PSI - Issue 61

Sandipan Baruah et al. / Procedia Structural Integrity 61 (2024) 180–187 Baruah and Singh/ Structural Integrity Procedia 00 (2024) 000 – 000

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incorporated into the constitutive relation to describe the decrease in material stiffness with increase in applied load. In the initial stages of development, damage models considered the local stresses and strains exhibited by the material points (Chaboche, 1981; Lemaitre, 1986). This was later found to be erroneous due to dependency of the results on the discretization of the spatial domain. Subsequently, a class of non-local damage models came into existence (Pijaudier-Cabot and Bažant , 1987), which considered that damage is driven by a non-local state variable such as stress or strain through a weighted integral averaging technique. Later on, a gradient-based averaging technique was formulated, which became popular due to full coupling between damage and elasticity (Peerlings et al., 1996; Geers et al., 1998; Peerlings, 1999). The procedure was extended further to elastoplastic analysis through modifications of the von Mises yield function and the material stiffness (Engelen et al., 2003; Sarkar et al., 2022). Despite being a reliable method, gradient-based elastoplastic damage models have not been significantly investigated for material behaviour at elevated temperatures. Among various damage-based works concerning high temperatures, Tang et al. (2016) proposed a modified damage law to account for elevated temperatures. They used classical continuum damage theories to predict the failure of steel sheets. Xuewen et al. (2019) developed a damage model for hot tensile test of chromium steels. Liu et al. (2020) proposed a continuum damage mechanics-based model for hot forging process where damage parameters were dependent on temperature. The deformation of titanium alloys in high-temperature tensile tests was studied through a damage model by Gao et al. (2020). Furthermore, various other continuum damage models have been proposed for high-temperature analysis of different engineering materials (Bong et al., 2021; Guo et al., 2021; Wu et al., 2022; Chen et al., 2023). However, almost all such damage models in literature are based on non-local averaging through an integral approach. The gradient-based method therefore requires to be investigated for elevated temperatures with an appropriate damage evolution law. Based on the above literature review, this paper presents a computational analysis of plastic softening of materials at elevated temperatures through gradient damage methodology. The highlights of the present work are: • A temperature-dependent damage law is presented to account for material softening through a yield function. • The gradient-enhanced damage method is used with appropriate material properties at elevated temperatures. • Tensile behaviour of two specimens at elevated temperatures is numerically simulated and validated with experimental evidences. • The present simulations are also compared with the simulations using classical elastoplasticity and are found better for capturing the material softening. This paper is divided into four sections. Section-1 covers the introductory part and the present research overview. Section-2 briefly describes the theoretical background of the gradient damage methodology for elastoplastic simulations. Section-3 describes the numerical examples and the results. Section-4 points out the conclusions drawn from the present work. 2. Numerical Methodology The gradient-enhanced damage methodology for elastoplasticity was formulated by Engelen et al. (2003) and incorporated into finite element analysis through damage laws dependent on the non-local equivalent strain. In the present work, a damage evolution law dependent on temperature and non-local equivalent strain is proposed. The primary governing equations are given below in Eq. (1) and Eq. (2), where ∇ is the divergence operator. . 0  + = σ b (1) 2 2 eq pl pl pl l    −  = (2) The constitutive equation for rate-independent elastoplasticity is given in Eq. (3), and the damage-dependent yield function is shown in Eq. (4) below.

(3) (4)

( ) = − σ C ε ε

pl

y (1 ) D 

F

 = − −

eq

The damage evolution law proposed in the present work is given in Eq. (5) below.