We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to $L^p$-perturbations of their fundamental forms: For a manifold ${\mathcal M}^d$ endowed with a reference metric and a reference shape operator, we show that a sequence of immersions $f_n:{\mathcal M}^d\to {\mathcal N}^{d+1}$, whose pullback metrics and shape operators are arbitrary close in $L^p$ to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold ${\mathcal N}$, removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.

We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti and De Lellis is generic in some sense. On this map, we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of $\mathbb {S}^2$-valued harmonic maps.

In this paper, we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider two types of problems: the displacement problem in which the outer boundary of the body is subjected to a Dirichlet-type boundary condition, and the one with zero traction on the boundary but with an internal pressure function. For a spherically symmetric body occupying the unit ball $\mathcal {B}\in \mathbb {R}^3$, the minimization is done within the class of radially symmetric deformations. We give conditions for the existence of such minimizers, for satisfaction of the Euler–Lagrange equations, and show that for large displacements or large internal pressures, the minimizer must develop a cavity at the centre. We discuss a numerical scheme for approximating the minimizers for the displacement problem, together with some simulations that show the dependence of the cavity radius and minimum energy on the displacement and mass density of the body.

We are concerned with a variant of the isoperimetric problem, which in our setting arises in a geometrically nonlinear two-well problem in elasticity. More precisely, we investigate the optimal scaling of the energy of an elastic inclusion of a fixed volume for which the energy is determined by a surface and an (anisotropic) elastic contribution. Following ideas from Conti and Schweizer (Commun. Pure Appl. Math. 59 (2006), 830–868) and Knüpfer and Kohn (Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 467 (2011), 695–717), we derive the lower scaling bound by invoking a two-well rigidity argument and a covering result. The upper bound follows from a well-known construction for a lens-shaped elastic inclusion.

The rigorous derivation of linear elasticity from finite elasticity by means of $\Gamma$-convergence is a well-known result, which has been extended to different models also beyond the elastic regime. However, in these results the applied forces are usually assumed to be dead loads, that is, their density in the reference configuration is independent of the actual deformation. In this paper we begin a study of the variational derivation of linear elasticity in the presence of live loads. We consider a pure traction problem for a nonlinearly elastic body subject to a pressure live load and we compute its linearization for small pressure by $\Gamma$-convergence. We allow for a weakly coercive elastic energy density and we prove strong convergence of minimizers.

We study ribbons of vanishing Gaussian curvature, i.e. flat ribbons, constructed along a curve in $\mathbb {R}^{3}$. In particular, we first investigate to which extent the ruled structure determines a flat ribbon: in other words, we ask whether for a given curve $\gamma$ and ruling angle (angle between the ruling line and the curve's tangent) there exists a well-defined flat ribbon. It turns out that the answer is positive only up to an initial condition, expressed by a choice of normal vector at a point. We then study the set of infinitely narrow flat ribbons along a fixed curve $\gamma$ in terms of energy. By extending a well-known formula for the bending energy of the rectifying developable, introduced in the literature by Sadowsky in 1930, we obtain an upper bound for the difference between the bending energies of two solutions of the initial value problem. We finally draw further conclusions under some additional assumptions on the ruling angle and the curve $\gamma$.

We study the $\Gamma$-convergence of nonconvex vectorial integral functionals whose integrands satisfy possibly degenerate growth and coercivity conditions. The latter involve suitable scale-dependent weight functions. We prove that under appropriate uniform integrability conditions on the weight functions, which shall belong to a Muckenhoupt class, the corresponding functionals $\Gamma$-converge, up to subsequences, to a degenerate integral functional defined on a limit weighted Sobolev space. The general analysis is then applied to the case of random stationary integrands and weights to prove a stochastic homogenization result for the corresponding functionals.

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

In this note, we provide an explicit formula for computing the quasiconvex envelope of any real-valued function W; SL(2) → ℝ with W(RF) = W(FR) = W(F) for all F ∈ SL(2) and all R ∈ SO(2), where SL(2) and SO(2) denote the special linear group and the special orthogonal group, respectively. In order to obtain our result, we combine earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal planar energy functions with a recent result on the equivalence between polyconvexity and rank-one convexity for objective and isotropic energies in planar incompressible nonlinear elasticity.

