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Background Since their discovery by Karl Langer in the 19th Century, Skin Tension Lines (STLs) have been used by surgeons to decide the location and orientation of an incision. Although these lines are patient-specific, most surgeons... more
Background
Since their discovery by Karl Langer in the 19th Century, Skin Tension Lines (STLs) have been used by surgeons to decide the location and orientation of an incision. Although these lines are patient-specific, most surgeons rely on generic maps to determine their orientation. Beyond the imprecise pinch test, there remains no accepted method for determining STLs in vivo.
Methods
(i) The speed of an elastic motion travelling radially on the skin of canine cadavers was measured with a commercial device called the Reviscometer®. (ii) Similar to the original experiments conducted by Karl Langer, circular excisions were made on the skin and the geometric changes to the resulting wounds and excised samples were used to determine the orientation of STLs.
Results
A marked anisotropy in the speed of the elastic wave travelling radially was observed. The orientation of the fastest wave was found to correlate with the orientation of the elongated wound (P<0.001, R2=74%). Similarly, the orientation of fastest wave was the same for both in vivo and excised isolated samples, indicating that the STLs have a structural basis. Resulting wounds expanded by an average area of 9% (+16% along STL and −10% across) while excised skin shrunk by an average area of 33% (23% along STL and 10% across).
Conclusion
Elastic surface wave propagation has been validated experimentally as a robust method for determining the orientation of STLs non-destructively and non-invasively. This study has implications for the identification of STLs and for the prediction of skin tension levels, both important factors in both human and veterinary reconstructive surgery.
Since their discovery by Karl Langer in the 19th Century, Skin Tension Lines (STLs) have been used by surgeons to decide the location and orientation of an incision. Although these lines are patient-specific, most surgeons rely on generic maps to determine their orientation. Beyond the imprecise pinch test, there remains no accepted method for determining STLs in vivo.
Methods
(i) The speed of an elastic motion travelling radially on the skin of canine cadavers was measured with a commercial device called the Reviscometer®. (ii) Similar to the original experiments conducted by Karl Langer, circular excisions were made on the skin and the geometric changes to the resulting wounds and excised samples were used to determine the orientation of STLs.
Results
A marked anisotropy in the speed of the elastic wave travelling radially was observed. The orientation of the fastest wave was found to correlate with the orientation of the elongated wound (P<0.001, R2=74%). Similarly, the orientation of fastest wave was the same for both in vivo and excised isolated samples, indicating that the STLs have a structural basis. Resulting wounds expanded by an average area of 9% (+16% along STL and −10% across) while excised skin shrunk by an average area of 33% (23% along STL and 10% across).
Conclusion
Elastic surface wave propagation has been validated experimentally as a robust method for determining the orientation of STLs non-destructively and non-invasively. This study has implications for the identification of STLs and for the prediction of skin tension levels, both important factors in both human and veterinary reconstructive surgery.
A body force concentrated at a point and moving at a high speed can induce shear-wave Mach cones in dusty-plasma crystals or soft materials, as observed experimentally and named the elastic Cherenkov effect (ECE). The ECE in soft... more
A body force concentrated at a point and moving at a high speed can induce shear-wave Mach cones in dusty-plasma crystals or soft materials, as observed experimentally and named the elastic Cherenkov effect (ECE). The ECE in soft materials forms the basis of the supersonic shear imaging (SSI) technique, an ultrasound-based dynamic elastography method applied in clinics in recent years. Previous studies on the ECE in soft materials have focused on isotropic material models. In this paper, we investigate the existence and key features of the ECE in anisotropic soft media, by using both theoretical analysis and finite element (FE) simulations, and we apply the results to the non-invasive and nondestructive
characterization of biological soft tissues. We also theoretically study the
characteristics of the shear waves induced in a deformed hyperelastic anisotropic soft material by a source moving with high speed, considering that contact between the ultrasound probe and the soft tissue may lead to finite deformation. On the basis of our theoretical analysis and numerical simulations, we propose an inverse approach to infer both the anisotropic and hyperelastic parameters of incompressible transversely isotropic (TI) soft materials. Finally, we investigate the properties of the solutions to the inverse problem by deriving the condition numbers in analytical form and performing numerical experiments. In Part II of the paper, both ex vivo and in vivo experiments are conducted to demonstrate the applicability of the inverse method in practical use.
characterization of biological soft tissues. We also theoretically study the
characteristics of the shear waves induced in a deformed hyperelastic anisotropic soft material by a source moving with high speed, considering that contact between the ultrasound probe and the soft tissue may lead to finite deformation. On the basis of our theoretical analysis and numerical simulations, we propose an inverse approach to infer both the anisotropic and hyperelastic parameters of incompressible transversely isotropic (TI) soft materials. Finally, we investigate the properties of the solutions to the inverse problem by deriving the condition numbers in analytical form and performing numerical experiments. In Part II of the paper, both ex vivo and in vivo experiments are conducted to demonstrate the applicability of the inverse method in practical use.
The equations governing the appearance of flexural static perturbations at the edge of a semi-infinite thin elastic isotropic plate, subjected to a state of homogeneous bi-axial pre-stress, are derived and solved. The plate is... more
The equations governing the appearance of flexural static perturbations at the edge of a semi-infinite thin elastic isotropic plate, subjected to a state of homogeneous bi-axial pre-stress, are derived and solved. The plate is incompressible and supported
by a Winkler elastic foundation with, possibly, wavenumber dependence. Small perturbations superposed onto the homogeneous state of pre-stress, within the three-dimensional elasticity theory, are considered. A series expansion of the plate kinematics in the plate thickness provides a consistent expression for the second variation of the potential energy, whose minimization gives the plate governing equations. Consistency considerations supplement a constraint on the scaling of the pre-stress so that the classical Kirchhoff–Love linear theory of pre-stretched elastic
plates is retrieved. Moreover, a scaling constraint for the foundation stiffness is also introduced. Edge wrinkling is investigated and compared with body wrinkling. We find that the former always precedes the latter in a state of uni-axial pre-stretch, regardless of the foundation stiffness. By contrast, a general bi-axial pre-stretch state may favour body wrinkling for moderate foundation stiffness. Wavenumber dependence significantly alters the predicted behaviour. The results may be especially relevant to modelling soft biological materials, such as skin or tissues, or stretchable organic thin-films, embedded in a compliant elastic matrix.
by a Winkler elastic foundation with, possibly, wavenumber dependence. Small perturbations superposed onto the homogeneous state of pre-stress, within the three-dimensional elasticity theory, are considered. A series expansion of the plate kinematics in the plate thickness provides a consistent expression for the second variation of the potential energy, whose minimization gives the plate governing equations. Consistency considerations supplement a constraint on the scaling of the pre-stress so that the classical Kirchhoff–Love linear theory of pre-stretched elastic
plates is retrieved. Moreover, a scaling constraint for the foundation stiffness is also introduced. Edge wrinkling is investigated and compared with body wrinkling. We find that the former always precedes the latter in a state of uni-axial pre-stretch, regardless of the foundation stiffness. By contrast, a general bi-axial pre-stretch state may favour body wrinkling for moderate foundation stiffness. Wavenumber dependence significantly alters the predicted behaviour. The results may be especially relevant to modelling soft biological materials, such as skin or tissues, or stretchable organic thin-films, embedded in a compliant elastic matrix.
