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This article highlights and corrects fundamental errors in the publication [A new restriction for initially stressed elastic solids, Quart. J. Mech. Appl. Math. 70, 455-478 (2017)] and related works.

The theory of superimposed incremental elastic deformations is applied to a de-formed con_guration of a residually-stressed circular cylindrical tube. The tube issubject to internal or external pressure and an axial load that maintain its circular cylindrical shape, and then bifurcations from this shape are analyzed. Detailed 1 Highlighted Revision governing equations and boundary conditions are provided for axisymmetric, pris- matic and asymmetric bifurcations for a general form of strain-energy function that incorporates radial and circumferential residual stress components. The theory is applied to a simple model strain-energy function with two material parameters and a parameter that reects the magnitude of the residual stress. Numerical computa-tions are used to illustrate the dependence of the results of the bifurcation analysis on these parameters and the axial and radial underlying deformation. It is shown that in general the presence of residual stress has a signi_cant e_ect compared with the corresponding results without residual stress.

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.

BACKGROUND RESULTS of replantation surgery following upper extremity traumatic amputation are extensively described in the literature, with success rates varying from 57 to 100 percent. The purpose of this study was to evaluate replantation success rate at a Level I trauma center over a 17-year period and to assess definable factors contributing to these results. METHODS A retrospective review of all digit and hand replantations at a Level I trauma center was performed using CPT codes from 2001 through 2018. Descriptive analyses, Mann-Whitney test, Kruskal-Wallis test, and logistic regressions were used. Significance was defined as p ≤ 0.05. RESULTS Analysis consisted of 76 patients with 101 amputated parts (93 digits and eight hands). Fifty-six single digit amputations (30 percent success rate), 37 multidigit injuries (22 percent digit success rate), and eight hand amputations (50 percent success rate) were attempted. The overall success rate was 25 of 76 patients (33 percent) and 29 of 101 parts (29 percent). The most common mechanism of injury was laceration (n = 56), followed by crush (n = 30), and avulsion (n = 11), with repair of laceration-type injuries having the greatest success rate (36 percent). CONCLUSIONS The authors report a lower success rate of hand and digit replantation than previously described in the literature. Whole hand and thumb replantations resulted in the highest survival rate in our series. Laceration mechanism showed a higher success rate than crush or avulsion-type injuries. The authors' modest results highlight the importance of effective internal auditing of low-volume replantation centers such as their own. Quality improvement measures are proposed for higher future success in replantation surgery at the authors' institution. CLINICAL QUESTION/LEVEL OF EVIDENCE Therapeutic, IV.

Growth in nature is associated with the development of residual stresses and is in general heterogeneous and anisotropic at all scales. Residual stress in an unloaded configuration of a growing material provides direct evidence of the mechanical regulation of heterogeneity and anisotropy of growth. The present study explores a model of stress-mediated growth based on the unloaded configuration that considers either the residual stress or the deformation gradient relative to the unloaded configuration as a growth variable. This makes it possible to analyze stress-mediated growth without the need to invoke the existence of a fictitious stress-free grown configuration. Furthermore, applications based on the proposed theoretical framework relate directly to practical experimental scenarios involving the “opening-angle” in arteries as a measure of residual stress. An initial illustration of the theory is then provided by considering the growth of a spherically symmetric thick-walled shell subjected to the incompressibility constraint.

