Exponents of the one-term Ogden model: insights from simple shear.

Isotropic one-term Ogden models are widely used to predict the mechanical response of both incompressible elastomers and soft tissue. Even though the exponent might be chosen to yield excellent agreement with some aspects of mechanical response, there is no guarantee that these models will be physically realistic in all situations. We show here that, in particular, the predictions of models with either negative or large positive exponents do not seem physically realistic in simple shear. The mechanical response of materials in shear should be physically realistic to ensure rational and reliable predictions for complex geometries and boundary conditions. We suggest that for problematic values of exponents of one-term models that extra Ogden invariants should necessarily be included in the model. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

The effect of fiber-matrix interaction on the kinking instability arising in the torsion of stretched fibrous biofilaments.

The response of fibrous soft tissues undergoing torsional deformations is a topic of current interest. Such deformations are common in ligaments and tendons and are also of particular interest in cardiac mechanics. The problem of torsion superimposed on extension of incompressible hyperelastic solid circular cylinders is a classic problem of nonlinear elasticity that has been considered by many authors in the context of rubber elasticity particularly for isotropic materials. A striking feature of such problems is the instability that arises with sufficiently large twist where a kink and then a knot suddenly appears. An energy approach to examining this instability when the extension and twist are prescribed was described by Gent and Hua (2004) and illustrated there for a neo-Hookean isotropic elastic material. The theoretical results were compared with experimental observations on natural rubber rods. Murphy (2015) has shown that the approach of Gent and Hua (2004) for isotropic materials can be simplified when the rods are assumed to be thin and this theory was applied to transversely isotropic materials by Horgan and Murphy (2016). In contrast with the case for isotropic materials, it was shown there that the kinking instability occurs even in the absence of stretch, i.e., for the case of pure torsion. Here we are concerned with the implications of this simplified thin rod instability theory for fiber-reinforced transversely isotropic materials that reflect fiber-matrix interaction. It is again shown that the kinking instability occurs even in the absence of stretch, i.e., for the case of pure torsion. The results are illustrated for a specific strain-energy density function that models fiber-matrix interaction. It is shown that the critical twist at which kinking occurs decreases as a measure of fiber-matrix interaction is increased so that the fiber-matrix interaction has a destabilizing effect. The results are illustrated using experimental data of other authors for skeletal muscles and for porcine brain white matter tissue.

Sex- and age-specific mechanical properties of liver tissue under dynamic loading conditions.

The liver is the most commonly injured abdominal organ following either blunt or penetrating impact. Current mechanical properties available in the literature are typically only measured at low strain rates, low strains, or use linear viscoelastic models. There is also a dearth of high-rate, large strain, viscoelastic data available for liver tissue which are required to model the deformation of the liver during high-rate impacts. Furthermore, the issue of whether mouse liver's mechanical properties are sex-dependent has not been addressed previously. Here, we present the first in vitro sex- and age-controlled mechanical characterisation of mixed-strain (C57BL and wild-type) mouse liver tissue at a localised length scale using large-deformation and high strain rate micro-indentation. We also investigated the effects of age on the mechanical properties of liver tissue. Force-relaxation experiments were performed on both male and female mouse livers up to 35% strain at 10/s and allowed to relax for 1s. The neo-Hookean based quasi-linear viscoelastic model was fitted to the experimental data to determine the large-strain behaviour of the tissue. A comprehensive statistical analysis was performed to determine whether any significant differences existed for (i) the short-term shear moduli and (ii) long-term shear moduli between 10 weeks-old male and female mouse livers, and (iii) the short-term and (iv) long-term shear moduli for 6, 10, and 56 weeks-old mouse livers. No significant differences were found between the mechanical properties in the sex groups. The 56 weeks-old liver tissue was found to be significantly stiffer than the 6 weeks-old liver tissue, but not the 10 weeks-old.

Mechanical characterisation of brain tissue up to 35% strain at 1, 10, and 100/s using a custom-built micro-indentation apparatus.

Understanding the behaviour of soft tissues under large strains and high loading rates is crucial in the field of biomechanics in order to investigate tissue behaviour during pathological processes such as traumatic brain injury (TBI). It is, therefore, necessary to characterise the mechanical properties of such tissues under large strain and high strain rates that are similar to those experienced during injury. However, there is a dearth of large strain and high rate mechanical properties for brain tissue. This is likely driven by the lack of commercially available equipment to perform such tests and the difficulties associated with developing appropriate custom-built apparatus. Here, we address this problem by presenting a novel, custom-built micro-indentation apparatus that is capable of characterising the mechanical properties of brain tissue up to 35% at 100/s with a spatial resolution of 250 µm. Indentations were performed on the cortex and cerebellum of five-week-old mouse brains up to 35% strain at 1, 10, and 100/s. Three hyperelastic models were fitted to the experimental data that demonstrate the strong rate-dependency of the tissue. The neo-Hookean shear modulus for the cortex tissue was calculated to be 2.36 ±â€¯0.46, 3.64 ±â€¯0.48, and 8.98 ±â€¯0.66 kPa (mean ±â€¯SD) for 1, 10, and 100/s, respectively. Similarly, the cerebellum shear modulus was calculated to be 1.12 ±â€¯0.26, 1.58 ±â€¯0.32, 3.10 ±â€¯0.70 kPa for 1, 10, and 100/s, respectively. Student's t-tests were used to show statistically significant differences between the cortex and cerebellum at each strain rate. Furthermore, we discuss the apparent strain-softening effect in the 100/s force-displacement curves for both regions after approximately 30% strain.

