Displacement-based multiscale modeling of fiber-reinforced composites by means of proper orthogonal decomposition

Advanced Modeling and Simulation in Engineering Sciences

Table 4 Relative displacement, accumulated plastic strain and stress deviations of the multiscale problem with a elastoplastic RVE for different numbers of POD modes after unloading

	\(u_x\)	\(u_y\)	\(u_z\)	\(\varepsilon _{pl}\)
\(m=6\)	\(2.1\cdot 10^{-1}\)	\(2.2\cdot 10^{-1}\)	289	\(1.1\cdot 10^{-1}\)
\(m=12\)	\(3.0\cdot 10^{-2}\)	\(1.4\cdot 10^{-2}\)	13.14	\(2.3\cdot 10^{-2}\)
\(m=24\)	\(3.0\cdot 10^{-2}\)	\(3.8\cdot 10^{-2}\)	18.23	\(2.7\cdot 10^{-2}\)

	\(\sigma _{xx}\)	\(\sigma _{yy}\)	\(\sigma _{zz}\)	\(\sigma _{xy}\)	\(\sigma _{xz}\)	\(\sigma _{yz}\)
\(m=6\)	\(7.6\cdot 10^{-2}\)	\(2.1\cdot 10^{-1}\)	\(2.8\cdot 10^{-2}\)	\(6.2\cdot 10^{-2}\)	43	1.99
\(m=12\)	\(7.0\cdot 10^{-3}\)	\(5.1\cdot 10^{-2}\)	\(3.8\cdot 10^{-2}\)	\(2.2\cdot 10^{-2}\)	5.6	0.165
\(m=24\)	\(1.0\cdot 10^{-2}\)	\(3.9\cdot 10^{-2}\)	\(3.9\cdot 10^{-2}\)	\(2.5\cdot 10^{-2}\)	1.28	0.81

The table gives the maximum values of the relative deviation \(|\Delta (*)|^*=|\overline{(*)} - (*)|/max(|(*)|)\) of each field \(\overline{(*)}\) computed in the FEPOD computation in comparison with the corresponding field \((*)\) of the reference FE\(^2\) computation for the multiscale example with elastoplastic material behavior and a different number of POD modes