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A remarkable feature of the elephant trunk is the pronounced wrinkling that enables its great flexibility. Here, we devise a general mathematical model that accounts for characteristic skin wrinkles formed during morphogenesis in elephant trunk. Using physically realistic parameters and operating within the theoretical framework of nonlinear morphoelasticity, we elucidate analytically and numerically the effect of skin thickness, relative stiffness and differential growth on the physiological pattern of transverse wrinkles distributed along the trunk. We conclude that, since the skin and muscle components have similar material properties, geometric parameters, such as curvature, play important roles. In particular, our model predicts that, in the proximal region close to the skull, where curvature is lower, fewer wrinkles form and sooner than in the distal narrower region where more wrinkles develop. Similarly, less wrinkling is found on the ventral side, which is flatter, compared to the dorsal side. In summary, the mechanical compatibility between the skin and the muscle enables them to grow seamlessly, while the wrinkled skin acts as a protective barrier that is both thicker and more flexible than the unwrinkled skin.

We model within the framework of finite elasticity two inherent instabilities observed in liquid crystal elastomers under uniaxial tension. First is necking which occurs when a material sample suddenly elongates more in a small region where it appears narrower than the rest of the sample. Second is shear striping, which forms when the in-plane director rotates gradually to realign and become parallel with the applied force. These phenomena are due to the liquid crystal molecules rotating freely under mechanical loading. To capture necking, we assume that the uniaxial order parameter increases with tensile stretch, as reported experimentally during polydomain-monodomain transition. To account for shear striping, we maintain the uniaxial order parameter fixed, as suggested by experiments. Our finite element simulations capture well these phenomena. As necking in liquid crystal elastomers has not been satisfactorily modelled before, our theoretical and numerical findings related to this effect can be of wide interest. Shear striping has been well studied, yet our computed examples also show how optimal stripe width increases with the nematic penetration depth measuring the competition between the Frank elasticity of liquid crystals and polymer elasticity. Although known theoretically, this result has not been confirmed numerically by previous nonlinear elastic models., Comment: 16 pages

Motivated by the need for new materials and green energy production and conversion processes, a class of mathematical models for liquid crystal elastomers integrated within a theoretical charge pump electrical circuit is considered. The charge pump harnesses the chemical and mechanical properties of liquid crystal elastomers transitioning from the nematic to isotropic phase when illuminated or heated to generate higher voltage from a lower voltage supplied by a battery. For the material constitutive model, purely elastic and neoclassical-type strain energy densities applicable to a wide range of monodomain nematic elastomers are combined, while elastic and photo-thermal responses are decoupled to make the investigation analytically tractable. By varying the model parameters of the elastic and neoclassical terms, it is found that liquid crystal elastomers are more effective than rubber when used as dielectric material within a charge pump capacitor.

We present a stochastic modeling framework for atomistic propagation of a Mode I surface crack, with atoms interacting according to the Lennard-Jones interatomic potential at zero temperature. Specifically, we invoke the Cauchy-Born rule and the maximum entropy principle to infer probability distributions for the parameters of the interatomic potential. We then study how uncertainties in the parameters propagate to the quantities of interest relevant to crack propagation, namely, the critical stress intensity factor and the lattice trapping range. For our numerical investigation, we rely on an automated version of the so-called numerical-continuation enhanced flexible boundary (NCFlex) algorithm., Comment: 20 pages, 10 figures

Stability is an important and fruitful avenue of research for liquid crystal elastomers. At constant temperature, upon stretching, the homogeneous state of a nematic body becomes unstable, and alternating shear stripes develop at very low stress. Moreover, these materials can experience classical mechanical effects, such as necking, void nucleation and cavitation, and inflation instability, which are inherited from their polymeric network. We investigate the following two problems: First, how do instabilities in nematic bodies change from those found in purely elastic solids? Second, how are these phenomena modified if the material constants fluctuate? To answer these questions, we present a systematic study of instabilities occurring in nematic liquid crystal elastomers, and examine the contribution of the nematic component and of fluctuating model parameters that follow probability laws. This combined analysis may lead to more realistic estimations of subsequent mechanical damage in nematic solid materials.

We derive a F\"{o}ppl-von K\'{a}rm\'{a}n-type constitutive model for solid liquid crystalline plates where the nematic director may or may not rotate freely relative to the elastic network. To obtain the reduced two-dimensional model, we rely on the deformation decomposition of a nematic solid into an elastic deformation and a natural shape change. The full solution to the resulting equilibrium equations consists of both the deformation displacement and stress fields. The model equations are applicable to a wide range of thin nematic bodies subject to optothermal stimuli and mechanical loads. For illustration, we consider certain reversible natural shape changes in simple systems which are stress free, and their counterparts, where the natural deformations are blocked and internal stresses appear. More general problems can be addressed within the same framework.

