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Bats fly using significantly different wing motions from other fliers, stemming from the complex interplay of their membrane wings' motion and structural properties. Biological studies show that many bats fly at Strouhal numbers, the ratio of flapping to flight speed, 50-150% above the range typically associated with optimal locomotion. We use high-resolution fluid-structure interaction simulations of a bat wing to independently study the role of kinematics and material/structural properties in aerodynamic performance and show that peak propulsive and lift efficiencies for a bat-like wing motion require flapping 66% faster than for a symmetric motion, agreeing with the increased flapping frequency observed in zoological studies. In addition, we find that reduced membrane stiffness is associated with improved propulsive efficiency until the membrane flutters, but that incorporating microstructural anisotropy arising from biological fibre reinforcement enables a tenfold reduction of the flutter energy while maintaining high aerodynamic efficiency. Our results indicate that animals with specialized flapping motions may have correspondingly specialized flapping speeds, in contrast to arguments for a universally efficient Strouhal range. Additionally, our study demonstrates the significant role that the microstructural constitutive properties of the membrane wing of a bat can have in its propulsive performance.

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BACKGROUND AND OBJECTIVE: This paper presents the development of a 3D physics-based numerical model of skin capable of representing the laser-skin photo-thermal interactions occurring in skin photorejuvenation treatment procedures. The aim of this model was to provide a rational and quantitative basis to control and predict temperature distribution within the layered structure of skin. Ultimately, this mathematical and numerical modelling platform will guide the design of an automatic robotic controller to precisely regulate skin temperature at desired depths and for specific durations. METHODS: The Pennes bioheat equation was used to account for heat transfer in a 3D multi-layer model of skin. The effects of blood perfusion, skin pigmentation and various convection conditions are also incorporated in the proposed model. The photo-thermal effect due to pulsed laser light on skin is computed using light diffusion theory. The physics-based constitutive model was numerically implemented using a combination of finite volume and finite difference techniques. Direct sensitivity routines were also implemented to assess the influence of constitutive parameters on temperature. A stability analysis of the numerical model was conducted. RESULTS: Finally, the numerical model was exploited to assess its ability to predict temperature distribution and thermal damage via a multi-parametric study which accounted for a wide array of biophysical parameters such as light coefficients of absorption for individual skin layers and melanin levels (correlated with ethnicity). It was shown how critical is the link between melanin content, laser light characteristics and potential thermal damage to skin. CONCLUSIONS: The developed photo-thermal model of skin-laser interactions paves the way for the design of an automated simulation-driven photorejuvenation robot, thus alleviating the need for inconsistent and error-prone human operators.

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PURPOSE: The lack of long-term patency of synthetic vascular grafts currently available on the market has directed research towards improving the performance of small diameter grafts. Improved radial compliance matching and tissue ingrowth into the graft scaffold are amongst the main goals for an ideal vascular graft. METHODS: Biostable polyurethane scaffolds were manufactured by electrospinning and implanted in subcutaneous and circulatory positions in the rat for 7, 14 and 28 days. Scaffold morphology, tissue ingrowth, and mechanical properties of the scaffolds were assessed before implantation and after retrieval. RESULTS: Tissue ingrowth after 24 days was 96.5 ± 2.3% in the subcutaneous implants and 77.8 ± 5.4% in the circulatory implants. Over the 24 days implantation, the elastic modulus at 12% strain decreased by 59% in direction of the fibre alignment whereas it increased by 1379% transverse to the fibre alignment of the highly aligned scaffold of the subcutaneous implants. The lesser aligned scaffold of the circulatory graft implants exhibited an increase of the elastic modulus at 12% strain by 77% in circumferential direction. CONCLUSION: Based on the observations, it is proposed that the mechanism underlying the softening of the highly aligned scaffold in the predominant fibre direction is associated with scaffold compaction and local displacement of fibres by the newly formed tissue. The stiffening of the scaffold, observed transverse to highly aligned fibres and for more a random fibre distribution, represents the actual mechanical contribution of the tissue that developed in the scaffold.