Author Correction: Region and species dependent mechanical properties of adolescent and young adult brain tissue.

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.

Region and species dependent mechanical properties of adolescent and young adult brain tissue.

Traumatic brain injuries, the leading cause of death and disability in children and young adults, are the result of a rapid acceleration or impact of the head. In recent years, a global effort to better understand the biomechanics of TBI has been undertaken, with many laboratories creating detailed computational models of the head and brain. For these models to produce realistic results they require accurate regional constitutive data for brain tissue. However, there are large differences in the mechanical properties reported in the literature. These differences are likely due to experimental parameters such as specimen age, brain region, species, test protocols, and fiber direction which are often not reported. Furthermore, there is a dearth of reported viscoelastic properties for brain tissue at large-strain and high rates. Mouse, rat, and pig brains are impacted at 10/s to a strain of ~36% using a custom-built micro-indenter with a 125 µm radius. It is shown that the resultant mechanical properties are dependent on specimen-age, species, and region, under identical experimental parameters.

Fiber orientation effects in simple shearing of fibrous soft tissues.

Fiber-reinforcement is a common feature of many soft biological tissues. Continuum mechanics modeling of the mechanical response of such tissues using transversely isotropic hyperelasticity is now well developed. The fundamental deformation of simple shear within this framework is examined here. It is well known that the normal stress effect characteristic of nonlinear elasticity plays a crucial role in maintaining a homogeneous deformation state in the bulk of the specimen. Here we consider the effect of anisotropy and fiber-orientation on the shear and normal stresses. It is shown that the confining traction that needs to be applied to the top and bottom faces of a block in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy and on the angle of orientation of the fibers. In the absence of such an applied traction, an unconfined sample tends to bulge outwards or contract inwards perpendicular to the direction of shear so that one has the possibility of both a positive or negative Poynting effect. The results are illustrated using experimental data obtained by other authors for porcine brain white matter. The general results obtained here are relevant to the development of accurate shear test protocols for the determination of constitutive properties of fibrous biological soft tissues.

Protection of cortex by overlying meninges tissue during dynamic indentation of the adolescent brain.

Traumatic brain injury (TBI) has become a recent focus of biomedical research with a growing international effort targeting material characterization of brain tissue and simulations of trauma using computer models of the head and brain to try to elucidate the mechanisms and pathogenesis of TBI. The meninges, a collagenous protective tri-layer, which encloses the entire brain and spinal cord has been largely overlooked in these material characterization studies. This has resulted in a lack of accurate constitutive data for the cranial meninges, particularly under dynamic conditions such as those experienced during head impacts. The work presented here addresses this lack of data by providing for the first time, in situ large deformation material properties of the porcine dura-arachnoid mater composite under dynamic indentation. It is demonstrated that this tissue is substantially stiffer (shear modulus, µ=19.10±8.55kPa) and relaxes at a slower rate (τ1=0.034±0.008s, τ2=0.336±0.077s) than the underlying brain tissue (µ=6.97±2.26kPa, τ1=0.021±0.007s, τ2=0.199±0.036s), reducing the magnitudes of stress by 250% and 65% for strains that arise during indentation-type deformations in adolescent brains. STATEMENT OF SIGNIFICANCE: We present the first mechanical analysis of the protective capacity of the cranial meninges using in situ micro-indentation techniques. Force-relaxation tests are performed on in situ meninges and cortex tissue, under large strain dynamic micro-indentation. A quasi-linear viscoelastic model is used subsequently, providing time-dependent mechanical properties of these neural tissues under loading conditions comparable to what is experienced in TBI. The reported data highlights the large differences in mechanical properties between these two tissues. Finite element simulations of the indentation experiments are also performed to investigate the protective capacity of the meninges. These simulations show that the meninges protect the underlying brain tissue by reducing the overall magnitude of stress by 250% and up to 65% for strains.

A viscoelastic analysis 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 within individual brain regions. The regional viscoelastic properties of P56 mouse brain tissue, up to 70µm displacement, are presented and discussed in the context of traumatic brain injury, particularly how the different regions of the brain respond to mechanical loads. Force-relaxation data obtained from micro-indentation measurements were fit to both linear and quasi-linear viscoelastic models to determine the time and frequency domain viscoelastic response of the pons, cortex, medulla oblongata, cerebellum, and thalamus. The damping ratio of each region was also determined. Each region was found to have a unique mechanical response to the applied displacement, with the pons and thalamus exhibiting the largest and smallest force-response, respectively. All brain regions appear to have an optimal frequency for the dissipation of energies which lies between 1 and 10Hz. STATEMENT OF SIGNIFICANCE: We present the first mechanical characterization of the viscoelastic response for different regions of mouse brain. Force-relaxation tests are performed under large strain dynamic micro-indentation, and viscoelastic models are used subsequently, providing time-dependent mechanical properties of brain tissue under loading conditions comparable to what is experienced in TBI. The unique mechanical properties of different brain regions are highlighted, with substantial variations in the viscoelastic properties and damping ratio of each region. Cortex and pons were the stiffest regions, while the thalamus and medulla were most compliant. The cerebellum and thalamus had highest damping ratio values and those of the medulla were lowest. The reported material parameters can be implemented into finite element computer models of the mouse to investigate the effects of trauma on individual brain regions.

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.

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.