A meshfree continuous–discontinuous approach for the ductile fracture modeling in explicit dynamics analysis

Abstract

This paper presents a combined continuous–discontinuous modeling technique for the dynamic ductile fracture analysis using an interactive particle enrichment algorithm and a strain-morphed nonlocal meshfree method. The strain-morphed nonlocal meshfree method is a nodel-integrated meshfree method which was recently proposed for the analysis of elastic-damage induced strain localization problems. In this paper, the strain-morphed nonlocal meshfree formulation is extended to the elastic–plastic-damage materials for the ductile fracture analysis. When the ductile material is fully degraded, the interactive particle enrichment scheme is introduced in the strain-morphed nonlocal meshfree formulation that permits a continuous-to-discontinuous failure modeling. The essence of the interactive particle enrichment algorithm is a particle insertion–deletion scheme that produces a visibility criterion for the description of a traction-free crack and leads to a better presentation of the ductile fracture process. Several numerical benchmarks are examined using the explicit dynamics analysis to demonstrate the effectiveness and accuracy of the proposed method.

Appendix

The combinations of isotropic damage and plasticity are widely used for ductile as well as semi-brittle fracture analyses. In this study, we follow a common approach [69–71] that combines the plasticity formulated on the effective stress $\bar{{{\varvec{\sigma }}}}$ [5, 6, 9, 33] with the strain-based damage.

For isotropic damage, the effective stress tensor $\bar{{{\varvec{\sigma }} }}$ is related to Cauchy stress tensor ${\varvec{\sigma }}$ by [5, 6, 9, 33]

$$\begin{aligned} \bar{{{\varvec{\sigma }} }}=\frac{{\varvec{\sigma }}}{\left( {1-d} \right) }={{\varvec{C}}^{e}:{\varvec{\varepsilon }} ^{e}}={{\varvec{C}}^{e}:\left( {{\varvec{\varepsilon }} - {\varvec{\varepsilon }} ^{p}} \right) } \end{aligned}$$

(52)

where ${\varvec{C}}^{e}$ is the fourth order elastic stiffness tensor, ${\varvec{\varepsilon }} ^{e}$ is the elastic strain, ${\varvec{\varepsilon }}$ is the total strain, ${\varvec{\varepsilon }} ^{p}$ is the plastic strain and d is the scale damage valuable. Note in engineering practice, the evolution equations for international valuable d and ${\varvec{\varepsilon }} ^{p}$ are phenomenological and material dependent. The algorithm for the stress update in Eq. (52) is divided into two stages [64, 65, 71]. First, the update of effective stress for the plastic part is carried out by an implicit algorithm; then, the damage part is evaluated from the plastic strain increment obtained in the first stage.

In the first stage, the yield function is also defined in the effective stress space as [64, 65]

$$\begin{aligned} \bar{{\phi }}^{p}\left( {\bar{{{\varvec{\tau }}}},e^{p}} \right) =\left\| {\bar{{{\varvec{\tau }}}}} \right\| -\sqrt{\frac{2}{3}}{\varvec{\sigma }} _y \left( {e^{p}} \right) \end{aligned}$$

(53)

where $\bar{{\tau }}$ is the effective or undamaged deviatoric stress tensor, and $\sigma _y$ is the flow stress. The local effective plastic strain $e^{p}$ is defined as usual by

$$\begin{aligned} e^{p}=\displaystyle \int _0^t {\sqrt{\frac{2}{3}}} \left\| {\dot{{\varvec{\varepsilon }} }^{p}\left( s \right) } \right\| ds \end{aligned}$$

(54)

with the classical flow rule of associative plasticity given by

$$\begin{aligned} \dot{{\varvec{\varepsilon }} }^{p}=\dot{\gamma }\frac{\partial \bar{{\phi }}^{p}}{\partial \bar{{{\varvec{\sigma }} }}}=\dot{\gamma }\frac{\bar{{\varvec{\tau }}}}{\left\| {\bar{{\varvec{\tau }}}} \right\| } \end{aligned}$$

(55)

where $\dot{{\varvec{\varepsilon }} }^{p}$ is the rate of plastic strain tensor and $\dot{\gamma }$ is the plastic multiplier which is consistent with the loading/unloading conditions by $\dot{\gamma }\ge 0,\bar{{\phi }}^{p}\le 0$ and $\dot{\gamma }\bar{{\phi }}^{p}=0$.

In the damage stage, the loading function is given by [64, 65]

$$\begin{aligned} \bar{{\phi }}^{d}\left( {e^{p}} \right) =\bar{{e}}^{p}-\kappa \end{aligned}$$

(56)

with the loading/unloading conditions by $\dot{\kappa }\ge 0,\bar{{\phi }}^{d}\le 0$ and $\dot{\kappa }\bar{{\phi }}^{d}=0$. In this study, a simple damage law is given in Eq. (50), and the nonlocal effective plastic strain $\bar{{e}}^{p}$ [64, 65] is obtained using the meshfree strain smoothing procedure [34, 58] by

$$\begin{aligned} \bar{{e}}^{p}=\int _\Omega {\tilde{\varPsi }^{c}\left( {{\varvec{X;X-\xi }} } \right) e^{p}d\varOmega } \end{aligned}$$

(57)

where $\tilde{\varPsi }^{c}$ is the strain smoothing function for regularization defined in Sect. 2. Note the strain smoothing function $\tilde{\varPsi }^{c}$ is defined in the reference configuration and is subjected to visibility criterion described in Sect. 3 using the interactive particle enrichment algorithm. Accordingly, the rate of the nonlocal effective plastic strain $\dot{\bar{{e}}}^{p}$ is computed by

$$\begin{aligned} \dot{\bar{{e}}}^{p}=\int _\Omega {\tilde{\varPsi }^{c}\left( {{\varvec{X;X-\xi }} } \right) \dot{e}^{p}d\varOmega } \end{aligned}$$

(58)

with

$$\begin{aligned} \dot{e}^{p}=\sqrt{\frac{2}{3}}\dot{\gamma } \end{aligned}$$

(59)

Finally, the Cauchy stress tensor ${\varvec{\sigma }}$ for the evaluation of internal force in Eq. (31) is computed by [64, 65]

$$\begin{aligned} {\varvec{\sigma }} =\left( {1-d\left( {\bar{{e}}^{p}} \right) } \right) \bar{{\varvec{\sigma }}} \end{aligned}$$

(60)

In this study, Eqs. (52)–(60) are utilized in the regular return mapping algorithm for explicit dynamics analysis.