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Key points The force in the half-sarcomere (hs), the functional unit of muscle, is due to the contributions of individual myosin motors arranged in parallel in the half-myosin filament and pulling on the opposing actin filament. According to a linear hs model, during an isometric contraction the force rises to its maximal steady value (T0) in proportion to the number of actin-attached motors, while the hs strain rises with a slope that depends on the compliance of the myofilaments. We measured the hs stiffness, superimposing small 4 kHz length oscillations on the development of isometric contraction, and found an elastic element in parallel to the myosin motors with a constant stiffness ∼1/20th that of the motor array at T0. The results support a structural model in which myosin motors are distributed in multiple substates, of which only the first ones are occupied during isometric force generation, causing a motor strain of ∼1.7 nm. Abstract Force in striated muscle is due to attachment of the heads of the myosin, the molecular motors extending from the myosin filament, to the actin filament in each half-sarcomere, the functional unit where myosin motors act in parallel. Mechanical and X-ray structural evidence indicates that at the plateau of isometric contraction (force T0), less than half of the elastic strain of the half-sarcomere is due to the strain in the array of myosin motors (s), with the remainder being accounted for by the compliance of filaments acting as linear elastic elements in series with the motor array. Early during the development of isometric force, however, the half-sarcomere compliance has been found to be less than that expected from the linear elastic model assumed above, and this non-linearity may affect the estimate of s. This question is investigated here by applying nanometre–microsecond-resolution mechanics to single intact fibres from frog skeletal muscle at 4°C, to record the mechanical properties of the half-sarcomere throughout the development of force in isometric contraction. The results are interpreted with mechanical models to estimate the compliance of the myosin motors. Our conclusions are as follows: (i) early during the development of an isometric tetanus, an elastic element is present in parallel with the myosin motors, with a compliance of ∼200 nm MPa−1 (∼20 times larger than the compliance of the motor array at T0); and (ii) during isometric contraction, s is 1.66 ± 0.05 nm, which is not significantly different from the value estimated with the linear elastic model.

An iterative solver is proposed to solve the family of linear equations arising from the numerical computation of non-linear problems. This solver relies on two quantities coming from previous steps of the computations: the preconditioning matrix is a matrix that has been factorized at an earlier step and previously computed vectors yield a reduced basis. The principle is to define an increment in two sub-steps. In the first sub-step, only the projection of the unknown on a reduced subspace is incremented and the projection of the equation on the reduced subspace is satisfied exactly. In the second sub-step, the full equation is solved approximately with the help of the preconditioner. Last, the convergence of the sequences is accelerated by a well-known method, the modified minimal polynomial extrapolation. This algorithm assessed by classical benchmarks coming from shell buckling analysis. Finally, its insertion in path following techniques is discussed. This leads to non-linear solvers with few matrix factorizations and few iterations. Copyright © 2009 John Wiley & Sons, Ltd.

In this contribution we introduce a novel space-time formulation for non-linear elasticity, able to calculate large deformations and displacements with high efficiency using structured and unstructured meshes in the space-time cylinder without changing the required regularity and thus, enabling the use of Lagrangian shape functions including tetrahedron and hypertetrahedron or tesseract elements. The common and indiscriminate treatment of spatial and temporal directions allows us to remove one of the major bottlenecks in parallel computations: The design of time-stepping schemes, which can neither been parallelized efficiently in time nor allow for local refinements as no information transfer backwards in time is possible. Moreover, stability of time-stepping schemes depend on the accumulation of local approximation errors in each time-step, in contrast to space-time formulation, characterized by enhanced stability and robustness. Eventually, we will demonstrate superiority in the convergence against classical time-stepping schemes using various examples including highly sensitive systems in the context of non-linear elasticity.

The equation of state of the triclinic compound 1,3,5-triamino-2,4,6-trinitrobenzene (TATB) as well as its second-order isothermal elastic tensor were computed through classical molecular dynamics simulations under various temperature and pressure conditions. Hydrostatic pressures similar to previous diamond anvil cell experiments were imposed within the range [0, 66] GPa and temperatures chosen between 100 and 900 K in conjunction with the most recent version of an all-atom fully-flexible molecule force field. The isothermal elastic constants were computed using the generalized Hooke's law by fitting Cauchy stress vs. linear strain curves. Along isobaric pathways, TATB single crystal stiffness is found to undergo linear softening, less pronounced at high pressure, while maintaining its elastic anisotropy. On the other hand, along an isothermal pathways, a non-linear increase is observed in the elastic constants with a significant decrease in anisotropy. Towards a precise mesoscopic modeling of TATB single crystal mechanical behavior, we provide "ready to plug-in" analytical formulations of the P,V,T equation of state and pressure-temperature dependent non-linear elasticity.

A symmetry-based thermodynamical model of third-order electro-elastic coupling is applied to zinc-blende semiconductors. Density functional theory is used to calculate the complete set of linear and non-linear coefficients. The entanglement of non-linear piezoelectricity, electrostriction and other non-linear phenomena, is studied in details at the same level of theory. Numerical results for InxGa1−xP/GaP [111] quantum wells show the predominance of the non-linear piezoelectricity contribution. However, for a coupled non-linear model, the electrostriction has also to be taken into account. Finally, the non-linear elasticity gives a significant correction on the strain calculation.

We present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle. The method is based on a variational approach, and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters, here those defining the support size of each basis function. We consider local maximum-entropy approximation schemes, which exhibit smooth basis functions with respect to both space and the discretization parameters (the node location and the locality parameters). We illustrate by the Poisson, linear and non-linear elasticity problems the effectivity of the method, which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions. Copyright © 2009 John Wiley & Sons, Ltd.

