Micromechanics of Fracture and Damage

Luc Dormieux is Professor at Ecole Nationale des Ponts et Chaussées (Laboratoire NAVIER) in Marne-la-Vallée, France. Djimédo Kondo is Professor at Sorbonne University (UPMC, Institut D'Alembert) in Paris, France.

• Notations  xiiiPreface xvPart 1. Elastic Solutions to Single Crack Problems  1Chapter 1. Fundamentals of Plane Elasticity 31.1. Complex representation of Airy’s biharmonic stress function 31.2. Force acting on a curve or an element of arc 71.3. Derivation of stresses  91.4. Derivation of displacements 111.5. General form of the potentials φ and ψ 121.6. Examples 151.6.1. Circular cavity under pressure 151.6.2. Circular cavity in a plane subjected to uniaxial traction at infinity 161.7. Conformal mapping 181.7.1. Application of conformal mapping to plane elasticity problems 181.7.2. The domain Σ is the unit disc |ζ| ≤ 1 201.7.3. The domain Σ is the complement Σ− of the unit disc 231.8. The anisotropic case 261.8.1. General features  261.8.2. Stresses, displacements and boundary conditions 281.9. Appendix: mathematical tools 291.9.1. Theorem 1  301.9.2. Theorem 2  311.9.3. Theorem 3  31Chapter 2. Fundamentals of Elasticity in View of Homogenization Theory  332.1. Green's function concept  332.2. Green’s function in two-dimensional conditions 342.2.1. The general anisotropic case 342.2.2. The isotropic case 352.3. Green’s function in three-dimensional conditions 382.3.1. The general anisotropic case 382.3.2. The isotropic case 392.4. Eshelby’s problems in linear microelasticity 412.4.1. The (elastic) inclusion problem 412.4.2. The Green operator of the infinite space 442.4.3. The Green operator of a finite domain 482.4.4. The inhomogeneity problem 502.4.5. The inhomogeneity problem with stress boundary conditions 512.4.6. The infinite heterogeneous elastic medium 522.5. Hill tensor for the elliptic inclusion 542.5.1. Properties of the logarithmic potential 542.5.2. Integration of the r,ir,l term 572.5.3. Components of the Hill tensor 592.6. Hill’s tensor for the spheroidal inclusion 602.6.1. Components of the Hill tensor 632.6.2. Series expansions of the components of the Hill tensor for flat spheroids 642.7. Appendix 652.8. Appendix: derivation of the χij 67Chapter 3. Two-dimensional Griffith Crack 713.1. Stress singularity at crack tip 723.1.1. Stress singularity in plane elasticity: modes I and II 733.1.2. Stress singularity in antiplane problems in elasticity: mode III 783.2. Solution to mode I problem 803.2.1. Solution of PI 823.2.2. Solution of PI 903.2.3. Displacement jump across the crack surfaces 913.3. Solution to mode II problem 923.3.1. Solution of PII 933.3.2. Solution of PII 963.3.3. Displacement jump across the crack surfaces 973.4. Appendix: Abel’s integral equation 983.5. Appendix: Neuber–Papkovitch displacement potentials 101Chapter 4. The Elliptic Crack Model in Plane Strains 1034.1. The infinite plane with elliptic hole 1034.1.3. Elliptic cavity in a plane subjected to a remote stress state at infinity 1074.1.4. Stress intensity factors 1084.1.5. Some remarks on unilateral contact 1114.2. Infinite plane with elliptic hole: the anisotropic case 1124.2.1. General properties 1124.2.2. Complex potentials for an elliptic cavity in the presence of traction at infinity 1154.2.3. Complex potentials for an elliptic cavity in the case of shear at infinity 1164.2.5. Displacement discontinuities 1214.2.6. Closed cracks 1234.3. Eshelby approach 1304.3.1. Mode I 1304.3.2. Mode II 133Chapter 5. Griffith Crack in 3D 1375.1. Griffith circular (penny-shaped) crack in mode I 1385.1.1. Solution of PI 1395.1.2. Solution of PI 1435.2. Griffith circular (penny-shaped) crack under shear loading 1445.2.1. Solution of PII 1465.2.2. Solution of PII 151Chapter 6. Ellipsoidal Crack Model: the Eshelby Approach 1556.1. Mode I 1566.2. Mode II 159Chapter 7. Energy Release Rate