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The measurement of the mechanical properties of skin (such as stiffness, extensibility and strength) is a key step in characterisation of both dermal ageing and disease mechanisms and in the assessment of tissue-engineered skin replacements. However, the biomechanical terminology and plethora of mathematical analysis approaches can be daunting to those outside the field. As a consequence, mechanical studies are often inaccessible to a significant proportion of the intended audience. Furthermore, devices for the measurement of skin function in vivo generate relative values rather than formal mechanical measures, therefore limiting the ability to compare studies. In this viewpoint essay, we discuss key biomechanical concepts and the influence of technical and biological factors (including the nature of the testing apparatus, length scale, donor age and anatomical site) on measured mechanical properties such as stiffness. Having discussed the current state-of-the-art in macro-mechanical and micromechanical measuring techniques and in mathematical and computational modelling methods, we then make suggestions as to how these approaches, in combination with 3D X-ray imaging and mechanics methods, may be adopted into a single strategy to characterise the mechanical behaviour of skin.

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The objective of this paper is to provide a review on some aspects of the mathematical and computational modelling of skin biophysics, with special focus on constitutive theories based on nonlinear continuum mechanics from elasticity, through anelasticity, including growth, to thermoelasticity. Microstructural and phenomenological approaches combining imaging techniques are also discussed. Finally, recent research applications on skin wrinkles will be presented to highlight the potential of physics-based modelling of skin in tackling global challenges such as ageing of the population and the associated skin degradation, diseases and traumas.

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Bone is a heterogeneous material and its mechanical properties vary within the body. Variations in the mechanical response of different bone samples taken from the body cannot be fully explained by only looking at local compositional information at the tissue level. Due to different states of the stress within bones, one might expect that mechanical properties change over the length of a bone; this has not been a matter of systematic research in previous studies. In this study, the distribution of the tissue elastic modulus along the bovine femur is investigated using three-point bending tests. Two bovine femora were split to seven and eight blocks from proximal to distal metaphysis, respectively and twenty beam shaped bone samples were extracted and tested from each block. Based on our findings, the longitudinal elastic modulus follows a gradient pattern along the bovine femur as it increases along the bone from the proximal metaphysis to mid-diaphysis and then decreases toward the distal metaphysis again. Considering long bones to be subjected to bending loads, this mechanism alters the bone structure to support load in the regions where it is needed; similar as outlined by Wolff's law. In another part of this study, microfocus X-ray computed tomography (µCT) was found unable to predict the same trend of changes for the elastic modulus via image-based or density-based elastic moduli calculations. This is insofar important as conventional finite element models of bone are often directly shaped from µCT data. Based on our findings, it seems that current computed tomography based finite element models generated in this manner may not adequately capture the local variation of material behavior of bone tissue, but this may be improved by considering the changes of the elastic modulus along the femur.

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Due to its multifactorial nature, skin friction remains a multiphysics and multiscale phenomenon poorly understood despite its relevance for many biomedical and engineering applications (from superficial pressure ulcers, through shaving and cosmetics, to automotive safety and sports equipment). For example, it is unclear whether, and in which measure, the skin microscopic surface topography, internal microstructure and associated nonlinear mechanics can condition and modulate skin friction. This study addressed this question through the development of a parametric finite element contact homogenisation procedure which was used to study and quantify the effect of the skin microstructure on the macroscopic skin frictional response. An anatomically realistic two-dimensional image-based multilayer finite element model of human skin was used to simulate the sliding of rigid indenters of various sizes over the skin surface. A corresponding structurally idealised multilayer skin model was also built for comparison purposes. Microscopic friction specified at skin asperity or microrelief level was an input to the finite element computations. From the contact reaction force measured at the sliding indenter, a homogenised (or apparent) macroscopic friction was calculated. Results demonstrated that the naturally complex geometry of the skin microstructure and surface topography alone can play as significant role in modulating the deformation component of macroscopic friction and can significantly increase it. This effect is further amplified as the ground-state Young's modulus of the stratum corneum is increased (for example, as a result of a dryer environment). In these conditions, the skin microstructure is a dominant factor in the deformation component of macroscopic friction, regardless of indenter size or specified local friction properties. When the skin is assumed to be an assembly of nominally flat layers, the resulting global coefficient of friction is reduced with respect to the local one. This seemingly counter-intuitive effect had already been demonstrated in a recent computational study found in the literature. Results also suggest that care should be taken when assigning a coefficient of friction in computer simulations, as it might not reflect the conditions of microscopic and macroscopic friction one intends to represent. The modelling methodology and simulation tools developed in this study go beyond what current analytical models of skin friction can offer: the ability to accommodate arbitrary kinematics (i.e. finite deformations), nonlinear constitutive properties and the complex geometry of the skin microstructural constituents. It was demonstrated how this approach offered a new level of mechanistic insight into plausible friction mechanisms associated with purely structural effects operating at the microscopic scale; the methodology should be viewed as complementary to physical experimental protocols characterising skin friction as it may facilitate the interpretation of observations and measurements and/or could also assist in the design of new experimental quantitative assays.

