# 4.37.1. Tearing Parameter

The tearing parameter model, proposed by Wellman [[1]], is implemented with the form,

(4.258)\[d = \frac{1}{d_{\text{crit}}} \int_0^{\bar{\varepsilon}^p} {\Big\langle \frac{{2\sigma_{\max}}}{{3\left( {\sigma_{\max} - p } \right)}}\Big\rangle } ^m d\bar{\varepsilon}^{p},\]

where \(\sigma_{ij}\) is the Cauchy stress tensor, \(p= \frac13 \sigma_{kk}\) is the mean hydrostatic stress, \(\sigma_{max}\) is the maximum principal stress, and \(\bar{\varepsilon}^{p}\) is the equivalent plastic strain. The two parameters of the model as \(m\), a fit exponent, and the critical failure (tearing) parameter, \(d_{\text{crit}}\). The angle brackets \(\langle \cdot \rangle\), denoting Macaulay brackets,

\[\begin{split}\langle x \rangle = \begin{cases} 0 \;\;\; \text{if } x \leq 0 \\ x \;\;\; \text{if } x > 0 \end{cases},\end{split}\]

are used to ensure that the failure process occurs only with tensile stress states and prevent “damage healing”. The failure process initiates once the integral term reaches the critical tearing parameter, such that \(d = 1\).

## 4.37.1.1. User Guide

    #
    # TEARING_PARAMETER Failure model definitions
    #
    TEARING PARAMETER EXPONENT = m

[1]

G. W. Wellman. A simple approach to modeling ductile failure. Technical Report SAND2012-1343, Sandia National Laboratories, Albuquerque, NM, June 2012. \href http://infoserve.sandia.gov/sand_doc/2012/121343.pdfpdf.