“…Isotropic hardening model for matrix phase used in this study is isotropic hardening as in Equation 1 and, for non-linear hardening, the most common equations used for isotropic hardening are power law (Ludwik law), Equation 2, and the exponential law (Voce law), Equation 3, which were equations used recently by several authors, for example, Chaboche, 1986, Simo and Hughes, 2006, Steglich et al, 2005, Allain and Bouaziz, 2008, Seifert and Schmidt, 2008, Seifert and Schmidt, 2009, Cardoso and Yoon, 2009, Taherizadeh et al, 2009, Cao et al, 2009, Leu and Li, 2012, Chaaba, 2013. Linear and non-linear evolutions of isotropic hardening, for the matrix phase only, can be calculated according to the equations below, power law or exponential law, respectively:…”