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The connection between macroscopic deformation and microscopic chain stretch is a key element in constitutive models for rubber-like materials that are based on the statistical mechanics of polymer chains. A new micro-macro chain stretch relation is proposed, using the Irving–Kirkwood–Noll procedure to construct a Cauchy stress tensor from forces along polymer chains. This construction assumes that the deformed polymer network remains approximately isotropic for low to moderate macroscopic stretches, a starting point recently adopted in the literature to propose a non-affine micro-macro chain stretch relation (Amores et al., 2021). Requiring the constructed Cauchy stress to be consistent with the stress tensor derived from the strain energy density results in a new chain stretch relation involving the exponential function. A hybrid chain stretch relation combining the new chain stretch with the well-known affine relation is then proposed to account for the whole range of stretches in experimental datasets. Comparison of the model predictions to experimental data in the literature shows that the two new micro-macro chain stretch relations in this work result in two-parameter constitutive models that outperform those based on existing chain stretches with no increase in the number of fitting parameters used. • New chain stretch relations are proposed for modelling rubber-like materials. • The Irving–Kirkwood–Noll procedure gives these relations a physical basis. • Full-network models using these relations capture experimental data accurately. • Improvements in accuracy are achieved when using only two model parameters. • The model parameters have molecular interpretations. [ABSTRACT FROM AUTHOR]

Ultrasound elastography performed under small strain conditions has been intensively studied. However, small deformations may be not sufficiently large to differentiate some abnormal tissues. By combining quasi-static and shear wave elastography, we developed a non-invasive method to estimate the localized stress- strain curve of materials. This method exerts progressive multistep uniaxial compression on the materials, and shear wave measurements were performed at every compression step. This method estimates the 2-D displacements between steps via a 2-D region growing motion tracking method and accumulates these displacements to obtain the large material displacements with respect to the initial configuration. At each step, the shear modulus and stress were calculated according to linear elastic theory. The proposed method was tested on custom-made tissue-mimicking phantoms. Mechanical compression testing was conducted on the samples made of the same material as the phantoms and taken as the reference. The stress-strain curves for the same material from the proposed method and from mechanical testing are in good agreement. The root mean square error (RMSE) and area percentage error (APE) of the stress-strain curve between ultrasound measurement and mechanical testing for soft materials ranged from 0.18 to 0.26 kPa and from 5.6% to 7.8%, respectively. The RMSE and APE for stiff materials ranged from 0.56 to 1.17 kPa and 8.0% to 17.9%. Therefore, our method was able to provide good estimates of the stress-strain curve for tissue-mimicking materials. [ABSTRACT FROM AUTHOR]

• Beading of soft tubes under surface tension is studied as a localization problem. • Bifurcation conditions are presented for a general strain-energy function. • Further insight into the post-bifurcation behaviour is provided. • Theoretical predictions are verified by numerical simulations. We investigate localised bulging or necking in an incompressible, hyperelastic cylindrical tube under axial stretching and surface tension. Three cases are considered in which the tube is subjected to different constraints. In case 1 the inner and outer surfaces are traction-free and under surface tension, whilst in cases 2 and 3 the inner and outer surfaces (respectively) are fixed to prevent radial displacement and surface tension. However, each free surface in these latter two cases is still under surface tension. We first state the analytical bifurcation conditions for localisation and then validate them numerically whilst determining whether localisation is preferred over bifurcation into periodic modes. It is shown that bifurcation into a localised solution is unattainable in case 1 but possible and favourable in cases 2 and 3. In contrast, in case 1 any bifurcation must necessarily take the form of a periodic mode with a non-zero wave number. Our results are validated using Finite Element Method (FEM) simulations. [ABSTRACT FROM AUTHOR]

Materials with a stochastic fiber network as the main structural constituent are broadly encountered in engineering and in biology. These materials are characterized by multiscale heterogeneity and hence their properties evaluated numerically or experimentally are generally dependent on the size of the sample considered. In this work we evaluate the size effect on the linear and non-linear mechanical response of three-dimensional stochastic fiber networks and determine its dependence on material parameters and on the degree of affinity of network deformation. The size effect is more pronounced in non-affine networks than in affine networks and decreases slowly when the model size increases. In order to eliminate this effect, models lager than can be effectively solved with current computers have to be considered. To address this issue, we propose a method that allows using relatively small models, while accurately predicting the small and large strain behaviors of the network. The method is based on the generalized boundary conditions introduced in (Glüge 2013, Computational Materials Science 79, 408–416), which are being adapted here to the requirements imposed by fibrous materials. [ABSTRACT FROM AUTHOR]

• We propose two particular fiber arrangements with the intention of realizing ultra-stiff and ultra-soft composites. • In particular, we also consider the case of curved fibers. • We introduce a model for these lattice plates that is able to describe the behavior of an in-plane orthotropic lamina within the framework of the strain-gradient elasticity theory. • Numerical simulations have been performed using the finite element method to investigate the mechanical behavior of the considered textures. Fiber-reinforced composites are materials that display a great potentiality in many applications for their multifaceted features. The employment of curved fibers contributes to enhancing this already wide range of possibilities further. In this paper, we propose two particular fiber arrangements with the intention of realizing ultra-stiff and ultra-soft composites. For this purpose, we explore the texture based on isostatic lines, which is bio-inspired to obtain a high strength composite and the pantographic pattern, which enables considerable displacement with very low resistance. To model these particular lattice plates, we consider an in-plane orthotropic lamina within the framework of the strain-gradient elasticity theory. Numerical simulations have been performed using the finite element method to investigate the mechanical behavior of the considered textures. A regular straight texture with fibers parallel to the edges of a rectangular sample has been taken into account as reference texture for comparison. Through some tests, we show that for the considered textures, namely the isostatic and the pantographic ones, the displacement force graphs differ by several orders of magnitude, justifying the characterization of their behavior. [ABSTRACT FROM AUTHOR]

