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Figure 15: Stress distribution within the RVE undergone volumetric expansion for both centric and off-centric inclusions when the inclusion is stiffer to the matrix (incl./matr. = 10). The distribution of the stress component [o11 + 022]/2 is shown throughout the RVE. Two volume fractions f = 15% and f = 30% are considered. CCA bounds on the overall bulk modulus are given for comparison. PBC always lies within TBC and DBC. Three main rows correspond to three boundary conditions and three main columns correspond to three different RVEs. For each boundary condition the centric and off-centric inclusion as well as two different volume fractions are examined.

Figure 15 Stress distribution within the RVE undergone volumetric expansion for both centric and off-centric inclusions when the inclusion is stiffer to the matrix (incl./matr. = 10). The distribution of the stress component [o11 + 022]/2 is shown throughout the RVE. Two volume fractions f = 15% and f = 30% are considered. CCA bounds on the overall bulk modulus are given for comparison. PBC always lies within TBC and DBC. Three main rows correspond to three boundary conditions and three main columns correspond to three different RVEs. For each boundary condition the centric and off-centric inclusion as well as two different volume fractions are examined.

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Table 1: Definitions of frequently used abbreviations and notations
Table 1: Definitions of frequently used abbreviations and notations
Figure 1: Complex RVE and its simplified counterparts. The inclusion volume fraction at the cutout could reach 100%. The cutout (left) shall be understood as the RVE. Three simplified RVEs are suggested to be sufficient to replace the cutout, namely tetragonal RVE, hexagonal RVE and circular RVE. For tetragonal RVE, the maximum reachable volume fraction is 78.53%. For hexagonal RVE this value is 90.59% and for circular RVE it could reach its maximum value, which is 100%. The cutout of the real micro-structure can eventually result in isotropic effective behavior. Both tetragonal and hexagonal packings are space-filling, hence intuitively are suitable candidates for a simplified RVE. However, only the circular RVE can furnish isotropic effective behavior resembling the real micro-structure. Furthermore, the circular RVE is the only simplified RVE that could reach the maximum volume fraction of a real cutout of the material.
Figure 1: Complex RVE and its simplified counterparts. The inclusion volume fraction at the cutout could reach 100%. The cutout (left) shall be understood as the RVE. Three simplified RVEs are suggested to be sufficient to replace the cutout, namely tetragonal RVE, hexagonal RVE and circular RVE. For tetragonal RVE, the maximum reachable volume fraction is 78.53%. For hexagonal RVE this value is 90.59% and for circular RVE it could reach its maximum value, which is 100%. The cutout of the real micro-structure can eventually result in isotropic effective behavior. Both tetragonal and hexagonal packings are space-filling, hence intuitively are suitable candidates for a simplified RVE. However, only the circular RVE can furnish isotropic effective behavior resembling the real micro-structure. Furthermore, the circular RVE is the only simplified RVE that could reach the maximum volume fraction of a real cutout of the material.
Figure 2: Computational homogenization graphical summary. The material configuration at the macro-scale Mg, with the boundary of Mg, is mapped to its spatial counterpart Mg, with the boundary of OMg,. The non-linear map My maps points MY from the material configuration to My in the spatial configuration. Line elements are mapped from the material configuration to the spatial configuration via MF. The domain B at the micro-scale corresponds to the RVE. In a strain-driven homogenization framework, the macroscopic deformation gradient is the input of the micro-problem and the macroscopic Piola stress is obtained by solving the boundary value problem at the micro-scale.
Figure 2: Computational homogenization graphical summary. The material configuration at the macro-scale Mg, with the boundary of Mg, is mapped to its spatial counterpart Mg, with the boundary of OMg,. The non-linear map My maps points MY from the material configuration to My in the spatial configuration. Line elements are mapped from the material configuration to the spatial configuration via MF. The domain B at the micro-scale corresponds to the RVE. In a strain-driven homogenization framework, the macroscopic deformation gradient is the input of the micro-problem and the macroscopic Piola stress is obtained by solving the boundary value problem at the micro-scale.
Shortly afterwards, Hashin [101] used the variational approach in [23] to derive bounds on the overall response of for the bounds for the shear modulus, for the first time, are given here as
Shortly afterwards, Hashin [101] used the variational approach in [23] to derive bounds on the overall response of for the bounds for the shear modulus, for the first time, are given here as
Voigt. The Hashin—Shtrikman bounds for the overall bulk and shear moduli read for stiffness ratios less than one. The upper and the lower bound switch for stiffness ratios more than one.
Voigt. The Hashin—Shtrikman bounds for the overall bulk and shear moduli read for stiffness ratios less than one. The upper and the lower bound switch for stiffness ratios more than one.
