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A mixed finite element formulation for ductile damage modeling of thermoviscoplastic metals accounting for void shearing

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Abstract

The modeling of ductile damage in engineering metallic materials is an essential step in a design process. In this paper, a mixed finite element formulation is developed to predict ductile damage in thermoviscoplastic porous metals. The novel aspect of the model is the enhancement of Gurson’s plasticity formulation with a void shearing mechanism capable describing thermoviscoplastic flow stress and thermal diffusion. Thus, the model accounts for void growth, nucleation and coalescence; strain and strain-rate hardening; thermal softening; heating by plastic work; thermal diffusion; localized shear banding; and material strength degradation. Associative plasticity and small strains are assumed. Both strong and weak forms describing the material complex behavior are presented. Time discretization by means of backward Euler and Newmark-\(\beta \) schemes is employed together with Galerkin finite element approximations, leading to a fully discrete set of nonlinear coupled algebraic equations. Two dynamic fracture problems involving ductile failure of plates under a plane strain assumption are numerically analyzed. The effects of the strain rate, thermal diffusion and void shearing mechanism are investigated in detail and shown to be significant. Results show that the present approach can reproduce plastically induced damage, localized shear banding, heating, porosity-induced stress degradation and crack-type damage evolution. The numerical performance is also reported in order to illustrate the convergence of the method.

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Acknowledgements

The authors appreciate all the support provided by the Department of Civil Engineering and Engineering Mechanics of the University of Columbia, and the financial aid granted by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).

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  1. Materials Engineering Department, Lorena School of Engineering, University of São Paulo, Lorena, SP, 12602-810, Brazil

    João Paulo Pascon

  2. Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY, 10027, USA

    Haim Waisman

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  1. João Paulo Pascon
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Appendix A: Matrices and force vectors

Appendix A: Matrices and force vectors

The mass matrices and force vectors defined in the system of Eqs. (53–57) are given by the following expressions in index notation:

$$\begin{aligned}&(\varvec{M}^u)_{ik} = \int _{\Omega } \left[ \frac{\rho }{\beta \Delta t^2} (\varvec{N}_u)_{ij} (\varvec{N}_u)_{kj} \right] dV \end{aligned}$$
(72)
$$\begin{aligned}&\quad (\varvec{f}^u_{int})_k = \int _{\Omega } \left[ (\nabla \varvec{N}_u)_{kij} (\hat{\varvec{\sigma }} \varvec{N}_{\sigma })_{ij} \right] dV \end{aligned}$$
(73)
$$\begin{aligned}&\quad (\varvec{f}^u_{ext})_k = \int _{\partial \Omega } \left[ (\varvec{N}_u)_{kj} (\bar{\varvec{t}})_j \right] dS \end{aligned}$$
(74)
$$\begin{aligned}&\quad (\varvec{M}^f)_{ik} = \int _{\Omega } \left[ (\varvec{N}_f)_i (\varvec{N}_f)_k \right] dV \end{aligned}$$
(75)
$$\begin{aligned}&\quad (\varvec{f}^f)_k = -\Delta t \int _{\Omega } \left[ g_f (\varvec{N}_f)_k \right] dV \end{aligned}$$
(76)
$$\begin{aligned}&\quad (\varvec{M}^T)_{ik} = \int _{\Omega } \left[ \rho {\hat{c}} (\varvec{N}_T)_i (\varvec{N}_T)_k \right] dV \end{aligned}$$
(77)
$$\begin{aligned}&\quad (\varvec{f}^T_{int})_k = \Delta t \int _{\Omega } \left[ \kappa ({\hat{T}})_m (\nabla \varvec{N}_T)_{ki} (\nabla \varvec{N}_T)_{mi} \right] dV \end{aligned}$$
(78)
$$\begin{aligned}&\quad (\varvec{f}^T_{ext})_k = \Delta t \int _{\Omega } \left[ \chi (1 - f) \sigma _Y g_p (\varvec{N}_T)_k \right] dV \nonumber \\&\qquad + \Delta t \int _{\partial \Omega } \left[ \kappa (\varvec{N}_T)_k \bar{\varvec{q}} \right] dS \end{aligned}$$
(79)
$$\begin{aligned}&\quad (\varvec{M}^{\sigma })_{ik} = \int _{\Omega } \left[ (\varvec{N}_{\sigma })_{imj} (\varvec{N}_{\sigma })_{kmj} \right] dV \end{aligned}$$
(80)
$$\begin{aligned}&\quad (\varvec{f}^{\sigma }_{int})_k = \int _{\Omega } \left[ (\varvec{N}_{\sigma })_{kij} (\varvec{\sigma }_{\epsilon _e})_{ij} \right] dV \end{aligned}$$
(81)
$$\begin{aligned}&\quad (\varvec{M}^{\bar{\gamma _p}})_{ik} = \int _{\Omega } \left[ (\varvec{N}_{\bar{\gamma _p}})_i (\varvec{N}_{\bar{\gamma _p}})_k \right] dV \end{aligned}$$
(82)
$$\begin{aligned}&\quad (\varvec{f}^{\bar{\gamma _p}})_k = -\Delta t \int _{\Omega } \left[ g_p (\varvec{N}_{\bar{\gamma _p}})_k \right] dV \end{aligned}$$
(83)

