Starting from three-dimensional non-linear elasticity under the restriction of incompressibility, we derive reduced models to capture the behaviour of strings in response to external forces. Our Γ-convergence analysis of the constrained energy functionals in the limit of shrinking cross-sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the Γ-limit is to establish recovery sequences that accommodate the non-linear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for 3d-1d dimension reduction problems.

In this paper, the Rayleigh wave propagation is investigated in rotating half-space of incompressible monoclinic elastic materials which are subjected to the impedance boundary conditions. In particular, the explicit secular equation of the Rayleigh wave is obtained. The main objective of this paper is to illustrate the dependence of dimensionless speed of Rayleigh wave on rotation, anisotropy and impedance parameters. An algorithm in MATLAB software is developed to solve the secular equation of Rayleigh wave. The speed of Rayleigh wave is plotted against rotation, anisotropy and impedance parameters.

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 treat the problem of constructing exact theories of rods and shells for thin incompressible bodies. We employ a systematic method that consists inimposing constraints to reduce the number of degrees of freedom of each crosssection to a finite number. We show that it is very difficult to produce theories that exactly preserve the incompressibility and we show that it is impossible to do so for naive theories. In particular, many exact theories have nonlocal effects.