Fracture Mechanics in Layered and Graded Solids: Analysis Using Boundary Element Methods

✍ Scribed by Tian Xiaohong; Quentin Zhong Qi Yue; Higher Education Press

• Publisher: De Gruyter
• Year: 2014
• Tongue: English
• Leaves: 317
• Category: Library

✦ Synopsis

Mechanical responses of solid materials are governed by their material properties. The solutions for estimating and predicting the mechanical responses are extremely difficult, in particular for non-homogeneous materials. Among these, there is a special type of materials whose properties are variable only along one direction, defined as graded materials or functionally graded materials (FGMs). Examples are plant stems and bones. Artificial graded materials are widely used in mechanical engineering, chemical engineering, biological engineering, and electronic engineering.

This work covers and develops boundary element methods (BEM) to investigate the properties of realistic graded materials. It is a must have for practitioners and researchers in materials science, both academic and in industry.

• Covers analysis of properties of graded materials.
• Presents solutions based methods for analysis of fracture mechanics.
• Presents two types of boundary element methods for layered isotropic materials and transversely isotropic materials.
• Written by two authors with extensive international experience in academic and private research and engineering.

✦ Table of Contents

Contents
Chapter 1 Introduction
1.1 Functionally graded materials
1.2 Methods for fracture mechanics
1.2.1 General
1.2.2 Analytical methods
1.2.3 Finite element method
1.2.4 Boundary element method
1.2.5 Meshless methods
1.3 Overview of the book
References
Chapter 2 Fundamentals of Elasticity and Fracture Mechanics
2.1 Introduction
2.2 Basic equations of elasticity
2.3 Fracture mechanics
2.3.1 General
2.3.2 Deformation modes of cracked bodies
2.3.3 Three-dimensional stress and displacement fields
2.3.4 Stress fields of cracks in graded materials and on the interface of bi-materials
2.4 Analysis of crack growth
2.4.1 General
2.4.2 Energy release rate
2.4.3 Maximum principal stress criterion
2.4.4 Minimum strain energy density criterion
2.4.5 The fracture toughness of graded materials
2.5 Summary
References
Chapter 3 Yue’s Solution of a 3D Multilayered Elastic Medium
3.1 Introduction
3.2 Basic equations
3.3 Solution in the transform domain
3.3.1 Solution formulation
3.3.2 Solution expressed in terms of g
3.3.3 Asymptotic representation of the solution matrices
Φ(ρ, z) and Ψ(ρ, z)
3.4 Solution in the physical domain
3.4.1 Solutions in the Cartesian coordinate system
3.4.2 Closed-form results for singular terms of the solution
3.5 Computational methods and numerical evaluation
3.5.1 General
3.5.2 Singularities of the fundamental solution
3.5.3 Numerical integration
3.5.4 Numerical evaluation and results
3.6 Summary
Appendix 1 The matrices of elastic coefficients
Appendix 2 The matrices in the asymptotic expressions of Φ(ρ, z) and Ψ(ρ, z)
Appendix 3 The matrices Gs[m, z,Φ] and Gt [m, z,Φ]
References
Chapter 4 Yue’s Solution-based Boundary Element Method
4.1 Introduction
4.2 Betti’s reciprocal work theorem
4.3 Yue’s solution-based integral equations
4.4 Yue’s solution-based boundary integral equations
4.5 Discretized boundary integral equations
4.6 Assembly of the equation system
4.7 Numerical integration of non-singular integrals
4.7.1 Gaussian quadrature formulas
4.7.2 Adaptive integration
4.7.3 Nearly singular integrals
4.8 Numerical integration of singular integrals
4.8.1 General
4.8.2 Weakly singular integrals
4.8.3 Strongly singular integrals
4.9 Evaluation of displacements and stresses at an internal point
4.10 Evaluation of boundary stresses
4.11 Multi-region method
4.12 Symmetry
4.13 Numerical evaluation and results
4.13.1 A homogeneous rectangular plate
4.13.2 A layered rectangular plate
4.14 Summary
References
Chapter 5 Application of the Yue’s Solution-based BEM toCrack Problems
5.1 Introduction
5.2 Traction-singular element and its numerical method
5.2.1 General
5.2.2 Traction-singular element
5.2.3 The numerical method of traction-singular elements
5.3 Computation of stress intensity factors
5.4 Numerical examples and results
5.5 Summary
References
Chapter 6 Analysis of Penny-shaped Cracks in Functionally Graded Materials
6.1 Introduction
6.2 Analysis methods for crack problems in a FGM system of infinite extent
6.2.1 The crack problem in a FGM