We find the strain energy function for isotropic incompressible solids exhibiting a linear relationship between shear stress and amount of shear, and between torque and amount of twist, when subject to large simple shear or torsion... more
We find the strain energy function for isotropic incompressible solids exhibiting a linear relationship between shear stress and amount of shear, and between torque and amount of twist, when subject to large simple shear or torsion deformations. It is inclusive of the well-known neo-Hookean and the Mooney–Rivlin models, but also can accommodate other terms, as certain arbitrary functions of the principal strain invariants. Effectively, the extra terms can be used to account for several non-linear effects observed experimentally but not captured by the neo-Hookean and Mooney–Rivlin models, such as strain stiffening effects due to limiting chain extensibility.
Living matter can functionally adapt to external physical factors by developing internal tensions, easily revealed by cutting experiments. Nonetheless, residual stresses intrinsically have a complex spatial distribution, and destructive... more
Living matter can functionally adapt to external physical factors by developing internal tensions, easily revealed by cutting experiments. Nonetheless, residual stresses intrinsically have a complex spatial distribution, and destructive techniques cannot be used to identify a natural stress-free configuration. This work proposes a novel elastic theory of pre-stressed materials. Imposing physical compatibility and symmetry arguments, we define a new class of free energies explicitly depending on the internal stresses. This theory is finally applied to the study of arterial remodelling, proving its potential for the non-destructive determination of the residual tensions within biological materials.
Many interesting shapes appearing in the biological world are formed by the onset of mechanical instability. In this work we consider how the build-up of residual stress can cause a solid to buckle. In all past studies a fictitious... more
Many interesting shapes appearing in the biological world are formed by the onset of mechanical instability. In this work we consider how the build-up of residual stress can cause a solid to buckle. In all past studies a fictitious (virtual) stress-free state was required to calculate the residual stress. In contrast, we use a model which is simple and allows the prescription of any residual stress field.
We specialize the analysis to an elastic tube subject to a two-dimensional residual stress, and find that incremental wrinkles can appear on its inner or its outer face, depending on the location of the highest value of the residual hoop stress. We further validate the predictions of the incremental theory with finite element simulations, which allow us to go beyond this threshold and predict the shape, number and amplitude of the resulting creases.
We specialize the analysis to an elastic tube subject to a two-dimensional residual stress, and find that incremental wrinkles can appear on its inner or its outer face, depending on the location of the highest value of the residual hoop stress. We further validate the predictions of the incremental theory with finite element simulations, which allow us to go beyond this threshold and predict the shape, number and amplitude of the resulting creases.
An initial stress within a solid can arise to support external loads or from processes such as thermal expansion in inert matter or growth and remodelling in living materials. For this reason, it is useful to develop a mechanical... more
An initial stress within a solid can arise to support external loads or from processes such as thermal expansion in inert matter or growth and remodelling in living materials. For this reason, it is useful to develop a mechanical framework of initially stressed solids irrespective of how this stress formed. An ideal way to do this is to write the free energy density Ψ in terms of initial stress τ and the elastic deformation gradient F, so we write Ψ = Ψ (F, τ). In this paper, we present a new constitutive condition for initially stressed materials, which we call the initial stress symmetry (ISS). We focus on two consequences of this condition. First, we examine how ISS restricts the possible choices of free energy densities Ψ = Ψ (F, τ) and present two examples of Ψ that satisfy the ISS. Second, we show that the initial stress can be derived from the Cauchy stress and the elastic deformation gradient. To illustrate, we take an example from biomechanics and calculate the optimal Cauchy stress within an artery subjected to internal pressure. We then use ISS to derive the optimal target residual stress for the material to achieve after remodelling, which links nicely with the notion of homeostasis.
Ray Ogden's work has had a major influence in the broad field of solid mechanics, within the context of continuum mechanics. It continues to do so as can be checked by looking at the exponential rise of his citation count, totalling... more
Ray Ogden's work has had a major influence in the broad field of solid mechanics, within the context of continuum mechanics. It continues to do so as can be checked by looking at the exponential rise of his citation count, totalling according to Google Scholar more than 15,000 to date, with an h-index of 51. Whatever value we attach to bibliometric indicators, these numbers clearly point to a deep and profound impact. Here, instead of presenting the long list of his achievements, awards and publications (to be found elsewhere), we prefer to highlight three of the themes for which his work has received the most attention. Needless to say, the spectrum of his abilities is far wider.
We use supersonic shear wave imaging (SSI) technique to measure not only the linear but also the nonlinear elastic properties of brain matter. Here, we tested six porcine brains ex vivo and measured the velocities of the plane shear waves... more
We use supersonic shear wave imaging (SSI) technique to measure not only the linear but also the nonlinear elastic properties of brain matter. Here, we tested six porcine brains ex vivo and measured the velocities of the plane shear waves induced by acoustic radiation force at different states of pre-deformation when the ultrasonic probe is pushed into the soft tissue. We relied on an inverse method based on the theory governing the propagation of small-amplitude acoustic waves in deformed solids to interpret the experimental data. We found that, depending on the subjects, the resulting initial shear modulus µ 0 varies from 1.8 to 3.2 kPa, the stiffening parameter b of the hyperelastic Demiray–Fung model from 0.13 to 0.73, and the third-(A) and fourth-order (D) constants of weakly nonlinear elasticity from −1.3 to −20.6 kPa and from 3.1 to 8.7 kPa, respectively. Paired t test performed on the experimental results of the left and right Yi Jiang and Guoyang Li have contributed equally to this study. lobes of the brain shows no significant difference. These values are in line with those reported in the literature on brain tis-sue, indicating that the SSI method, combined to the inverse analysis, is an efficient and powerful tool for the mechanical characterization of brain tissue, which is of great importance for computer simulation of traumatic brain injury and virtual neurosurgery.
We show that porcine brain matter can be modelled accurately as a very soft rubber-like material using the Mooney-Rivlin strain energy function, up to strains as high as 60%. This result followed from simple shear experiments performed on... more
We show that porcine brain matter can be modelled accurately as a very soft rubber-like material using the Mooney-Rivlin strain energy function, up to strains as high as 60%. This result followed from simple shear experiments performed on small rectangular fresh samples (2.5 cm^3 and 1.1 cm^3) at quasi-static strain rates. They revealed a linear shear stress-shear strain relationship (R^2>0.97), characteristic of Mooney-Rivlin materials at large strains. We found that porcine brain matter is about 30 times less resistant to shear forces than a silicone gel. We also verified experimentally that brain matter exhibits the positive Poynting effect of non-linear elasticity, and numerically that the stress and strain fields remain mostly homogeneous throughout the thickness of the samples in simple shear.
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... more
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.
We revisit an iconic deformation of non-linear elasticity: the inflation of a rubber spherical thin shell. We use the 3-parameter Mooney and Gent-Gent (GG) phenomenological models to explain the stretch–strain curve of a typical... more
We revisit an iconic deformation of non-linear elasticity: the inflation of a rubber spherical thin shell. We use the 3-parameter Mooney and Gent-Gent (GG) phenomenological models to explain the stretch–strain curve of a typical inflation, as these two models cover a wide spectrum of known models for rubber, including the Varga, Mooney–Rivlin, one-term Ogden, Gent-Thomas and Gent models. We find that the basic physics of inflation exclude the Varga, one-term Ogden and Gent-Thomas models. We find the link between the exact solution of non-linear elasticity and the membrane and Young–Laplace theories often used a priori in the literature. We compare the performance of both models on fitting the data for experiments on rubber balloons and animal bladder. We conclude that the GG model is the most accurate and versatile model on offer for the modelling of rubber balloon inflation.