Current clinical practice for aneurysmatic interventions is often based on the maximum diameter of the vessel and/or on the growth rate, although rupture can occur at any diameter and growth rate, leading to fatality. For 27 medial samples obtained from 12 non-aneurysmatic (control) and 9 aneurysmatic human descending thoracic aortas we examined: the mechanical responses up to rupture using uniaxial extension tests of circumferential and longitudinal specimens; the structure of these tissues using second-harmonic imaging and histology, in particular, the content proportions of collagen, elastic fibers and smooth muscle cells in the media. It was found that the mean failure stresses were higher in the circumferential directions (Control-C 1474 kPa; Aneurysmatic-C 1446 kPa), than in the longitudinal directions (Aneurysmatic-L 735 kPa; Control-L 579 kPa). This trend was the opposite to that observed for the mean collagen fiber directions measured from the loading axis (Control-L > Aneurysmatic-L > Aneurysmatic-C > Control-C), thus suggesting that the trend in the failure stress can in part be attributed to the collagen architecture. The difference in the mean values of the out-of-plane dispersion in the radial/longitudinal plane between the control and aneurysmatic groups was significant. The difference in the mean values of the mean fiber angle from the circumferential direction was also significantly different between the two groups. Most specimens showed delamination zones near the ruptured region in addition to ruptured collagen and elastic fibers. This study provides a basis for further studies on the microstructure and the uniaxial failure properties of (aneurysmatic) arterial walls towards realistic modeling and prediction of tissue failure. Statement of Significance A data set relating uniaxial failure properties to the microstructure of non-aneurysmatic and aneurysmatic human thoracic aortic medias under uniaxial extension tests is presented for the first time. It was found that the mean failure stresses were higher in the circumferential directions, than in the longitudinal directions. The general trend for the failure stresses was Control-C > Aneurysmatic-C > Aneurysmatic-L > Control-L, which was the opposite of that observed for the mean collagen fiber direction relative to the loading axis (Control-L > Aneurysmatic-L > Aneurysmatic-C > Control-C) suggesting that the trend in the failure stress can in part be attributed to the collagen architecture. This study provides a first step towards more realistic modeling and prediction of tissue failure.

The stability of an electroelastic dielectric elastomer plate with compliant electrodes on its major surfaces under an applied potential difference is examined on the basis of the incremental theory of electroelastic fields. The Stroh method of analysis of the governing equations is used with the material constitutive law given in terms of the energy function Ω*(F, DL), where F is the deformation gradient and DL is the Lagrangian electric displacement field. For a particular class of energy functions, explicit bifurcation equations are obtained for antisymmetric and symmetric modes of instability and the results are illustrated for a Gent electroelastic material model with different values of the Gent parameter. This work confirms previous results obtained in terms of the energy function Ω(F, EL), where EL is the Lagrangian electric field.

Generalised Structure Tensors (GSTs) are used to formulate constitutive models for anisotropic fibre-reinforced materials in which fibres are dispersed. The GST approach has been applied so far to models based on invariants I 4 and I 5 ( I 6 and I 7 ). These anisotropic invariants capture the effect of deformation on each fibre family in isolation, unlike the invariant I 8 , which couples two fibre families. We extend the GST approach to models based on the invariant I 8 . We consider two different formulations and for each model derive expressions for stress and elasticity tensors in both the general case and for axisymmetric distributions. We apply the proposed formulation to the hyperelastic Holzapfel–Ogden model for myocardium and obtain a modified model, in which fibre dispersion is consistently accounted for in every term of the strain-energy function. We demonstrate that when accounting for fibre dispersion in the coupling term, the effect on the predicted material response can be significant and may also reduce material symmetry.

Residual stresses in an unloaded configuration of an elastic material have a significant influence on the response of the material from that configuration, but the effect of residual stress on the stability of the material, whether loaded or unloaded, has only been addressed to a limited extent. In this paper we consider the level of residual stress that can be supported in a thick-walled circular cylindrical tube of non-linearly elastic material without loss of stability when subjected to fixed axial stretch and either internal or external pressure. In particular, we consider the tube to have radial and circumferential residual stresses, with a simple form of elastic constitutive law that accommodates the residual stress, and incremental deformations restricted to the cross section of the tube. Results are described for a tube subject to a level of (internal or external) pressure characterized by the internal azimuthal stretch. Subject to restrictions imposed by the strong ellipticity condition, the emergence of bifurcated solutions is detailed for their dependence on the level of residual stress and mode number.

