. Green’s function in a half space
5. Green’s function of two-dimensional anisotropic plates containing an elliptic hole
6. Contact problem under a punch
7. Hertzian Contact solution in bonded dissimilar materials in presence of a loading
8. Fully open crack between dissimilar anisotropic composites
9. Formulation of the integral equation
10. The Comninou interface crack
11. Method of Solution

5. Complete solution to three-dimensional indentation and crack problems in isotropic elastic media
1. Introduction
2. Circular crack and Punch problem
3. Point dislocation in front of a crack
4. Dislocation outside a circular punch
5. Elastic field around a circular crack and punch: Fabrikant’s method
7. Basic solutions in three-dimensional Contact problem in isotropic elastic media
8. Formulation of the integral equation and its solution for the contact problem
9. Alternative method of solution
10. Complete solutions of the elastic field inside the elastic half space
12. Stresses on the axis of symmetry
13. Surface displacement for elliptic contact
14. Circular contact: A Particular case
15. Line contact
16. Tangential indentation
17. Elliptic crack in an isotropic elastic medium
18. Indentation stress field for Hertzian contact
19. Features of Hertzian fracture

6. Three-dimensional interface crack in isotropic and anisotropic elastic media
1. Introduction
2. Formulation of the problem
3. Analytical Solution of the pair of integral equations
4. Energy release rate
5. Interface crack in anisotropic medium
6. Constant normal pressure
7. General observation

7. Three-dimensional elliptic indentation and crack problem in piezoelectric media
1. Introduction
2. Basic solution in piezoelectric medium
3. Formulation of contact and crack problem under normal loading
4. Integral equations for contact and crack problem
5. Formulation of the integral equations
6. Method of solution for contact problem
7. Total mechanical load and electric charge
8. Limiting case of transversely isotropic media
9. Elliptic crack in piezoelectric medium under shear loading
10. Complete solution in the medium
11. Complete field
12. Crack tip field
13. Crack in piezoelectric medium
14. Stress intensity factor for constant shear loading
15. General observation and discussion

8. Crack-microcrack interaction and crack and punch in plate and layered media
1. Introduction
2. Two-dimensional Crack-microcrack interaction
3. Kachanov’s method for two-dimensional crack interaction problem
4. Three-dimensional crack interaction
5. Interaction between circular cracks: Kachanov’s method
6. Interaction between circular cracks under shear loading - Kachanov’s method
7. Summary of numerical results: Interaction between circular cracks
8. Interaction between elliptic cracks
9. Interaction between equal coplanar elliptic cracks subjected to normal loading
10. Interaction between circular and elliptic cracks

9. Weight function theory
1. Introduction
2. Basic theory
3. Application
4. Axisymmetric weight function for a circular crack
5. Crack face weight functions for circular crack
6. Crack face weight functions for half plane crack
7. Weight function theory for an elliptic crack in an infinite medium
8. Determination of the potentials
9. Approximate method for the determination of the weight function
10. The Petroski Achenbach method
11. Discussion and some applications of the weight function theory

10. Surface displacement in an elastic half space due to an earthquake source on an inclined fault plane
1. Introduction
2. Statement of the problem
3. Reduction by Cagniard’s technique
4. Reduction in case of S - wave
5. Complete form of surface displacement
6. Discussion

11. Earth response to uniform self similar crack motion
1. Introduction
2. Formulation
3. Formulation of the problem
4. Method of homogeneous solution
5. Body force equivalents and surface displacement
6. Discussion

12. Growth of a semi-infinite crack at a varying velocity
1. Introduction
2. Growth of a half plane infinite crack at a varying velocity
3. Wiener -Hopf Method
4. Reduction of the integral equation

12.5 Discussion

13. Dynamic response of elliptical footings
1. Introduction
2. Basic solutions for forced vibration of elliptic disc

14. Two-dimensional low frequency scattering of acoustic wave by a rough surface
1. Introduction
2. Statement of the scattering problem
3. Scattering cross section
4. Examples

15. Scattering and impact response of a half plane crack in transversely isotropic and isotropic media
1. Introduction
2. Formulation of the problem
3. Limiting case: Isotropic medium
4. Diffraction by a line crack in a transversely isotropic medium
5. Line crack in an isotropic medium
6. Stopping of a line crack

16. Scattering from an elliptic crack
1. Introduction
2. Formulation of the problem
3. Low frequency case
4. Mid frequency case
5. Effective elastic moduli and attenuation coefficient
6. Numerical results and general discussion
7. Dynamic crack opening displacement
8. Dynamic stress intensity factor
9. Scattering Cross-Section and Back-Scattered Displacement