The remaining matrices that appear in the fully discrete linearized system (64) are also provided in index notation:

$$\begin{aligned}&(\varvec{K}^{u \sigma })_{km} = \int _{\Omega } \left[ (\nabla \varvec{N}_u)_{kij} (\varvec{N}_{\sigma })_{mij} \right] dV \end{aligned}$$
(84)
$$\begin{aligned}&\quad (\varvec{G}_f^f)_{km} = -\Delta t \int _{\Omega } \left[ \frac{\partial g_f}{\partial f} (\varvec{N}_f)_k (\varvec{N}_f)_m \right] dV \end{aligned}$$
(85)
$$\begin{aligned}&\quad (\varvec{G}_f^T)_{km} = -\Delta t \int _{\Omega } \left[ \frac{\partial g_f}{\partial T} (\varvec{N}_f)_k (\varvec{N}_T)_m \right] dV \end{aligned}$$
(86)
$$\begin{aligned}&\quad (\varvec{G}_f^{\sigma })_{km} = -\Delta t \int _{\Omega } \left[ \frac{\partial g_f}{\partial \varvec{\sigma }} (\varvec{N}_f)_k (\varvec{N}_{\sigma })_m \right] dV \end{aligned}$$
(87)
$$\begin{aligned}&\quad (\varvec{G}_f^{\bar{\gamma _p}})_{km} = -\Delta t \int _{\Omega } \left[ \frac{\partial g_f}{\partial \bar{\gamma _p}} (\varvec{N}_f)_k (\varvec{N}_{\bar{\gamma _p}})_m \right] dV \end{aligned}$$
(88)
$$\begin{aligned}&\quad (\varvec{L}_T^f)_{km} = \Delta t \int _{\Omega } \left[ -\chi (\varvec{N}_f)_m \sigma _Y g_p (\varvec{N}_T)_k \right. \nonumber \\&\qquad + \chi (1 - f) \sigma _Y\nonumber \\&\qquad \left. \frac{\partial g_p}{\partial f} (\varvec{N}_T)_k (\varvec{N}_f)_m + \chi (1 - f) g_p \frac{\partial \sigma _Y}{\partial f} (\varvec{N}_T)_k (\varvec{N}_f)_m \right] dV \end{aligned}$$
(89)
$$\begin{aligned}&\quad (\varvec{L}_T^T)_{km} = \Delta t \int _{\Omega } \left[ \chi (1 - f) \sigma _Y \frac{\partial g_p}{\partial T} (\varvec{N}_T)_k (\varvec{N}_T)_m \right] dV \nonumber \\&\qquad + \Delta t \int _{\Omega } \left[ \kappa (\nabla \varvec{N}_T)_{ki} (\nabla \varvec{N}_T)_{mi} \right] dV \end{aligned}$$
(90)
$$\begin{aligned}&\quad (\varvec{L}_T^{\sigma })_{km} = \Delta t \int _{\Omega } \left[ \chi (1 - f) \sigma _Y \frac{\partial g_p}{\partial \varvec{\sigma }} (\varvec{N}_T)_k (\varvec{N}_{\sigma })_m \right. \nonumber \\&\quad \left. + \chi (1 - f) g_p \frac{\partial \sigma _Y}{\partial \varvec{\sigma }} (\varvec{N}_T)_k (\varvec{N}_{\sigma })_m \right] dV \end{aligned}$$
(91)
$$\begin{aligned}&\quad (\varvec{L}_T^{\bar{\gamma _p}})_{km} = \Delta t \int _{\Omega } \left[ \chi (1 - f) \sigma _Y \frac{\partial g_p}{\partial \bar{\gamma _p}} (\varvec{N}_T)_k (\varvec{N}_{\bar{\gamma _p}})_m \right] dV \end{aligned}$$
(92)