Collaborative research between the disciplines of forensic pathology and biomechanics was undertaken to investigate the hyperelastic properties of human skin, to determine the force required for sharp instrument penetration of skin, and... more
Collaborative research between the disciplines of forensic pathology and biomechanics was undertaken to investigate the hyperelastic properties of human skin, to determine the force required for sharp instrument penetration of skin, and to develop a finite element model, which reflects the mechanisms of sharp instrument penetration. These studies have led to the development of a “stab metric,” based on simulations, to describe the force magnitudes in stabbing incidents. Such a metric should, in time, replace the crudely quantitative descriptors of stabbing forces currently used by forensic pathologists.
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... more
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).
One of the least studied universal deformations of incompressible nonlinear elasticity, namely the straightening of a sector of a circular cylinder into a rectangular block, is revisited here and, in particular, issues of existence and... more
One of the least studied universal deformations of incompressible nonlinear elasticity, namely the straightening of a sector of a circular cylinder into a rectangular block, is revisited here and, in particular, issues of existence and stability are addressed. Particular attention is paid to the system of forces required to sustain the large static deformation, including by the application of end couples. The influence of geometric parameters and constitutive models on the appearance of wrinkles on the compressed face of the block is also studied. Different numerical methods for solving the incremental stability problem are compared and it is found that the impedance matrix method, based on the resolution of a matrix Riccati differential equation, is the more precise.
We consider the elastic deformation of a circular cylindrical sector composed of an incompressible isotropic soft solid when it is straightened into a rectangular block. In this process, the circumferential line elements on the original... more
We consider the elastic deformation of a circular cylindrical sector composed of an incompressible isotropic soft solid when it is straightened into a rectangular block. In this process, the circumferential line elements on the original inner face of the sector are stretched while those on the original outer face are contracted. We investigate the geometrical and physical conditions under which the latter line elements can be contracted to the point where a localized incremental instability develops. We provide a robust algorithm to solve the corresponding two-point boundary value problem, which is stiff numerically. We illustrate the results with full incremental displacement fields in the case of Mooney–Rivlin materials and also perform an asymptotic analysis for thin sectors.
The application of pure torsion to a long and thin cylindrical rod is known to provoke a twisting instability, evolving from an initial kink to a knot. In the torsional parallel-plate rheometry of stubby cylinders, the geometrical... more
The application of pure torsion to a long and thin cylindrical rod is known to provoke a twisting instability, evolving from an initial kink to a knot. In the torsional parallel-plate rheometry of stubby cylinders, the geometrical constraints impose zero displacement of the axis of the cylinder, preventing the occurrence of such twisting instability. Under these experimental conditions, wrinkles occur on the cylinder’s surface at a given critical angle of torsion. Here we investigate this subclass of elastic instability—which we call torsion instability—of soft cylinders subject to a combined finite axial stretch and torsion, by applying the theory of incremental elastic deformation superimposed on finite strains. We formulate the incremental boundary elastic problem in the Stroh differential form, and use the surface impedance method to build a robust numerical procedure for deriving the marginal stability curves. We present the results for a Mooney–Rivlin material and study the influence of the material parameters on the elastic bifurcation.
Mechanical characterization of brain tissue at high loading velocities is crucial for modeling Traumatic Brain Injury (TBI). During severe impact conditions, brain tissue experiences compression, tension and shear. Limited experimental... more
Mechanical characterization of brain tissue at high loading velocities is crucial for modeling Traumatic Brain Injury (TBI). During severe impact conditions, brain tissue experiences compression, tension and shear. Limited experimental data is available for brain tissue in extension at dynamic strain rates. In this research, a High Rate Tension Device (HRTD) was developed to obtain dynamic properties of brain tissue in extension at strain rates of ≤90/s. In vitro tensile tests were performed to obtain properties of brain tissue at strain rates of 30, 60 and 90/s up to 30% strain. The brain tissue showed a stiffer response with increasing strain rates, showing that hyperelastic models are not adequate. Specifically, the tensile engineering stress at 30% strain was 3.1±0.49 kPa, 4.3±0.86 kPa, 6.5±0.76 kPa (mean±SD) at strain rates of 30, 60 and 90/s, respectively. Force relaxation tests in tension were also conducted at different strain magnitudes (10–60% strain) with the average rise time of 24 ms, which were used to derive time dependent parameters. One-term Ogden, Fung and Gent models were used to obtain material parameters from the experimental data. Numerical simulations were performed using a one-term Ogden model to analyze hyperelastic behavior of brain tissue up to 30% strain. The material parameters obtained in this study will help to develop biofidelic human brain finite element models, which can subsequently be used to predict brain injuries under impact conditions and as a reconstruction and simulation tool for forensic investigations.
During severe impact conditions, brain tissue experiences a rapid and complex deformation, which can be seen as a mixture of compression, tension and shear. Diffuse axonal injury (DAI) occurs in animals and humans when both the strains... more
During severe impact conditions, brain tissue experiences a rapid and complex deformation, which can be seen as a mixture of compression, tension and shear. Diffuse axonal injury (DAI) occurs in animals and humans when both the strains and strain rates exceed 10% and 10/s, respectively. Knowing the mechanical properties of brain tissue in shear at these strains and strain rates is thus of particular importance, as they can be used in finite element simulations to predict the occurrence of brain injuries under different impact conditions. However, very few studies in the literature provide this information. In this research, an experimental setup was developed to perform simple shear tests on porcine brain tissue at strain rates ≤120/s. The maximum measured shear stress at strain rates of 30, 60, 90 and 120/s was 1.15±0.25 kPa, 1.34±0.19 kPa, 2.19±0.225 kPa and 2.52±0.27 kPa, (mean±SD), respectively at the maximum amount of shear, K=1. Good agreement of experimental, theoretical (Ogden and Mooney–Rivlin models) and numerical shear stresses was achieved (p=0.7866–0.9935). Specimen thickness effects (2.0–10.0 mm thick specimens) were also analyzed numerically and we found that there is no significant difference (p=0.9954) in the shear stress magnitudes, indicating a homogeneous deformation of the specimens during simple shear tests. Stress relaxation tests in simple shear were also conducted at different strain magnitudes (10–60% strain) with the average rise time of 14 ms. This allowed us to estimate elastic and viscoelastic parameters (initial shear modulus, μ=4942.0 Pa, and Prony parameters: g1=0.520, g2=0.3057, τ1=0.0264 s, and τ2=0.011 s) that can be used in FE software to analyze the non-linear viscoelastic behavior of brain tissue. This study provides new insight into the behavior in finite shear of brain tissue under dynamic impact conditions, which will assist in developing effective brain injury criteria and adopting efficient countermeasures against traumatic brain injury.