SUMMARY Conservative formulations of the governing laws of elastoplastic solid media have distinct advantages when solved using high-order shock capturing methods for simulating processes involving large deformations and shock waves. In this paper one such model is considered where inelastic deformations are accounted for via conservation laws for elastic strain with relaxation source terms. Plastic deformations are governed by the relaxation time of tangential stresses. Compared with alternative Eulerian conservative models, the governing system consists of fewer equations overall. A numerical scheme for the inhomogeneous system is proposed based upon the temporal splitting. In this way the reduced system of non-linear elasticity is solved explicitly, with convective fluxes evaluated using high-order approximations of Riemann problems locally throughout the computational mesh. Numerical stiffness of the relaxation terms at high strain rates is avoided by utilizing certain properties of the governing model and performing an implicit update. The methods are demonstrated using test cases involving large deformations and high strain rates in one-, two-, and three-dimensions. Copyright q 2009 John Wiley & Sons, Ltd. Received 25 February 2009; Revised 27 May 2009; Accepted 29 May 2009

The anterior median artery is an unpaired vessel that leaves the anterior end of the lobster (Homarus americanus) heart and supplies hemolymph to the brain, the antennae, and the eyestalks. This vessel has a trilaminar organization, consisting of a tunica interna with elastic fibrils, a tunica intermedia represented by a bilayered cell mass, and a tunica externa with collagen fibrils. The exposed ends of the medial cells in the tunica intermedia exhibit small, diffuse bundles of microfilaments that are penetrated by microtubules. These bundles have a circumferential or a slightly oblique orientation relative to the lumen of the vessel. The precise role of the microfilaments is unresolved. If the irregularly shaped bundles are static structures, they might contribute to the non-linear elasticity of the artery. Alternatively, if they generate force, a coordinated contraction of the medial cells might reduce the luminal diameter of the artery and, thus, retard hemolymph flow. Microfilaments of the medial cells anchor to subplasmalemmal filamentous mats, some of which are integral to intermediate junctions and some of which are associated with unbounded cell membranes (hemi-intermediate junctions). Contraction of the microfilament-bearing cells would have to occur without the benefit of nervous innervation or the participation of communicating (gap) junctions. If cell contractility is confirmed, a reclassification of the anterior median artery, from capacitance vessel to resistance vessel, is in order, and the bilayered cell mass in the tunica intermedia would likely qualify as the first unstriated muscle found in crustaceans.

The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non-linear elasticity and the calculus of variations. In this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm. For instance, we show the following. Suppose that $f_o$ is a homeomorphism of the circle with $f_{o}^{-1} \in {\cal W}^{1/2, 2}$. Then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non-vanishing Jacobian determinant. The condition on $f_o$ is sharp. Classically the mapping $f_o$ is assumed to be quasisymmetric. Then there is an extremal quasiconformal mapping with boundary values $f_o$, but it is not always unique and it is seldom smooth. Indeed, even when $f_o$ is quasisymmetric, the ${\cal L}^1$-minimiser for the distortion function will almost never be quasiconformal. We further find that there are many new and unexpected phenomena concerning existence, uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex. These seem to differ markedly from phenomena observed when studying multi-well type functionals.

In the present contribution, the problem of non-linear deformation of materials with stochastically distributed anisotropic inclusions is considered on the basis of the methods of mechanics of stochastically non-homogeneous media. The homogenization model of materials of stochastic structure with physically non-linear components is developed for the case of a matrix which is strengthened by unidirectional ellipsoidal inclusions. It is assumed that the matrix is isotropic, deforms non-linearly; inclusions are linear-elastic and have transversally-isotropic symmetry of physical and mechanical properties. Stochastic differential equations of physically non-linear elasticity theory form the underlying equations. Transformation of these equations into integral equations by using the Green’s function and application of the method of conditional moments allow us to reduce the problem to a system of non-linear algebraic equations. This system of non-linear algebraic equations is solved by the Newton-Raphson method. On the analytical as well as the numerical basis, the algorithm for determination of the non-linear effective characteristics of such a material is introduced. The non-linear behavior of such a material is caused by the non-linear matrix deformations. On the basis of the numerical solution, the dependences of homogenized Poisson’s coefficients on macro-strains and the non-linear stress-strain diagrams for a material with randomly distributed unidirectional ellipsoidal pores are predicted and discussed for different volume fractions of pores.

A formal analogy between the equations of pure plastic and viscoplastic flow theory for void‐containing metals and those of standard non‐linear elasticity is presented. The formulation is particularized for the analysis of axisymmetric sheet metal forming problems using simple two node linear finite elements. Details of the treatment of friction and strain hardening phenomena, time increment computation and elastic effects are also given. Examples of the effect of void porosity on the hemispherical stretching of a circular sheet are presented.

Excised sheets of canine myocardium were subjected to cyclic loading and unloading in the predominant fibre and cross-fibre directions to determine passive mechanical properties. Myocardium under biaxial loading exhibits both non-linear elasticity and viscoelasticity with some strain-rate dependence in the position of the stress-strain relations, but very little rate dependence in the area enclosed by the loading and unloading portions of the stress-strain loops. Fibre and cross-fibre directions demonstrate anisotropic behaviour, with both the degree and direction of the anisotropy being dependent upon the region of the heart from which specimens are obtained. In the same specimen biaxial as compared with uniaxial loading yields different interpretations as to the material properties.