Abstract Deriving the amplitude equation for a buckling mode is an important issue in non-linear elasticity. Focusing on pattern formations in growing tubular tissues, many existing literature often adopted the numerical methods. In this paper, we propose a semi-analytical approach for the bilayer tubular structures under growth to derive the amplitude equation of a single wrinkling mode, from which a transition between supercritical and subcritical bifurcations can be determined. In the framework of finite elasticity, a weakly non-linear analysis is carried out and the amplitude equation is deduced by the virtual work method. A semi-analytical solution is obtained, with an analytical expression whose exact coefficients are determined numerically. Then a parametric study is carried out by use of the semi-analytical solution. When the total growth factor is prescribed, the critical mode number governs the amplitude, and a lower mode corresponds to a higher amplitude. When the incremental growth factor after bifurcation is fixed, it turns out that the dependence of the amplitude on the modulus ratio is non-monotonic if the thicknesses of the two layers are specified. For a given geometry, when the modulus ratio ξ>5, the wrinkled amplitude is mainly dominated by the critical mode number, and a smaller critical mode number deepens the wrinkle. However, the amplitude is a decreasing function of ξ when ξ < 5. The obtained analytical solutions are also validated by the corresponding numerical solutions based on the finite element method. The proposed semi-analytical approach is applicable for most variable coefficient problems arising from cylindrical and spherical structures. [ABSTRACT FROM AUTHOR]

In nature, limbless locomotion is adopted by a wide range of organisms at various length scales. Interestingly, undulatory, crawling and inching/looping gait constitutes a fundamental class of limbless locomotion and is often observed in many species such as caterpillars, earthworms, leeches, larvae, and C. elegans. In this work, we developed a computationally efficient 3D Finite Element (FE) unified framework for the locomotion of limbless organisms on frictional substrates. Muscle activity is simulated with a multiplicative decomposition of the deformation gradient, which allows mimicking a broad range of locomotion patterns in 3D contractile elastic solids. We propose a two-field FE formulation based on positions and velocities. Governing partial differential equations are transformed into equivalent time-continuous differential–algebraic equations (DAEs). Next, the optimal locomotion strategies are studied in the framework of optimal control theory. We resort to adjoint-based methods and deduce the first-order optimality conditions, that yield a system of DAEs with two-point end conditions. Hidden symplectic structure is discussed, and Symplectic Euler time integration of optimality conditions are employed. The resulting discrete first-order optimality conditions form a non-linear programming problem that is solved efficiently with the Forward Backward Sweep Method. Finally, some numerical examples are provided to demonstrate the comprehensiveness of the proposed computational framework and investigate the energy-efficient optimal strategy out of distinct locomotion patterns adopted by limbless organisms. [Display omitted] • A deformation gradient decomposition is proposed for modelling 3D contractile bodies. • Optimal body locomotion on frictional substrates are deduced and discretised with FE. • Optimality equations are numerically solved with Forward Backward Sweep Method. • Different gait strategies found in Nature are compared and its efficiency discussed. [ABSTRACT FROM AUTHOR]

Highlights • Local stability analysis for incompressible elastic solids under dead-load surface tractions. • Explicit necessary and sufficient algebraic local stability conditions are found. • The special case of a Gent-Thomas incompressible, isotropic elastic parallelepiped is considered. • The response of the parallelepiped as the material parameters vary is described. Abstract We study the equilibrium and the local stability for an incompressible elastic solid under arbitrary dead-load surface tractions on the boundary. We particularize the analysis to homogeneous deformations of a homogeneous, isotropic parallelepiped, finding necessary and sufficient algebraic local stability conditions consistent with the further requirement of the zero moment condition. We specialize the study for a uniform distribution of dead-load surface tractions s > 0 on two pairs of faces and –s on the remaining two faces, and we find that two classes of equilibrium solutions may occur: symmetric and asymmetric solutions, respectively. For the symmetric solutions we also determine local stability inequalities. For the special case of a Gent-Thomas material we show that both equilibrium symmetric and asymmetric solutions may occur if the material parameters satisfy certain inequalities. Then, we completely describe the response of the parallelepiped in a loading process starting from the unloaded state for five ranges of the values of the material parameters. In particular, for one of these ranges we show that symmetric solutions are the unique locally stable homogenous equilibrium deformations until a critical value s cr of the load; at s cr , symmetric solutions lose their uniqueness, and a bifurcation into asymmetric solutions may occur. [ABSTRACT FROM AUTHOR]

• Fracture toughness and non-linear elastic properties evaluated in a single test. • Three-dimensional displacement fields obtained by digital volume correlation of in situ laboratory X-ray computed tomographs. • Tensile softening obtained by inverse finite element analysis of the strain field. • Strain energy release rate obtained by contour integral method in a finite element model. • The model uses the derived non-linear properties and measured displacement field. • The fracture toughness was constant with crack length. Effective small specimen tests are needed to obtain fracture toughness and elastic properties as the limited availability of irradiated graphite restricts the quantity and dimensions of test specimens. Both properties have evaluated simultaneously in a crack propagation test with the double cleavage drilled compression (DCDC) specimen geometry of a fine-grained graphite (SNG742) that has non-linear elastic properties in the unirradiated condition. Three-dimensional displacement fields were obtained by digital volume correlation of in situ laboratory X-ray computed tomographs, and the 3D crack geometry (crack tip position, crack opening displacements and angle) was determined objectively by a wavelet variance method. The tensile softening of the Young modulus was determined by inverse analysis of the strain field using the finite element model updating (FEMU) method. The strain energy release rate of the quasi-static propagating crack was calculated using the contour integral method in a finite element model with the derived non-linear elastic properties and the measured displacements as boundary conditions. The critical strain energy release rate was constant with crack length (118 ± 14 J m−2) and equivalent to a fracture toughness of 1.13 ± 0.07 MPa m1/2. [ABSTRACT FROM AUTHOR]

