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# Microscopic and macroscopic instabilities in hyperelastic fiber composites

V. Slesarenko and S. Rudykh, Microscopic and macroscopic instabilities in hyperelastic fiber composites. *J. Mech. Phys. Solids*, **99**, 471 – 482 (2017)

**Abstract**:

In this paper, we study the interplay between macroscopic and microscopic instabilities in 3D periodic fiber reinforced composites undergoing large deformations. We employ the Bloch-Floquet analysis to determine the onset of microscopic instabilities for composites with hyperelastic constituents. We show that the primary mode of buckling in the fiber composites is determined by the volume fraction of fibers and the contrast between elastic moduli of fiber and matrix phases. We find that for composites with volume fraction of fibers exceeding a threshold value, which depends on elastic modulus contrast, the primary buckling mode corresponds to the long wave or macroscopic instability. However, composites with a lower amount of fibers experience microscopic instabilities corresponding to wavy or helical buckling shapes. Buckling modes and critical wavelengths are shown to be highly tunable by material composition. A comparison between the instability behavior of 3D fiber composites and laminates, subjected to uniaxial compression, reveals the significant differences in critical strains, wavelengths, and transition points from macro- to microscopic instabilities in these composites.

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## Comments

## very interesting material, has this ever been observed ?

dear Stephan

very interesting work! However, this sounds so basic that I am surprised if it has not been observed before! Did you try any experiment?

regards

mike

## Thank you Mike! There are

Thank you Mike! There are many expermental obserevations of wavy patterns in similar hyperelastic layered materials in 2D (for example, Li et al, Adv. Eng. Mater. 2013, and Rudykh and Slesarenko, Soft Matter 2016). In 3D, there are similar observations, for example, in the system of a Nitinol slender rod embedded in an elastomeric matrix by Su et al., Soft Matter 2014. We have some experiments on heperelastic fiber composites, and I plan to post these results when we summarize them.

## but from a practical design perspective, what does this suggest?

Composites is a rich area of course, and one can "tune" properties, perhaps inspired by nature, and who knows what more. From a practical point of view, your higher mode instatibilities are more dangerous and inexpected than the first mode one?

For example, in aeronautical structures, buckling is well known as structures are thin (despite often they are sandwiches, so there is bigger inertia than just a laminate of composites), and people design also in post-buckling regime, called sometimes crippling.

Crippling of stiffeners

Course subject(s) 8. CRIPPLING OF STIFFENERS

In this lecture Prof. C. Kassapoglou talks about local failure modes of stiffeners, which do not cause the structure to lose all load carrying capabilities. This behavior is called crippling, and is preferable over total failure modes such as column buckling since your structure retains some strength after initial failure.

He starts by writing out general equations for a stiffener on an elastic foundation, for a multitude of boundary conditions, followed by an example of using graphite pins under an angle with respect to the facing as core of a sandwich material. He then proceeds to tell about the main topic of the lecture, the crippling of stiffeners. He makes a distinction between segments of the stiffener cross section which have either one or no edge free, and compares the analytic results with actual testing data.

After the break, a case of an aircraft fire is discussed, and what considerations are when repairing composite structures. At the end of the lecture the practical implications of radius regions in the cross section are discussed, and examples are given of how to deal with them.

AE4509 Crippling of stiffeners

Lecture_8_-_Flange_crippling.pdf

## crippling is a "local" buckling unrelated to length of the beam

p.s. I'd be interested to know if you have given any thought of the problem of crippling, because its wavelength is unrelated to length of the beam. In principle, in experiments, how we can distinguish if we are observing your higher order modes, or a crippling mode?

And in general, what is the connection / distinction between your mode of buclking, and crippling? This would have some seriour implication in composite industry.

## Thanks again Mike! What we

Thanks again Mike! What we analyze is actually material instabilities, and these are not "higher" modes. These instabilities can develop at different length scales. To identify at which length scale it starts to develop first, we scan diffrent wave numbers. Once the instability is detected, we know the corresponding wave length for that. Usually each composition has a unique critical wavelength. Alternatively, one can check only the long-wave limit and obtain an estimate for onset of instability -- this approach works well for large volume fractions of fibers.

As for the experiments, we indeed have finite size of the sample, and the actual boundary conditions can afffect the result; so, typically we would need samples that will include a few critical wavelength to reduce the influence of the finite sample size

## of course, the parallel with crippling is loose

thanks Stephan

of course the parallel with crippling is loose, but I was hoping it would serve you as an example from industry. In crippling, the failure mode is not global instability (buckling, which depends on size of the specimen), but a local one, which is not strictly a "higher mode". Therefore, there is a parallel with your material instability which also turns out independent on the size of the specimen.

I am not expert of your theory (and not even much of crippling), so I cannot make precise statements. It may be that your methods could be applied to crippling too, with significant interest from aeronautical companies, since that is a dominant failure mode, also in composites, and is not well understood.

mike