Keywords
Diffuse axonal injury (DAI); Ogden; Mooney–Rivlin; Traumatic brain injury (TBI); Homogeneous; Viscoelastic; Relaxation
Keywords
Diffuse axonal injury (DAI); Ogden; Mooney–Rivlin; Traumatic brain injury (TBI); Homogeneous; Viscoelastic; Relaxation
The magnitude of force used in a stabbing incident can be difficult to quantify, although the estimate given by forensic pathologists is often seen as ‘critical’ evidence in medico-legal situations. The main objective of this study is to... more
The magnitude of force used in a stabbing incident can be difficult to quantify, although the estimate given by forensic pathologists is often seen as ‘critical’ evidence in medico-legal situations. The main objective of this study is to develop a quantitative measure of the force associated with a knife stabbing biological tissue, using a combined experimental and numerical technique. A series of stab-penetration tests were performed to quantify the force required for a blade to penetrate skin at various speeds and using different ‘sharp’ instruments. A computational model of blade penetration was developed using ABAQUS/EXPLICIT, a non-linear finite element analysis (FEA) commercial package. This model, which incorporated element deletion along with a suitable failure criterion, is capable of systematically quantifying the effect of the many variables affecting a stab event. This quantitative data could, in time, lead to the development of a predictive model that could help indicate the level of force used in a particular stabbing incident.
We give conditions on the strain–energy function of nonlinear anisotropic hyperelastic materials that ensure compatibility with the classical linear theories of anisotropic elasticity. We uncover the limitations associated with the... more
We give conditions on the strain–energy function of nonlinear anisotropic hyperelastic materials that ensure compatibility with the classical linear theories of anisotropic elasticity. We uncover the limitations associated with the volumetric–deviatoric separation of the strain–energy used, for example, in many Finite Element (FE) codes in that it does not fully represent the behavior of anisotropic materials in the linear regime. This limitation has important consequences. We show that, in the small deformation regime, a FE code based on the volumetric–deviatoric separation assumption predicts that a sphere made of a compressible anisotropic material deforms into another sphere under hydrostatic pressure loading, instead of the expected ellipsoid. For finite
The modelling of off-axis simple tension experiments on transversely isotropic nonlinearly elastic materials is considered. A testing protocol is proposed where normal force is applied to one edge of a rectangular specimen with the... more
The modelling of off-axis simple tension experiments on transversely isotropic nonlinearly elastic materials is considered. A testing protocol is proposed where normal force is applied to one edge of a rectangular specimen with the opposite edge allowed to move laterally but constrained so that no vertical displacement is allowed. Numerical simulations suggest that this deformation is likely to remain substantially homogeneous throughout the specimen for moderate deformations. It is therefore further proposed that such tests can be modelled adequately as a homogenous deformation consisting of a triaxial stretch accompanied by a simple shear. Thus the proposed test should be a viable alternative to the standard biaxial tests currently used as material characterisa-tion tests for transversely isotropic materials in general and, in particular, for soft, biological tissue. A consequence of the analysis is a kinematical universal relation for off-axis testing that results when the strain-energy function is assumed to be a function of only one isotropic and one anisotropic invariant, as is typically the case. The universal relation provides a simple test of this assumption, which is usually made for mathematical convenience. Numerical simulations also suggest that this universal relation is unlikely to agree with experimental data and therefore that at least three invariants are necessary to fully capture the mechanical response of transversely isotropic materials.
We present examples of body wave and surface wave propagation in deformed solids where the slowest and the fastest waves do not travel along the directions of least and greatest stretch, respectively. These results run counter to commonly... more
We present examples of body wave and surface wave propagation in deformed solids where the slowest and the fastest waves do not travel along the directions of least and greatest stretch, respectively. These results run counter to commonly accepted theory, practice, and implementation of the principles of acousto-elasticity in initially isotropic solids. For instance, we find that in nickel and steel the fastest waves are along the direction of greatest compression, not greatest extension (and vice-versa for the slowest waves), as soon as those solids are deformed. Further, we find that when some materials are subject to a small-but-finite deformation, other extrema of wave speeds appear in non-principal directions. Examples include nickel, steel, polystyrene, and a certain hydrogel. The existence of these ''oblique'', non-principal extremal waves complicates the protocols for the non-destructive determination of the directions of extreme strains.
The large variability in experimentally measured mechanical properties of brain tissue is due to many factors including heterogeneity, anisotropy, age dependence and post-mortem time. Moreover, differences in test protocols also influence... more
The large variability in experimentally measured mechanical properties of brain tissue is due to many factors including heterogeneity, anisotropy, age dependence and post-mortem time. Moreover, differences in test protocols also influence these measured properties. This paper shows that the temperature at which porcine brain tissue is stored or preserved prior to testing has a significant effect on the mechanical properties of brain tissue, even when tests are conducted at the same temperatures. Three groups of brain tissue were stored separately for at least 1 h at three different preservation temperatures, i.e., ice cold, room temperature (22 1C) and body temperature (37 1C), prior to them all being tested at room temperature ($ 22 1C). Significant differences in the corresponding initial elastic shear modulus m (Pa) (at various amounts of shear, 0 r K r1.0) were observed. The initial elastic moduli were 1043 7 271 Pa, 714 7 210 Pa and 497 7156 Pa (mean 7SD) at preservation temperatures of ice cold, 22 1C and 37 1C, respectively. Based on this investigation, it is strongly recommended that brain tissue samples must be preserved at an ice-cold temperature prior to testing in order to minimize the difference between the measured in vitro test results and the in vivo properties. A by-product of the study is that simple shear tests allow for large, almost perfectly homogeneous deformation of brain matter.
We propose two toy-models to describe, predict and interpret the wrinkles appearing on the surface of skin when it is sheared. With the first model, we account for the lines of greatest tension present in human skin by subjecting a layer... more
We propose two toy-models to describe, predict and interpret the wrinkles appearing on the surface of skin when it is sheared. With the first model, we account for the lines of greatest tension present in human skin by subjecting a layer of soft tissue to a pre-stretch, and for the epidermis by endowing one of the layer's faces with a surface tension. For the second model, we consider an anisotropic model for the skin, to reflect the presence of stiff collagen fibres in a softer elastic matrix. In both cases, we find an explicit bifurcation criterion, linking geometrical and material parameters to a critical shear deformation accompanied by small static wrinkles, with decaying amplitudes normal to the free surface of skin.
Research Interests:
Research Interests:
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.... more
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.
On the basis of the general non-linear theory of a hyperelastic material with initial stress, initially without consideration of the origin of the initial stress, we determine explicit expressions for the stress-dependent tensor of... more
On the basis of the general non-linear theory of a hyperelastic material with initial stress, initially without consideration of the origin of the initial stress, we determine explicit expressions for the stress-dependent tensor of incremental elastic moduli. In considering three special cases of initial stress within the general framework, namely hydrostatic stress, uniaxial stress and planar shear stress, we then elucidate in general form the dependence of various elastic moduli on the initial stress. In each case, the effect of initial stress on the wave speed of homogeneous plane waves is studied and it is shown how various special theories from the earlier literature fit within the general framework. We then consider the situation in which the initial stress is a pre-stress associated with a finite deformation and, in particular, we discuss the specialization to the second-order theory of elasticity and highlight connections between several classical approaches to the topic, again with special reference to the influence of higher-order terms on the speed of homogeneous plane waves. Some discrepancies arising in the earlier literature are noted.