This paper revisits stiffness optimization of non-linear elastic structures. Due to the non-linearity, several possible stiffness measures can be identified and in this work conventional compliance, i.e. secant stiffness designs are compared to tangent stiffness designs. The optimization problem is solved by the method of moving asymptotes and the sensitivities are calculated using the adjoint method. For the tangent cost function it is shown that although the objective involves the third derivative of the strain energy an efficient formulation for calculating the sensitivity can be obtained. Loss of convergence due to large deformations in void regions is addressed by using a fictitious strain energy such that small strain linear elasticity is approached in the void regions. A well posed topology optimization problem is formulated by using restriction which is achieved via a Helmholtz type filter. The numerical examples provided show that for low load levels, the designs obtained from the different stiffness measures coincide whereas for large deformations significant differences are observed. [ABSTRACT FROM AUTHOR]

Application of seismic interferometry in buildings gained interest in the recent years for structural health monitoring. It allows us to derive the shear wave velocity for an equivalent homogeneous medium representing the structure. Previous authors suggested using a shear beam model to compute the fundamental frequency of the structure out of this velocity. This model is however not adapted to a large part of existing buildings having different behaviors. In this paper, we propose a correction factor from shear beam and based on the Timoshenko beam to link the fundamental frequencies with the observed pulse velocity and the relative effects of shear and bending. This factor provides corrections up to 60% in frequency with respect to the shear beam model. The proposed correction factor shows that the higher velocities observed in the literature for shear wall buildings compared to frame buildings is compensated by their bending flexibility, resulting in resonance frequencies scaling similarly with building height. This model is applied to an 18-story reinforced concrete shear wall building (Ophite tower). The observed pulse velocity obtained by seismic interferometry in this building was about 500 m/s correlated to the resonance frequency by the correction factor. We show that variations of velocity in this structure and the well-studied Factor building (California), related to non-linear behavior at low-strains (down to 10 −5 ), can be retrieved with seismic interferometry, demonstrating that this method is sensitive enough for structural health monitoring. [ABSTRACT FROM AUTHOR]

A sequence of polynomial expressions have been shown to describe the strained surface energy of low-index hexagonal and square transition metal surfaces. Distinguishable functions describe the hexagonal FCC (1 1 1) and HCP (0 0 0 1) surfaces, but a single function describes the FCC (1 0 0) and BCC (1 0 0) surfaces. A far weaker dependence exists between the strained surface energy and the electronic state of the surface, and the competition between geometric and electronic states across is discussed. [ABSTRACT FROM AUTHOR]

We study the equilibrium homogeneous deformations of a homogeneous parallelepiped made of an arbitrary incompressible, isotropic elastic material and subject to a distribution of dead-load surface tractions corresponding to an equibiaxial tensile stress state accompanied by an orthogonal uniaxial compression of the same amount. We show that only two classes of homogeneous equilibrium solutions are possible, namely symmetric deformations, characterized by two equal principal stretches, and asymmetric deformations, with all different principal stretches. Following the classical energy-stability criterion, we then find necessary and sufficient conditions for both symmetric and asymmetric equilibrium deformations to be weak relative minimizers of the total potential energy. Finally, we analyze the mechanical response of a parallelepiped made of an incompressible Mooney–Rivlin material in a monotonic dead loading process starting from the unloaded state. As a major result, we model the actual occurrence of a bifurcation from a primary branch of locally stable symmetric deformations to a secondary, post-critical branch of locally stable asymmetric solutions. [ABSTRACT FROM AUTHOR]

Shales serve essential functions as seals for conventional hydrocarbon reservoirs, waste storage repositories and unconventional plays. Exploration and evaluation of shales rely on the characterization of their multiscale elastic properties. However, there are limited studies on the geologic controls of elastic stress sensitivity in organic shales. This study investigated dry Lokpanta, Bakken, and Niobrara shales of different thermal maturity stages. First, we used a pulse-transmission ultrasonic setup to conduct multidirectional (0o, 45o and 90o) hydrostatic-stress sensitivity (0–27.58 MPa) tests to measure elastic waves and derive other elastic properties. Furthermore, we inverted microscopic parameters from the elastic measurements based on effective medium theories such as damage parameters (α ij and β ijkl), crack density (ρ), normal to tangential microcrack compliance (B), specific tangential crack compliance (s n B T) and microcracks orientation anisotropy (η). Finally, we studied the effects of organic matter maturity and clay diagenesis on the multi-scaled elastic stress sensitivity. Under applied hydrostatic stress, the non-linear elastic response in the shales is consistent with the closure of aligned microcracks. Microcrack closure decreased the elastic anisotropy at lower stress (<10 MPa), while aligned clays and organic matter control the stress-insensitive anisotropy at higher stress. As organic matter matures and smectite transforms to illite, orthogonal velocity stress sensitivity increases and extrinsic contribution to elastic anisotropy increases. There is also a corresponding increase in orthogonal damage second-rank tensor (α 33) and a decrease in orthogonal damage fourth-rank tensor (β 3333) with organic matter-clay thermal maturity. These relationships are due to the thermally-induced development of new and/or extensional roughly-textured microcracks parallel to the laminated microfabric, which is of higher compliance in the shear than compression. Microcrack closure also decreases ρ under applied pressure. Most of the samples had B < 0.5, suggesting the presence of non-idealized roughly textured microcracks. Generally, s n B T broadly decreases and η increases with clay diagenesis and organic matter maturity also because of the generation and/or enhancement of aligned microcracks resulting in higher tangential stiffness. • Aligned microcracks closure govern the non-linear elasticity in the organic shales. • Microcracks formation is due to organic matter and clay diagenesis. • Roughly-textured microcracks caused a non-linear decrease in crack density. • Organic matter and clay diagenesis affect microscopic-micromechanical properties. [ABSTRACT FROM AUTHOR]