17. Two-dimensional crack and contact problems – Transform method
1. Introduction
2. Formulation
3. Anti plane fracture analysis of a functionally graded piezoelectric layer on a substrate
4. Discussions

18. Effective moduli of elastic inclusion and inhomogeneity
1. Introduction
2. Ellipsodal inclusion
3. Eshelby tensor
4. Equivalent inclusion method - Ellipsoidal inhomogeneity
5. Wu’s result
6. Self consistent scheme - Energy equivalent method
7. Effective medium theory of composites
8. Self consistent theory : Various approximate schemes
9. Mori Tanaka’s method and Kuster Toks model
10. Kuster Toks
11. Differential effective medium theory
12. Effective dynamic elastic moduli of a random distribution of inclusion
13. Propagation of elastic waves in composites with random set of spherical inclusions (EMM Version)
14. General Remark

19. Numerical method in elasto-static and elasto-dynamic crack problem
1. Introduction
2. Three-dimensional elasto-static case
3. Derivation of singular integral equation from body force method
4. CPV and hypersingular integral equation
5. Numerical implementation
6. Discussion of the results obtained by various workers
7. Boundary integral method in elastic wave scattering problem
8. Formulation of BIM
9. Discretisation and regularisation technique for BIE
10. Alternate method
11. Zhang and Achenbach’s method for two-dimensional BI
12. Alternate BIM for anisotropic piezoelectric media
13. Two dimensional BIE for anisotropic media
14. Details of numerical scheme
15. Stress intensity factor evaluation
16. Element free BIM
17. Discussion

PART TWO

20. Large Amplitude Free Vibration of Rotating Non-Homogeneous Beams with Non- Linear Spring and Mass System
1. Introduction
2. Formulation of the Problem
3. Solution Methodology
4. Linear solution
5. Non-linear solution
6. Results and discussions
7. Conclusion
8. Appendix

21. Stability of an Anisotropic Right-Angled Isosceles Triangular Plate under Large Deflection
1. Introduction
2. Constitutive Equations
3. Governing Equations for an Anisotropic Right-Angled Triangular Plate
4. Stability Analysis of an Anisotropic Right-Angled Isosceles Triangular Plate underLarge Deflection

22. Large Amplitude Free Vibrations of Irregular Plates using Complex Variable Technique
1. Introduction
2. Governing Equation
3. Applications of the Method
4. Experimental Verification
5. Discussion on Numerical and Experimental Results
6. Conclusion

23. Large Amplitude Vibrations of Thin Elastic Plates using Conformal Transformation
1. Introduction
2. Governing Equations
3. Applications of the Method
4. Results and Conclusions

24. Large Deflection of a Circular Plate on Elastic Foundation
1. Introduction
2. Governing Equations
3. Solution for a Circular Plate under Transverse Load
4. Numerical Results and Discussions

25. A Modified Approach to the Nonlinear Analysis of Thin Elastic Plates
1. Introduction
5. Large Deflection of Elastic Plates under Uniform Load
6. Large Deflection of Circular Elastic Plates under a Concentrated Load at the Centre

26. Large Amplitude Free Vibration of Parabolic Plates
1. Introduction
2. Governing Equations
3. Transverse Vibration of Parabolic Plates
4. Solution of the Problem
5. Numerical Results
6. Observations and Conclusions

27. Large Amplitude Free Vibration of Sandwich Parabolic Plates
1. Introduction
2. Governing Equations
3. Equation for Sandwich Parabolic Plate
4. Solution of the Problem
5. Numerical Results and Discussions

28. Large Amplitude Vibration of Orthotropic Sandwich Elliptic Plates
1. Introduction
2. Governing Equations
3. Stress-Strain Relations for each Face-Sheet of the Sandwich Plate
4. Strain and Displacement Relations of the Sandwich Elliptic Plate
5. Derivation of Strain Energy of the Sandwich Plate
6. Vibration of an Orthotropic Sandwich Elliptic Plate
7. Solution of the Problem
8. Numerical Results and Discussions
9. Conclusion

29. Large Amplitude Vibration of Heated Orthotropic Sandwich Elliptic Plates
1. Introduction
2. Governing Equations
3. Stress-Strain-Temperature Relations for each Face-Sheet of the Heated Sandwich Plate
4. Strain and Displacement Relations of the Sandwich Plate
5. Strain Energy of a Heated Sandwich Plate
6. Governing Equation for the Heated Sandwich Elliptic Plate
7. Solution of the Problem
8. Numerical Results and Discussions
9. Conclusion