For monodomain nematic elastomers, we construct generalised elastic-nematic constitutive models combining purely elastic and neoclassical-type strain-energy densities. Inspired by recent developments in stochastic elasticity, we extend these models to stochastic-elastic-nematic forms where the model parameters are defined by spatially-independent probability density functions at a continuum level. To investigate the behaviour of these systems and demonstrate the effects of the probabilistic parameters, we focus on the classical problem of shear striping in a stretched nematic elastomer for which the solution is given explicitly. We find that, unlike in the neoclassical case where the inhomogeneous deformation occurs within a universal interval that is independent of the elastic modulus, for the elastic-nematic models, the critical interval depends on the material parameters. For the stochastic extension, the bounds of this interval are probabilistic, and the homogeneous and inhomogeneous states compete in the sense that both have a a given probability to occur. We refer to the inhomogeneous pattern within this interval as "likely striping".

Cavitation in solids can be caused by tensile dead-load traction or impulse traction. The two different types of boundary conditions lead to different static and dynamic solutions. In addition, if the material is stochastic, i.e., the model parameters are represented by probability distributions, the expected behaviour is more complicated to describe. Here, following the first instalment of this work, we examine the static and dynamic cavitation of a stochastic material under a uniform tensile impulse traction in different spherical geometries. We find that the critical load at which a cavity forms at the centre of the sphere is the same as for the homogeneous sphere composed entirely of the material found at its centre, while the post-cavitation radial motion is non-oscillatory. However, there are some important differences in the nonlinear elastic responses. Specifically, subcritical bifurcation, with snap cavitation, is obtained in a static sphere of stochastic neo-Hookean material and also in a radially inhomogeneous sphere, whereas for composite spheres, a supercritical bifurcation, with stable cavitation, is possible as well. Given the non-deterministic material parameters, the results are characterised by probability distributions., Comment: This is part 2 of a two-part article, which has been merged with part 1 to form a single-part article entitled: Likely cavitation and radial motion of stochastic elastic spheres arXiv:1906.10514. Therefore, there is no need for a separate Part 2

The cavitation of solid elastic spheres is a classical problem of continuum mechanics. Here, we study this problem within the context of "stochastic elasticity" where the constitutive parameters are characterised by probability density functions. We consider homogeneous spheres of stochastic neo-Hookean material, composites with two concentric stochastic neo-Hookean phases, and inhomogeneous spheres of locally neo-Hookean material with a radially varying parameter. In all cases, we show that the material at the centre determines the critical load at which a spherical cavity forms there. However, while under dead-load traction, a supercritical bifurcation, with stable cavitation, is obtained in a static sphere of stochastic neo-Hookean material, for the composite and radially inhomogeneous spheres, a subcritical bifurcation, with snap cavitation, is also possible. For the dynamic spheres, oscillatory motions are produced under suitable dead-load traction, such that a cavity forms and expands to a maximum radius, then collapses again to zero periodically, but not under impulse traction. Under a surface impulse, a subcritical bifurcation is found in a static sphere of stochastic neo-Hookean material and also in an inhomogeneous sphere, whereas in composite spheres, supercritical bifurcations can occur as well. Given the non-deterministic material parameters, the results can be characterised in terms of probability distributions., Comment: arXiv admin note: text overlap with arXiv:1901.06145

It has recently been shown that for a Cauchy stress response induced by a strictly rank-one convex hyperelastic energy potential, a homogeneous Cauchy stress tensor field cannot correspond to a non-homogeneous deformation if the deformation gradient has discrete values, i.e. if the deformation is piecewise affine linear and satisfies the Hadamard jump condition. In this note, we expand upon these results and show that they do not hold for arbitrary deformations by explicitly giving an example of a strictly rank-one convex energy and a non-homogeneous deformation such that the induced Cauchy stress tensor is constant. In the planar case, our example is related to another previous result concerning criteria for generalized convexity properties of conformally invariant energy functions, which we extend to the case of strict rank-one convexity.

Stochastic homogeneous hyperelastic solids are characterised by strain-energy densities where the parameters are random variables defined by probability density functions. These models allow for the propagation of uncertainties from input data to output quantities of interest. To investigate the effect of probabilistic parameters on predicted mechanical responses, we study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material, formulated as quasi-equilibrated motions where the system is in equilibrium at every time instant. Additionally, we study finite shear oscillations of a cuboid, which are not quasi-equilibrated. We find that, for hyperelastic bodies of stochastic neo-Hookean or Mooney-Rivlin material, the amplitude and period of the oscillations follow probability distributions that can be characterised. Further, for cylindrical tubes and spherical shells, when an impulse surface traction is applied, there is a parameter interval where the oscillatory and non-oscillatory motions compete, in the sense that both have a chance to occur with a given probability. We refer to the dynamic evolution of these elastic systems, which exhibit inherent uncertainties due to the material properties, as `likely oscillatory motions'.

