Extreme softness of brain matter in simple shear¹¹1 dedicated to Martine Ben Amar in esteem and friendship

Modelling the mechanical properties of brain matter is quite a straightforward process once it is accepted that a small enough sample can be considered to be isotropic, homogeneous, nonlinear and viscoelastic. However, testing its mechanical behaviour in order to evaluate its constitutive parameters is not a straightforward matter at all, and is fraught with potential modelling mistakes and experimental pitfalls.

For example, standard tensile tests cannot be performed properly (i.e. cannot follow standard practice, e.g. [1]) on brain matter, because it fractures easily in that regime. As a consequence of that brittleness, it is not practical to cut dog-bone shaped specimens and one must then resort to using cylindrical or prismatic samples, to be glued at their extremities to the cross-heads of a tensile machine, for simple tension or simple compression tests. However, because the faces are glued, end effects intervene early in the deformation, see [2]. They make the deformation field strongly inhomogeneous with the consequence that the stress-strain relationship becomes impossible to determine analytically.

Another standard testing protocol exists [3] which has received a lot less attention for brain matter, namely the simple shear test. This is a most illuminating homogeneous deformation, which brain matter seems to be able to withstand quite well, see Fig.1 where the amount of shear is $K=1.0$, corresponding to a maximal stretch of $K/2+\sqrt{1+K^{2}/4}=1.618$, i.e. an extension of 62%.

In this paper we show that careful modelling and experimental data acquisition lead to an almost complete characterisation of brain matter as an incompressible isotropic nonlinear elastic material. Hence, we show experimentally that forces normal to the platens develop during large simple shear, and this effect allows us to rule out the entire class of generalised neo-Hookean solids. Also, we obtain a linear shear-stress/amount-of-shear relationship, a property characteristic of the Mooney-Rivlin material, with strain energy density

where $C_{1}$, $C_{2}$ are constants and $I_{1}$, $I_{2}$ are the first two principal invariants of the left Cauchy-Green deformation tensors.

The conclusion is that it is thus sufficient to determine only two material parameters in order to fully characterise the (quasi-static) mechanical behaviour of brain matter, up to at least 60%, which is more than adequate for practical purposes, including the study of diffuse axonal injury (DAI), which is believed to occur at macroscopic shear strains of approximately 10%–50% (see e.g. [4, 5]).

Other advances put forth by this work are the exhibition of the normal force effects generated by simple shear in brain matter, the numerical verification that the experimental protocol does indeed produce almost homogeneous fields, and the comparison of the stiffness of brain matter with that of a silicone gel at quasi-static speeds. We found that at quasi-static strain rates, the former is about 30 times softer in shear than the latter.

Let us start by recalling that in simple shear, $W=W(K^{2})$ so that $\sigma_{12}$ given by (2.1) is always an odd function of $K$. This is in line with the physics of the deformation, as the shear stress necessary to shear the block by amount $K$ is the opposite of that necessary to shear it by an amount $-K$. In particular it follows that $\sigma_{12}$ cannot be of the form $aK^{b}$ where $a$, $b$ are curve-fitting constants as in [10].

The proof that brain matter behaves as a Mooney-Rivlin material as long as stretches are below $60\%$ is reached here through the verification of the linearity between $\sigma_{12}$ and $K$. This result is not that surprising, because there exist a well-known [6, 11] connection between the Mooney-Rivlin strain energy density and that of the most general nonlinear, third-order elasticity, isotropic, incompressible material,

where $\mu$ is the shear modulus, $A$ is a third-order elastic constant, and $\bm{E}$ is the Green strain. At the same order of approximation, (1.1) and (4.1) are equivalent, and the constants are connected through [12]

Note that shear force data only provides a value for $\mu=2(C_{1}+C_{2})$, and normal force data or another testing protocol are required to complete its material characterization.

The normal forces at play during simple shear have been checked qualitatively in this study, through experiments and numerical simulations. Recalling that in linear elasticity, $\sigma_{11}=\sigma_{22}=0$, then the observation of non-zero normal forces for tests confirms that the behaviour of the brain matter is indeed nonlinear. This is the well-known (positive) Poynting effect. The presence of this phenomenon also allows us to conclude that brain matter does not belong to the so-called ”generalized neo-Hookean class”, where $W=W(I_{1})$ only, because if it did, then $\sigma_{22}$ would be zero according to (2.1)₂. In effect, observing the bulging out of the brain matter tells us that $C_{2}>0$.

The comparison of the experimental slopes (i.e. of the shear moduli $\mu$) between the Sylgard gel and the brain samples show that brain matter is an extremely soft solid, at least 30 times less resistant to shearing forces than a silicone gel, when sheared at quasi-static speeds. We note that at higher strain rates, the shear modulus is much increased, allowing the brain to resist better to shearing tractions during an impact for instance. Hence we found [15] that it was 10 to 20 times higher at strain rates occurring in traumatic brain injury incidents. Specifically we measured it to be $1.157\pm 0.25$ kPa, $1.347\pm 0.19$ kPa, $2.197\pm 0.225$ kPa and $2.527\pm 0.27$ kPa at 30, 60, 90 and 120/s rates, respectively, again for destructive simple shear tests on porcine brains.

The numerical simulation of simple shear finally confirms that outside localized edge areas, the stress distribution is largely homogeneous throughout the sample. It makes experimental simple shear a good candidate to extract material properties based on reliable simple analytical models. As explained earlier, we chose the numerical values of $C_{1}$ and $C_{2}$ such that $2(C_{1}+C_{2})=165$ Pa, in line with the experimental determination of the shear modulus $\mu$. We picked $C_{2}$ much larger than $C_{1}$ to emphasise the visual bulging of the brain matter through the hole. Lower values of $C_{2}$ also showed the same effect, as expected.

A shortcoming of our experimental protocol is that it only provides access to the value of $C_{1}+C_{2}$ and to the sign of $C_{2}$, but not to the actual values of $C_{1}$ and $C_{2}$ separately. To access these values we would need to measure $\sigma_{22}$, which is not a trivial task. An alternate, non-destructive, method is to use the large acousto-elastic effect [13], which has recently been applied to porcine brains [14].

Partial funding by a “New Foundations” award from the Irish Research Council is gratefully acknowledged by the first author. We thank Mélanie Otténio (Université de Lyon) for her most useful input.