Traumatic brain injury (TBI) occurs when local mechanical load exceeds certain tolerance levels for brain tissue. Extensive research has been done previously for brain matter experiencing compression at quasistatic loading; however, limited data is available to model TBI under dynamic impact conditions. In this research, an experimental setup was developed to perform unconfined compression tests and stress relaxation tests at strain rates≤90/s. The brain tissue showed a stiffer response with increasing strain rates, showing that hyperelastic models are not adequate. Specifically, the compressive nominal stress at 30% strain was 8.83±1.94, 12.8±3.10 and 16.0±1.41 kPa (mean±SD) at strain rates of 30, 60 and 90/s, respectively. Relaxation tests were also conducted at 10%- 50% strain with the average rise time of 10 ms, which can be used to derive time dependent parameters. Numerical simulations were performed using one-term Ogden model with initial shear modulusµ o =6.06±1.44, 9.44±2.427 and 12.64±1.227 kPa (mean±SD) at strain rates of 30, 60 and 90/s, respectively. A separate set of bonded and lubricated tests were also performed under the same test conditions to estimate the friction coefficient µ, by adopting combined experimental-computational approach. The values ofµwere 0.1±0.03 and 0.15±0.07 (mean±SD) at 30 and 90/s strain rates, respectively, indicating that pure slip conditions cannot be achieved in unconfined compression tests even under fully lubricated test conditions. The material parameters obtained in this study will help to develop biofidelic human brainfinite element models, which can subsequently be used to predict brain injuries under impact conditions. c

http://www.maths.nuigalway.ie/~destrade/Publis/destrade_90.pdf

Mechanical characterization of brain tissue in compression at dynamic strain rates.

The Holzapfel-Gasser-Ogden (HGO) model for anisotropic hyperelastic behaviour of collagen fibre reinforced materials was initially developed to describe the elastic properties of arterial tissue, but is now used extensively for modelling a variety of soft biological tissues. Such materials can be regarded as incompressible, and when the incompressibility condition is adopted the strain energy Ψ of the HGO model is a function of one isotropic and two anisotropic deformation invariants. A compressible form (HGO-C model) is widely used in finite element simulations whereby the isotropic part of Ψ is decoupled into volumetric and isochoric parts and the anisotropic part of Ψ is expressed in terms of isochoric invariants. Here, by using three simple deformations (pure dilatation, pure shear and uniaxial stretch), we demonstrate that the compressible HGO-C formulation does not correctly model compressible anisotropic material behaviour, because the anisotropic component of the model is insensitive to volumetric deformation due to the use of isochoric anisotropic invariants. In order to correctly model compressible anisotropic behaviour we present a modified anisotropic (MA) model, whereby the full anisotropic invariants are used, so that a volumetric anisotropic contribution is represented. The MA model correctly predicts an anisotropic response to hydrostatic tensile loading, whereby a sphere deforms into an ellipsoid. It also computes the correct anisotropic stress state for pure shear and uniaxial deformations. To look at more practical applications, we developed a finite element user-defined material subroutine for the simulation of stent deployment in a slightly compressible artery. Significantly higher stress triaxiality and arterial compliance are computed when the full anisotropic invariants are used (MA model) instead of the isochoric form (HGO-C model).

A robust anisotropic hyperelastic formulation for the modelling of soft tissue.

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The Non Linear Field Theories Of Mechanics

The mechanical response of a homogeneous isotropic linearly elastic material can be fully characterized by two physical constants, the Young’s modulus and the Poisson’s ratio, which can be derived by simple tensile experiments. Any other linear elastic parameter can be obtained from these two constants. By contrast, the physical responses of nonlinear elastic materials are generally described by parameters which are scalar functions of the deformation, and their particular choice is not always clear. Here, we review in a unified theoretical framework several nonlinear constitutive parameters, including the stretch modulus, the shear modulus and the Poisson function, that are defined for homogeneous isotropic hyperelastic materials and are measurable under axial or shear experimental tests. These parameters represent changes in the material properties as the deformation progresses, and can be identified with their linear equivalent when the deformations are small. Universal relations between certain of these parameters are further established, and then used to quantify nonlinear elastic responses in several hyperelastic models for rubber, soft tissue and foams. The general parameters identified here can also be viewed as a flexible basis for coupling elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2017.0607

How to characterize a nonlinear elastic material? A review on nonlinear constitutive parameters in isotropic finite elasticity.

