# Name: 		_README.txt fr Ordner "\02_Mesoscopic-Scale\02\02"

# Content:		This folder contains all data for the example in Sec. 7.2.2.2 of [1]
				
				The example investigates different layered RVEs using the proposed periodic boundary conditions. The RVE consists
				of three layers (2 face layers and 1 core layer), where the core fraction rho_c and the stiffness ratio alpha_M
				of the layers is varied. 
					rho_C=0.2, 0.4, 0.6, 0.8 and alpha_M = 0.1, 0.5, 10
				The RVE size is chosen to be lx x ly x h = (1x1x1) and linear elastic material behaviour is used for both layers.
					EF = 100 N/mm2, EC=1000 N/mm2, nu = 0.3
				
				A reference solution is obtained from the literature [2] and is given by the following functions: 
					for alpha_M=0.1:  0.4444444*(0.9*x^3 - 1.0)^2)/((0.006666667*(x - 1.0)^3*(3.0*x^2 + 9.0*x + 8.0) - x*((x^2 - 1.0)^2 - 0.1333333*x^2*(x^2 - 1.0) + 0.005333333*x^4))*(9.0*x - 10.0))
					for alpha_M=0.5: (0.4444444*(0.5*x^3 - 1.0)^2)/((x - 2.0)*(0.03333333*(x - 1.0)^3*(3.0*x^2 + 9.0*x + 8.0) - x*((x^2 - 1.0)^2 - 0.6666667*x^2*(x^2 - 1.0) + 0.1333333*x^4)))
					for alpha_M=10: -(0.4444444*(9*x^3 + 1)^2)/((0.9*x + 0.1)*(0.6666667*(x - 1)^3*(3*x^2 + 9*x + 8) - x*((x^2 - 1)^2 - 13.33333*x^2*(x^2 - 1) + 53.33333*x^4)))
				
				Furthermore, for the case of alpha_M=0.1, the moment reduction constraint (MRC) is employed in a 
				homogeneous manner (hMRC), i.e. it is not accounted for stiffness jumps. 
				
				As a comparison the solution obtained by a second order homogenisation approach for thick shells, following [3],
				is incorporated in the figure.
												
				Abbreviation of the boundary conditions:
				pbc - periodic boundary conditions
				sbc - shell boundary conditions
				tbc - traction boundary conditions
				
				Detailed description may be found in [1].

# Datum:		28.08.2024

# Author:		Leonie Mester

# Files:		Fig_7-9.pdf		- Figure 7.9 from [1], illustration of layered RVE and dimensioning
				Fig_7-11.pdf 	- Figure 7.11 from [1], Shear correction factor kappa of a layered RVE with different
								  core fractions rho_C and stiffness ratios \alpha_M
				alpha=0,1.dat 			- Data for obtained shear correction factor Kappa using periodic boundary conditions for different 
										  values of rho and alpha_M=0.1 and data for shear correction factor using homogeneous 
										  moment reduction constraint (hMRC)
				alpha=0,5.dat 			- Data for obtained shear correction factor Kappa using periodic boundary conditions for different 
										  values of rho and alpha_M=0.5 and data for shear correction factor using homogeneous 
										  moment reduction constraint (hMRC)				
				alpha=10.dat  			- Data for obtained shear correction factor Kappa using periodic boundary conditions for different 
										  values of rho and alpha_M=10 and data for shear correction factor using homogeneous 
										  moment reduction constraint (hMRC)
				Hii2022_alpha=0,1.dat	- Data for obtained shear correction factor Kappa obtained by second order homogenisation approach
										  for thick shells, from [3].
				
# References:	[1]  	Leonie Mester, Computational Homogenisation and Multiscale Modelling Employing an Image-based
						Approach for the Structural Analysis of Shells, RWTH Aachen, PhD Thesis (2024)
				[2]		Stefanos Vlachoutsis. Shear correction factors for plates and shells. International
						Journal for Numerical Methods in Engineering, 33(7):15371552, 1992.
						DOI: 10.1002/nme.1620330712.
				[3]		A. K. Hii and B. El Said. A kinematically consistent second-order computational homogenisation 
						framework for thick shell models. Computer Methods in Applied Mechanics and Engineering, 
						398:115136, 2022. DOI: 10.1016/j.cma.2022.115136.