Unconfined compression tests are more convenient to perform on cylindrical samples of brain tissue than tensile tests in order to estimate mechanical properties of the brain tissue because they allow homogeneous deformations. The... more
Unconfined compression tests are more convenient to perform on cylindrical samples of brain tissue than tensile tests in order to estimate mechanical properties of the brain tissue because they allow homogeneous deformations. The reliability of these tests depends significantly on the amount of friction generated at the specimen/platen interface. Thus, there is a crucial need to find an approximate value of the friction coefficient in order to predict a possible overestimation of stresses during unconfined compression tests. In this study, a combined experimental–computational approach was adopted to estimate the dynamic friction coefficient m of porcine brain matter against metal platens in compressive tests. Cylindrical samples of porcine brain tissue were tested up to 30% strain at variable strain rates, both under bonded and lubricated conditions in the same controlled environment. It was established that m was equal to 0.0970.03, 0.1870.04, 0.1870.04 and 0.2070.02 at strain rates of 1, 30, 60 and 90/s, respectively. Additional tests were also performed to analyze brain tissue under lubricated and bonded conditions, with and without initial contact of the top platen with the brain tissue, with different specimen aspect ratios and with different lubricants (Phosphate Buffer Saline (PBS), Polytetrafluor-oethylene (PTFE) and Silicone). The test conditions (lubricant used, biological tissue, loading velocity) adopted in this study were similar to the studies conducted by other research groups. This study will help to understand the amount of friction generated during unconfined compression of brain tissue for strain rates of up to 90/s.
Mechanical characterization of brain tissue has been investigated extensively by various research groups over the past 50 years. These properties are particularly important for modeling Traumatic Brain Injury (TBI). In this research, we... more
Mechanical characterization of brain tissue has been investigated extensively by various research groups over the past 50 years. These properties are particularly important for modeling Traumatic Brain Injury (TBI). In this research, we present the design and calibration of a High Rate Tension Device (HRTD) capable of performing tests up to a maximum strain rate of 90/s. We use experimental and numerical methods to investigate the effects of inhomogeneous deformation of porcine brain tissue during tension at different specimen thicknesses (4.0–14.0 mm), by performing tension tests at a strain rate of 30/s. One-term Ogden material parameters (l = 4395.0 Pa, a = À2.8) were derived by performing an inverse finite element analysis to model all experimental data. A similar procedure was adopted to determine Young's modulus (E = 11200 Pa) of the linear elastic regime. Based on this analysis, brain specimens of aspect ratio (diameter/thickness) S 6 1.0 are required to minimise the effects of inhomogeneous deformation during tension tests.
Extensive research has been carried out for at least 50 years to understand the mechanical properties of brain tissue in order to understand the mechanisms of traumatic brain injury (TBI). The observed large variability in experimental... more
Extensive research has been carried out for at least 50 years to understand the mechanical properties of brain tissue in order to understand the mechanisms of traumatic brain injury (TBI). The observed large variability in experimental results may be due to the inhomo-geneous nature of brain tissue and to the broad range of test conditions. However, test temperature is also considered as one of the factors influencing the properties of brain tissue. In this research, the mechanical properties of porcine brain have been investigated at 22 1C (room temperature), and at 37 1C (body temperature) while maintaining a constant preservation temperature of approximately 4–5 1C. Unconfined compression tests were performed at dynamic strain rates of 30 and 50 s À1 using a custom made test apparatus. There was no significant difference (p ¼0.8559–0.9290) between the average engineering stresses of the brain tissue at the two different temperature conditions. The results of this study should help to understand the behavior of brain tissue at different temperature conditions, particularly in unconfined compression tests.
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... more
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.
Collagen fibres play an important role in the mechanical behaviour of many soft tissues. Modelling of such tissues now often incorporates a collagen fibre distribution. However, the availability of accurate structural data has so far... more
Collagen fibres play an important role in the mechanical behaviour of many soft tissues. Modelling of such tissues now often incorporates a collagen fibre distribution. However, the availability of accurate structural data has so far lagged behind the progress of anisotropic constitutive modelling. Here, an automated process is developed to identify the orientation of collagen fibres using inexpensive and relatively simple techniques. The method uses established histological techniques and an algorithm implemented in the MATLAB image processing toolbox. It takes an average of 15 s to evaluate one image, compared to several hours if assessed visually. The technique was applied to histological sections of human skin with different Langer line orientations and a definite correlation between the orientation of Langer lines and the preferred orientation of collagen fibres in the dermis ðp<0:001; R 2 ¼ 0:95Þ was observed. The structural parameters of the Gasser–Ogden– Holzapfel (GOH) model were all successfully evaluated. The mean dispersion factor for the dermis was j ¼ 0:1404AE 0:0028: The constitutive parameters l, k 1 and k 2 were evaluated through physically-based, least squares curve-fitting of experimental test data. The values found for l, k 1 and k 2 were 0.2014 MPa, 243.6 and 0.1327, respectively. Finally, the above model was implemented in ABAQUS/ Standard and a finite element (FE) computation was performed of uniaxial extension tests on human skin. It is expected that the results of this study will assist those wishing to model skin, and that the algorithm described will be of benefit to those who wish to evaluate the collagen dispersion of other soft tissues.
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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,... more
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 modulus µ 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 of µ were 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 brain finite element models, which can subsequently be used to predict brain injuries under impact conditions.
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The mechanical properties of skin are important for a number of applications including surgery, dermatology, impact biomechanics and forensic science. In this study, we have investigated the influence of location and orientation on the... more
The mechanical properties of skin are important for a number of applications including surgery, dermatology, impact biomechanics and forensic science. In this study, we have investigated the influence of location and orientation on the deformation characteristics of 56 samples of excised human skin. Uniaxial tensile tests were carried out at a strain rate of 0.012 s −1 on skin from the back. Digital Image Correlation was used for 2D strain measurement and a histological examination of the dermis was also performed. The mean ultimate tensile strength (UTS) was 21.6 ± 8.4 MPa, the mean failure strain 54% ± 17%, the mean initial slope 1.18 ± 0.88 MPa, the mean elastic modulus 83.3 ± 34.9 MPa and the mean strain energy was 3.6 ± 1.6 MJ/m 3. A multivariate analysis of variance has shown that these mechanical properties of skin are dependent upon the orientation of the Langer lines (P < 0.0001 − P = 0.046). The location of specimens on the back was also found to have a significant effect on the UTS (P = 0.0002), the elastic modulus (P = 0.001) and the strain energy (P = 0.0052). The histological investigation concluded that there is a definite correlation between the orientation of the Langer lines and the preferred orientation of collagen fibres in the dermis (P < 0.001). The data obtained in this study will provide essential information for those wishing to model the skin using a structural constitutive model.
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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... more
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 451 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.