Inverse non-linear stress–strain relations between the logarithmic Hencky strain tensor and the Cauchy stress tensor are derived for incompressible isotropic materials. The constitutive model is based on a new type of complementary energy potential that considered to be the function of the second and third invariants of the deviatoric Cauchy stress tensor in a power-law form. A proper scaling of the third deviatoric stress invariant allows separate fitting to uniaxial tension and equibiaxial tension (or uniaxial compression) data. The predictive capabilities of a four- and a six-parameter model are presented through a parameter fitting procedure using Treloar's experimental data for rubber. The performance of the new constitutive model is also demonstrated by solving the inflation problem of a spherical shell. • Complementary energy-based constitutive model in power-law form • The formulation applies the invariants of the deviatoric Cauchy stress tensor • Explicit strain–stress relations between the Hencky strain and the Cauchy stress • The model allows separate fits to uniaxial and equibiaxial measurement data • Numerical solutions and comparisons for the inflation of a spherical rubber shell. [ABSTRACT FROM AUTHOR]

In the powder metallurgy (PM) pressing process the mechanical properties of the green body are highly dependent on the material density. During the ejection stage of the pressing process the elastic behaviour is important especially for the crack formation in the powder compact. Experiments show a non-linear and also stress dependent elastic behaviour of green bodies. In this study diametral compression tests have been used to study elastic deformation during crack formation in a tensile fracture process of metal powder disc compacts. The powder material used for the experiments was a press-ready premix containing Distaloy AE, 0.5% graphite (uf-4) and 0.6% Kenolube. Tensile strength is used as a failure condition and limits the stress in the fracture interface. To control the tensile fracture, a cohesive zone model is used. The softening rate of the fracture model is obtained from the corresponding rate of the dissipated energy. The deformation of the powder material is modelled with an elastic–plastic cap model where an easy-to-use model for non-linearity in the elastic state due to stress is presented. The model is implemented in a finite element code and tested in simulation of a diametral compression testing. Results from simulations correlate well with experimental results and demonstrate the importance of including the non-linear elastic effect of the powder compacts. Results also show the necessity to accurate model the elasticity in the tooling to correct capture force–displacement response and fracturing processes. [ABSTRACT FROM AUTHOR]

We present a coupled Eulerian–Lagrangian (CEL) formulation aimed at modeling the moving interface of hyperelastic materials undergoing large to extreme deformations. This formulation is based on an Eulerian description of kinematics of deformable bodies together with an updated Lagrangian formulation for the transport of the deformation gradient tensor. The extended finite element method (XFEM) is used to discretize the mechanical equilibrium and deformation gradient transport equations in a two-phase domain. A mixed interpolation scheme (biquadratic for the velocity and bilinear for the deformation gradient) is adopted to improve the accuracy of the numerical formulation. The interface describing the deformed shape of the body is represented by the level set function and is evolved using the grid based particle method. The performance of the scheme is explored in two-dimensions in the compressible regime. For an adequate spatial and temporal discretization, our numerical results are in good agreement with theory and with numerical results from the traditional Lagrangian formulation (in Abaqus). The advantage of the proposed formulation is that material motion is not coupled with that of the mesh; this eliminates the issues of mesh distortion and the need for remeshing associated with Lagrangian formulations when bodies undergo very large distortions. It is therefore well adapted to describe the motion of complex fluids and soft matter whose physical properties are intermediate between conventional liquids and solids. [ABSTRACT FROM AUTHOR]

We revisit the hybridizable discontinuous Galerkin method for non-linear elasticity introduced by S.-C. Soon (2008). We show that it can be recast as a minimization problem of a non-linear functional over a space of discontinuous approximations to the displacement. The functional can be written as the sum over the elements of the classic potential energy plus a new energy associated to the inter-element jumps of the displacement. We then show that if this new energy is not properly weighted, the minimizers might not converge to the exact solution. We construct an example illustrating this phenomenon and show how to overcome it by suitably increasing the weight of the energy of the inter-element jumps. Finally, we explore the performance of the method for the case of piecewise-linear approximations in rather demanding situations in both two-dimensional and, for the first time, three-dimensional situations. They include almost incompressible materials, large deformations with large-shear layers, and cavitation. We also compare the method with the continuous Galerkin method and a previously explored discontinuous Galerkin method, and show that, when using piecewise-linear approximations and a moderate number of degrees of freedom, the current method turns out to be more efficient for the computation of the gradient. [ABSTRACT FROM AUTHOR]