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Electro-spun biodegradable polymer fibrous structures exhibit anisotropic mechanical properties dependent on the degree of fibre alignment. Degradation and mechanical anisotropy need to be captured in a constitutive formulation when computational modelling is used in the development and design optimisation of such scaffolds. Biodegradable polyester-urethane scaffolds were electro-spun and underwent uniaxial tensile testing in and transverse to the direction of predominant fibre alignment before and after in vitro degradation of up to 28 days. A microstructurally-based transversely isotropic hyperelastic continuum constitutive formulation was developed and its parameters were identified from the experimental stress-strain data of the scaffolds at various stages of degradation. During scaffold degradation, maximum stress and strain in circumferential direction decreased from 1.02 ± 0.23 MPa to 0.38 ± 0.004 MPa and from 46 ± 11 % to 12 ± 2 %, respectively. In longitudinal direction, maximum stress and strain decreased from 0.071 ± 0.016 MPa to 0.010 ± 0.007 MPa and from 69 ± 24 % to 8 ± 2 %, respectively. The constitutive parameters were identified for both directions of the non-degraded and degraded scaffold for strain range varying between 0% and 16% with coefficients of determination r(2)>0.871. The six-parameter constitutive formulation proved versatile enough to capture the varying non-linear transversely isotropic behaviour of the fibrous scaffold throughout various stages of degradation.

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The study of skin biophysics has largely been driven by consumer goods, biomedical and cosmetic industries which aim to design products that efficiently interact with the skin and/or modify its biophysical properties for health or cosmetic benefits. The skin is a hierarchical biological structure featuring several layers with their own distinct geometry and mechanical properties. Up to now, no computational models of the skin have simultaneously accounted for these geometrical and material characteristics to study their complex biomechanical interactions under particular macroscopic deformation modes. The goal of this study was, therefore, to develop a robust methodology combining histological sections of human skin, image-processing and finite element techniques to address fundamental questions about skin mechanics and, more particularly, about how macroscopic strains are transmitted and modulated through the epidermis and dermis. The work hypothesis was that, as skin deforms under macroscopic loads, the stratum corneum does not experience significant strains but rather folds/unfolds during skin extension/compression. A sample of fresh human mid-back skin was processed for wax histology. Sections were stained and photographed by optical microscopy. The multiple images were stitched together to produce a larger region of interest and segmented to extract the geometry of the stratum corneum, viable epidermis and dermis. From the segmented structures a 2D finite element mesh of the skin composite model was created and geometrically non-linear plane-strain finite element analyses were conducted to study the sensitivity of the model to variations in mechanical properties. The hybrid experimental-computational methodology has offered valuable insights into the simulated mechanics of the skin, and that of the stratum corneum in particular, by providing qualitative and quantitative information on strain magnitude and distribution. Through a complex non-linear interplay, the geometry and mechanical characteristics of the skin layers (and their relative balance), play a critical role in conditioning the skin mechanical response to macroscopic in-plane compression and extension. Topographical features of the skin surface such as furrows were shown to act as an efficient means to deflect, convert and redistribute strain-and so stress-within the stratum corneum, viable epidermis and dermis. Strain reduction and amplification phenomena were also observed and quantified. Despite the small thickness of the stratum corneum, its Young׳s modulus has a significant effect not only on the strain magnitude and directions within the stratum corneum layer but also on those of the underlying layers. This effect is reflected in the deformed shape of the skin surface in simulated compression and extension and is intrinsically linked to the rather complex geometrical characteristics of each skin layer. Moreover, if the Young׳s modulus of the viable epidermis is assumed to be reduced by a factor 12, the area of skin folding is likely to increase under skin compression. These results should be considered in the light of published computational models of the skin which, up to now, have ignored these characteristics.