When an elastic tube reinforced with helical fibres is inflated, its ends rotate. In large deformations, the amount and chirality of rotation is highly non-trivial, as it depends on the choice of strain-energy density and the arrangements of the fibres. For anisotropic hyperelastic tubes where the material parameters are single-valued constants, the problem has been satisfactorily addressed. However, in many systems, the material parameters are not precisely known, and it is therefore more appropriate to treat them as random variables. The problem is then to understand chirality in a probabilistic framework. Here, we develop a procedure for examining the elastic responses of a hyperelastic cylindrical tube of stochastic anisotropic material, where the material parameters are spatially-independent random variables defined by probability density functions. The tube is subjected to uniform dead loading consisting of internal pressure, axial tension and torque. Assuming that the tube wall is thin and that the resulting deformation is the combined inflation, extension and torsion from the reference circular cylindrical configuration to a deformed circular cylindrical state, we derive the probabilities of radial expansion or contraction, and of right-handed or left-handed torsion. We refer to these stochastic behaviours as `likely inflation' and `likely chirality', respectively., Comment: International Journal of Non-Linear Mechanics 2019

We revisit the classic problem of elastic cavitation within the framework of stochastic elasticity. For the deterministic elastic problem, involving homogeneous isotropic incompressible hyperelastic spheres under radially symmetric tension, there is a critical dead-load traction at which cavitation can occur for some materials. In addition to the well-known case of stable cavitation post-bifurcation at the critical dead load, we show the existence of unstable snap cavitation for some isotropic materials satisfying Baker-Ericksen inequalities. For the stochastic problem, we derive the probability distribution of the deformations after bifurcation. In this case, we find that, due to the probabilistic nature of the material parameters, there is always a competition between the stable and unstable states. Therefore, at a critical load, stable or unstable cavitation occurs with a given probability, and there is also a probability that the cavity may form under smaller or greater loads than the expected critical value. We refer to these phenomena as `likely cavitation'. Moreover, we provide examples of homogeneous isotropic incompressible materials exhibiting stable or unstable cavitation together with their stochastic equivalent.

In large deformations, internally pressurised elastic spherical shells and tubes may undergo a limit-point, or inflation, instability manifested by a rapid transition in which their radii suddenly increase. The possible existence of such an instability depends on the material constitutive model. Here, we revisit this problem in the context of stochastic incompressible hyperelastic materials, and ask the question: what is the probability distribution of stable radially symmetric inflation, such that the internal pressure always increases as the radial stretch increases? For the classic elastic problem, involving isotropic incompressible materials, there is a critical parameter value that strictly separates the cases where inflation instability can occur or not. By contrast, for the stochastic problem, we show that the inherent variability of the probabilistic parameters implies that there is always competition between the two cases. To illustrate this, we draw on published experimental data for rubber, and derive the probability distribution of the corresponding random shear modulus to predict the inflation responses for a spherical shell and a cylindrical tube made of a material characterised by this parameter., Comment: arXiv admin note: text overlap with arXiv:1808.01263

The problem of the Rivlin cube is to determine the stability of all homogeneous equilibria of an isotropic incompressible hyperelastic body under equitriaxial dead loads. Here, we consider the stochastic version of this problem where the elastic parameters are random variables following standard probability laws. Uncertainties in these parameters may arise, for example, from inherent data variation between different batches of homogeneous samples, or from different experimental tests. As for the deterministic elastic problem, we consider the following questions: what are the likely equilibria and how does their stability depend on the material constitutive law? In addition, for the stochastic model, the problem is to derive the probability distribution of deformations, given the variability of the parameters.

In isotropic finite elasticity, unlike in the linear elastic theory, a homogeneous Cauchy stress may be induced by non-homogeneous strains. To illustrate this, we identify compatible non-homogeneous three-dimensional deformations producing a homogeneous Cauchy stress on a cuboid geometry, and provide an example of an isotropic hyperelastic material, which is not rank-one convex, and for which the homogeneous stress and the associated non-homogeneous strains on a domain similar to those analysed are given explicitly., Comment: arXiv admin note: substantial text overlap with arXiv:1608.05040

In this note, we show that the Cauchy stress tensor $\sigma$ in nonlinear elasticity is injective along rank-one connected lines provided that the constitutive law is strictly rank-one convex. This means that $\sigma(F+\xi\otimes \eta)=\sigma(F)$ implies $\xi\otimes \eta=0$ under strict rank-one convexity.

Two mathematical models are developed within the theoretical framework of large strain elasticity for the determination of upper and lower bounds on the total strain energy of a finitely deformed hyperelastic body in unilateral contact with a rigid surface or with an elastic substrate. The model problems take the form of two continuous optimization problems with inequality constraints, the solutions of which are used to provide an enclosure on the uniform external load acting on the body's surface away from the contact zone., Comment: in press

We discuss whether homogeneous Cauchy stress implies homogeneous strain in isotropic nonlinear elasticity. While for linear elasticity the positive answer is clear, we exhibit, through detailed calculations, an example with inhomogeneous continuous deformation but constant Cauchy stress. The example is derived from a non rank-one convex elastic energy.