Continuum modeling of a free-standing graphene monolayer, viewed as a two dimensional 2-lattice, requires specifications of the components of the shift vector that act as an auxiliary variable. The field equations are then the equations ruling the shift vector, together with momentum and moment of momentum equations. We present an analysis of simple loading histories such as axial, biaxial tension/compression and simple shear for a range of problems of increasing difficulty. We start by laying down the equations of a simplified model which can still capture bending effects. Initially, we ignore out of plane deformations. For this case, we solve analytically the equations ruling the auxiliary variables in terms of the shift vector; these equations are algebraic when the loading is specified. As a next step, still working on the simplified model, out-of-plane deformations are taken into account and the equations complicate dramatically. We describe how wrinkling/buckling can be introduced into the model and apply the Cauchy–Kowalevski theorem to get existence and uniqueness in terms of the shift vector for some characteristic cases. Finally, for the treatment of the most general problem, we classify the equations of momentum and give conditions for the Cauchy–Kowalevski theorem to apply. [ABSTRACT FROM AUTHOR]

When estimating climate change impact on crop yield, a typical assumption is constant elasticity of yield with respect to a climate variable even though the elasticity may be inconstant. After estimating both constant and inconstant elasticities with respect to temperature and precipitation based on provincial panel data in China 1980–2008, our results show that during that period, the temperature change contributes positively to total yield growth by 1.3% and 0.4% for wheat and rice, respectively, but negatively by 12% for maize. The impacts of precipitation change are marginal. We also compare our estimates with other studies and highlight the implications of the inconstant elasticities for crop yield, harvest and food security. We conclude that climate change impact on crop yield would not be an issue in China if positive impacts of other socio-economic factors continue in the future. [ABSTRACT FROM AUTHOR]

This study concerns the local behavior of the solutions of the governing equations of non-linear elastostatics in the vicinity of the corner of a wedge-shaped region of angle α ∈ ( 0 , 2 π ] . It contains an asymptotic investigation in the plane strain regime – using a subclass of non-linear harmonic elastic solids that have a proper asymptotic constitutive structure in the vicinity of the corner – of the deformation field near the corner point that separates a free from an adjoining fixed segment of the boundary. It is well-known that, as the corner point is approached, the singular field behavior predicted by the classical linear theory of elasticity yields oscillatory deformations that are not injective. This anomalous behavior is related to a spurious self-intersection anomaly. Using a proper constitutive structure within a subclass of non-linear harmonic materials in the vicinity of the corner, we obtain an asymptotic expansion of the deformation field for which the foregoing anomalous behavior is avoided. The conditions on this field which guarantee injectivity lead to an unexpected behavior of the deformed free surface that is physically possible. In the particular case of α = π , the second-order expansion of the deformation field agrees with its counterpart found elsewhere in the literature. For another particular case, concerning α = π / 2 , the second-order expansion is an improvement over its counterpart found in the literature. [ABSTRACT FROM AUTHOR]

Abstract: Starting from a constrained three-dimensional analysis, we propose a new model for an initially curved Cosserat rod with two independent directors, in which the cross-section can inflate and ovalize. The usage of unconstrained quaternions gets rid of the non-linear constraints and singularities in the parametrization of objective strains, related to the orthonormality of rotation matrices. Furthermore, the classic definitions of curvatures and twist have been modified and tuned for a linear constitutive law (i.e. a linear relationship between one-dimensional stresses and strains). The capabilities and the limitations of the model in describing the non-linear effects of cross-section deformation on bending and torsion, and usually handled with bi- and three-dimensional theories, are then discussed. [Copyright &y& Elsevier]

Abstract: Depending on material and details of cyclic loading conditions, fatigue damage develops and spreads by various modes of crack initiation and subsequent crack growth. In order to gain a deeper understanding of the underlying microstructural mechanisms, many details must be considered. The present study focuses on just a few specific aspects, namely the role of cyclic slip irreversibilities, the significance of fatigue crack initiation in high and ultrahigh cycle fatigue (HCF, UHCF) and, finally the analysis of non-linear effects of elasticity and compliance of fatigued specimens, as a promising non-destructive in situ fatigue damage diagnosis tool. Selected experimental data obtained on different materials will be considered to illustrate the issues named above. [Copyright &y& Elsevier]

Abstract: The mechanical behavior of soft matter is characterized by large shape changes, often accompanied by small changes of elastic energy; non-linear elasticity, with large, inhomogeneous, and anisotropic distortions, that may evolve in time, proved to be an effective modeling tool for many of such soft materials. Here, we deal with the definition of an appropriate strain measure, called the elastic metric, upon which the elastic energy density can be defined. Moreover, we discuss two key issues about distortions: one deals with the notion of compatible distortions, that is, distortion fields yielding a global configuration without any change of the elastic energy; the other concerns the symmetries of the material responses. We also present few selected examples of non-linear, anisotropic, elastic response where distortions play a key role. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the effect of residual stress on the elastic behaviour of materials undergoing finite elastic deformations. The theory is based on a general constitutive framework for hyperelastic materials with residual stress. Several simple problems, whose solutions are known in the situation where there is no residual stress, are analyzed in order to elucidate the influence of residual stress, and the results are illustrated for two prototype constitutive laws. [Copyright &y& Elsevier]

Abstract: The propagation of finite-amplitude time-harmonic shear horizontal waves, in a pre-stressed compressible elastic layer of finite thickness embedded between two identical compressible elastic half-spaces, is investigated. This is accomplished by combining finite-amplitude linearly polarized inhomogeneous transverse plane wave solutions in the half-spaces and finite-amplitude linearly polarized unattenuated transverse plane wave solutions in the layer. The layer and half-spaces are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. The special case where the interfaces between the layer and the half-spaces are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the shear horizontal wave speed with the pre-stress and the propagation angle. [Copyright &y& Elsevier]