and Conditions for Crack Propagation 1637.1. Driving force of crack propagation 1637.2. Stress intensity factor and energy release rate 167Part 2. Homogenization of Microcracked Materials 173Chapter 8. Fundamentals of Continuum Micromechanics 1758.1. Scale separation 1758.2. Inhomogeneity model for cracks 1778.2.1. Uniform strain boundary conditions 1778.2.2. Uniform stress boundary conditions 1818.2.3. Linear elasticity with uniform strain boundary conditions 1828.2.4. Linear elasticity with uniform stress boundary conditions 1858.3. General results on homogenization with Griffith cracks 1878.3.1. Hill’s lemma with Griffith cracks 1878.3.2. Uniform strain boundary conditions 1888.3.3. Uniform stress boundary conditions 1908.3.4. Derivation of effective properties in linear elasticity: principle of the approach 1908.3.5. Appendix 194Chapter 9. Homogenization of Materials Containing Griffith Cracks 1979.1. Dilute estimates in isotropic conditions 1979.1.1. Stress-based dilute estimate of stiffness  1999.1.2. Stress-based dilute estimate of stiffness with closed cracks 2029.1.3. Strain-based dilute estimate of stiffness with opened cracks 2049.1.4. Strain-based dilute estimate of stiffness with closed cracks 2059.2. A refined strain-based scheme 2069.3. Homogenization in plane strain conditions for anisotropic materials 2089.3.1. Opened cracks 2089.3.2. Closed cracks 211Chapter 10. Eshelby-based Estimates of Strain Concentration and Stiffness  21310.1. Dilute estimate of the strain concentration tensor: general features 21310.1.1. The general case 21310.2. The particular case of opened cracks 21510.2.1. Spheroidal crack 21510.2.2. Elliptic crack 21610.2.3. Crack opening change 21810.3. Dilute estimates of the effective stiffness for opened cracks 22010.3.1. Opened parallel cracks 22210.3.2. Opened randomly oriented cracks 22410.4. Dilute estimates of the effective stiffness for closed cracks 22610.4.1. Closed parallel cracks 22810.4.2. Closed randomly oriented cracks 22810.5. Mori–Tanaka estimate of the effective stiffness 22910.5.1. Opened cracks 23110.5.2. Closed cracks 233Chapter 11. Stress-based Estimates of Stress Concentration and Compliance 23511.1. Dilute estimate of the stress concentration tensor 23511.2. Dilute estimates of the effective compliance for opened cracks 23611.2.1. Opened parallel cracks 23711.2.2. Opened randomly oriented cracks 23911.2.3. Discussion 23911.3. Dilute estimate of the effective compliance for closed cracks 24011.3.1. 3D case 24111.3.2. 2D case 24211.3.3. Stress concentration tensor 24311.3.4. Comparison with other estimates 24411.4. Mori–Tanaka estimates of effective compliance 24411.4.1. Opened cracks 24611.4.2. Closed cracks 24611.5. Appendix: algebra for transverse isotropy and applications 246Chapter 12. Bounds 25112.1. The energy definition of the homogenized stiffness 25212.2. Hashin–Shtrikman’s bound 25512.2.1. Hashin–Shtrikman variational principle 25512.2.2. Piecewise constant polarization field 25912.2.3. Random microstructures 26112.2.4. Application of the Ponte-Castaneda and Willis (PCW) bound to microcracked media 270Chapter 13. Micromechanics-based Damage Constitutive Law and Application 27313.1. Formulation of damage constitutive law 27313.1.1. Description of damage level by a single scalar variable 27413.1.2. Extension to multiple cracks 27613.2. Some remarks concerning the loss of uniqueness of the mechanical response in relation to damage 27713.3. Mechanical fields and damage in a hollow sphere subjected to traction 28013.3.1. General features 28013.3.2. Case of damage model based on the dilute estimate 28413.3.3. Complete solution in the case of the damage model based on PCW estimate  28513.4. Stability of the solution to damage evolution in a hollow sphere 29613.4.1. The MT damage model 29813.4.2. The general damage model [13.44] 300Bibliography 305Index 309