We prove that, in the limit of vanishing thickness, equilibrium configurations of inhomogeneous, three-dimensional non-linearly elastic rods converge to equilibrium configurations of the variational limit theory. More precisely, we show that, as $h\searrow 0$, stationary points of the energy , for a rod $\Omega _h\subset {\open R}^3$ with cross-sectional diameter h, subconverge to stationary points of the Γ-limit of , provided that the bending energy of the sequence scales appropriately. This generalizes earlier results for homogeneous materials to the case of materials with (not necessarily periodic) inhomogeneities.

In this paper we are interested in the microscopic modelling of a two-dimensional two-well problem that arises from the square-to-rectangular transformation in (two-dimensional) shape-memory materials. In this discrete set-up, we focus on the surface energy scaling regime and further analyse the Hamiltonian that was introduced by Kitavtsev et al. in 2015. It turns out that this class of Hamiltonians allows for a direct control of the discrete second-order gradients and for a one-sided comparison with a two-dimensional spin system. Using this and relying on the ideas of Conti and Schweizer, which were developed for a continuous analogue of the model under consideration, we derive a (first-order) continuum limit. This shows the emergence of surface energy in the form of a sharp-interface limiting model as well the explicit structure of the minimizers to the latter.

This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

We consider a simple body that is hyperelastic in the large-strain regime until the 3-covector defining the first Piola–Kirchhoff stress, once it has been projected on the appropriate second-rank tensor space, reaches a threshold indicating critical states. No information is given on the post-critical behaviour. We determine the existence of equilibrium configurations according to the constraint. Such configurations can have a concentration of strain in regions with vanishing volume. The related stress appears naturally as a measure over the deformation graph. Once it is restricted to the regular part of the deformation, such a measure determines the first Piola–Kirchhoff stress tensor and may also be concentrated over sets with vanishing volume projections on the reference and current placements. These configurations in space can be interpreted as dislocations or dislocation walls. We analyse explicitly specific cases.

We prove the existence of weak solutions for the strongly nonlinear parabolic problem

in the anisotropic Sobolev space , where the data f are assumed to be in the dual, and the nonlinear term g(x, t, s) has growth and sign conditions on s.

We consider a class of non-quasi-convex frame indifferent energy densities that includes Ogden-type energy densities for nematic elastomers. For the corresponding geometrically linear problem, we provide an explicit minimizer of the energy functional satisfying a non-trivial boundary condition. Other attainment results, both for the nonlinear and the linearised model, are obtained by using the theory of convex integration introduced by Müller and Šverák in the context of crystalline solids.

We use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the “Impulse method” which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.

Although in recent years bone piezoelectricity has been normally neglected, lately a new interest has appeared to show the importance of bone piezoelectricity in wet bone's complex response to loading. Here we numerically study a problem, including a strain-adaptive bone remodelling and the piezoelectricity. Its variational formulation leads to a coupled system composed of two linear variational equations for displacements and electric potential, and a parabolic variational inequality for the apparent density. Fully discrete approximations are now introduced by using the finite element method to approximate spatial variable and the explicit Euler scheme to discretise time derivatives. Some a priori error estimates are proved and the linear convergence of the algorithm is deduced under additional regularity conditions. Finally, some one- and two-dimensional numerical simulations are described to show the accuracy of the proposed algorithm and the behaviour of the solution.

An isolated cylindrical void is located inside an incompressible nonlinear-elastic medium whose constitutive behaviour is governed by a neo-Hookean strain energy function. In-plane hydrostatic pressure is applied in the far-field so that the void changes its radius and an inhomogeneous region of deformation arises in the vicinity of the void. We consider scattering from the void in the deformed configuration due to an incident field (of small amplitude) generated by a horizontally polarized shear (SH) line source, a distance r0 (R0) away from the centre of the void in the deformed (undeformed) configuration. We show that the scattering coefficients of this scattered field are unaffected by the pre-stress (initial deformation). In particular, they depend not on the deformed void radius a or distance r0, but instead on the original void size A and original distance R0.

An axisymmetric formulation for modeling three-dimensional deformation of structures of revolution is presented. The axisymmetric deformation model is described using the cylindrical coordinate system. Large displacement effects and material nonlinearities and anisotropy are accommodated by the formulation. Mathematical derivation of the formulation is given, and an example is presented to demonstrate the capabilities and efficiency of the technique compared to the full three-dimensional model.

There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental, including uses in civil engineering, the food industry, land mine detection and ultrasonic imaging. Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials. We begin with a brief introduction of some basic terminology and relationships in continuum mechanics, and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms. To complete the set of equations, we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain proposed in the literature for both elastic and viscoelastic materials. In addition, we discuss some applications for these constitutive equations. Finally, we give a computational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.