A rectangular plate of dielectric elastomer exhibiting gradients of material properties through its thickness will deform inhomogeneously when a potential difference is applied to compliant electrodes on its major surfaces, because each plane parallel to the major surfaces will expand or contract to a different extent. Here we study the voltage-induced bending response of a functionally graded dielectric plate on the basis of the nonlinear theory of electroelasticity, when both the elastic shear modulus and the electric permittivity change with the thickness coordinate. The theory is illustrated for a neo-Hookean electroelastic energy function with the shear modulus and permittivity varying linearly across the thickness. In general the bending angle increases with the potential difference, and this enables the material inhomogeneity to be tuned to control the bending shape. We derive the Hessian criterion that ensures stability of the bent configurations in respect of a general form of electroelastic constitutive law specialized for the considered geometry. This requires that the Hessian remains positive. For the considered model we show that the bent configuration is stable until the voltage reaches the value for which the cross section of the bent configuration forms a complete circle.

Two-phase piecewise homogeneous plane deformations are examined in respect of a neo-Hookean matrix material reinforced with embedded aligned fibres characterized by a single stiffness parameter. The deformations are interpreted in terms of fibre kinking and fibre splitting. Previous work has shown that such a transversely isotropic material can lose ellipticity if the reinforcing stiffness is sufficiently large and the fibre direction is sufficiently compressed. In particular, it was shown that the associated failure modes are characterised by the emergence of weak surfaces of discontinuity that are normal to the fibre direction (the onset of fibre kinking) or parallel to the fibre direction (the onset of fibre splitting). Here, the analysis of strong surfaces of discontinuity, developing from weak ones, is studied. The considered model can give rise to piecewise smooth plane deformations separated by a plane stationary surface of discontinuity, interpreted as either kinking or splitting. Attention is restricted to (plane) deformations in which, on one side of the surface of discontinuity, the load axis is aligned with the fibre axis. Then the fibre stretch on this side of the discontinuity is a natural load parameter. The ellipticity status of the two-phase piecewise homogeneous plane deformations is shown to span all four possible ellipticity/non-ellipticity permutations. If both deformation states are elliptic, then a suitable intermediate deformation is shown to be non-elliptic. Moreover, it is shown that the mechanism is dissipative, and maximally dissipative quasi-static failure motion is examined in respect of both kinking and splitting. It follows that, firstly, surfaces of discontinuity perpendicular to the fibre direction, associated with fibre kinking, are nucleated followed by surfaces of discontinuity parallel to the fibre direction, associated with fibre splitting. With respect to kinking, such maximally dissipative kinks nucleate only in compression as weak surfaces of discontinuity, with the subsequent motion converting non-elliptic deformation to elliptic deformation.

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.

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 deformations, the commonly adopted assumption that fibres cannot support compression is incorrectly implemented in current FE codes and leads to the unphysical result that under hydrostatic tension a sphere of compressible anisotropic material deforms into a larger sphere. peer-reviewed

BackgroundIn March 2020, an influx of admissions in COVID-19 positive patients threatened to overwhelm healthcare facilities in East Baton Rouge Parish, Louisiana. Exacerbating this problem was a shortage of diagnostic testing capability, resulting in a delay in time-to-result return. An improvement in diagnostic testing availability and timeliness was necessary to improve the allocation of resources and ultimate throughput of patients. The management of a COVID-19 positive patient or patient under investigation requires infection control measures that can quickly consume personal protective equipment (PPE) stores and personnel available to treat these patients. Critical shortages of both PPE and personnel also negatively impact care in patients admitted with non-COVID-19 illnesses.MethodsA multisectoral partnership of healthcare providers, facilities and academicians created a molecular diagnostic lab within an academic research facility dedicated to testing inpatients and healthcare personnel for SARS-CoV-2. The purpose of the laboratory was to provide a temporary solution to the East Baton Rouge Parish healthcare community until individual facilities were self-sustaining in testing capabilities. We describe the partnership and the impacts of this endeavor by developing a model derived from a combination of data sources, including electronic health records, hospital operations, and state and local resources.FindingsOur model demonstrates two important principles: the impact of reduced turnaround times (TAT) on potential differences in inpatient population numbers for COVID-19 and savings in PPE attributed to the more rapid TAT.InterpretationOverall, we provide rationale for and demonstration of the utility of multisectoral partnerships when responding to public health emergencies.