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Considerable progress has been made in understanding implant wear and developing numerical models to predict wear for new orthopaedic devices. However any model of wear could be improved through a more accurate representation of the biomaterial mechanics, including time-varying dynamic and inelastic behaviour such as viscosity and plastic deformation. In particular, most computational models of wear of UHMWPE implement a time-invariant version of Archard's law that links the volume of worn material to the contact pressure between the metal implant and the polymeric tibial insert. During in-vivo conditions, however, the contact area is a time-varying quantity and is therefore dependent upon the dynamic deformation response of the material. From this observation one can conclude that creep deformations of UHMWPE may be very important to consider when conducting computational wear analyses, in stark contrast to what can be found in the literature. In this study, different numerical modelling techniques are compared with experimental creep testing on a unicondylar knee replacement system in a physiologically representative context. Linear elastic, plastic and time-varying visco-dynamic models are benchmarked using literature data to predict contact deformations, pressures and areas. The aim of this study is to elucidate the contributions of viscoelastic and plastic effects on these surface quantities. It is concluded that creep deformations have a significant effect on the contact pressure measured (experiment) and calculated (computational models) at the surface of the UHMWPE unicondylar insert. The use of a purely elastoplastic constitutive model for UHMWPE lead to compressive deformations of the insert which are much smaller than those predicted by a creep-capturing viscoelastic model (and those measured experimentally). This shows again the importance of including creep behaviour into a constitutive model in order to predict the right level of surface deformation on a tibial insert. At high compressive loads, inelastic deformation mechanisms (creep and plasticity) dominate the mechanical response of UHMWPE components by altering the surface geometry (penetration depth and so contact area) and therefore the contact pressure. Although generic creep models can provide a good first approximation of material characteristics, for best accuracy both viscous and plastic effects must be captured, and model parameters must be founded upon specific experimental test data. Such high-fidelity numerical creep models will provide a better foundation for the next generation of more robust and accurate in-silico wear prediction tools.

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Bone is multi-scale hierarchical composite material making the prediction of fragility, as well as pinning it to a certain cause, complicated. For proper mechanical simulation and reflection of bone properties in models, microscopic structural features of bone tissue need to be included. This study sets out to gain a mechanistic insight into the role of various microstructural features of bone tissue in particular cement lines and interlamellar areas. Further the hypothesis that compliant interlamellar areas and cement lines within osteonal bone act as strain amplifiers was explored. To this end, a series of experimentally-based micromechanical finite element models of bovine osteonal bone were developed. Different levels of detail for the bone microstructure were considered and combined with the results of physical three-point bending tests and an analytical composite model of a single osteon. The objective was to examine local and global effects of interface structures. The geometrical and microstructural characteristics of the bone samples were derived from microscopy imaging. Parametric finite element studies were conducted to determine optimal values of the elastic modulus of interstitial bone and interlamellar areas. The average isotropic elastic modulus of interfaces suggested in this study is 88.5MPa. Based on the modelling results, it is shown that interfaces are areas of accumulated strain in bone and are likely to act as potential paths for crack propagation. The strain amplification capability of interface structures in the order of 10 predicted by the models suggests a new explanation for the levels of strain required in bone homoeostasis for maintenance and adaptation.

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Coronary stents are tubular type scaffolds that are deployed, using an inflatable balloon on a catheter, most commonly to recover the lumen size of narrowed (diseased) arterial segments. A common differentiating factor between the numerous stents used in clinical practice today is their geometric design. An ideal stent should have high radial strength to provide good arterial support post-expansion, have high flexibility for easy manoeuvrability during deployment, cause minimal injury to the artery when being expanded and, for drug eluting stents, should provide adequate drug in the arterial tissue. Often, with any stent design, these objectives are in competition such that improvement in one objective is a result of trade-off in others. This study proposes a technique to parameterize stent geometry, by varying the shape of circumferential rings and the links, and assess performance by modelling the processes of balloon expansion and drug diffusion. Finite element analysis is used to expand each stent (through balloon inflation) into contact with a representative diseased coronary artery model, followed by a drug release simulation. Also, a separate model is constructed to measure stent flexibility. Since the computational simulation time for each design is very high (approximately 24 h), a Gaussian process modelling approach is used to analyse the design space corresponding to the proposed parameterization. Four objectives to assess recoil, stress distribution, drug distribution and flexibility are set up to perform optimization studies. In particular, single objective constrained optimization problems are set up to improve the design relative to the baseline geometry-i.e. to improve one objective without compromising the others. Improvements of 8, 6 and 15% are obtained individually for stress, drug and flexibility metrics, respectively. The relative influence of the design features on each objective is quantified in terms of main effects, thereby suggesting the design features which could be altered to improve stent performance. In particular, it is shown that large values of strut width combined with smaller axial lengths of circumferential rings are optimal in terms of minimizing average stresses and maximizing drug delivery. Furthermore, it is shown that a larger amplitude of the links with minimum curved regions is desirable for improved flexibility, average stresses and drug delivery.