Abstract: A small but not negligible non-linear elastic behaviour was detected when investigating cyclic uniaxial push–pull experiments on a high strength bainitic steel. Cyclic torsion experiments led to the conclusion that the shear modulus was relatively constant. A non-linear elastic model was implemented where the bulk modulus was extended with a second order term related to the elastic dilatation and where the shear modulus was constant. The material presented a strength differential effect (SDE), with larger yield stress in compression than in tension. Consequently, the non-linear elastic model was combined with a plasticity model that incorporated a Drucker–Prager yield surface, non-associated flow rule and combined non-linear hardening. Expressions that include non-linear elasticity were derived for the elastic–plastic hardening and the compliance tensors. The extended material model predicted the elastic–plastic results from cyclic push–pull experiments. Also, a phenomenological analysis of the cyclic elastic response showed isotropic damage in the elastic moduli. The steady-state damage increased linearly with the cyclic plastic strain range. [Copyright &y& Elsevier]

Abstract: Finite element simulations of different shear deformations in non-linear elasticity are presented. We pay particular attention to the Poynting effects in hyperelastic materials, complementing recent theoretical findings by showing these effects manifested by specific models. As the finite element method computes uniform deformations exactly, for simple shear deformation and pure shear stress, the Poynting effect is represented exactly, while for the generalised shear and simple torsion, where the deformation is non-uniform, the solution is approximated efficiently and guaranteed computational bounds on the magnitude of the Poynting effect are obtained. The numerical results further indicate that, for a given elastic material, the same sign effect occurs under different shearing mechanisms, showing the genericity of the Poynting effect under a variety of shearing loads. In order to derive numerical models that exhibit either the positive or the negative Poynting effect, the so-called generalised empirical inequalities, which are less restrictive than the usual empirical inequalities involving material parameters, are assumed. [Copyright &y& Elsevier]

Theoretically the auscultatory method using Korotkoff sounds is more related to the maximum artery closure status, while the oscillometric method is more related to the overall artery closure status under the cuff. Therefore, the latter is less accurate than the former. This work introduces a new method, which is more accurate than the oscillometric method and suitable for automatic devices. To monitor the maximum artery closure status, a piezoelectric film sensor is attached to the skin just above the brachial artery and under the central section of the cuff where maximum cuff pressure is transferred to the arm. Using the waveform features obtained by this sensor, measurement errors of 0.7 ± 2.5 and 1.27 ± 4.53 mmHg were obtained for the systolic and diastolic pressure, respectively. These reflect small deviations from auscultatory clinical data. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with numerical simulations of three-dimensional finite deformation of a thick-walled circular elastic tube subject to internal or external pressure and zero displacement on its ends. We formulate the system of equations that can accommodate large strain and displacement for the incompressible isotropic neo-Hookean material. The fully non-linear governing equations are solved using the C++ based object-oriented finite element library libMesh. A Lagrangian mesh is used to discretize the governing equations, and a weighted residual Galerkin method and Newton iteration solver are used in the numerical scheme. To overcome the sensitivity of the fully non-linear system to small changes in the iterations, the analytical form of the Jacobian matrix is derived, which ensures a fast and better numerical convergence than using a numerically approximated Jacobian matrix. Results are presented for different parameters in terms of wall thickness/radius ratio, and length/radius ratio, as well as internal/external pressure. Validation of the model is achieved by the excellent agreement with the results obtained using the commercial package Abaqus. Comparison is also made with the previous work on axisymmetric version of the same system (Zhu et al., 2008 [34]; Zhu et al. 2010 [43]), and interesting fully three-dimensional post-buckling deformations are highlighted. The success of the current approach paves the way for fluid–structure interaction studies with potential application to collapsible tube flows and modeling of complex physiological systems. [Copyright &y& Elsevier]

Abstract: Strain measures consistent with the linear, infinitesimal form of the strain-energy function are obtained within the context of isotropic, homogeneous, compressible, and non-linear elasticity. It will be shown that there are two distinct families of such measures. One family has already been much studied in the literature, the most important member being the strains whose principal values are a function only of the corresponding principal stretches. The second family of strains appears new. The motivation for studying such strains is the intuitive expectation that, for at least moderate deformations, a good fit with experimental data from material characterisation tests will be obtained with the corresponding strain-energy functions. In particular, there is the expectation that such models could prove useful for the modelling of biological soft tissue, whose physiological response is characterised by moderate strains. It will be shown that this is indeed the case for simple tension tests on porcine brain tissue. [Copyright &y& Elsevier]

Abstract: Morphoelastic theories have demonstrated that elastic instabilities can occur during the growth of soft materials, initiating the transition toward complex patterns. Within the framework of non-linear elasticity, the theory of incremental elastic deformations is classically employed for solving stability problems with finite strains. In this work, we define a variational method to study the bifurcation of growing cylinders with circular section. Accounting for a constant axial pre-stretch, we define a set of canonical transformations in mixed polar coordinates, providing a locally isochoric mapping. Introducing a generating function to derive an implicit gradient form of the mixed variables, the incompressibility constraint for the elastic deformation is solved exactly. The canonical representation allows to transform a generic boundary value problem, characterized by conservative body forces and surface traction loads, into a completely variational formulation. The proposed variational method gives a straightforward derivation of the linear stability analysis, which would otherwise require lengthy manipulations on the governing incremental equations. The definition of a generating function can also account for the presence of local singularities in the elastic solution. Bifurcation analysis is performed for few constrained growth problems of biomechanical interests, such as the mucosal folding of tubular tissues and surface instabilities in tumor growth. In a concluding section, the theoretical results are discussed for clarifying how anisotropy, residual strains and external constraints can affect the stability properties of soft tissues in growth and remodeling processes. [Copyright &y& Elsevier]