$$\begin{aligned}&\quad (\varvec{H}_{\sigma }^u)_{km} = \int _{\Omega } \left[ (\varvec{N}_{\sigma })_{kij} \frac{\partial (\varvec{\sigma }_{\epsilon _e})_{ij}}{\partial \varvec{u}} (\varvec{N}_u)_m \right] dV \end{aligned}$$
(93)
$$\begin{aligned}&\quad (\varvec{H}_{\sigma }^f)_{km} = \int _{\Omega } \left[ (\varvec{N}_{\sigma })_{kij} \frac{\partial (\varvec{\sigma }_{\epsilon _e})_{ij}}{\partial f} (\varvec{N}_f)_m \right] dV \end{aligned}$$
(94)
$$\begin{aligned}&\quad (\varvec{H}_{\sigma }^T)_{km} = \int _{\Omega } \left[ (\varvec{N}_{\sigma })_{kij} \frac{\partial (\varvec{\sigma }_{\epsilon _e})_{ij}}{\partial T} (\varvec{N}_T)_m \right] dV \end{aligned}$$
(95)
$$\begin{aligned}&\quad (\varvec{H}_{\sigma }^{\sigma })_{km} = \int _{\Omega } \left[ (\varvec{N}_{\sigma })_{kij} \frac{\partial (\varvec{\sigma }_{\epsilon _e})_{ij}}{\partial \varvec{\sigma }} (\varvec{N}_{\sigma })_m \right] dV \end{aligned}$$
(96)
$$\begin{aligned}&\quad (\varvec{H}_{\sigma }^{\bar{\gamma _p}})_{km} = \int _{\Omega } \left[ (\varvec{N}_{\sigma })_{kij} \frac{\partial (\varvec{\sigma }_{\epsilon _e})_{ij}}{\partial \bar{\gamma _p}} (\varvec{N}_{\bar{\gamma _p}})_m \right] dV \end{aligned}$$
(97)
$$\begin{aligned}&\quad (\varvec{P}_{\bar{\gamma _p}}^f)_{km} = -\Delta t \int _{\Omega } \left[ \frac{\partial g_p}{\partial f} (\varvec{N}_f)_m (\varvec{N}_{\bar{\gamma _p}})_k \right] dV \end{aligned}$$
(98)
$$\begin{aligned}&\quad (\varvec{P}_{\bar{\gamma _p}}^T)_{km} = -\Delta t \int _{\Omega } \left[ \frac{\partial g_p}{\partial T} (\varvec{N}_T)_m (\varvec{N}_{\bar{\gamma _p}})_k \right] dV \end{aligned}$$
(99)
$$\begin{aligned}&\quad (\varvec{P}_{\bar{\gamma _p}}^{\sigma })_{km} = -\Delta t \int _{\Omega } \left[ \frac{\partial g_p}{\partial \varvec{\sigma }} (\varvec{N}_{\sigma })_m (\varvec{N}_{\bar{\gamma _p}})_k \right] dV \end{aligned}$$
(100)
$$\begin{aligned}&\quad (\varvec{P}_{\bar{\gamma _p}}^{\bar{\gamma _p}})_{km} = -\Delta t \int _{\Omega } \left[ \frac{\partial g_p}{\partial \bar{\gamma _p}} (\varvec{N}_{\bar{\gamma _p}})_m (\varvec{N}_{\bar{\gamma _p}})_k \right] dV \end{aligned}$$
(101)

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Pascon, J.P., Waisman, H. A mixed finite element formulation for ductile damage modeling of thermoviscoplastic metals accounting for void shearing. Comput Mech 67, 1307–1330 (2021). https://doi.org/10.1007/s00466-021-02000-w

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