Micromechanical Models of Ductile Damage and Fracture

Abstract

Two classes of micromechanics-based models of void enlargement are presented succinctly with their fundamental hypotheses and synopsis of derivation highlighted. The first class of models deals with conventional void growth, i.e., under conditions of generalized plastic flow within the elementary volume. The second class of models deals with void coalescence, i.e., an accelerated void growth process in which plastic flow is highly localized. The structure of constitutive relations pertaining to either class of models is the same but their implications are different. With this as basis, two kinds of integrated models are presented which can be implemented in a finite-element code and used in ductile fracture simulations, in particular for metal forming processes. This chapter also describes elements of material parameter identification and how to use the integrated models.

Appendices

Appendix A. GLD Criterion Parameters

There are six parameters which depend on the microstructural variables f and w: C, g, k , η, and α ₂, listed by order of appearance in criterion (17), and α ₁, which mainly appears in the evolution law of w:

$$ g=0\kern1.5em \left(\mathrm{p}\right);\kern2em g=\frac{e_2^3}{\sqrt{1-{e}_2^2}}=f\frac{e_1^3}{\sqrt{1-{e}_1^2}}=f\frac{{\left(1-{w}^2\right)}^{\frac{3}{2}}}{w}\left(\mathrm{o}\right) $$

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where (p) and (o) are a shorthand notation for prolate and oblate, respectively. We recall that e ₁ and e ₂ are the eccentricities of the void and the outer boundary of the RVE, respectively. Both are implicit functions of f and w.

$$ \kappa =\left\{\begin{array}{ll}{\left[\frac{1}{\sqrt{3}}+\frac{1}{\mathrm{In}\;f}\left(\left(\sqrt{3}-2\right)\mathrm{In}\frac{e_1}{e_2}\right)\right]}^{-1}\hfill & \left(\mathrm{p}\right)\hfill \\ {}\frac{3}{2}{\left[1+\frac{\left({g}_f-{g}_1\right)+\frac{4}{5}\left({g}_f^{5/2}-{g}_1^{5/2}\right)-\frac{3}{5}\left({g}_f^5-{g}_1^5\right)}{\mathrm{In}\frac{g_f}{g_1}}\right]}^{-1}\hfill & \left(\mathrm{o}\right)\hfill \end{array}\right. $$

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where

$$ {g}_f\equiv \frac{g}{g+f},\kern2em {g}_1\equiv \frac{g}{g+1} $$

$$ {\alpha}_2=\left\{\begin{array}{ll}\frac{\left(1+{e}_2^2\right)}{{\left(1+{e}_2^2\right)}^2+2{\left(1-{e}_2^2\right)}^2}\hfill & \left(\mathrm{p}\right)\hfill \\ {}\frac{\left(1-{e}_2^2\right)\left(1-2{e}_2^2\right)}{{\left(1-2{e}_2^2\right)}^2+2\left(1-{e}_2^2\right)}\hfill & \left(\mathrm{o}\right)\hfill \end{array}\right. $$

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$$ \begin{array}{l}\eta =-\frac{2}{3}\frac{\kappa {Q}^{*}\left(g+1\right)\left(g+f\right)\mathrm{sh}}{{\left(g+1\right)}^2+{\left(g+f\right)}^2+\left(g+1\right)\left(g+f\right)\left[\kappa {H}^{*}\mathrm{sh}-2\mathrm{ch}\right]},\\ {}C=-\frac{2}{3}\frac{\kappa \left(g+1\right)\left(g+f\right)\mathrm{sh}}{\left({Q}^{*}+\frac{3}{2}\eta {H}^{*}\right)\eta },\kern2em \mathrm{sh}\equiv \sinh \left(\kappa {H}^{*}\right),\kern2em \mathrm{ch}\equiv \cosh \left(\kappa {H}^{*}\right)\end{array} $$

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where H* = 2( α ₁ – α ₂) and Q ^* ≡ (1 – f).

$$ {\alpha}_1=\left\{\begin{array}{ll}\left[{e}_1-\left(1-{e}_1^2\right){ \tanh}^{-1}{e}_1\right]/\left(2{e}_1^3\right)\hfill & \left(\mathrm{p}\right)\hfill \\ {}\left[-{e}_1\left(1-{e}_1^2\right)+\sqrt{1-{e}_1^2}{ \sin}^{-1}{e}_1\right]/\left(2{e}_1^3\right)\hfill & \left(\mathrm{o}\right)\hfill \end{array}\right. $$