This paper is concerned with the bifurcation analysis of a pressurized electroelastic spherical shell with compliant electrodes on its inner and outer boundaries. The theory of small incremental electroelastic deformations superimposed on a radially finitely deformed electroelastic thick-walled spherical shell is used to determine those underlying configurations for which the superimposed deformations do not maintain the perfect spherical shape of the shell. Specifically, axisymmetric bifurcations are analyzed, and results are obtained for three different electroelastic energy functions, namely electroelastic counterparts of the neo-Hookean, Gent and Ogden elastic energy functions. For the neo-Hookean energy function it was reported previously that for the purely mechanical case axisymmetric bifurcations are possible under external pressure only, no bifurcation solutions being possible for internally pressurized spherical shells. In the case of an electroelastic neo-Hookean model bifurcation under internal pressure becomes possible when the potential difference between the electrodes exceeds a certain value, which depends on the ratio of inner to outer undeformed radii. Results obtained for the three classes of model are significantly different and are illustrated for a range of fixed values of the potential difference. Although of less practical significance, results are also shown for fixed charges, and these are both different between the models and different from the case of fixed potential difference.

We examine the stability of a soft dielectric plate deformed by the coupled effects of a mechanical pre-stress applied on its lateral faces and an electric field applied through its thickness under charge control. The electric field is created by spraying charges on the major faces of the plate: although in practice this mode of actuation is harder to achieve than a voltage-driven deformation, here we find that it turns out to be much more stable in theory and in simulations.First we show that the electromechanical instability based on the Hessian criterion associated with the free energy of the system does not occur at all for charge-driven dielectrics for which the electric displacement is linear in the electric field. Then we show that the geometric instability associated with the formation of small-amplitude wrinkles on the faces of the plate that arises under voltage control does not occur either under charge control. This is in complete contrast to voltage-control actuation, where Hessian and wrinkling instabilities can occur once certain critical voltages are reached.For the mechanical pre-stresses, two modes that can be implemented in practice are used: equi-biaxial and uni-axial. We confirm the analytical and numerical stability results of homogeneous deformation modes with Finite Element simulations of real actuations, where inhomogeneous fields may develop. We find complete agreement in the equi-biaxial case, and very close agreement in the uni-axial case, when the pre-stress is due to a dead-load weight. In the latter case, the simulations show that small inhomogeneous effects develop near the clamps, and eventually a compressive lateral stress emerges, leading to a breakdown of the numerics. (C) 2020 Elsevier Ltd. All rights reserved. This work is supported by a Government of Ireland Postgraduate Scholarship from the Irish Research Council (Project GOIPG/2016/712). We thank Giacomo Moretti, Yipin Su and Giuseppe Zurlo for most helpful inputs. peer-reviewed 2022-04-06

We propose a mechanical model to account for progressive damage in collagen fibres within fibrous soft tissues. The model has a similar basis to the pseudoelastic model that describes the Mullins effect in rubber but it also accounts for the effect of cross-links between collagen fibres. We show that the model is able to capture experimental data obtained from rat tail tendon fibres, and the combined effect of damage and collagen cross-links is illustrated for a simple shear test. The proposed three-dimensional framework allows a straightforward implementation in finite-element codes, which are needed to analyse more complex boundary-value problems for soft tissues under supra-physiological loading or tissues weakened by disease.

Peter Chadwick studied mathematics as an undergraduate at the University of Manchester, graduating with first-class honours in 1952, from where he moved to Cambridge and completed a PhD on the thermal history of the Earth in the Department of Geodesy and Geophysics under the supervision of Dr Robert Stoneley. His research then developed to focus primarily on the propagation of waves, and he made a major contribution to the mathematical theory of elastic wave propagation and became a world-leading authority in this area. He also made fundamental advances in the modelling of the thermo-elastic properties of rubberlike materials. At the University of East Anglia, where he was a professor for 26 years, he was the driving force behind the development of a research group in theoretical mechanics in the School of Mathematics and Physics, leading by example and supporting and encouraging fellow faculty members, especially the younger staff, academic visitors and students. He gave considerable service to the University of East Anglia in a number of capacities, including a period as Dean of the School, and to the scientific community, through substantial journal editorial activities and as a member of several national and international committees.

