Jeremiah G. Murphy

Simple shearing of soft biological tissues

Shearing is induced in soft tissues in numerous physiological settings. The limited experimental data available suggest that a severe strain-stiffening effect occurs in the shear stress when soft biological tissues are subjected to simple shear in certain directions. This occurs at relatively small amounts of shear (when compared with the simple shear of rubbers). This effect is modelled within the framework of nonlinear elasticity by consideration of a class of incompressible anisotropic materials. Owing to the large stresses generated for relatively small amounts of shear, particular care must be exercised in order to maintain a homogeneous deformation state in the bulk of the specimen. The results obtained are relevant to the development of accurate shear test protocols for the determination of constitutive properties of soft tissues. It is also demonstrated that there is a fundamental ambiguity in determining the normal stresses in simple shear when soft tissues are modelled as incompressible hyperelastic materials owing to the arbitrary nature of the hydrostatic pressure term. Two physically well-motivated approaches to determining the pressure are presented here, and the resulting hydrostatic stresses are compared and contrasted. The possible generation of cavitational damage owing to critical hydrostatic stress levels is briefly discussed.

Nonmonotonic cavity growth in finite, compressible elasticity

We obtain closed form solutions to the problems of cylindrical and spherical cavitation when a stretch is prescribed on the outer boundary for a compressible elastic material. The strain energy function is quite general and contains an arbitrary function of a linear combination of the principal stretch invariants. The cavitating solution is shown to be preferred to the corresponding homogeneous deformation. Three modes of cavitation are identified for the spherical problem whereas only one is possible in the cylindrical case. The most general strain energy function for which the cavitating solution is possible is briefly discussed.

Strain Energy Functions for a Poisson Power Law Function in Simple Tension of Compressible Hyperelastic Materials

The most general strain energy function that yields a power law relationship between the principal stretches in the simple tension of nonlinear, elastic, homogeneous, compressible, isotropic materials is obtained. The approach taken generalises that used by Blatz and Ko. The strain energy function obtained depends on the choice of two stretch invariants. The forms of the strain energy function for a number of such choices are obtained. Finally, some consequences of the choice of strain energy function on the stress–strain relationship for uniaxial tension are investigated.

Constitutive modeling for moderate deformations of slightly compressible rubber

A constitutive model based on an assumption regarding the general response to hydrostatic pressure is proposed for the moderate deformations of slightly compressible (or nearly incompressible) rubber. It is shown that an excellent fit is obtained with the available experimental data for a particularly simple form of strain-energy density. The data considered are from those material characterization tests that involve only moderate deformations.

Some Remarks on Kinematic Modeling of Limiting Chain Extensibility

A kinematic model is proposed for materials whose polymeric molecular chains have a maximum achievable length. A widely used model for this effect which imposes a bound on the first strain invariant is examined and is shown that the difference between the models can be up to 23% for a limiting stretch value proposed in the literature. The closeness of fit is also examined for other limiting stretch values. A family of approximations is introduced and shown to converge to the proposed model. Convergence is shown to be fast for an important limiting stretch value.

Mechanical characterization of the P56 mouse brain under large-deformation dynamic indentation

The brain is a complex organ made up of many different functional and structural regions consisting of different types of cells such as neurons and glia, as well as complex anatomical geometries. It is hypothesized that the different regions of the brain exhibit significantly different mechanical properties, which may be attributed to the diversity of cells and anisotropy of neuronal fibers within individual brain regions. The regional dynamic mechanical properties of P56 mouse brain tissue in vitro and in situ at velocities of 0.71–4.28 mm/s, up to a deformation of 70 μm are presented and discussed in the context of traumatic brain injury. The experimental data obtained from micro-indentation measurements were fit to three hyperelastic material models using the inverse Finite Element method. The cerebral cortex elicited a stiffer response than the cerebellum, thalamus, and medulla oblongata regions for all velocities. The thalamus was found to be the least sensitive to changes in velocity, and the medulla oblongata was most compliant. The results show that different regions of the mouse brain possess significantly different mechanical properties, and a significant difference also exists between the in vitro and in situ brain.

Dynamic mechanical properties of murine brain tissue using micro-indentation

In the past 50 years significant advances have been made in determining the macro-scale properties of brain tissue in compression, tension, shear and indentation. There has also been significant work done at the nanoscale using the AFM method to characterise the properties of individual neurons. However, there has been little published work on the micro-scale properties of brain tissue using an appropriate indentation methodology to characterise the regional differences at dynamic strain rates. This paper presents the development and use of a novel micro-indentation device to measure the dynamic mechanical properties of brain tissue. The device is capable of applying up to 30/s strain rates with a maximum indentation area of 2500 μm(2). Indentation tests were carried out to determine the shear modulus of the cerebellum (2.11 ± 1.26 kPa, 3.15 ± 1.66 kPa, 3.71 ± 1.23 kPa) and cortex (4.06 ± 1.69 kPa, 6.14 ± 3.03 kPa, 7.05 ± 3.92 kPa) of murine brain tissue at 5, 15, and 30/s up to 14% strain. Numerical simulations were carried out to verify the experimentally measured force-displacement results.

Uniform transmural strain in pre-stressed arteries occurs at physiological pressure

Residual deformation (strain) exists in arterial vessels, and has been previously proposed to induce homogeneous transmural strain distribution. In this work, we present analytical formulations that predict the existence of a finite internal (homeostatic) pressure for which the transmural deformation is homogenous, and the corresponding stress field. We provide evidence on the physical existence of homeostatic pressure when the artery is modeled as an incompressible tube with orthotropic constitutive strain-energy function. Based on experimental data of rabbit carotid arteries and porcine coronary arteries, the model predicts a homeostatic mean pressure of ~90 mmHg and 70-120 mmHg, respectively. The predictions are well within the physiological pressure range. Some consequences of this strain homogeneity in the physiological pressure range are explored under the proposed assumptions.

A method to model simple tension experiments using finite elasticity theory with an application to some polyurethane foams

A framework for the modelling of simple tension data for homogeneous, isotropic, non-linear, unconstrained elastic materials is given. This framework is then used to develop a method for the modelling of such data. Specifically, if the data suggest a power law relationship between the axial and lateral stretches, an approach that is a generalisation of the one used by Blatz and Ko [Trans. Soc. Rheol. 6 (1962) 223] is recommended. If the power law relationship does not hold, a new approach is suggested. This procedure is then applied to experimental data for two polyurethane foams.

PLANE STRESS PROBLEMS FOR COMPRESSIBLE MATERIALS

Within the context of unconstrained, finite elasticity, exact solutions will be obtained for a number of plane stress boundary-value problems. The inflation and azimuthal shearing of cylinders of the so-called Varga material will be considered. Additionally, a simple, static interpretation of the strong ellipticity condition will be given for these materials.