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Finally, the parameter α ^G₁ which enters the evolution law (20) of the void shape parameter is given by

$$ {\alpha}_1^{\mathrm{G}}=\left\{\begin{array}{ll}1/\left(3-{e}_1^2\right)\hfill & \left(\mathrm{p}\right)\hfill \\ {}\left(1-{e}_1^2\right)/\left(3-2{e}_1^2\right)\hfill & \left(\mathrm{o}\right)\hfill \end{array}\right. $$

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Appendix B. KB Criterion Parameters

There are six parameters which depend on the microstructural variables f and w and on the anisotropy tensor : C, g, κ, η, and α ₂, listed by order of appearance in criterion (23) and α ₁, which appears in the evolution law of W:

$$ g=0\kern2em \left(\mathrm{p}\right);\kern1.5em g=\frac{e_2^3}{\sqrt{1-{e}_2^2}}=f\frac{e_1^3}{\sqrt{1-{e}_1^2}}=f\frac{{\left(1-{w}^2\right)}^{\frac{3}{2}}}{w}\kern1em \left(\mathrm{o}\right) $$

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We recall that e ₁ and e ₂ are the eccentricities of the void and the outer boundary of the RVE, respectively. Both are implicit functions of f and w. Next, the full expression of κ was provided by Keralavarma and Benzerga (2010) but can be simplified into

$$ \kappa =\left\{\begin{array}{ll}\frac{3}{h}{\left\{1+\frac{h_{\mathrm{t}}}{h^2\mathrm{In}\;f}\mathrm{In}\frac{1-{e}_2^2}{1-{e}_1^2}\right\}}^{-1/2}\hfill & \left(\mathrm{p}\right)\hfill \\ {}\frac{3}{h}{\left\{1+\frac{\left({g}_f-g1\right)+\frac{4}{5}\left({g}_f^{5/2}-{g}_1^{5/2}\right)-\frac{3}{5}\left({g}_f^5-{g}_1^5\right)}{\mathrm{In}\left({g}_f/{g}_1\right)}\right\}}^{-1}\hfill & \left(\mathrm{o}\right)\hfill \end{array}\right. $$

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where shorthand notations are used for

$$ {g}_f\equiv \frac{g}{g+f},\kern2em {g}_1\equiv \frac{g}{g+1} $$

$$ {\alpha}_2=\left\{\begin{array}{ll}\frac{\left(1+{e}_2^2\right)}{{\left(1+{e}_2^2\right)}^2+2\left(1-{e}_2^2\right)}\hfill & \left(\mathrm{p}\right)\hfill \\ {}\frac{\left(1-{e}_2^2\right)\left(1-2{e}_2^2\right)}{{\left(1-2{e}_2^2\right)}^2+2\left(1-{e}_2^2\right)}\hfill & \left(\mathrm{o}\right)\hfill \end{array}\right. $$

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$$ \begin{array}{l}\eta =-\frac{2}{3{h}_{\mathrm{q}}}\frac{\kappa {Q}^{*}\left(g+1\right)\left(g+f\right)\mathrm{sh}}{{\left(g+1\right)}^2+{\left(g+f\right)}^2+\left(g+1\right)\left(g+f\right)\left[\kappa {H}^{*}\mathrm{sh}-2\mathrm{ch}\right]},\\ {}C=-\frac{2}{3}\frac{\kappa \left(g+1\right)\left(g+f\right)\mathrm{sh}}{\left({Q}^{*}+\frac{3}{2}{h}_{\mathrm{q}}\eta {H}^{*}\right)\eta },\kern2em \mathrm{sh}\equiv \sinh \left(\kappa {H}^{*}\right),\kern2em \mathrm{ch}\equiv \cosh \left(\kappa {H}^{*}\right)\end{array} $$

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where ${H}^{*}\equiv 2\sqrt{h_{\mathrm{q}}}\left({\alpha}_1-{\alpha}_2\right)\kern1em \mathrm{and}\kern1em {Q}^{*}\equiv \sqrt{h_{\mathrm{q}}}\left(1-f\right).$

$$ {\alpha}_1=\left\{\begin{array}{ll}\left[{e}_1-\left(1-{e}_1^2\right){ \tanh}^{-1}{e}_1\right]/\left(2{e}_1^3\right)\hfill & \left(\mathrm{p}\right)\hfill \\ {}\left[-{e}_1\left(1-{e}_1^2\right)+\sqrt{1-{e}_1^2}{ \sin}^{-1}{e}_1\right]/\left(2{e}_1^3\right)\hfill & \left(\mathrm{o}\right)\hfill \end{array}\right. $$

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Note that the expressions of α ₂ and α ₁ are identical to those given by Gologanu et al. (1997) for isotropic matrices.