It is apparent from the literature that the density of cross-links in collagenous tissue has a stiffening effect on the mechanical response of the tissue. This paper represents an initial attempt to characterize this effect on the elastic response, specifically in respect of arterial tissue. Two approaches are presented. First, a simple phenomenological continuum model with a cross-link-dependent stiffness is considered, and the influence of the cross-link density on the response in uniaxial tension is illustrated. In the second approach, a 3D model is developed that accounts for the relative orientation and stiffness of (two families of) collagen fibers and cross-links and their coupling using an invariant-based strain-energy function. This is also illustrated for uniaxial tension, and the influence of different cross-link arrangements and material parameters is detailed. Specialization of the model for plane strain is then used to show the effect of the cross-link orientation (relative to the fibers) and cross-link density on the shear stress versus the amount of shear deformation response. The elasticity tensor for the general (3D) case is provided with a view to subsequent finite element implementation. © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license. (http://creativecommons.org/licenses/by/4.0/)

In two recent papers, conditions for which axisymmetric incremental bifurcation could arise for a circular cylindrical tube subject to axial extension and radial inflation in the presence of an axial load, internal pressure and a radial electric field were examined, the latter being effected by a potential difference between compliant electrodes on the inner and outer radial surfaces of the tube. The present paper takes this work further by considering the incremental deformations to be time-dependent. In particular, both the axisymmetric vibration of a tube of finite length with appropriate end conditions and the propagation of axisymmetric waves in a tube are investigated. General equations and boundary conditions governing the axisymmetric incremental motions are obtained and then, for purposes of numerical evaluation, specialized for a Gent electroelastic model. The resulting system of equations is solved numerically and the results highlight the dependence of the frequency of vibration and wave speed on the tube geometry, applied deformation and electrostatic potential. In particular, the bifurcation results obtained previously are recovered as a special case when the frequency vanishes. Specification of an incremental potential difference in the present work ensures that there is no incremental electric field exterior to the tube. Results are also illustrated for a neo-Hookean electroelastic model and compared with those previously obtained for the case in which no incremental potential difference (or charge) is specified and an external field is required.

In this paper, the influence of a radial electric field generated by compliant electrodes on the curved surfaces of a tube of dielectric electroelastic material subject to radially symmetric finite deformations is analyzed within the framework of the general theory of nonlinear electroelasticity. The analysis is illustrated for two constitutive equations based on the neo-Hookean and Gent elasticity models supplemented by an electrostatic energy term with a deformation dependent permittivity.

The microstructure of arteries, consisting, in particular, of collagen, elastin, and vascular smooth muscle cells, plays a very significant role in their biomechanical response during a cardiac cycle. In this article, we highlight the microstructure and the contributions of each of its components to the overall mechanical behavior. We also describe the changes of the microstructure that occur as a result of abdominal aortic aneurysms and disease, such as atherosclerosis. We also focus on how the passive and active constituents are incorporated into a mathematical model without going into detail of the mathematical formulation. We conclude by mentioning open problems toward a better characterization of the biomechanical aspects of arteries that will be beneficial for a better understanding of cardiovascular pathophysiology.

Dispersed collagen fibers in fibrous soft biological tissues have a significant effect on the overall mechanical behavior of the tissues. Constitutive modeling of the detailed structure obtained by using advanced imaging modalities has been investigated extensively in the last decade. In particular, our group has previously proposed a fiber dispersion model based on a generalized structure tensor. However, the fiber tension–compression switch described in that study is unable to exclude compressed fibers within a dispersion and the model requires modification so as to avoid some unphysical effects. In a recent paper we have proposed a method which avoids such problems, but in this present study we introduce an alternative approach by using a new general invariant that only depends on the fibers under tension so that compressed fibers within a dispersion do not contribute to the strain-energy function. We then provide expressions for the associated Cauchy stress and elasticity tensors in a decoupled form. We have also implemented the proposed model in a finite element analysis program and illustrated the implementation with three representative examples: simple tension and compression, simple shear, and unconfined compression on articular cartilage. We have obtained very good agreement with the analytical solutions that are available for the first two examples. The third example shows the efficacy of the fibrous tissue model in a larger scale simulation. For comparison we also provide results for the three examples with the compressed fibers included, and the results are completely different. If the distribution of collagen fibers is such that it is appropriate to exclude compressed fibers then such a model should be adopted. © 2017. This is the authors’ accepted and refereed manuscript to the article. Locked until 14.9.2019 due to copyright restrictions. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