Abstract: Here we consider finite-amplitude wave motions in Mooney–Rivlin elastic materials which are first subjected to a static homogeneous deformation (prestrain). We assume that the time-dependent displacement superimposed on the prestrain is along a principal axis of the prestrain and depends on two spatial variables in the principal plane orthogonal to this axis. Thus all waves considered here are linearly polarized along this axis. After retrieving known results for a single homogeneous plane wave propagating in a principal plane, a superposition of an arbitrary number of sinusoidal homogeneous plane waves is shown to be a solution of the equations of motion. Also, inhomogeneous plane wave solutions with complex wave vector in a principal plane and complex frequency are obtained. Moreover, appropriate superpositions of such inhomogeneous waves are also shown to be solutions. In each case, expressions are obtained for the energy density and energy flux associated with the wave motion. [Copyright &y& Elsevier]

Abstract: The failure behavior of isotropic non-linear elastic materials is macroscopically studied in terms of elastic strain energy density generalizing the Coulomb criterion. This generalization is based on a rigorous mathematical substrate developed on the principle of conservation of the total elastic strain energy. In the general case of loading the behavior of a material is described with regard to the secant elastic moduli depending on both first strain and second deviatoric strain invariants. This dependence enlightens, in physical terms, the different reaction of materials in normal and shear stresses. Besides, these two moduli establish two constitutive equations for the complete description of any material, instead of the usual one. A theoretical application is given and the failure surfaces which are obtained in stress space are being commented. Predictions obtained in tension of steel under pressure from Bridgman''s experiments and some of his observations for the failure behavior of steels are explained on the existence of a universal criterion with the present approach. [Copyright &y& Elsevier]

Abstract: Soft biological tissues with collagen reinforcement can be represented by a porous matrix saturated by a fluid and reinforced by a network of statistically oriented, impermeable fibres. This paper aims at determining the effect of the fibres on both the elastic properties and the permeability of the system, under large deformations, and represents the unification and generalisation of previous works in which the elasticity was studied for a pure solid, in the absence of the pore fluid, and the permeability was studied in the neighbourhood of the undeformed configuration. Throughout this work, the saturation constraint is assumed to hold, and the solid and fluid phases are assumed to be intrinsically incompressible. These hypotheses imply that the volumetric fraction of one of the two phases (e.g., the solid) is sufficient to determine the distribution of solid and fluid mass at every point of the homogenised porous medium. Overall incompressibility is achieved at compaction, i.e., when pores are closed and all the fluid has escaped. A new form of the elastic strain energy potential is proposed, based on the sum of a given “base” potential and a “correction” potential, function of the volumetric deformation, which serves solely to impose the incompressibility constraint at compaction. The large-strain overall permeability is obtained by employing a pull-back of the structure tensor to the reference configuration. This gives rise to an integral form of the permeability that needs to be calculated at each increment of deformation. The presentation is entirely covariant, so that general curvilinear coordinates can immediately be employed, if needed. [Copyright &y& Elsevier]

Abstract: This paper presents a 3D finite element upper arm model, validated by experiments as well as clinical data, used to study the error introduced in blood pressure measurements due to variability of arm tissue mechanical properties. The model consists of three separate cylindrical parts: soft tissue, bone and brachial artery. The artery volume changes under the cuff are used to represent the cuff pressure oscillations for analyzing blood pressure measurements. These oscillation trends are identical to observed clinical data. Also an upper arm simulator is designed and built for model validation. The model shows that the variation of soft tissue compressibility introduces an error up to 5% in blood pressure measurements. It is also revealed that the variation of the brachial artery and arm tissue stiffness has an insignificant effect on oscillometric blood pressure measurement method. [Copyright &y& Elsevier]

Abstract: A general implicit solution for determining volume-preserving transformations in the n-dimensional Euclidean space is obtained in terms of a set of 2n generating functions in mixed coordinates. For n=2, the proposed representation corresponds to the classical definition of a potential stream function in a canonical transformation. For n=3, the given solution defines a more general class of isochoric transformations, when compared to existing methods based on multiple potentials. Illustrative examples are discussed both in rectangular and in cylindrical coordinates for applications in mechanical problems of incompressible continua. Solving exactly the incompressibility constraint, the proposed representation method is suitable for determining three-dimensional isochoric perturbations to be used in bifurcation theory. Applications in non-linear elasticity are envisaged for determining the occurrence of complex instability patterns for soft elastic materials. [Copyright &y& Elsevier]

Abstract: We present a simplified heart model with the aim of introducing a mechanical point of view in the interpretation of the pressure–volume loops. In particular, we propose a mechanical description of muscle contraction, and discuss its implications with reference to a specific pressure–volume loop representative of a normal human patient. [Copyright &y& Elsevier]

Abstract: We show how the compound matrix method can be extended to give eigenfunctions as well as generalised eigenvalues to bifurcation problems in non-linear elasticity. When the incremental problem is formulated in terms of displacements only there are significant difficulties that arise from the non-trivial boundary conditions. In order to avoid these problems we adopt a Stroh formulation of the incremental problem. This then produces trivial boundary conditions for the compound matrix eigenvalue problem and more importantly known initial conditions for the compound matrix eigenfunction problem. This results in a straightforward and robust calculation for the eigenfunctions. [Copyright &y& Elsevier]