Figure 3: Packing network for each RVE. Tetragonal RVE packing (left). Hexagonal RVE packing (center) and the circular RVE packing (right). The boundaries of the tetragonal and hexagonal RVEs completely match.
Figure 3: Packing network for each RVE. Tetragonal RVE packing (left). Hexagonal RVE packing (center) and the circular RVE packing (right). The boundaries of the tetragonal and hexagonal RVEs completely match.
Figure 4: Maximum reachable volume fraction for each RVE. For the circular RVE, the inclusion can embed the whole RVE due its geometry of incl./matr. — 0, the inclusion represents a void. On the contrary, the stiffness ratio more than one corresponds to
Figure 4: Maximum reachable volume fraction for each RVE. For the circular RVE, the inclusion can embed the whole RVE due its geometry of incl./matr. — 0, the inclusion represents a void. On the contrary, the stiffness ratio more than one corresponds to
Figure 5: Different stiffness ratios for various RVE types. The columns correspond to RVE types and the rows correspond to the stiffness ratios. Matrix parameters remain the same throughout the examples while inclusion parameters vary in order to produce the required stiffness ratios. The stiffness of the inclusion is identified by a color scheme varying from light to dark.
Figure 5: Different stiffness ratios for various RVE types. The columns correspond to RVE types and the rows correspond to the stiffness ratios. Matrix parameters remain the same throughout the examples while inclusion parameters vary in order to produce the required stiffness ratios. The stiffness of the inclusion is identified by a color scheme varying from light to dark.
Figure 6: Effective bulk modulus versus volume fraction f. Zoom boxes are included to provide more clarity when the results are less distinguish- able. The numerical results are shown by lines with points on top of them whereas the analytical results are shown using lines solely. HSB stands for Hashin—Shtrikman bounds. Composite cylinder assemblage is denoted as CCA. Figure 7 shows the effective shear modulus Mu versus the volume fraction for different RVEs as well as stiffness ratios. It is observed that different boundary conditions, in general, provide more distinguishable effective values
Figure 6: Effective bulk modulus versus volume fraction f. Zoom boxes are included to provide more clarity when the results are less distinguish- able. The numerical results are shown by lines with points on top of them whereas the analytical results are shown using lines solely. HSB stands for Hashin—Shtrikman bounds. Composite cylinder assemblage is denoted as CCA. Figure 7 shows the effective shear modulus Mu versus the volume fraction for different RVEs as well as stiffness ratios. It is observed that different boundary conditions, in general, provide more distinguishable effective values
Figure 7: Effective shear modulus versus volume fraction f. Zoom boxes are included to provide more clarity when the results are less distinguish- able. The numerical results are shown by lines with points on top of them whereas the analytical results are shown using lines solely. HSB stands for Hashin—Shtrikman bounds. Composite cylinder assemblage is denoted as CCA. compared to the previous case. Nevertheless, as we move from the tetragonal RVE towards the circular RVE, PBC moves towards DBC and ultimately coincides with it. Thus, prescribing PBD and TBC to the circular RVE yield
Figure 7: Effective shear modulus versus volume fraction f. Zoom boxes are included to provide more clarity when the results are less distinguish- able. The numerical results are shown by lines with points on top of them whereas the analytical results are shown using lines solely. HSB stands for Hashin—Shtrikman bounds. Composite cylinder assemblage is denoted as CCA. compared to the previous case. Nevertheless, as we move from the tetragonal RVE towards the circular RVE, PBC moves towards DBC and ultimately coincides with it. Thus, prescribing PBD and TBC to the circular RVE yield
Figure 8: Overall moduli versus stiffness ratio in a logarithmic scale. Two specific volume fractions are considered that are represented in each row. A good agreement of the results is observed for the volume fraction f = 25%. Increasing the volume fraction to f = 75% leads to more distinguishable results. More accordance between the results is observed for the bulk modulus Mx compared to the shear modulus M,. The zoom boxes provide for further clarity.
Figure 8: Overall moduli versus stiffness ratio in a logarithmic scale. Two specific volume fractions are considered that are represented in each row. A good agreement of the results is observed for the volume fraction f = 25%. Increasing the volume fraction to f = 75% leads to more distinguishable results. More accordance between the results is observed for the bulk modulus Mx compared to the shear modulus M,. The zoom boxes provide for further clarity.