Effective communication is fundamental to the quality and safety of healthcare. In the U.S., language and cultural differences combined with low health literacy are challenges that physicians must overcome. The Spanish-speaking patient population continues to grow and is disproportionately affected by this, with 25% speaking little to no English. In our hand and upper extremity surgery practice, a common problem arises both with verbal Spanish communication and with the physical exam. The hand and upper extremity examination requires a high level of patient involvement to elucidate subtle derangements in individual tendon or nerve function. Our purpose is to summarize a model upper extremity clinical encounter in Spanish and provide a guide for patient-physician interaction, particularly during the physical exam where visual aids can be more useful than simple verbal commands. This guide is a simple step towards enhanced communication and understanding between providers and patients, with the goal of providing higher quality care with greater efficiency and satisfaction. The purpose is not to replace the professional interpreter, but to augment the physician-patient interaction during the upper extremity clinical encounter. Incorporating a cross-language template may yield improved patient understanding, enhanced participation in the examination leading to better diagnosis, and improved satisfaction for both patient and provider. Key words: Communication, Language, Patient satisfaction, Spanish, Upper extremity

This study compares two models that are used to describe the elastic properties of fiber-reinforced materials with dispersed fibers, in particular some soft biological tissues such as arterial walls and cartilages. The two model approaches involve different constitutive frameworks, one being based on a generalized structure tensor (GST) and the other on the method of angular integration (AI). By using two representative examples, with the same number of parameters for each model, it is shown that the predictions of the two models are virtually identical for a significant range of large deformations, which contradicts conclusions contained in several papers that are based on faulty analysis. Additionally, each of the models is fitted to sets of uniaxial data from the circumferential and axial directions of the adventitia of a human aorta, both models providing excellent agreement with the data. While the predictions of the two models are comparable and exclusion of compressed fibers can be accommodated by either model, it is well known that the AI model requires more computational time than the GST model when used within a finite element environment, in particular if compressed fibers are excluded.

Detailed information on the three-dimensional dispersion of collagen fibres within layers of healthy and diseased soft biological tissues has been reported recently. Previously we have proposed a constitutive model for soft fibrous solids based on the angular integration approach which allows the exclusion of any compressed collagen fibre within the dispersion. In addition, a computational implementation of that model in a general purpose finite element program has been investigated and verified with the standard fibre-reinforcing model for fibre contributions. In this study, we develop the proposed fibre dispersion model further using an exponential form of the strain-energy function for the fibre contributions. The finite element implementation of this model with a rotationally symmetrical dispersion of fibres is also presented. This includes explicit expressions for the stress and elasticity tensors. The performance and implementation of the new model are demonstrated by means of a uniaxial extension test, a simple shear test, and an extension–inflation simulation of a residually stressed carotid artery segment. In each example we have obtained good agreement between the finite element solution and the analytical or experimental results. © 2017. This is the authors' accepted and refereed manuscript to the article. The final authenticated version is available online at: https://doi.org/10.1177%2F1081286517712077

This first chapter of the volume on Constitutive Modelling of Solid Continua sets out briefly the main concepts of general continuum mechanics without reference to specific material behaviour as a backdrop for the detailed descriptions of different types of material behaviour that are contained in the remaining chapters. The focus is therefore on the fundamental ideas of kinematics of a continuum (deformation and motion), the global balance equations governing the motion of a general continuum, stress and the energy balance equation and the derivation of the local governing equations, including mass conservation and the equation of motion. For the most part, detailed proofs are not provided and reference is made to standard texts for more details since this volume is aimed at researchers who have a reasonable background in continuum mechanics and wish to widen the scope of their knowledge in different areas of solid mechanics.

