Size effects in nonlinear periodic materials exhibiting reversible pattern transformations

This paper focuses on size effects in periodic mechanical metamaterials driven by reversible pattern transformations due to local elastic buckling instabilities in their microstructure. Two distinct loading cases are studied: compression and bending, in which the material exhibits pattern transformation in the whole structure or only partially. The ratio between the height of the specimen and the size of a unit cell is defined as the scale ratio. A family of shifted microstructures, corresponding to all possible arrangements of the microstructure relative to the external boundary, is considered in order to determine the ensemble averaged solution computed for each scale ratio. In the compression case, the top and the bottom edges of the specimens are fully constrained, which introduces boundary layers with restricted pattern transformation. In the bending case, the top and bottom edges are free boundaries resulting in compliant boundary layers, whereas additional size effects emerge from imposed strain gradient. For comparison, the classical homogenization solution is computed and shown to match well with the ensemble averaged numerical solution only for very large scale ratios. For smaller scale ratios, where a size effect dominates, the classical homogenization no longer applies.


Introduction
compressive load reaches a critical value. As a result of this transformation, the incremen-8 tal effective properties of the material change dramatically. Fig. 1 shows the deformation 9 pattern of such a material under combined compression and shear. 10 One of the earlier works on such porous elastomers is presented by Mullin et al. (2007), 11 where it is shown that pattern transformations are triggered by a reversible elastic instabil-12 ity. The onset of instabilities in materials with arbitrary microstructures for finitely strained, 13 rate-independent solids can be modelled through Bloch analysis (see Geymonat et al., 1993). of such foams decreases with decreasing sample size. Such a behavior was attributed to the 59 relative increase in thickness of the weak boundary layers located at the stress-free edges. 60 Chen and Fleck (2002) studied aluminum foams sandwiched between metallic substrates. 61 They showed that under constrained deformation, the yield strength increases by almost a 62 factor of two as the height to width ratio of the foam is decreased from 20 to 3. It was also 63 reported that regular hexagonal honeycomb foams do not show this type of size effect. 64 In contrast to the above listed works focusing mainly on foams and honeycombs, nei-65 ther qualitative nor quantitative study of size effects in transforming porous elastomers (and 66 metamaterials based hereon) has been provided in the literature. In order to fill this gap, 67 we provide in this paper a systematic study of precisely those kinds of microstructures and 68 their size effects. In particular, a regular periodic mechanical metamaterial which undergoes 69 reversible pattern transformations is considered. Transformed patterns induce a character-70 istic fluctuation in the deformation field, which is responsible for its anomalous behavior. A 71 hyperelastic material containing periodically arranged circular holes is adopted. Two dis-  The contents of this paper is divided into three sections. The first one defines the problem 95 to be studied, including its geometry, material properties, and the boundary conditions used 96 for the two loading cases. It also details the ensemble averaging scheme used to define the 97 homogenized solution for a range of scale ratios, and the numerical implementation of the 98 model by finite element method. The next section, Section 3, reports the detailed results 99 obtained for the case of compression and bending. Finally, our paper closes with a summary 100 and conclusions in Section 4. For completeness, Appendix A summarizes basic ideas of the 101 first-order computational homogenization.

102
Throughout this paper, the following notation conventions are used: In this equation, X ∈ Ω, X = X 1 e 1 + X 2 e 2 , is the position vector in the plane of the problem 134 for the undeformed geometry. F = (∇ 0 x) c defines the deformation gradient where x gives the 135 current position vector and ∇ 0 denotes the gradient operator with respect to the reference 136 coordinate frame. J denotes the determinant of F , and I 1 and I 2 are the invariants of the 137 right Cauchy-Green deformation tensor C = F c · F , given as The adopted material parameters are given by m 1 = 0.55 MPa, m 2 = 0.3 MPa, and bulk where X BC and x BC denote position vectors of corresponding material points located on  For the bending case, the lateral edges AD and BC are subjected to a relative rota-156 tion θ, combined with periodic boundary conditions, as indicated schematically in Fig. 2b.

