Micromechanics of Fracture and Damage

<p>Notations&nbsp; xiii</p>
<p>Preface xv</p>
<p>Part 1. Elastic Solutions to Single Crack Problems&nbsp; 1</p>
<p>Chapter 1. Fundamentals of Plane Elasticity 3</p>
<p>1.1. Complex representation of Airy s biharmonic stress function 3</p>
<p>1.2. Force acting on a curve or an element of arc 7</p>
<p>1.3. Derivation of stresses&nbsp; 9</p>
<p>1.4. Derivation of displacements 11</p>
<p>1.5. General form of the potentials and 12</p>
<p>1.6. Examples 15</p>
<p>1.6.1. Circular cavity under pressure 15</p>
<p>1.6.2. Circular cavity in a plane subjected to uniaxial traction at infinity 16</p>
<p>1.7. Conformal mapping 18</p>
<p>1.7.1. Application of conformal mapping to plane elasticity problems 18</p>
<p>1.7.2. The domain is the unit disc | | 1 20</p>
<p>1.7.3. The domain is the complement of the unit disc 23</p>
<p>1.8. The anisotropic case 26</p>
<p>1.8.1. General features&nbsp; 26</p>
<p>1.8.2. Stresses, displacements and boundary conditions 28</p>
<p>1.9. Appendix: mathematical tools 29</p>
<p>1.9.1. Theorem 1&nbsp; 30</p>
<p>1.9.2. Theorem 2&nbsp; 31</p>
<p>1.9.3. Theorem 3&nbsp; 31</p>
<p>Chapter 2. Fundamentals of Elasticity in View of Homogenization Theory&nbsp; 33</p>
<p>2.1. Green′s function concept&nbsp; 33</p>
<p>2.2. Green s function in two–dimensional conditions 34</p>
<p>2.2.1. The general anisotropic case 34</p>
<p>2.2.2. The isotropic case 35</p>
<p>2.3. Green s function in three–dimensional conditions 38</p>
<p>2.3.1. The general anisotropic case 38</p>
<p>2.3.2. The isotropic case 39</p>
<p>2.4. Eshelby s problems in linear microelasticity 41</p>
<p>2.4.1. The (elastic) inclusion problem 41</p>
<p>2.4.2. The Green operator of the infinite space 44</p>
<p>2.4.3. The Green operator of a finite domain 48</p>
<p>2.4.4. The inhomogeneity problem 50</p>
<p>2.4.5. The inhomogeneity problem with stress boundary conditions 51</p>
<p>2.4.6. The infinite heterogeneous elastic medium 52</p>
<p>2.5. Hill tensor for the elliptic inclusion 54</p>
<p>2.5.1. Properties of the logarithmic potential 54</p>
<p>2.5.2. Integration of the r,ir,l term 57</p>
<p>2.5.3. Components of the Hill tensor 59</p>
<p>2.6. Hill s tensor for the spheroidal inclusion 60</p>
<p>2.6.1. Components of the Hill tensor 63</p>
<p>2.6.2. Series expansions of the components of the Hill tensor for flat spheroids 64</p>
<p>2.7. Appendix 65</p>
<p>2.8. Appendix: derivation of the ij 67</p>
<p>Chapter 3. Two–dimensional Griffith Crack 71</p>
<p>3.1. Stress singularity at crack tip 72</p>
<p>3.1.1. Stress singularity in plane elasticity: modes I and II 73</p>
<p>3.1.2. Stress singularity in antiplane problems in elasticity: mode III 78</p>
<p>3.2. Solution to mode I problem 80</p>
<p>3.2.1. Solution of PI 82</p>
<p>3.2.2. Solution of PI 90</p>
<p>3.2.3. Displacement jump across the crack surfaces 91</p>
<p>3.3. Solution to mode II problem 92</p>
<p>3.3.1. Solution of PII 93</p>
<p>3.3.2. Solution of PII 96</p>
<p>3.3.3. Displacement jump across the crack surfaces 97</p>
<p>3.4. Appendix: Abel s integral equation 98</p>
<p>3.5. Appendix: Neuber Papkovitch displacement potentials 101</p>
<p>Chapter 4. The Elliptic Crack Model in Plane Strains 103</p>
<p>4.1. The infinite plane with elliptic hole 103</p>
<p>4.1.3. Elliptic cavity in a plane subjected to a remote stress state at infinity 107</p>
<p>4.1.4. Stress intensity factors 108</p>
<p>4.1.5. Some remarks on unilateral contact 111</p>
<p>4.2. Infinite plane with elliptic hole: the anisotropic case 112</p>
<p>4.2.1. General properties 112</p>
<p>4.2.2. Complex potentials for an elliptic cavity in the presence of traction at infinity 115</p>
<p>4.2.3. Complex potentials for an elliptic cavity in the case of shear at infinity 116</p>
<p>4.2.5. Displacement discontinuities 121</p>
<p>4.2.6. Closed cracks 123</p>
<p>4.3. Eshelby approach 130</p>
<p>4.3.1. Mode I 130</p>
<p>4.3.2. Mode II 133</p>
<p>Chapter 5. Griffith Crack in 3D 137</p>
<p>5.1. Griffith circular (penny–shaped) crack in mode I 138</p>
<p>5.1.1. Solution of PI 139</p>
<p>5.1.2. Solution of PI 143</p>
<p>5.2. Griffith circular (penny–shaped) crack under shear loading 144</p>
<p>5.2.1. Solution of PII 146</p>
<p>5.2.2. Solution of PII 151</p>
<p>Chapter 6. Ellipsoidal Crack Model: the Eshelby Approach 155</p>
<p>6.1. Mode I 156</p>
<p>6.2. Mode II 159</p>
<p>Chapter 7. Energy Release Rate and Conditions for Crack Propagation 163</p>
<p>7.1. Driving force of crack propagation 163</p>