This chapter provides the framework for the development of constitutive theories of solids by focusing on constitutive laws for nonlinearly elastic solids. These exemplify the general principles of constitutive theory that should be applied to all types of material behaviour, in particular, the notions of objectivity and material symmetry, including the important symmetries of isotropy, transverse isotropy and orthotropy based in part on deformation invariants. Details are given for the various general stress–deformation relations for each case of symmetry in respect of hyperelastic materials (which are characterized by a strain-energy function), with or without an internal constraint such as incompressibility, and these are illustrated by particular prototype models. The notion of residual stress (in an unloaded configuration) is discussed and the form of strain-energy function required to accommodate residual stress in the material response is developed.

In this chapter, the equations governing the mechanical behavior of electroelastic solids capable of finite deformations are summarized with particular reference to the development of constitutive equations describing the electromechanical interactions in soft dielectric materials. Following a brief summary of some background from continuum mechanics and nonlinear elasticity theory, the equations of electrostatics are given in Eulerian form and then re-cast in Lagrangian form. The electroelastic constitutive equations are based on the existence of a so-called total energy function, which may be regarded as a function of the Lagrangian form of either the electric field or electric displacement as the independent electric variable, together with the deformation gradient. For each form of the total energy function, corresponding expressions for the (total) nominal and Cauchy stress tensors are provided, both in full generality and for their isotropic specializations for unconstrained and incompressible materials. The general formulas are then applied to the basic problem of homogeneous deformation of a rectangular plate, and illustrated by the choice of a simple specific example of constitutive equation. As a further illustration, the theory is applied to the analysis of a nonhomogeneous deformation of a circular cylindrical tube subject to an axial force, a torsional moment, and a radial electric field generated by a potential difference between flexible electrodes covering its inner and outer curved surfaces.

A universal relation is an algebraic relation between stress and strain that holds for any material within a certain class, irrespective of the exact form of the material response function and parameter values. Classical universal relations, such as Rivlin’s famous relation for simple shear, apply to stress components produced by one and the same deformation. We present a family of relations that connect stress components under different deformations, which we call para-universal relations to highlight this difference. The proposed para-universal relations hold for any orthotropic material whose response function is additively decomposed into terms, each of which possesses a symmetry with respect to one of the axes of orthotropy. Using basic properties of the permutation group S3, we demonstrate that such an additive decomposition implies the proposed identities. The established para-universal relations hold for an arbitrary local deformation and, like classical universal relations, are linked to material symmetry and apply to a wide class of materials. Since the proposed para-universal relations do not hold for all orthotropic material models, they present a convenient way to test for the suitability of additively split strain-energy functions, which are often used to model the nonlinearly elastic response of soft tissues. Such a test can be performed on collected experimental data prior to choosing an exact form of the response function and fitting its parameters. We use published experimental data for human myocardium and also synthetic data to illustrate this.

Fiber dispersion in collagenous soft tissues has an important influence on the mechanical response, and the modeling of the collagen fiber architecture and its mechanics has developed significantly over the last few years. The purpose of this paper is twofold, first to develop a method for excluding compressed fibers within a dispersion for the generalized structure tensor (GST) model, which several times in the literature has been claimed not to be possible, and second to draw attention to several erroneous and misleading statements in the literature concerning the relative values of the GST and the angular integration (AI) models. For the GST model we develop a rather simple method involving a deformation dependent dispersion parameter that allows the mechanical influence of compressed fibers within a dispersion to be excluded. The theory is illustrated by application to simple extension and simple shear in order to highlight the effect of exclusion. By means of two examples we also show that the GST and the AI models have equivalent predictive power, contrary to some claims in the literature. We conclude that from the theoretical point of view neither of these two models is superior to the other. However, as is well known and as we now emphasize, the GST model has proved to be very successful in modeling the data from experiments on a wide range of tissues, and it is easier to analyze and simpler to implement than the AI approach, and the related computational effort is much lower.