On the basis of the nonlinear theory of elasticity, the general constitutive equation for an isotropic hyperelastic solid in the presence of initial stress is derived. This derivation involves invariants that couple the deformation with... more
On the basis of the nonlinear theory of elasticity, the general constitutive equation for an isotropic hyperelastic solid in the presence of initial stress is derived. This derivation involves invariants that couple the deformation with the initial stress and in general, for a compressible material, it requires 10 invariants, reducing to 9 for an incompressible material. Expressions for the Cauchy and nominal stress tensors in a finitely deformed configuration are given along with the elasticity tensor and its specialization to the initially stressed undeformed configuration. The equations governing infinitesimal motions superimposed on a finite deformation are then used to study the combined effects of initial stress and finite deformation on the propagation of homogeneous plane waves in a homogeneously deformed and initially stressed solid of infinite extent. This general framework allows for various different specializations, which make contact with earlier works. In particular, connections with results derived within Biot's classical theory are highlighted. The general results are also specialized to the case of a small initial stress and a small pre-deformation, i.e. to the evaluation of the acoustoelastic effect. Here the formulas derived for the wave speeds cover the case of a second-order elastic solid without initial stress and subject to a uniaxial tension [Hughes and Kelly, Phys. Rev. 92 (1953) 1145] and are consistent with results for an undeformed solid subject to a residual stress [Man and Lu, J. Elasticity 17 (1987) 159]. These formulas provide a basis for acoustic evaluation of the second-and third-order elasticity constants and of the residual stresses. The results are further illustrated in respect of a prototype model of nonlinear elasticity with initial stress, allowing for both finite deformation and nonlinear dependence on the initial stress.
We establish a connection between the general equations of nonlinear elastodynamics and the nonlinear ordinary differential equation of Pinney [Proc Amer Math Soc 1950; 1: 681]. As a starting point, we use the exact travelling wave... more
We establish a connection between the general equations of nonlinear elastodynamics and the nonlinear ordinary differential equation of Pinney [Proc Amer Math Soc 1950; 1: 681]. As a starting point, we use the exact travelling wave solutions of nonlinear elasticity discovered by Carroll [Acta Mechanica 1967; 3: 167]. The connection provides a method for finding new exact and approximate dynamic solutions for neo-Hookean and Mooney–Rivlin solids, and for the general third-and fourth-order elasticity models of incompressible solids.
In this paper, in the context of the quasi-magnetostatic approximation, we examine incremental motions superimposed on a static finite deformation of a magneto-elastic material in the presence of an applied magnetic field. Explicit... more
In this paper, in the context of the quasi-magnetostatic approximation, we examine incremental motions superimposed on a static finite deformation of a magneto-elastic material in the presence of an applied magnetic field. Explicit expressions are obtained for the associated magneto-acoustic (or magneto-elastic moduli) tensors in the case of an incompressible isotropic magneto-elastic material, and these are then used to study the propagation of incremental plane waves. The propagation condition is derived in terms of a generalized acoustic tensor and the results are illustrated by obtaining explicit formulas in two special cases: first, when the material is undeformed but subject to a uniform bias field and, second for a prototype model of magneto-elastic interactions in the finite deformation regime. The results provide a basis for the experimental determination of the material parameters of a magneto-sensitive elastomer from measurements of the speed of incremental waves for different pre-strains, bias magnetic fields, and directions of propagation.
Euler's celebrated buckling formula gives the critical load N for the buckling of a slender cylindrical column with radius B and length L as N/(π 3 B 2) = (E/4)(B/L) 2 , where E is Young's modulus. Its derivation relies on the assumptions... more
Euler's celebrated buckling formula gives the critical load N for the buckling of a slender cylindrical column with radius B and length L as N/(π 3 B 2) = (E/4)(B/L) 2 , where E is Young's modulus. Its derivation relies on the assumptions that linear elasticity applies to this problem, and that the slenderness (B/L) is an infinitesimal quantity. Here we ask the following question: What is the first non-linear correction in the right hand-side of this equation when terms up to (B/L) 4 are kept? To answer this question, we specialize the exact solution of incremental non-linear elasticity for the homogeneous compression of a thick compressible cylinder with lubricated ends to the theory of third-order elasticity. In particular, we highlight the way second-and third-order constants—including Poisson's ratio—all appear in the coefficient of (B/L) 4 .
The stress–strain relationship of biological soft tissues affected by Marfan's syndrome is believed to be nonconvex. More specifically, Haughton and Merodio recently proposed a strain energy density leading to localized strain-softening,... more
The stress–strain relationship of biological soft tissues affected by Marfan's syndrome is believed to be nonconvex. More specifically, Haughton and Merodio recently proposed a strain energy density leading to localized strain-softening, in order to model the unusual mechanical behavior of these isotropic, incompressible tissues. Here we investigate how this choice of strain energy affects the results of some instabilities studies, such as those concerned with the compression of infinite and semi-infinite solids, slabs, and cylinders, or with the bending of blocks, and draw comparisons with known results established previously for the case of a classical neo-Hookean solid. We find that the localized strain-softening effect leads to early instability only when instability occurs at severe compression ratios for neo-Hookean solids, as is the case for bulk, surface, and bending instabilities.
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Continuum Mechanics deals with the physical properties of all kinds of materials, be they solids, fluids, or gases. In this mathematical theory, the difference between states of matter is only a question of constitutive equations. In a... more
Continuum Mechanics deals with the physical properties of all kinds of materials, be they solids, fluids, or gases. In this mathematical theory, the difference between states of matter is only a question of constitutive equations. In a not-so-distant past, scientists engaged in continuum mechanics were interested in any kind of macroscopic substance. A perfect example of these savants is Claude-Louis Navier [1785–1836], who formulated the general theory of linear elasticity and what is now known as the Navier–Stokes equations of fluid mechanics. Nowadays the situation is completely different. The hyper-specialization of knowledge has produced a generation of researchers who concentrate on the small courtyard of a subject. It is now usual to know ''everything'' about a special material, for a special class of deformations, for a special kind of problems. Of course, exceptio probat regulam in casibus non exceptis, and among the fellows currently engaged in research in Continuum Mechanics, K.R. Rajagopal is a notable exception to this state of affairs. Professor Rajagopal is a savant moderne not only for his culture and for his skill, but also for his aptitude to consider solid mechanics and fluid mechanics as the two sides of the same coin. This is one of the peculiarity of all his research in Mechanics , not to mention all his eminent accomplishments in several other fields such as Mathematics, Philosophy, Applied Dynamical Systems and many more. This special issue of IJES is the Festschrift celebrating the 60th birthday of K.R. Rajagopal. It was clearly a challenge to find an outlet capable of honoring a Savant with so many interests. First, we encountered a space problem, because we ended up receiving more than 60 submissions for our special volume. Second, the range of topics reflecting the interests of Rajagopal is so wide that it is was hard to find a journal able to cover all this spectrum. Thankfully, all the accepted contributions could fit within the scope of this Journal. For these reasons we have express our deepest thanks and gratitude to all the Elsevier staff which assisted us in managing what seemed to be an endless flow of manuscripts, referee reports, and revisions. We also have to thank Mark Kachanov, Editor-in-Chief of the International Journal of Engineering Science, for allowing us to host this special issue. We are most grateful to Alan Wineman for writing the biographical sketch and of course to all the contributors and reviewers for their patience and efficiency with meeting the requirements of a rigorous and time-constrained review process. We are proud to have put together such a large collection of papers of excellent quality and variety in the field.