<p>7.2. Stress intensity factor and energy release rate 167</p>
<p>Part 2. Homogenization of Microcracked Materials 173</p>
<p>Chapter 8. Fundamentals of Continuum Micromechanics 175</p>
<p>8.1. Scale separation 175</p>
<p>8.2. Inhomogeneity model for cracks 177</p>
<p>8.2.1. Uniform strain boundary conditions 177</p>
<p>8.2.2. Uniform stress boundary conditions 181</p>
<p>8.2.3. Linear elasticity with uniform strain boundary conditions 182</p>
<p>8.2.4. Linear elasticity with uniform stress boundary conditions 185</p>
<p>8.3. General results on homogenization with Griffith cracks 187</p>
<p>8.3.1. Hill s lemma with Griffith cracks 187</p>
<p>8.3.2. Uniform strain boundary conditions 188</p>
<p>8.3.3. Uniform stress boundary conditions 190</p>
<p>8.3.4. Derivation of effective properties in linear elasticity: principle of the approach 190</p>
<p>8.3.5. Appendix 194</p>
<p>Chapter 9. Homogenization of Materials Containing Griffith Cracks 197</p>
<p>9.1. Dilute estimates in isotropic conditions 197</p>
<p>9.1.1. Stress–based dilute estimate of stiffness&nbsp; 199</p>
<p>9.1.2. Stress–based dilute estimate of stiffness with closed cracks 202</p>
<p>9.1.3. Strain–based dilute estimate of stiffness with opened cracks 204</p>
<p>9.1.4. Strain–based dilute estimate of stiffness with closed cracks 205</p>
<p>9.2. A refined strain–based scheme 206</p>
<p>9.3. Homogenization in plane strain conditions for anisotropic materials 208</p>
<p>9.3.1. Opened cracks 208</p>
<p>9.3.2. Closed cracks 211</p>
<p>Chapter 10. Eshelby–based Estimates of Strain Concentration and Stiffness&nbsp; 213</p>
<p>10.1. Dilute estimate of the strain concentration tensor: general features 213</p>
<p>10.1.1. The general case 213</p>
<p>10.2. The particular case of opened cracks 215</p>
<p>10.2.1. Spheroidal crack 215</p>
<p>10.2.2. Elliptic crack 216</p>
<p>10.2.3. Crack opening change 218</p>
<p>10.3. Dilute estimates of the effective stiffness for opened cracks 220</p>
<p>10.3.1. Opened parallel cracks 222</p>
<p>10.3.2. Opened randomly oriented cracks 224</p>
<p>10.4. Dilute estimates of the effective stiffness for closed cracks 226</p>
<p>10.4.1. Closed parallel cracks 228</p>
<p>10.4.2. Closed randomly oriented cracks 228</p>
<p>10.5. Mori Tanaka estimate of the effective stiffness 229</p>
<p>10.5.1. Opened cracks 231</p>
<p>10.5.2. Closed cracks 233</p>
<p>Chapter 11. Stress–based Estimates of Stress Concentration and Compliance 235</p>
<p>11.1. Dilute estimate of the stress concentration tensor 235</p>
<p>11.2. Dilute estimates of the effective compliance for opened cracks 236</p>
<p>11.2.1. Opened parallel cracks 237</p>
<p>11.2.2. Opened randomly oriented cracks 239</p>
<p>11.2.3. Discussion 239</p>
<p>11.3. Dilute estimate of the effective compliance for closed cracks 240</p>
<p>11.3.1. 3D case 241</p>
<p>11.3.2. 2D case 242</p>
<p>11.3.3. Stress concentration tensor 243</p>
<p>11.3.4. Comparison with other estimates 244</p>
<p>11.4. Mori Tanaka estimates of effective compliance 244</p>
<p>11.4.1. Opened cracks 246</p>
<p>11.4.2. Closed cracks 246</p>
<p>11.5. Appendix: algebra for transverse isotropy and applications 246</p>
<p>Chapter 12. Bounds 251</p>
<p>12.1. The energy definition of the homogenized stiffness 252</p>
<p>12.2. Hashin Shtrikman s bound 255</p>
<p>12.2.1. Hashin Shtrikman variational principle 255</p>
<p>12.2.2. Piecewise constant polarization field 259</p>
<p>12.2.3. Random microstructures 261</p>
<p>12.2.4. Application of the Ponte–Castaneda and Willis (PCW) bound to microcracked media 270</p>
<p>Chapter 13. Micromechanics–based Damage Constitutive Law and Application 273</p>
<p>13.1. Formulation of damage constitutive law 273</p>
<p>13.1.1. Description of damage level by a single scalar variable 274</p>
<p>13.1.2. Extension to multiple cracks 276</p>
<p>13.2. Some remarks concerning the loss of uniqueness of the mechanical response in relation to damage 277</p>
<p>13.3. Mechanical fields and damage in a hollow sphere subjected to traction 280</p>
<p>13.3.1. General features 280</p>
<p>13.3.2. Case of damage model based on the dilute estimate 284</p>
<p>13.3.3. Complete solution in the case of the damage model based on PCW estimate&nbsp; 285</p>
<p>13.4. Stability of the solution to damage evolution in a hollow sphere 296</p>
<p>13.4.1. The MT damage model 298</p>
<p>13.4.2. The general damage model [13.44] 300</p>
<p>Bibliography 305</p>
<p>Index 309</p>