Constraining magnitudes of mechanical and thermo-mechanical parameters of rocks and shear zones are the important goals in structural geology and tectonics (Talbot in J Struct Geol 21:949–957, 1999). Such parameters aid dynamic scaling of analogue tectonic models (Ramberg in Gravity, deformation and the Earth’s crust in theory, experiments and geological applications, 2nd edn. Academic Press, London, 1981), which are useful to unravel tectonics in further details (Schultz-Ela and Walsh in J Struct Geol 24:247–275, 2002). The channel flow extrusion of the Higher Himalayan Shear Zone (HHSZ, = Higher Himalaya) can be explained by a top-to-S/SW simple shear (i.e. the D2 deformation) in combination with a pressure gradient induced flow against gravity. Presuming its Newtonian incompressible rheology with parallel inclined boundaries, the viscosity (μ) of this shear zone along a part of the Himalayan chain through India, Nepal and Bhutan is estimated to vary widely between ~1016 and 1023 Pa s, and its Prandtl number (Pr) within ~1021–1028. The estimates utilized ranges of known thickness (6–58 km) of the HHSZ, that of its top subzone of ductile shear of normal shear sense (STDSU: 0.35–9.4 km), total rate of slip of its two boundaries (0.7–131 mm year−1), pressure gradient (0.02–6 kb km−1), density (2.2–3.1 g cm−3) and thermal diffusivity (0.5 × 10−6–2.1 × 10−6 m s−2) along the orogenic trend. Considering most of the parameters specifically for the Sutlej section (India), the calculated viscosity (μ) and the Prandtl number (Pr) of the HHSZ are deduced to be μ: ~1017–1023 Pa s and Pr ~ 1022–1028. The upper limits of the estimated viscosity ranges are broadly in conformity with a strong Tibetan mid-crust from where a part of the HHSZ rocks extruded. On the other hand, their complete ranges match with those for its constituent main rock types and partly with those for the superstructure and the infrastructure. The estimated mechanical and thermo-mechanical parameters of the HHSZ will help to build dynamically scaled analogue models for the Himalayan deformation of the D2–phase.

Channel flow extrusion model to constrain dynamic viscosity and Prandtl number of the Higher Himalayan Shear Zone

For homogeneous, isotropic, non-linearly elastic materials, the form of the homogeneous deformation consistent with the application of a Cauchy shear stress is derived here for both compressible and incompressible materials. It is shown that this deformation is not simple shear, in contrast to the situation in linear elasticity. Instead, it consists of a triaxial stretch superposed on a classical simple shear deformation, for which the amount of shear cannot be greater than 1. In other words, the faces of a cubic block cannot be slanted by an angle greater than 45° by the application of a pure shear stress alone. The results are illustrated for those materials for which the strain-energy function does not depend on the principal second invariant of strain. For the case of a block deformed into a parallelepiped, the tractions on the inclined faces necessary to maintain the derived deformation are calculated.