30. Stability Analysis of Thermal Bending and Buckling of Plates due toLargeDeflection
1. Introduction
2. Governing Equations
3. Solution for Simply Supported Rectangular Plate
4. Solution for Clamped Circular Plate
5. Solution for Clamped Elliptic Plate

31. Stability of Thin Plates under Edge Thrust due to Large Deflections, Buckling being Resisted by a Force Proportional to the Displacement
1. Introduction
2. Governing Equations
3. Solution for Simply Supported Rectangular Plate
4. Solution for Clamped Circular Plate
5. Conclusion

32. Large Deflection of Clamped Cylindrical Shell
1. Introduction
2. Nonlinear Analysis of Clamped Cylindrical Shells under Static Load
3. Large Amplitude Free Vibration of Clamped Cylindrical Shells
4. Discussion

33. Large Deflection of Heated Orthotropic Thin Cylindrical Shell
1. Introduction
2. Governing Equations
3. Solution of the Problem
4. Numerical Computations and Discussion
5. Observation and Conclusion

34. Nonlinear Vibration and Stability of an Orthotropic Sandwich Shell of Double Curvature with Orthotropic Core
1. Introduction
2. Governing Equations
3. Stability of a Shallow Sandwich Shell
4. Solution for movable edge
6.

7. Numerical Results and Discussions

35. Nonlinear Vibrations of a Heated Orthotropic Sandwich Shell　of Double Curvature with Orthotropic Core
1. Introduction
4. Appendix

36. Nonlinear Vibration of Spherical Shells of Variable Thickness
1. Introduction
2. Governing Equations
3. Solution for Spherical Shell of Variable Thickness
4. Numerical Computations and Graphs
5. Conclusions

### About the Authors

Prof. Arabinda Roy graduated from Presidency College, followed by M.Sc.degree from the University of Calcutta .He has two Doctoral degrees to his credit, one from Calcutta University (1969) and the other from the University of Cambridge (U.K) in 1972 under Dr. E.R. Lapwood. He was awarded Commonwealth Research Fellowship to work at Emmanuel College, Cambridge (1969-72).He started his professional career in 1974 at Geological Survey of India as Senior Geophysicist. He later joined his alma mater the Department of Applied Mathematics, University of Calcutta from where he retired as a Professor in 2007 after serving for more than three decades. He was a visiting Professor at I.I.M.A.S., UNAM for a year in 1981. He worked as Principal investigator in a UGC research project. Four students worked under him for Ph.D. degree. Roy’s research interests primarily include theoretical Seismology, wave propagation, vibration and scattering problems, contact and crack theory and associated fields. He has oft-quoted publications in journals of international repute.

Rasajit Kumar Bera

Prof. Rasajit Kumar Bera is a Gold Medalist in Applied Mathematics from University of Calcutta. He received his Ph.D. from Jadavpur University in 1968. Previously, he was a faculty in Presidency College, Kolkata, Bengal Engineering College, Shibpur and joined as Professor & Head of the Department of Science in NITTTR, Kolkata in 1993 from where he retired. He was then invited to act as Professor& Head of the Department of Mathematics in Heritage Institute of Technology, Kolkata where he taught for more than ten years. He guided ten students for Ph.D. Degrees in different topics of Applied Mathematics including Fractional Calculus. 130 Research Papers of him have been published in National and International Journals. He has also contributed in more than ten books. Mathematical Physics for Engineers published by New Age International Publishers and Encyclopaedia of Thermal Stresses edited by Prof. R.B. Hetnarski, published by Springer are among them. Prof. Bera is an associate editor of International Journal of Applied and Computational Mathematics – A Springer Scopus indexed journal. His research interest includes Mathematical Theory of Linear and Non-Linear Elasticity, Generalized Thermo-Elasticity, Thermo-Elasticity in Random Media, Numerical Methods and Computation, Fractional Calculus. Among many Research Projects completed by him, a Major Project from Bhaba Atomic Research Centre(BARC) on Fractional Calculus applied to describe Reactor Kinetics & Flux Matching is worth mentioning.

### Subject Categories

##### BISAC Subject Codes/Headings:
TEC005000
TECHNOLOGY & ENGINEERING / Construction / General
TEC009020
TECHNOLOGY & ENGINEERING / Civil / General
TEC009070
TECHNOLOGY & ENGINEERING / Mechanical