Experimental Testing of Multiscale Similitude in Mechanical Metamaterials

Summer Student, Bethia Hardy

Project lead, Dr Raul Ochoa Cabrero (Manchester Metropolitan University)

Introduction

Metamaterials exhibit changes in its properties based on the design of its structure. One example is the auxetic rotating squares design theorised by Grima and Evans, 2000 [1]. The rotating nature of the internal structure results in the system being theorised to have a Poisson’s ratio of -1, with experimental work confirming this result [2].
Micropolar elasticity incorporates the internal rotation of the points within a material as well as the translation of points [3]. Lakes and Drugan, 2015 [4] conducted bending tests on a bar of foam and compared the resulting deformation to both the classical theory of elasticity and micropolar elasticity. They found a sigmoidal curve, with different magnitude dependent on the pore size, suggesting a local length scale effect that is consistent with micropolar elasticity. Investigation of scale effects in mechanical metamaterials requires novel modelling approaches and therefore finite similitude is proposed as a candidate to account for size effects in the physical modelling of the material.

This report explores the behaviour of a foam bar with the auxetic rotating square structure and the size effects produced by internal features to inform and validate the application of finite similitude to mechanical metamaterials.

Methods

The auxetic rotating squares (Grima and Evans, 2000) for laser cutting were implemented with a hinge. The centre of rotation was positioned in the middle of the hinge. The squares were cut at a 45⁰C angle to reduce the amount of internal contact between them.
Samples of three different square sizes were cut: 20mm (1/1), 10mm (1/2) and 6.67mm (1/3), scaling features accordingly.
The samples use LD60 foam (Very Firm 60Kg/m³), 12mm thick. (Total thickness of a tested sample 24mm) [6]

Three sets of samples corresponding to each size (1,1/2,1/3) were laser cut:

Unit square format	Hinge size (mm)
3×12 (c)	2.5	2	1.67	2.5
2×8 (b)	5	4	2	2.5
1×4 (a)	7.5	6	4	2.5

Figure 1: Samples. a – 1×4 unit squares arrangement with each square 20mm in length and the hinge 6mm wide.  b – 2×8 unit squares arrangement, each square 10mm in length and the hinge 4mm wide. c – 3×12 unit square arrangement with each square 6.67mm in length and the hinge 2mm wide.

A pattern (dots) was added to the samples for DIC analysis during 4-point bending test:

Figure 2: four-point bending set-up with axis.

The sample was displaced using by 13mm at 26mm/sec, and the test recorded perpendicular for 2D  analysis.

The video of the four-point bending test was then processed in GOM correlate to find the x and y position data of each dotted points.
The data was then used to find the engineering strain in the x-axis across the middle of the sample, in symmetrical points:

This was calculated for each pair of points when the sample was displaced in the y-direction by 5mm (as calculated in GOM correlate). The strain could then be plotted against the y-position data, setting the neutral axis in the middle of the sample, tilt was removed from the results. The engineering strain was utilised to compare the behaviour of the different material configurations and characterise the scale effects.

Results and Discussion

Figure 3: Graph of engineering strain against y position of the points of samples a b and c with hinge thickness of 6mm, 4mm and 2mm respectively. 8a show the strain. 8b shows the engineering strain with the tilt removed, compared to the classical strain.

Figure 4: Graph of the three samples with the tilt removed. The y position has been adjusted for comparison of the equivalent unit squares within in each sample. The y-axis is each of the points on the sample. Points 12-9 is the first unit square at the top of the sample, points 8-5 are the second unit square, and point 4-1 are the third unit square.

As shown by Fig. 4, each sample follows a sinusoidal pattern, the total length of which depends on the number of unit squares in the y direction. Increasing the hinge size reduces the overall strain on the sample since the microrotation/microdeformation is highly dependent on hinge thickness. There is a much greater variation in the strain between the smaller values of hinge thickness.  Similarly, the sinusoidal patten is present when analysing results according to unit cell size changes, with size effects accounting for the variation in the amplitude and frequency of the graphs. Further plots supporting these findings can be found in the Supplementary Materials file.

Conclusions

The work undertaken has documented and characterised the size effects associated to the internal features of mechanical metamaterials (rotating squares) under bending. It can be concluded that:

• results exhibit a sinusoidal behaviour conforming to micropolar predictions;
• scale effects are attributed to length changes in the internal features of the metamaterial; and
• cell size/number and hinge thickness ratios are suitable invariants to target using the finite similitude theory.

References

[1] Grima J. N. and Evans K. E. (2000) Auxetic Behavior from Rotating Squares. Available at: 10.1023/A:1006781224002 (Accessed: 16 June 2025)

[2] Chen M. et al. (2022) Elastic Properties of Lightweight Rotating Square Structures. Available at: https://doi.org/10.1016/j.mtcomm.2022.104256 (Accessed: 18 June 2025)

[3] Lakes R. S. (2016) Physical Meaning of Elastic Constants in Cosserat, Void,  and Microstretch Elasticity. Available at: 10.2140/jomms.2016.11.217 (Accessed 18 June 2025)

[4] Lakes R. S. and Drugan W. J. (2015) Bending of a Cosserat Elastic Bar of Square Cross  Section – Theory and Experiment. Available at: 10.1115/1.4030626 (Accessed 18 June 2025)

[5] Frenzel T., Kadic. M. and Wegener M. (2017) Three-Dimensional Mechanical Metamaterials with a Twist. Available at: 10.1126/science.aao4640 (Accessed 18 June 2025)

[6] Algeos (2025) Plastazote foam sheets. available at: https://www.algeos.com/plastazote-sheets (Accessed 6 August 2025)