Figure 9: Illustration of the insensitivity of the Hashin—Shtrikman bounds to micro-structure’s constituents. Similarly, Voigt and Reuss bounds do not distinguish between the fiber and matrix The numerical results depicted in this figure correspond to the circular RVE. CCA solution totally agrees with the numerical results. The first column shows incl./matr = 0.1 where the inclusion and the matrix Lamé parameters are set to 1 and 10, respectively and the fiber’s volume fraction is f. The second column corresponds to incl./matr = 10 where the inclusion and the matrix Lamé parameters are set to 10 and 1, respectively and the x-axis shows 1 — f instead of f. The third column shows the subtraction of the results in the first and the second column.
Figure 9: Illustration of the insensitivity of the Hashin—Shtrikman bounds to micro-structure’s constituents. Similarly, Voigt and Reuss bounds do not distinguish between the fiber and matrix The numerical results depicted in this figure correspond to the circular RVE. CCA solution totally agrees with the numerical results. The first column shows incl./matr = 0.1 where the inclusion and the matrix Lamé parameters are set to 1 and 10, respectively and the fiber’s volume fraction is f. The second column corresponds to incl./matr = 10 where the inclusion and the matrix Lamé parameters are set to 10 and 1, respectively and the x-axis shows 1 — f instead of f. The third column shows the subtraction of the results in the first and the second column.
Figure 10: Illustration of different moduli versus volume fraction for two different stiffness ratios. The numerical results depicted in this figure correspond to the circular RVE. Each column corresponds to a specific material property and each row stands for specific stiffness ratio. Only numerical results are included for the Poisson ratio since analytical estimates do not provide physically meaningful results.
Figure 10: Illustration of different moduli versus volume fraction for two different stiffness ratios. The numerical results depicted in this figure correspond to the circular RVE. Each column corresponds to a specific material property and each row stands for specific stiffness ratio. Only numerical results are included for the Poisson ratio since analytical estimates do not provide physically meaningful results.
Figure 11: Illustration of the evolution of the effective properties versus the degree of periodicity and randomness for periodic and random macro structures. Three different boundary conditions of DBC, PBC and TBC are imposed to two different micro-structures having different distribution o inclusions for stiffness ratio of 0.1. The volume fraction is set to 15%. The size of the RVE remains constant as we increase the level of periodicit: or randomness. The horizontal axis shows the degree of periodicity and randomness for the periodic and random microstructure, respectively. Th micro-structures for some levels are depicted at the bottom of the figure. Analytical bounds, as well as numerical results for circular RVE ar included to provide further information.
Figure 11: Illustration of the evolution of the effective properties versus the degree of periodicity and randomness for periodic and random macro structures. Three different boundary conditions of DBC, PBC and TBC are imposed to two different micro-structures having different distribution o inclusions for stiffness ratio of 0.1. The volume fraction is set to 15%. The size of the RVE remains constant as we increase the level of periodicit: or randomness. The horizontal axis shows the degree of periodicity and randomness for the periodic and random microstructure, respectively. Th micro-structures for some levels are depicted at the bottom of the figure. Analytical bounds, as well as numerical results for circular RVE ar included to provide further information.
Figure 12: Illustration of the evolution of the effective properties versus the degree of periodicity and randomness for periodic and random macro structures. Three different boundary conditions of DBC, PBC and TBC are imposed to two different micro-structures having different distributiot of inclusions for stiffness ratio of 10. The volume fraction is set to 15%. The size of the RVE remains constant as we increase the level of periodicit: or randomness. The horizontal axis shows the degree of periodicity and randomness for the periodic and random microstructure, respectively. Th micro-structures for some levels are depicted at the bottom of the figure. Analytical bounds, as well as numerical results for circular RVE ar included to provide further information.
Figure 12: Illustration of the evolution of the effective properties versus the degree of periodicity and randomness for periodic and random macro structures. Three different boundary conditions of DBC, PBC and TBC are imposed to two different micro-structures having different distributiot of inclusions for stiffness ratio of 10. The volume fraction is set to 15%. The size of the RVE remains constant as we increase the level of periodicit: or randomness. The horizontal axis shows the degree of periodicity and randomness for the periodic and random microstructure, respectively. Th micro-structures for some levels are depicted at the bottom of the figure. Analytical bounds, as well as numerical results for circular RVE ar included to provide further information.