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Characterising and modelling the mechanical behaviour of biological soft tissues is an essential step in the development of predictive computational models to assist research for a wide range of applications in medicine, biology, tissue engineering, pharmaceutics, consumer goods, cosmetics, transport or military. It is therefore critical to develop constitutive models that can capture particular rheological mechanisms operating at specific length scales so that these models are adapted for their intended applications. Here, a novel mesoscopically-based decoupled invariant-based continuum constitutive framework for transversely isotropic and orthotropic biological soft tissues is developed. A notable feature of the formulation is the full decoupling of shear interactions. The constitutive model is based on a combination of the framework proposed by Lu and Zhang [Lu, J., Zhang, L., 2005. Physically motivated invariant formulation for transversely isotropic hyperelasticity. International Journal of Solids and Structures 42, 6015-6031] and the entropic mechanics of tropocollagen molecules and collagen assemblies. One of the key aspects of the formulation is to use physically-based nanoscopic quantities that could be extracted from experiments and/or atomistic/molecular dynamics simulations to inform the macroscopic constitutive behaviour. This effectively couples the material properties at different levels of the multi-scale hierarchical structure of collagenous tissues. The orthotropic hyperelastic model was shown to reproduce very well the experimental multi-axial properties of rabbit skin. A new insight into the shear response of a skin sample subjected to a simulated indentation test was obtained using numerical direct sensitivity analyses.

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We present here a multi-objective and multi-disciplinary coronary stent design optimization paradigm. Coronary stents are tubular, often mesh-like, structures which are deployed in diseased (stenosed) artery segments to provide a scaffolding feature that compresses atheromatus plaque, hence restoring luminal area and maintaining vessel patency. A three variable geometry parameterisation of a CYPHER (Cordis Corporation, Johnson & Johnson co.) type stent is proposed to explore the functionality of a sequence of circumferential rings connected by 'n' shaped links. The performance of each design is measured by six figures of merit (objectives/metrics) representing (i) acute recoil, (ii) tissue stresses, (iii) haemodynamic disturbance, (iv) drug delivery, (v) uniformity of drug distribution, and (vi) flexibility. These metrics are obtained from computational simulations of (i) structural deformation through balloon inflated expansion of a stent into contact with a stenosed vessel, (ii) pulsatile flow over the deformed stent embedded in the vessel wall, (iii) steady-state drug distribution into the tissue, and (iv) flexibility of a stent in response to an applied moment. Design improvement is obtained by a multi-objective surrogate modelling approach using a non-dominated sorting genetic algorithm (NSGA-II) to search for an optimal family of designs. A number of trade-offs between the different objectives are identified. In particular a conflict between pairs of the following objectives are shown -- (a) volume average stress vs. recoil, (b) volume average drug vs. volume average stress, (c) flexibility vs. volume average stress, (d) flexibility vs. haemodynamic disturbance, (e) volume average drug vs. haemodynamic disturbance, and (f) uniformity of drug vs. volume average stress. Different paradigms to choose the optimal designs from the obtained Pareto fronts are presented and under each such paradigm, the optimal designs and there relative positions with respect to a representative CYPHER stent are shown. The methodology and the results of this work could potentially be useful in further optimisation studies and development of a family of stents with increased resistance to in-stent restenosis and thrombosis.