157
Mathematically, this can be written as where R(θ) is the rotation tensor. The material points in the deformed configuration x AD and x BC are expressed relative to the center of rotation P , e.g. P = (0, −P X 2 ) T . For im-160 plementation purposes, however, and in order to eliminate the dependence on P , Eq. (5) is, 161 after discretization, written for each node located on the edges BC and AD, and subsequently

178
To describe the relative positioning of the microstructure with respect to the specimen 179 geometry, we introduce a constant translation vector ζ ∈ Ω, where ζ h and ζ v denote normalized shifts along the horizontal and vertical directions relative 181 to the unit cell size . Since the two loading cases considered are periodic in the horizon-182 tal direction, horizontally shifting the microstructure has no effect other than shifting the 183 solution correspondingly. As a consequence, statistical ensemble average over all horizontal 184 shifts can effectively be obtained as a simple spatial average, i.e. 185 mean P ζv 11 (X 2 ) = 1 2 − P ζv 11 (X 1 , X 2 ) dX 1 , −L/2 ≤ X 2 ≤ L/2, which is independent of the horizontal coordinate X 1 . In Eq. (7), the P 11 component of the where the P 11 stress component has been used as an example again.

194
All quantities reported below such as the first Piola-Kirchhoff stress P or the deformation where nint[•] denotes the nearest integer to •. Note that ensemble averaging does not 208 introduce any new length scales, unlike methods like moving volume averaging, which makes 209 it an apt method especially for the cases with relatively small L/ ratios.

377
Before presenting the results for the bending case, the normalization employed throughout 378 this section is first clarified. Graphs of nominal stress and strain will be presented, which 379 follow from the standard theory of Bernoulli beams. As a rotation angle θ is prescribed 380 to the modeled specimen (recall Fig. 2(b)), the corresponding strain at a given vertical 381 coordinate X 2 reads as where κ = θ/(2 ) is the curvature. The where M is the bending moment, I X 3 is the second moment of area, N the reaction normal are available (cf. Fig. 17a for instance).  : Comparison between horizontally free (case (i)) vs fixed (case (ii)) bending for a specimen with scale ratio L/ = 5 and ζ v = 0. The deformed shapes are shown on the left, whereas the nominal stresses due to bending (6M/L 2 ) and tension (N/L) vs the applied nominal strain (θL/(4 )) are depicted on the right.   gradient is clearly visible, and a similar pattern to the one observed in the compression case 444 develops below the neutral axis (recall Fig. 6). For small scale ratios, especially for L/ = 5 445 shown in Fig. 16c, a distorted shape results, as the strain gradient is too strong compared 446 to the microstructural length . Least squares fit (recall Fig. 8). limits only mildly, the peak difference being less than 4 % in both the linear as well as bifur-485 cated regime. Naturally, extreme values occur for small scale ratios, especially for L/ ≤ 10.

486
For higher scale ratios, the ensemble averaged stress approaches its homogenized asymptote 487 from below, as opposed to the compression case presented in Fig. 11. This behavior is a   For the compression case, the nominal stress F/(2 ) is used to quantify the size effect.

522
The thickness of the boundary layer relative to the height of the specimen is studied across    The presented results constitute a rich basis for developing advanced homogenization 546 schemes, to be explored in future work.  large based on prior insights from full scale simulations, although in general identification of 590 the correct size is a rather delicate problem on its own, cf. e.g. Saiki et al. (2002). Boundary 591 conditions applied on ∂Ω m are chosen to be periodic. As a consequence of the separation of 592 scales, the strain field obtained from the first-order computational homogenization is constant 593 in the case of compression (recall Section 3.1), ignoring any boundary layers. This means 594 that all macroscopic points X M (and hence RVEs) experience the same state of deformation 595 and only one RVE suffices to carry out the simulation. In the case of bending the situation 596 changes due to the presence of a strain gradient, cf. Fig. A.23 and Section 3.2, meaning that 597 individual RVEs experience different states of deformation. In order to avoid any bias due 598 to discretization, a mesh sensitivity study has been performed to identify the macroscopic 599 element size h M that yields accurate results (128 elements per specimen height L), cf. also