1 Introduction
2 Second-order asymptotic homogenization method based on strain gradient theory
2.1 The macroscale energy for an RVE
2.2 The microscale energy for an RVE
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in the order of \(\epsilon ^{-2}\)$$\begin{aligned} \frac{\partial }{\partial y_j} \Big ( C_{ijkl}^\text {m} \frac{\partial \overset{0}{u}_k}{\partial y_l} \Big ) = 0 \ ; \end{aligned}$$(10)
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in the order of \(\epsilon ^{-1}\)$$\begin{aligned} \Big ( C_{ijkl}^\text {m} \frac{\partial \overset{0}{u}_k}{\partial y_l} \Big )_{,j} + \frac{\partial }{\partial y_j} \big ( C_{ijkl}^\text {m} \overset{0}{u}_{k,l} \big ) + \frac{\partial }{\partial y_j} \Big ( C_{ijkl}^\text {m} \frac{\partial \overset{1}{u}_k}{\partial y_l} \Big ) = 0 \ ; \end{aligned}$$(11)
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in the order of \(\epsilon ^{0}\)$$\begin{aligned} \big ( C_{ijkl}^\text {m} \overset{0}{u}_{k,l} \big )_{,j} + \Big ( C_{ijkl}^\text {m} \frac{\partial \overset{1}{u}_k}{\partial y_l} \Big )_{,j} + \frac{\partial }{\partial y_j} \big ( C_{ijkl}^\text {m} \overset{1}{u}_{k,l} \big ) + \frac{\partial }{\partial y_j} \Big ( C_{ijkl}^\text {m} \frac{\partial \overset{2}{u}_k}{\partial y_l} \Big ) + f_i = 0 \ . \end{aligned}$$(12)
2.3 The equivalence of the macro- and microscale strain energy for an RVE
3 Computational study
3.1 Examination of size effects for different boundary conditions
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In the torsion test, the direct computation shows a deviation (stiffening) from the homogenized Cauchy continuum. This difference is known as a (stiffening) size effect.
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No significant size effects are observed in extension and shear tests. Homogenized Cauchy continua are accurate enough for these cases.
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With the homogenized strain gradient continuum, the size effect can be fully captured.
Unit cell | \(V_f\) | l mm | t mm | \(\mathbf{C}\) in MPa | \(\mathbf{D}\) in N |
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19 % | 1.0 | 0.1 | \( \begin{bmatrix} 11.177 &{} 0.555 &{} 0 \\ 0.555 &{} 11.177 &{} 0 \\ 0 &{} 0 &{} 0.060 &{} \end{bmatrix} \) | \( \begin{bmatrix} 0.005 &{} 0.042 &{} -0.048 \\ 0.042 &{} 1.598 &{} 0.076 \\ -0.048 &{} 0.076 &{} 0.033 &{} \end{bmatrix} \) |
3.2 Examination of size effects for different aspect ratios of the samples
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With increasing aspect ratio, the microstructural effects become more important and the size effects become more pronounced.
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Strain gradient continua show good agreement with direct computation, and the size effects can be grasped well even for the large aspect ratio value of 50.
3.3 Examination of size effects for different unit cells (closed or open cells)
Unit cell | \(V_f\) | l mm | t mm | \(\mathbf{C}\) MPa | \(\mathbf{D}\) N |
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19 % | 1.0 | 0.1 | \( \begin{bmatrix} 11.177 &{} 0.555 &{} 0 \\ 0.555 &{} 11.177 &{} 0 \\ 0 &{} 0 &{} 0.060 &{} \end{bmatrix} \) | \( \begin{bmatrix} 0.004 &{} -0.019 &{} 0.001 \\ -0.019 &{} -0.909 &{} -0.007 \\ 0.001 &{} -0.007 &{} 0.027 &{} \end{bmatrix} \) |
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Structures constructed from closed cells show a stiffening size effect, and open cell constructs show a softening effect.
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Strain gradient continua are able to grasp both stiffening and softening effects.
3.4 Examination of size effects for different topologies of microstructures
3.4.1 Numerical tests under prescribed displacements
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Stiffening size effects are found for all three lattices. Strain gradient continua are able to grasp the stiffening effects accurately in all cases.
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The stiffening size effects gradually decay from the square lattice, the mixed triangle A lattice, to the mixed triangle B lattice.
Unit cell | \(V_f\) | t mm | l mm | \(\mathbf{C}\) MPa | \(\mathbf{D}\) N |
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36 % | 0.200 | 1.0 | \( \begin{bmatrix} 22.849 &{} 2.293 &{} 0 \\ 2.293 &{} 22.849 &{} 0 \\ 0 &{} 0 &{} 0.534 \end{bmatrix} \) | \( \begin{bmatrix} 0.028 &{} 0.153 &{} -0.129 \\ 0.153 &{} 2.779 &{} 0.322 \\ -0.129 &{} 0.322 &{} 0.168 &{} \end{bmatrix} \) | |
36 % | 0.083 | 1.0 | \( \begin{bmatrix} 17.150 &{} 7.445 &{} 0 \\ 7.445 &{} 17.150 &{} 0 \\ 0 &{} 0 &{} 6.510 \end{bmatrix} \) | \( \begin{bmatrix} 0.050 &{} -0.007 &{} 0.38 \\ -0.007 &{} 1.427 &{} 0.453 \\ 0.38 &{} 0.453 &{} 0.045 \end{bmatrix} \) | |
36 % | 0.058 | 1.0 | \( \begin{bmatrix} 18.810 &{} 5.790 &{} 0 \\ 5.790 &{} 18.810 &{} 0 \\ 0 &{} 0 &{} 4.853 \end{bmatrix} \) | \( \begin{bmatrix} 0.032 &{} -0.007 &{} 0.307 \\ -0.007 &{} 0.483 &{} 0.343 \\ 0.307 &{} 0.343 &{} 0.022 \end{bmatrix} \) |
3.4.2 Numerical tests for prescribed tractions
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The direct computations for lattices show smaller total displacements and larger total strain energies than the Cauchy continuum. This stiffening size effect is observed for all three lattices.
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A good match between strain gradient continua and direct computations is visible for both total displacements and total strain energies. Strain gradient continua are able to grasp stiffening effects accurately in all of the cases.
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The topologies of the microstructures have a big influence on the size effects.
4 Discussions
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Beam bending effects
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Zigzagging edge arrangements contributing less stiffness than the interior.
5 Conclusions
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A comparative study will be undertaken between the presented computational homogenization method and that shown in [23, 31]. Indeed, in [31], the constitutive parameters of macro-homogenized model are identified by a fitting procedure to get the best possible agreement with micromodel. The homogenization is made “globally” and conjecturing the macro-continuum model “a priori.” On the other hand, in the current work, the structure of the homogenized description and its parameters are explicitly deduced from an RVE. It will be interesting to investigate differences of the identified parameters by means of these two methods.