Abstract: We consider the numerical simulation of contact problems in elasticity with large deformations. The non-penetration condition is described by means of a signed distance function to the obstacle''s boundary. Techniques from level set methods allow for an appropriate numerical approximation of the signed distance function preserving its non-smooth character. The emerging non-convex optimization problem subject to non-smooth inequality constraints is solved by a non-smooth multiscale SQP method in combination with a non-smooth multigrid method as interior solver. Several examples in three space dimensions including applications in biomechanics illustrate the capability of our methods. [Copyright &y& Elsevier]

Abstract: It is proven that magnetizable quasicrystals undergoing large deformations admit elastic ground states characterized by a net of linear topological defects for the magnetic spin field. [ABSTRACT FROM AUTHOR]

Abstract: Packed pebble beds are particular granular systems composed of a large amount of small particles, arranged in irregular lattices and surrounded by a gas filling interstitial spaces. Due to their heterogeneous structure, pebble beds have non-linear and strongly coupled thermal and mechanical behaviours whose constitutive models seem limited, being not suitable for fusion-relevant design-oriented applications. Within the framework of the modelling activities promoted for the lithiated ceramics and beryllium pebble beds foreseen in the Helium-Cooled Pebble Bed breeding blanket concept of DEMO, at the Department of Nuclear Engineering of the University of Palermo (DIN) a thermo-mechanical constitutive model has been set-up assuming that pebble beds can be considered as continuous, homogeneous and isotropic media. The present paper deals with the DIN non-linear elasticity constitutive model, based on the assumption that during the reversible straining of a pebble bed its effective logarithmic bulk modulus depends on the equivalent pressure according to a modified power law and its effective Poisson modulus remains constant. In these hypotheses the functional dependence of the effective tangential and secant bed deformation moduli on either the equivalent pressure or the volumetric strain have been derived in a closed analytical form. A procedure has been, then, defined to assess the model parameters for a given pebble bed from its oedometric test results and it has been applied to both polydisperse lithium orthosilicate and single size beryllium pebble beds. [ABSTRACT FROM AUTHOR]

Abstract: Within the framework of the direct approach to the plate theory we consider the infinitesimal deformations of a plate made of hyperelastic materials taking into account the non-homogeneously distributed initial stresses. Here we consider the plate as a material surface with 5 degrees of freedom (3 translations and 2 rotations). Starting from the equations of the non-linear elastic body and describing the small deformations superposed on the finite deformation we present the two-dimensional constitutive equations for a plate. The influence of initial stresses in the bulk material on the plate behavior is considered. [ABSTRACT FROM AUTHOR]

Abstract: The mechanical response of an inflated spherical membrane–fluid structure in contact with rigid parallel planes is studied. The membrane is assumed to be a two-dimensional non-linear elastic and isotropic structure, while no assumption is imposed on the fluid. A numerical procedure is employed to compute the equilibrium configurations of the membrane–fluid structure. This study provides information regarding the contact force, stress distribution and pressure in the membrane and in the enclosed fluid, respectively. It was observed that a transition between unwrinkled to partially wrinkled configurations of the membrane occurs subjected to the loading conditions. Further investigation of the wrinkled configurations is presented. [Copyright &y& Elsevier]

Abstract: A study of direction-based models for the representation of isotropic and anisotropic hyperelastic behaviour of rubber-like materials is proposed. The interest in such models is sustained by their ability to account for the Mullins effect induced anisotropy. For such a purpose, the directional models should be initially isotropic and representative of the hyperelastic behaviour of rubber-like materials. Various models were defined according to different sets of directions. Models were tested in terms of their initial anisotropy and their ability to reproduce the classic full-network hyperelastic behaviour. Various models were proved to perform very well. [Copyright &y& Elsevier]

Abstract: A discontinuous Galerkin finite element method (DGFEM) for incompressible, non-linear elasticity is derived. One of the limitations of the continuous Galerkin finite element method (CGFEM) when applied to incompressible elasticity is that the accuracy of the numerical solution may be adversely affected by the phenomenon known as locking—this may prevent the use of a low order polynomial approximation to the displacement on each element. We demonstrate using simulations that the DGFEM presented here does not suffer from this drawback. Two further advantages in this setting of this DGFEM over CGFEMs are that: (i) highly anisotropic meshes—meshes containing elements with a very high aspect ratio—may be used without significantly degrading the accuracy of the solution; and (ii) discontinuities in the elasticity parameters are handled more effectively. [Copyright &y& Elsevier]

Abstract: A framework is developed for electromechanical behavior of dielectric crystalline solids subjected to finite deformations. The theory is formulated in the context of electrostatics; however, vacancies in the lattice may carry an electric charge, and their concentrations may be large. The material is treated as a continuous body with a continuous distribution of point vacancies, but volumes and charges of individual defects enter the description. The deformation gradient is decomposed multiplicatively into terms accounting for recoverable thermoelasticity and irreversible volume changes associated with vacancies. Thermodynamic arguments lead to constitutive relations among electromechanical quantities framed in the elastically unloaded intermediate configuration, with the Cauchy stress tensor consistently non-symmetric as a result of electrostatic effects. The requirement of non-negative dissipation imposes constraints on vacancy migration. Following postulation of a quadratic form for the free energy potential, a kinetic equation for vacancy flux is derived in the intermediate configuration, with diffusion driven by gradients of vacancy concentration, electrostatic potential, hydrostatic pressure, and crystal structure. Effects of geometric non-linearity (i.e. finite elastic strains and large vacancy concentrations) are found to affect vacancy diffusion in a body subjected to biaxial lattice strain, for example a film device with lattice mismatch at its interfaces. [Copyright &y& Elsevier]