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Within the context of finite deformation elasticity theory the problem of deforming an open sector of a thick-walled circular cylindrical tube into a complete circular cylindrical tube is analyzed. The analysis provides a means of... more
Within the context of finite deformation elasticity theory the problem of deforming an open sector of a thick-walled circular cylindrical tube into a complete circular cylindrical tube is analyzed. The analysis provides a means of estimating the radial and circumferen-tial residual stress present in an intact tube, which is a problem of particular concern in dealing with the mechanical response of arteries. The initial sector is assumed to be unstressed and the stress distribution resulting from the closure of the sector is then calculated in the absence of loads on the cylindrical surfaces. Conditions on the form of the elastic strain-energy function required for existence and uniqueness of the deformed configuration are then examined. Finally, stability of the resulting finite deformation is analyzed using the theory of incremental deformations superimposed on the finite deformation , implemented in terms of the Stroh formulation. The main results are that convex-ity of the strain energy as a function of a certain deformation variable ensures existence and uniqueness of the residually-stressed intact tube, and that bifurcation can occur in the closing of thick, widely opened sectors, depending on the values of geometrical and physical parameters. The results are illustrated for particular choices of these parameters, based on data available in the biomechanics literature.
Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress–strain relation can then be inverted... more
Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress–strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incom-pressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incom-pressible elasticity l, A, and D are of the same order of magnitude. Several examples of application of the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation.
We study incremental wave propagation for what is seemingly the simplest boundary value problem, namely that constituted by the plane interface of a semi-infinite solid. With a view to model loaded elastomers and soft tissues, we focus on... more
We study incremental wave propagation for what is seemingly the simplest boundary value problem, namely that constituted by the plane interface of a semi-infinite solid. With a view to model loaded elastomers and soft tissues, we focus on incompressible solids, subjected to large homogeneous static deformations. The resulting strain-induced anisotropy complicates matters for the incremental boundary value problem, but we transpose and take advantage of powerful tech- niques and results from the linear anisotropic elastodynamics theory. In particular we cover several situations where fully explicit secular equations can be derived, including Rayleigh and Stoneley waves in principal directions, and Rayleigh waves polarized in a principal plane or propagating in any direction in a principal plane. We also discuss the merits of polynomial secular equations with respect to more robust, but less transparent, exact secular equations.
We provide a simple introduction to wave propagation in the frame- work of linear elastodynamics. We discuss bulk waves in isotropic and anisotropic linear elastic materials and we survey several families of surface and interface waves.... more
We provide a simple introduction to wave propagation in the frame- work of linear elastodynamics. We discuss bulk waves in isotropic and anisotropic linear elastic materials and we survey several families of surface and interface waves. We conclude by suggesting a list of books for a more detailed study of the topic.
Keywords: linear elastodynamics, anisotropy, plane homogeneous waves, bulk waves, interface waves, surface waves.
Keywords: linear elastodynamics, anisotropy, plane homogeneous waves, bulk waves, interface waves, surface waves.
The general theory of nonlinear anisotropic elasticity is extended to describe small-amplitude motions and static de- formations that can be superimposed on large pre-strains of fibre- reinforced solids. The linearised governing equations... more
The general theory of nonlinear anisotropic elasticity is extended to describe small-amplitude motions and static de- formations that can be superimposed on large pre-strains of fibre- reinforced solids. The linearised governing equations of incremental motion are derived. Then they are solved for some illustrative situations which reveal a wide spectrum of possible behaviours compared to the case of initially isotropic materials. Particular attention is paid to the propagation of homogeneous waves and to the formation of static wrinkles. These objects prove useful in the investigation of the issues of material (in the bulk) and geometrical (at boundaries) stability. Attempts are also made at modelling some experimental observations made on (isotropic) silicone and (anisotropic) biological soft tissues.
Keywords: biological soft tissue; characterization; constitutive model; experiments; strain rate dependence
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We prove theoretically that when a soft solid is subjected to an extreme deformation, wrinkles can form on its surface at an angle that is oblique to a principal direction of stretch. These oblique wrinkles occur for a strain that is... more
We prove theoretically that when a soft solid is subjected to an extreme deformation, wrinkles can form on its surface at an angle that is oblique to a principal direction of stretch. These oblique wrinkles occur for a strain that is smaller than the one required to obtain wrinkles normal to the direction of greatest compression. We go on to explain why they will probably never be observed in real-world experiments. This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’
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We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete... more
We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asymptotic solutions for thin sectors. When the deformations are severe enough, the compressed side of the elastic material may buckle and wrinkles could then develop. We analyse, in detail, the onset of this instability for the Mooney–Rivlin strain energy, which covers the cases of the neo-Hookean model in exact nonlinear elasticity and of third-order elastic materials in weakly nonlinear elasticity. In particular, the associated theoretical and numerical treatment allows us to predict the number and wavelength of the wrinkles. Guided by experimental observations, we finally look at the development of creases, which we simulate through advanced finite-element computations. In some cases, the linearized analysis allows us to predict correctly the number and the wavelength of the creases, which turn out to occur only a few per cent of strain earlier than the wrinkles.
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Traumatic brain injuries and damage are major causes of death and disability. Whereas recent experimental evidence has uncovered mechanical phenomena accompanying the neural activity, the mechanism by which mechanical impact affects... more
Traumatic brain injuries and damage are major causes of death and disability. Whereas recent experimental evidence has uncovered mechanical phenomena accompanying the neural activity, the mechanism by which mechanical impact affects neuronal impairment remains unclear. We propose a 3D model of a nerve bundle to understand the electrophysiological changes due to trauma. Here, the electrical and mechanical phenomena are simulated simultaneously by using electro-thermal equivalences in the finite element software Abaqus CAE 6.13-3. This model provides a unique framework which combines a real-time fully coupled electro-mechanical, modulated threshold for spiking activation and damage as a function of strain and strain rate. Results show the alteration of electrostriction and neural activity due to damage as observed in experiments. One of the key findings is the distribution of residual stresses and strains at the membrane of each fibre due to mechanically-induced electrophysiological impairments.
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We investigate how two finite-amplitude, transverse, plane body waves may be superposed to propagate in a deformed hyperelastic incompressible solid. We find that the equations of motion reduce to a well-determined system of partial... more
We investigate how two finite-amplitude, transverse, plane body waves may be superposed to propagate in a deformed hyperelastic incompressible solid. We find that the equations of motion reduce to a well-determined system of partial differential equations, making the motion controllable for all solids. We find that in deformed Mooney–Rivlin materials, they may travel along any direction and be polarised along any transverse direction, an extension of a result by Boulanger and Hayes (Quart. J. Mech. Appl. Math. 45 (1992) 575). Furthermore, their motion is governed by a linear system of partial differential equations, making the Mooney–Rivlin special in that respect. We select another model to show that for other materials, the equations are nonlinear. We use asymptotic equations to reveal the onset of nonlinearity for the waves, paying particular attention to how close the propagation direction is to the principal axes of pre-deformation.
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We investigate the theoretical nonlinear response, Hessian stability, and possible wrinkling behaviour of a voltage-activated dielectric plate immersed in a tank filled with silicone oil. Fixed rigid electrodes are placed on the top and... more
We investigate the theoretical nonlinear response, Hessian stability, and possible wrinkling behaviour of a voltage-activated dielectric plate immersed in a tank filled with silicone oil. Fixed rigid electrodes are placed on the top and bottom of the tank, and an electric field is generated by a potential difference between the electrodes. We solve the associated incremental boundary value problem of superimposed, inhomogeneous small-amplitude wrinkles, signalling the onset of instability. We decouple the resulting bifurcation equation into symmetric and antisymmetric modes. For a neo-Hookean dielectric plate, we show that a potential difference between the electrodes can induce a thinning of the plate and thus an increase of its planar area, similar to the scenarios encountered when there is no silicone oil. However, we also find that, depending on the material and geometric parameters, an increasing applied voltage can also lead to a <i>thickening</i> of the plate, and thus a shrinking of its area. In that scenario, Hessian instability and wrinkling bifurcation may then occur spontaneously once some critical voltages are reached.
