Nucleation under multi-axial loading in variational phase-field models of brittle fracture

Abstract

Phase-field models of brittle fracture can be regarded as gradient damage models including an intrinsic internal length. This length determines the stability threshold of solutions with homogeneous damage and thus the strength of the material, and is often tuned to retrieve the experimental strength in uniaxial tensile tests. In this paper, we focus on multiaxial stress states and show that the available energy decompositions, introduced to avoid crack interpenetration and to allow for unsymmetric fracture behavior in tension and compression, lead to multiaxial strength surfaces of different but fixed shapes. Thus, once the length scale is tailored to recover the experimental tensile strength, it is not possible to match the experimental compressive or shear strength. We propose a new energy decomposition that enables the straightforward calibration of a multi-axial failure surface of the Drucker-Prager type. The new decomposition, which hinges upon the theory of structured deformations, encompasses the volumetric-deviatoric and the no-tension models as special cases. Preserving the variational structure of the model, it includes an additional free parameter that can be calibrated based on the experimental ratio of the compressive to the tensile strength (or, if possible, of the shear to the tensile strength), as successfully demonstrated on two data sets taken from the literature.

Appendices

The spectral strain energy decomposition

The stress-strain relationship is again given by Eq.(17a17b) with

$$\begin{aligned} \varvec{\sigma }_{D}=\lambda \mathrm {tr}^{+}\left( \varvec{\varepsilon }\right) {\mathbf {I}}+2\mu \varvec{\varepsilon }^{+}\quad \varvec{\sigma }_{R}=\lambda \mathrm {tr}^{-}\left( \varvec{\varepsilon }\right) {\mathbf {I}}+2\mu \varvec{\varepsilon }^{-} \end{aligned}$$

which can be inverted most easily in component form. Thus, whereas the strain domain is directly obtained as

$$\begin{aligned} {\mathcal {R}}\left( \alpha \right)= & {} \left\{ \varvec{\varepsilon }\in \mathrm {Sym}:\frac{1}{2}\lambda \left[ \mathrm {tr}^{+}\left( \varvec{\varepsilon }\right) \right] ^{2}\right. \\&\left. +\,\,\mu \varvec{\varepsilon }^{+}\cdot \varvec{\varepsilon }^{+}\le -\frac{w_{1}w'\left( \alpha \right) }{a'\left( \alpha \right) }\right\} \end{aligned}$$

the stress domain must be found componentwise by distinguishing several cases. Assuming, without loss of generality, $\varepsilon _{1}\ge \varepsilon _{2}\ge \varepsilon _{3}$, ${\mathcal {R}}^{*}\left( \alpha \right)$ is obtained as the set of $\varvec{\sigma }\in Sym$ such that

• if $\sigma _{3}-\nu \left( \sigma _{1}+\sigma _{2}\right) \ge 0$

  $$\begin{aligned} \frac{1}{18\kappa }\mathrm {tr}^{2}\left( \varvec{\sigma }\right) +\frac{1}{4\mu }\left\| \varvec{\sigma }_{\mathrm {dev}}\right\| ^{2} \le \frac{w_{1}w'\left( \alpha \right) }{s'\left( \alpha \right) } \end{aligned}$$

• else if $\left[ \left( 1+a\left( \alpha \right) \right) \lambda +2\mu \right] \sigma _{2}-\lambda \sigma _{1}-a\left( \alpha \right) \lambda \sigma _{3}\ge 0$ and $\sigma _{1}+\sigma _{2}+a\left( \alpha \right) \sigma _{3}\ge 0$