Figure 13: Stress distribution within the RVE undergone simple shear for both centric and off-centric inclusions when the inclusion is more compliant to the matrix (incl./matr. = 0.1). The distribution of the stress component [011 + 022]/2 is shown throughout the RVE. Two volume fractions f = 15% and f = 30% are considered. CCA bounds on the overall bulk modulus are given for comparison. PBC always lies within TBC and DBC. Three main rows correspond to three boundary conditions and three main columns correspond to three different RVEs. For each boundary condition the centric and off-centric inclusion as well as two different volume fractions are examined. serve asa comparison. The difference between the results is more distinguishable for f = 30% than f = 15%. As expected, the effective is prescribed, the off-centric inclusion leads to a stiffer response compared to the centric one. In contrast, if TBC s imposed, the effective modulus corresponding to the RVE with off-centric inclusion underestimates the one with
Figure 13: Stress distribution within the RVE undergone simple shear for both centric and off-centric inclusions when the inclusion is more compliant to the matrix (incl./matr. = 0.1). The distribution of the stress component [011 + 022]/2 is shown throughout the RVE. Two volume fractions f = 15% and f = 30% are considered. CCA bounds on the overall bulk modulus are given for comparison. PBC always lies within TBC and DBC. Three main rows correspond to three boundary conditions and three main columns correspond to three different RVEs. For each boundary condition the centric and off-centric inclusion as well as two different volume fractions are examined. serve asa comparison. The difference between the results is more distinguishable for f = 30% than f = 15%. As expected, the effective is prescribed, the off-centric inclusion leads to a stiffer response compared to the centric one. In contrast, if TBC s imposed, the effective modulus corresponding to the RVE with off-centric inclusion underestimates the one with
Figure 14: Stress distribution within the RVE undergone simple shear for both centric and off-centric inclusions when the inclusion is more compliant to the matrix (incl./matr. = 0.1). The distribution of the stress component 012 is shown throughout the RVE. Two volume fractions f = 15% and f = 30% are considered. CCA bounds on the overall shear modulus are given for comparison. PBC always lies within TBC and DBC. Three main rows correspond to three boundary conditions and three main columns correspond to three different RVEs. For each boundary condition the centric and off-centric inclusion as well as two different volume fractions are examined.
Figure 14: Stress distribution within the RVE undergone simple shear for both centric and off-centric inclusions when the inclusion is more compliant to the matrix (incl./matr. = 0.1). The distribution of the stress component 012 is shown throughout the RVE. Two volume fractions f = 15% and f = 30% are considered. CCA bounds on the overall shear modulus are given for comparison. PBC always lies within TBC and DBC. Three main rows correspond to three boundary conditions and three main columns correspond to three different RVEs. For each boundary condition the centric and off-centric inclusion as well as two different volume fractions are examined.
Figure 16: Stress distribution within the RVE undergone simple shear for both centric and off-centric inclusions when the inclusion is stiffer to the matrix (incl./matr. = 10). The distribution of the stress component 12 is shown throughout the RVE. Two volume fractions f = 15% and f = 30% are considered. CCA bounds on the overall shear modulus are given for comparison. PBC always lies within TBC and DBC. Three main rows correspond to three boundary conditions and three main columns correspond to three different RVEs. For each boundary condition the centric and off-centric inclusion as well as two different volume fractions are examined.
Figure 16: Stress distribution within the RVE undergone simple shear for both centric and off-centric inclusions when the inclusion is stiffer to the matrix (incl./matr. = 10). The distribution of the stress component 12 is shown throughout the RVE. Two volume fractions f = 15% and f = 30% are considered. CCA bounds on the overall shear modulus are given for comparison. PBC always lies within TBC and DBC. Three main rows correspond to three boundary conditions and three main columns correspond to three different RVEs. For each boundary condition the centric and off-centric inclusion as well as two different volume fractions are examined.
Figure 17: Macro Piola stress with respect to the volume fraction f for various types of RVE. The first row is associated with the volumetric expansion and thus the macroscopic stress component of interest is Mp... The bottom row corresponds to the simple shear deformation and thus the macroscopic stress component of interest is ““P,y. Each column corresponds to specific RVE. At each plot, different boundary conditions for a specific RVE are examined.
Figure 17: Macro Piola stress with respect to the volume fraction f for various types of RVE. The first row is associated with the volumetric expansion and thus the macroscopic stress component of interest is Mp... The bottom row corresponds to the simple shear deformation and thus the macroscopic stress component of interest is ““P,y. Each column corresponds to specific RVE. At each plot, different boundary conditions for a specific RVE are examined.
Figure 18: Macro Piola stress with respect to the volume fraction f for boundary conditions. The first row is associated with the volumetric expansion and thus the macroscopic stress component of interest is Mp,... The bottom row corresponds to the simple shear deformation and thus the macroscopic stress component of interest is “’P,y. Each column corresponds to specific boundary condition. At each plot, different RVE types for a specific boundary condition are examined.
Figure 18: Macro Piola stress with respect to the volume fraction f for boundary conditions. The first row is associated with the volumetric expansion and thus the macroscopic stress component of interest is Mp,... The bottom row corresponds to the simple shear deformation and thus the macroscopic stress component of interest is “’P,y. Each column corresponds to specific boundary condition. At each plot, different RVE types for a specific boundary condition are examined.
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