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The first objective of this computational study was to assess the strain magnitude and distribution within the three-dimensional (3D) trabecular bone structure around an osseointegrated dental implant loaded axially. The second objective was to investigate the relative micromotions between the implant and the surrounding bone. The work hypothesis adopted was that these virtual measurements would be a useful indicator of bone adaptation (resorption, homeostasis, formation). In order to reach these objectives, a microCT-based finite element model of an oral implant implanted into a Berkshire pig mandible was developed along with a robust software methodology. The finite element mesh of the 3D trabecular bone architecture was generated from the segmentation of microCT scans. The implant was meshed independently from its CAD file obtained from the manufacturer. The meshes of the implant and the bone sample were registered together in an integrated software environment. A series of non-linear contact finite element (FE) analyses considering an axial load applied to the top of the implant in combination with three sets of mechanical properties for the trabecular bone tissue was devised. Complex strain distribution patterns are reported and discussed. It was found that considering the Young's modulus of the trabecular bone tissue to be 5, 10 and 15GPa resulted in maximum peri-implant bone microstrains of about 3000, 2100 and 1400. These results indicate that, for the three sets of mechanical properties considered, the magnitude of maximum strain lies within an homeostatic range known to be sufficient to maintain/form bone. The corresponding micro-motions of the implant with respect to the bone microstructure were shown to be sufficiently low to prevent fibrous tissue formation and to favour long-term osseointegration.

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This study is devoted to the development of a non-linear anisotropic model for the human periodontal ligament (PDL). A thorough knowledge of the behaviour of the PDL is vital in understanding the mechanics of orthodontic tooth mobility, soft tissue response and proposed treatment plans. There is considerable evidence that the deformation of the PDL is the key factor determining the orthodontic tooth movement. The paper focuses on the biomechanical aspect of the behaviour of the PDL. In terms of continuous mechanics, the PDL may be treated as an anisotropic poro-visco-hyperelastic fibre-reinforced compressible material which is subject to large deformations and has an essentially non-linear behaviour. Furthermore, there are issues related to the non-linear tooth and PDL geometry. A new constitutive model for the PDL is proposed. The macroscopic continuum approach is used. The model is based on the non-linear large deformation theory, involving the Lagrangian description. The material is assumed to be compressible, visco-hyperelastic and transversely isotropic. A free-energy function is suggested that incorporates the properties. It also takes into account that the PDL behaves differently in tension and compression. The free-energy function and the associated constitutive equations involve several material parameters, which are to be evaluated from experimental strain-stress data available from the literature and the tooth movement experiments conducted by our team using novel optical motion analysis techniques.

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Further to our previous work on the development of a general constitutive framework for transversely isotropic viscohyperelasticity (Limbert, G, Middleton, J. A transversely isotropic viscohyperelastic material. Application to the modelling of biological soft connective tissues. Int J Solids Struct 2004;41(15):4237-60.), we propose a phenomenological constitutive law to describe the anisotropic viscohyperelastic behaviour of the human posterior cruciate ligament (PCL) at high strain rates. The mechanical formulation is based on the definition of a Helmholtz free energy function containing a hyperelastic and a viscous potential. The equations are valid for arbitrary kinematics and satisfy elemental thermodynamic principles. Identification of the constitutive model with experimental data obtained from human PCL specimens was performed and showed the ability of the model to capture accurately the mechanical characteristics of the PCL at various strain rates. Influence of the isotropic and directional viscous stress responses on the global mechanical response are discussed in connection with the modelling hypotheses. This work was motivated by the need to provide an accurate constitutive model of the PCL to be used in finite element analyses of human occupants in car crash simulations. Besides uniaxial tests along the natural fibre orientation of the PCL, additional tests such as equibiaxial, strip biaxial compression-tension and shear tests were also performed in order to assess the physical response of the model in different loading situations. It was found that the model performed as well in these conditions.

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In this study, a constitutive law based on a nearly incompressible transversely isotropic hyperelastic potential is proposed to describe the mechanical behaviour of the anterior cruciate ligament (ACL). The constitutive formulation is valid for arbitrary kinematics (finite elasticity) and is thermodynamically admissible. Based on anatomic measurements performed on a human cadaveric knee specimen, a three-dimensional continuum finite element model of the ACL was developed. The numerical model was used to simulate clinical procedures such as the Lachman and drawer tests, which are performed to assess the existence and severity of an ACL injury. Finite element analyses showed that the two procedures have distinct effects on the behaviour of the ACL and provided new insights into the stress distributions. Moreover, good qualitative and quantitative agreement was found between the present study and results obtained experimentally in comparable conditions.

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This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.

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