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We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete... more
We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asymptotic solutions for thin sectors. When the deformations are severe enough, the compressed side of the elastic material may buckle and wrinkles could then develop. We analyse, in detail, the onset of this instability for the Mooney–Rivlin strain energy, which covers the cases of the neo-Hookean model in exact nonlinear elasticity and of third-order elastic materials in weakly nonlinear elasticity. In particular, the associated theoreti...
Research Interests: Engineering, Physics, Materials Science, Nonlinear Elasticity, Linear Elasticity, and 15 moreFinite element method, Medicine, Instability, Mathematical Sciences, Physical sciences, Bending, Compressibility, Hyperelastic material, Nonlinear system, Bifurcations, Ogden, Finite, Isotropy, buckle, and eversion
Research Interests: Engineering, Physics, Materials Science, Nonlinear Elasticity, Linear Elasticity, and 15 moreFinite element method, Medicine, Instability, Mathematical Sciences, Physical sciences, Bending, Compressibility, Hyperelastic material, Nonlinear system, Bifurcations, Ogden, Finite, Isotropy, buckle, and eversion
Research Interests: Materials Engineering, Mechanical Engineering, Physics, Materials Science, Biomedical Engineering, and 15 moreBehavior, Medicine, Relaxation, Composite Material, Shear Stress, In Vivo, Compression, Head injury, Constitutive model, Simple shear, Shear Modulus, Viscoelastic, Ogden, Homogeneous, and Shear Rate
Research Interests: Materials Engineering, Materials Science, Biomedical Engineering, Behavior, Biology, and 15 moreElasticity, Finite Element Analysis, Brain, Humans, Compressive Strength, Animals, Impact, Friction, In Vivo, Head injury, Hyperelastic material, Constitutive model, Friction Coefficient, Biomechanical Phenomena, and Brain injuries
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In vivo measurement of the mechanical properties of thin-walled soft tissues (e.g., mitral valve, artery and bladder) and in situ mechanical characterization of thin-walled artificial soft biomaterials in service are of great challenge.... more
In vivo measurement of the mechanical properties of thin-walled soft tissues (e.g., mitral valve, artery and bladder) and in situ mechanical characterization of thin-walled artificial soft biomaterials in service are of great challenge. Those thin-walled structures are usually pre-stressed to achieve and/or improve their functional performance, which further complicate the inverse analysis to identify the mechanical properties. In this study, we investigate the properties of guided waves generated by focused acoustic radiation force in immersed pre-stressed plates and tubes, and show that they can address this challenge.
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Mechanical stresses across different length scales play a fundamental role in understanding biological systems’ functions and engineering soft machines and devices. However, it is challenging to noninvasively probe local mechanical... more
Mechanical stresses across different length scales play a fundamental role in understanding biological systems’ functions and engineering soft machines and devices. However, it is challenging to noninvasively probe local mechanical stresses in situ, particularly when the mechanical properties are unknown. We propose an acoustoelastic imaging–based method to infer the local stresses in soft materials by measuring the speeds of shear waves induced by custom-programmed acoustic radiation force. Using an ultrasound transducer to excite and track the shear waves remotely, we demonstrate the application of the method by imaging uniaxial and bending stresses in an isotropic hydrogel and the passive uniaxial stress in a skeletal muscle. These measurements were all done without the knowledge of the constitutive parameters of the materials. The experiments indicate that our method will find broad applications, ranging from health monitoring of soft structures and machines to diagnosing diseases that alter stresses in soft tissues.
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We place the Ogden model of rubber elasticity, published inProceedings of the Royal Society50 years ago, in the wider context of the theory of nonlinear elasticity. We then follow with a short interview of Ray Ogden FRS and introduce the... more
We place the Ogden model of rubber elasticity, published inProceedings of the Royal Society50 years ago, in the wider context of the theory of nonlinear elasticity. We then follow with a short interview of Ray Ogden FRS and introduce the papers collected for this Theme Issue.This article is part of the theme issue ‘The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity’.
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Limiting chain extensibility is a characteristic that plays a vital role in the stretching of highly elastic materials. The Gent model has been widely used to capture this behaviour, as it performs very well in fitting stress-stretch data... more
Limiting chain extensibility is a characteristic that plays a vital role in the stretching of highly elastic materials. The Gent model has been widely used to capture this behaviour, as it performs very well in fitting stress-stretch data in simple tension, and involves two material parameters only. Recently, Anssari-Benam and Bucchi (Int. J. Non. Linear. Mech. 128:103626, 2021) introduced a different form of generalised neo-Hookean model, focusing on the molecular structure of elastomers, and showed that their model encompasses all ranges of deformations, performing better than the Gent model in many respects, also with only two parameters. Here we investigate the nonlinear vibration and stability of a dielectric elastomer balloon modelled by that strain energy function. We derive the deformation field in spherical coordinates and the governing equations by the Euler-Lagrange method, assuming that the balloon retains its spherical symmetry as it inflates. We consider in turn that t...
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We investigate the theoretical nonlinear response, Hessian stability, and possible wrinkling behaviour of a voltage-activated dielectric plate immersed in a tank filled with silicone oil. Fixed rigid electrodes are placed on the top and... more
We investigate the theoretical nonlinear response, Hessian stability, and possible wrinkling behaviour of a voltage-activated dielectric plate immersed in a tank filled with silicone oil. Fixed rigid electrodes are placed on the top and bottom of the tank, and an electric field is generated by a potential difference between the electrodes. We solve the associated incremental boundary value problem of superimposed, inhomogeneous small-amplitude wrinkles, signalling the onset of instability. We decouple the resulting bifurcation equation into symmetric and antisymmetric modes. For a neo-Hookean dielectric plate, we show that a potential difference between the electrodes can induce a thinning of the plate and thus an increase of its planar area, similar to the scenarios encountered when there is no silicone oil. However, we also find that, depending on the material and geometric parameters, an increasing applied voltage can also lead to a thickening of the plate, and thus a shrinking o...
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Skin tension lines are natural lines of tension that occur within the skin as a result of growth and remodeling mechanisms. Researchers have been aware of their existence and their surgical implications for over 150 years. Research in the... more
Skin tension lines are natural lines of tension that occur within the skin as a result of growth and remodeling mechanisms. Researchers have been aware of their existence and their surgical implications for over 150 years. Research in the twentieth century showed clearly, through destructive mechanical testing, that the orientation of skin tension lines greatly affects the mechanical response of skin in situ. More recent work has determined that this anisotropic response is, at least in part, due to the structural arrangement of collagen fibres within the dermis. This observation can be incorporated into mathematical and mechanical models using the popular Gasser-Ogden-Holzapfel constitutive equation. Advances in non-invasive measurement techniques for the skin, such as those based on elastic wave propagation, have enabled patient-specific identification of skin tension lines in an accurate and rapid manner. Using this technique on humans, we show that there is considerable variatio...
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Intuition suggests that twisting a cylinder will shorten it, but here it is shown that for a cylinder of an incompressible material, like rubber, twisting will always produce elongation.