  $$\begin{aligned}&\frac{1}{4a^{2}\left( \alpha \right) \mu \left[ \left( 2+a\left( \alpha \right) \right) \lambda +2\mu \right] ^{2}}\\&\qquad \cdot \left\{ 4\mu ^{2}\left( \sigma _{1}^{2}+\sigma _{2}^{2}\right) +2\lambda \mu \left[ \left( 3+2a\left( \alpha \right) \right) \sigma _{1}^{2}\right. \right. \\&\left. \qquad -2\sigma _{1}\sigma _{2}+\left( 3+2a\left( \alpha \right) \right) \sigma _{2}^{2}+a^{2}\left( \alpha \right) \sigma _{3}^{2}\right] \\&\qquad +\lambda ^{2}\left[ \left( 2+2a\left( \alpha \right) +a^{2}\left( \alpha \right) \right) \left( \sigma _{1}^{2}+\sigma _{2}^{2}\right) \right. \\&\qquad -2a^{2}\left( \alpha \right) \sigma _{2}\sigma _{3}+2a^{2}\left( \alpha \right) \sigma _{3}^{2}\\&\qquad \left. \left. -2\sigma _{1}\left( 2\sigma _{2}+2a\left( \alpha \right) \sigma _{2}+a^{2}\left( \alpha \right) \sigma _{3}\right) \right] \right\} \\&\quad \le -\frac{w_{1}w'\left( \alpha \right) }{a'\left( \alpha \right) } \end{aligned}$$

• else if $\left[ \left( 1+a\left( \alpha \right) \right) \lambda +2a\left( \alpha \right) \mu \right] \sigma _{2}-\lambda \sigma _{1}-a\left( \alpha \right) \lambda \sigma _{3}\ge 0$ and $\sigma _{1}+\sigma _{2}+a\left( \alpha \right) \sigma _{3}\le 0$

  Table 4 Tensile, compressive and shear strengths according to all considered models (we assume here $\nu >-1/4$)

  $$\begin{aligned}&\frac{1}{4a^{2}\left( \alpha \right) \mu \left[ \left( 2+a\left( \alpha \right) \right) \lambda +2a\left( \alpha \right) \mu \right] ^{2}}\\&\quad \cdot \left\{ \left[ \left( \lambda +a\left( \alpha \right) \lambda +2a\left( \alpha \right) \mu \right) \sigma _{1}-\lambda \sigma _{2}-a\left( \alpha \right) \lambda \sigma _{3}\right] ^{2}\right. \\&\qquad +\left. \left[ \left( \lambda \!+\!a\left( \alpha \right) \lambda \!+\!2a\left( \alpha \right) \mu \right) \sigma _{2}\!-\!\lambda \sigma _{1}\!-\!a\left( \alpha \right) \lambda \sigma _{3}\right] ^{2}\right\} \\&\quad \le -\frac{w_{1}w'\left( \alpha \right) }{a'\left( \alpha \right) } \end{aligned}$$

• else if $2\mu \sigma _{1}+a\left( \alpha \right) \lambda \left( 2\sigma _{1}-\sigma _{2}-\sigma _{3}\right) \ge 0$ and $\sigma _{1}+a\left( \alpha \right) \left( \sigma _{2}+\sigma _{3}\right) \ge 0$

  $$\begin{aligned}&\frac{1}{4a^{2}\left( \alpha \right) \mu \left[ \left( 1+2a\left( \alpha \right) \right) \lambda +2\mu \right] ^{2}}\\&\quad \cdot \left\{ \left[ \left( 8a\left( \alpha \right) +2\right) \lambda \mu +4\mu ^{2}\right] \sigma _{1}^{2}\right. \\&\quad +\left. a^{2}\left( \alpha \right) \lambda \left[ 2\mu \left( \sigma _{2}\!+\!\sigma _{3}\right) ^{2}\!+\!\lambda \left( \!-2\sigma _{1}\!+\!\sigma _{2}\!+\!\sigma _{3}\right) ^{2}\right] \right\} \\&\le -\frac{w_{1}w'\left( \alpha \right) }{a'\left( \alpha \right) } \end{aligned}$$

• else if $2\left( \lambda +\mu \right) \sigma _{1}-\lambda \left( \sigma _{2}+\sigma _{3}\right) \ge 0$ and $\sigma _{1}+a\left( \alpha \right) \left( \sigma _{2}+\sigma _{3}\right) \le 0$

  $$\begin{aligned}&\frac{1}{4\mu \left[ \left( 1+2a\left( \alpha \right) \right) \lambda +2a\left( \alpha \right) \mu \right] ^{2}}\\&\quad \cdot \left[ 2\left( \lambda +\mu \right) \sigma _{1}-\lambda \left( \sigma _{2}+\sigma _{3}\right) \right] ^{2}\le -\frac{w_{1}w'\left( \alpha \right) }{a'\left( \alpha \right) } \end{aligned}$$

Tensile, compressive and shear strengths according to all models considered in this paper, including the newly proposed generalized one.