6.2.2 The multi-region method for crack problems of infinite extent
6.2.3 The layered discretization technique for FGMs
6.2.4 Numerical verifications
6.3 The SIFs for a crack parallel to the FGM interlayer
6.3.1 General
6.3.2 A crack subjected to uniform compressive stresses
6.3.3 A crack subjected to uniform shear stresses
6.4 The growth of the crack parallel to the FGM interlayer
6.4.1 The strain energy density factor of an elliptical crack
6.4.2 Crack growth under a remotely inclined tensile loading
6.5 The SIFs for a crack perpendicular to the FGM interlayer
6.5.1 General
6.5.2 Numerical verifications
6.5.3 The SIFs for a crack subjected to uniform compressive stresses
6.5.4 The SIFs for a crack subjected to uniform shear stresses
6.6 The growth of the crack perpendicular to the FGM interlayer
6.6.1 The crack growth under a remotely inclined tensile loading
6.6.2 The crack growth under a remotely inclined compressive loading
6.7 Summary
References
Chapter 7 Analysis of Elliptical Cracks in Functionally Graded Materials
7.1 Introduction
7.2 The SIFs for an elliptical crack parallel to the FGM interlayer
7.2.1 General
7.2.2 Elliptical crack under a uniform compressive stress
7.2.3 Elliptical crack under a uniform shear stress
7.3 The growth of an elliptical crack parallel to the FGM interlayer
7.4 The SIFs for an elliptical crack perpendicular to the FGM interlayer
7.4.1 General
7.4.2 Elliptical crack under a uniform compressive stress
7.4.3 Elliptical crack under a uniform shear loading
7.5 The growth of an elliptical crack perpendicular to the FGM interlayer
7.5.1 Crack growth under a remotely inclined tensile loading
7.5.2 Crack growth under a remotely inclined compressive loading
7.6 Summary
References
Chapter 8 Yue’s Solution-based Dual Boundary Element Method
8.1 Introduction
8.2 Yue’s solution-based dual boundary integral equations
8.2.1 The displacement boundary integral equation
8.2.2 The traction boundary integral equation
8.2.3 The dual boundary integral equations for crack problems
8.3 Numerical implementation
8.3.1 Boundary discretization
8.3.2 The discretized boundary integral equation
8.4 Numerical integrations
8.4.1 Numerical integrations for the displacement BIE
8.4.2 Numerical integrations for the traction BIE
8.5 Linear equation systems for the discretized dual BIEs
8.6 Numerical verifications
8.6.1 Calculation of stress intensity factors
8.6.2 The effect of different meshes and the coefficientD on the SIF values
8.7 Summary
Appendix 4 Finite-part integrals and Kutt’s numerical quadrature
A4.1 Introduction
A4.2 Kutt’s numerical quadrature
References
Chapter 9 Analysis of Rectangular Cracks in the FGMs
9.1 Introduction
9.2 A square crack in FGMs of infinite extent
9.2.1 General
9.2.2 A square crack parallel to the FGM interlayer
9.2.3 A square crack having a 45◦
angle with the FGM interlayer
9.2.4 A square crack perpendicular to the FGM interlayer
9.3 A square crack in the FGM interlayer
9.4 A rectangular crack in FGMs of infinite extent
9.4.1 General
9.4.2 A rectangular crack parallel to the FGM interlayer
9.4.3 A rectangular crack with long sides perpendicular to the FGM interlayer
9.4.4 A rectangular crack with short sides perpendicular to the FGM interlayer
9.5 A square crack in a FGM of finite extent
9.6 Square cracks in layered rocks
9.6.1 General
9.6.2 The crack dimensions and the parameters of layered rocks
9.6.3 A square crack subjected to a uniform compressive load
9.6.4 A square crack subjected to a non-uniform compressive load
9.7 Rectangular cracks in layered rocks
9.7.1 General
9.7.2 A rectangular crack subjected to a linear compressive load
9.7.3 A rectangular crack subjected to a nonlinear compressive load
9.8 Summary
References
Chapter 10 Boundary element analysis of fracturemechanics in transversely isotropic bi-materials
10.1 Introduction
10.2 Multi-region BEM analysis of cracks in transversely isotropic bi-materials
10.2.1 General
10.2.2 Calculation of the stress intensity factors
10.2.3 A penny-shaped crack perpendicular to the interface of transversely isotropic bi-materials
10.2.4 An elliptical crack perpendicular to the interface of transversely isotropic bi-materials
10.3 Dual boundary element analysis of a square crack in transversely isotropic bi-materials
10.3.1 General
10.3.2 Numerical verification
10.3.3 Numerical results and discussions
10.4 Summary
Appendix 5 The fundamental solution of transversely isotropic bi-materials
References