/pdf/simple-shear-is-not-so-simple-rgmajf90k5.pdf

Simple shear is not so simple

Incompressible nonlinearly hyperelastic materials are rarely simulated in Finite Element numerical experiments as being perfectly incompressible because of the numerical difficulties associated with globally satisfying this constraint. Most commercial Finite Element packages therefore assume that the material is slightly compressible. It is then further assumed that the corresponding strain-energy function can be decomposed additively into volumetric and deviatoric parts. We show that this decomposition is not physically realistic, especially for anisotropic materials, which are of particular interest for simulating the mechanical response of biological soft tissue. The most striking illustration of the shortcoming is that with this decomposition, an anisotropic cube under hydrostatic tension deforms into another cube instead of a hexahedron with non-parallel faces. Furthermore, commercial numerical codes require the specification of a `compressibility parameter' (or `penalty factor'), which arises naturally from the flawed additive decomposition of the strain-energy function. This parameter is often linked to a `bulk modulus', although this notion makes no sense for anisotropic solids; we show that it is essentially an arbitrary parameter and that infinitesimal changes to it result in significant changes in the predicted stress response. This is illustrated with numerical simulations for biaxial tension experiments of arteries, where the magnitude of the stress response is found to change by several orders of magnitude when infinitesimal changes in `Poisson's ratio' close to the perfect incompressibility limit of 1/2 are made.

Deficiencies in numerical models of anisotropic nonlinearly elastic materials.

The classical flexure problem of non-linear incompressible elasticity is revisited assuming that the bending angle suffered by the block is specified instead of the usual applied moment. The general moment-bending angle relationship is then obtained and is shown to be dependent on only one non-dimensional parameter: the product of the aspect ratio of the block and the bending angle. A Maclaurin series expansion in this parameter is then found. The first-order term is proportional to $\mu$, the shear modulus of linear elasticity; the second-order term is identically zero, because the moment is an odd function of the angle; and the third-order term is proportional to $\mu(4\beta -1)$, where $\beta$ is the non-linear shear coefficient, involving third-order and fourth-order elasticity constants. It follows that bending experiments provide an alternative way of estimating this coefficient, and the results of one such experiment are presented. In passing, the coefficients of Rivlin's expansion in exact non-linear elasticity are connected to those of Landau in weakly (fourth-order) non-linear elasticity.

Onset of Non-Linearity in the Elastic Bending of Blocks

We identify three distinct shearing modes for simple shear deformations of transversely isotropic soft tissue which allow for both positive and negative Poynting effects (that is, they require compressive and tensile lateral normal stresses, respectively, in order to maintain simple shear). The positive Poynting effect is that usually found for isotropic rubber. Here, specialisation of the general results to three strain-energy functions which are quadratic in the anisotropic invariants, linear in the isotropic strain invariants and consistent with the linear theory suggests that there are two Poynting effects which can accompany the shearing of soft tissue: a dominant negative effect in one mode of shear and a relatively small positive effect in the other two modes. We propose that the relative inextensibility of the fibres relative to the matrix is the primary mechanism behind this large negative Poynting effect.

Dominant negative Poynting effect for soft tissues.

Finite element simulations of rubbers and biological soft tissue usually assume that the material being deformed is slightly compressible. It is shown here that, in shearing deformations, the corresponding normal stress distribution can exhibit extreme sensitivity to changes in Poisson’s ratio. These changes can even lead to a reversal of the usual Poynting effect. Therefore, the usual practice of arbitrarily choosing a value of Poisson’s ratio when numerically modelling rubbers and soft tissue will, almost certainly, lead to a significant difference between the simulated and actual normal stresses in a sheared block because of the difference between the assumed and actual value of Poisson’s ratio. The worrying conclusion is that simulations based on arbitrarily specifying Poisson’s ratio close to 1=2 cannot accurately predict the normal stress distribution even for the simplest of shearing deformations. It is shown analytically that this sensitivity is caused by the small volume changes, which inevitably accompany all deformations of rubber-like materials. To minimise these effects, great care should be exercised to accurately determine Poisson’s ratio before simulations begin. Copyright © 2011 John Wiley & Sons, Ltd.

/pdf/slight-compressibility-and-sensitivity-to-changes-in-poisson-5a55s6b3op.pdf

Jerry G. Murphy

Papers

Simple shear is not so simple

Deficiencies in numerical models of anisotropic nonlinearly elastic materials.

Onset of Non-Linearity in the Elastic Bending of Blocks

Dominant negative Poynting effect for soft tissues.

Slight compressibility and sensitivity to changes in Poisson's ratio