Residual stresses are generated in living tissues by processes of growth and adaptation and they significantly influence the mechanical behaviour of the tissues. Thus, to effectively model the elastic response of the tissues relative to a residually stressed configuration the residual stresses need to be incorporated into the constitutive equations. The purposes of this paper are (a) to summarise a general elastic constitutive formulation that includes residual stress, (b) to specify the tensors needed for the three-dimensional implementation of the theory in a nonlinear finite element code, and (c) to use the theory and its implementation to evaluate the wall stress distribution in an abdominal aortic aneurysm (AAA) using patient specific geometry and material model parameters. The considered material is anisotropic with two preferred directions indicating the orientation of the collagen fibres in the aortic tissue. The method described in this paper is general and can be used, by specifying appropriate energy functions, to investigate other residually stressed biological systems.

In this paper the propagation of small amplitude surface waves guided by a layer with a finite thickness on an incompressible half-space is studied. The layer and half-space are both assumed to be initially stressed. The combined effect of initial stress and finite deformation on the speed of Rayleigh waves is analyzed and illustrated graphically. With a suitable simple choice of constitutive law that includes initial stress, it is shown that in many cases, as is to be expected, the effect of a finite deformation (with an associated pre-stress) is very similar to that of an initial stress (without an accompanying finite deformation). However, by contrast, when the finite deformation and initial stress are considered together independently with a judicious choice of material parameters different features are found that do not appear in the separate finite deformation or initial stress situations on their own.

Within the framework of the quasi-electrostatic approximation, the theory of the superposition of infinitesimal deformations and electric fields on a finite deformation with an underlying electric field is employed to examine the problem of the reflection of small amplitude homogeneous electroelastic plane waves from the boundary of an incompressible finitely deformed electroactive half-space. The theory is applied to the case of two-dimensional incremental motions and electric fields with an underlying biassing electric field normal to the half-space boundary, and the general incremental governing equations are obtained for this specialization. For illustration, the equations are then applied to a simple prototype electroelastic model for which it is found that only a single reflected wave is possible and a surface wave is in general generated for each angle of incidence. Explicit formulas are obtained for the wave speed and the reflection and surface wave coefficients in terms of the deformation, magnitude of the electric (displacement) field, the electromechanical coupling parameters, and the angle of incidence, and the results are illustrated graphically.

This paper is concerned with the bifurcation analysis of a pressurized electroelastic circular cylindrical tube with closed ends and compliant electrodes on its curved boundaries. The theory of small incremental electroelastic deformations superimposed on a finitely deformed electroelastic tube is used to determine those underlying configurations for which the superimposed deformations do not maintain the perfect cylindrical shape of the tube. First, prismatic bifurcations are examined and solutions are obtained which show that for a neo-Hookean electroelastic material prismatic modes of bifurcation become possible under inflation. This result contrasts with that for the purely elastic case for which prismatic bifurcation modes were found only for an externally pressurized tube. Second, axisymmetric bifurcations are analyzed, and results for both neo-Hookean and Mooney–Rivlin electroelastic energy functions are obtained. The solutions show that in the presence of a moderate electric field the electroelastic tube becomes more susceptible to bifurcation, i.e., for fixed values of the axial stretch axisymmetric bifurcations become possible at lower values of the circumferential stretches than in the corresponding problems in the absence of an electric field. As the magnitude of the electric field increases, however, the possibility of bifurcation under internal pressure becomes restricted to a limited range of values of the axial stretch and is phased out completely for sufficiently large electric fields. Then, axisymmetric bifurcation is only possible under external pressure.

When a transversely isotropic circular cylindrical tube is subject to axial extension and inflation, the governing equations of equilibrium can lose ellipticity under certain combinations of deformation and direction of transverse isotropy. In this paper, it is shown how the inclusion of an axial shear deformation moderates the loss of ellipticity condition. In particular, this condition is analysed for a material model consisting of an isotropic neo-Hookean matrix within which are embedded fibres whose properties are characterized by the addition to the strain-energy function of a reinforcing model depending on the local fibre direction.

We describe a case of cerebral trichomoniasis in a neonate who developed seizures and multi-organ failure during treatment for staphylococcal sepsis. Brain abscesses were identified on cranial sonography. Trichomonas vaginalis was isolated from cerebrospinal fluid. There was a fatal outcome despite metronidazole therapy. This is the first report of T. vaginalis brain abscess in a neonate.