Journal of Elasticity最新文献

The electroelastic problems of one-dimensional hexagonal piezoelectric quasicrystal materials with a periodic distribution of slant mode-III cracks under anti-plane shear and electromechanical loading are analyzed in this paper. Based on the three electrical boundary conditions at the crack surfaces, electrically permeable, electrically semi-permeable and electrically impermeable condition, the problems are classified as solving singular integral equations by using screw dislocation solutions. For two special cases of coplanar and parallel periodic crack arrays, the closed form solutions for the electroelastic fields, including stress fields, electric fields and tearing displacements, have been determined. The solutions of the singular integral equations for slant cracks can be transformed into the solutions of algebraic equations, the field intensity factors and mechanical strain energy release rates have been determined. The numerical solutions show that the normalized mechanical strain energy release rates increase under the influence of phonon field stress, phason field stress as well as electric fields, indicating that cracks are more likely to propagate in piezoelectric quasicrystal materials. In addition, it is found that the stress fields at the crack tips exhibited singularity, and the variation law of the total energy release rates with the applied electrical loading are also obtained.

Using general solutions of the equilibrium equations of linearized stability theory, transcendental equations for determining the critical strains corresponding to the onset of wrinkling of a thin coating film located on a semibounded substrate are obtained. The substrate/film (bilayer) system is assumed to be under plane strain conditions, and the body materials are nonlinearly elastic with arbitrary structures of elastic potentials. Two variants of the boundary conditions at the interface are considered: perfectly bonded layers and perfectly lubricated layers, corresponding to the “strongest” and “weakest” types of bonding between the bilayer components. Numerical results for determining the critical values of the wrinkling strain are presented for the harmonic potential (compressible bodies, large strains), the quadratic potential (compressible bodies, small strains), the Treloar potential, and the Bartenev–Khazanovich potential (incompressible bodies, large strains). The nature of the dependence of the critical strain and critical wavelength on the elastic constants of the substrate and film materials and on the type of elastic potential was studied. A comparison of the obtained results with known theoretical and experimental results was carried out.

We consider the general theory of 6-parameter shells, in which material points on the midsurface are endowed with 3 translational and 3 rotational degrees of freedom. In this framework, we derive quasiconvexity conditions and rank-one convexity conditions. These inequalities represent necessary conditions for energy minimizers; they are the two-dimensional counterparts of the well-known relaxed convexity conditions in three-dimensional finite elasticity. As a specific feature, the quasiconvexity inequality for shells contains the gradients in the tangent plane of the variation fields associated to deformation and microrotation. Finally, we also deduce the Legendre-Hadamard condition for shells, as a consequence of the rank-one convexity inequality.

Due to the geometric feature and information, increasing investigations have been focused on tangled systems with rich mechanics behaviors. But most of them are limited to single physical intertwining or mathematical knot, which makes it difficult to illustrate the interaction between mechanics and geometry. This work proposes different kinds of elastic tangles with three ropes, in which the mechanics components and geometric properties are obtained. Five topological parameters are derived to show the relation between tangle type and mechanics characteristic. Based on the geometry knot theory, the strength and stability of tangles can be predicted by geometry rules. Moreover, numerical calculation and experiment are performed to support the theoretical prediction. This work wishes to provide guidance for the design and control of systems with complex entanglements and further inspire geometric mechanics of slender flexible elastomers.

Liquid crystal elastomers (LCEs) marry the large deformation response of a cross-linked polymer network with the nematic order of liquid crystals pendent to the network. Of particular interest is the actuation of LCE sheets where the nematic order, modeled by a unit vector called the director, is specified heterogeneously in the plane of the sheet. Heating such a sheet leads to a large spontaneous deformation, coupled to the director design through a metric constraint that is now well-established by the literature. Here we go beyond the metric constraint and identify the full plate theory that underlies this phenomenon. Starting from a widely used bulk model for LCEs, we derive a plate theory for the pure bending deformations of patterned LCE sheets in the limit that the sheet thickness tends to zero using the framework of (Gamma )-convergence. Specifically, after dividing the bulk energy by the cube of the thickness to set a bending scale, we show that all limiting midplane deformations with bounded energy at this scale satisfy the aforementioned metric constraint. We then identify the energy of our plate theory as an ansatz-free lower bound of the limit of the scaled bulk energy, and construct a recovery sequence that achieves this plate energy for all smooth enough midplane deformations. We conclude by applying our plate theory to a variety of examples.

The celebrated experiments of J. H. Poynting in 1909 have given rise to a vast literature regarding an interesting feature of the nonlinear response of soft solids. Poynting conducted a series of experiments on metal wires and found that loaded wires lengthen when twisted. Thus to maintain a constant length in such experiments, a compressive axial force would need to be applied at the ends of the specimen. This is the classical (positive) Poynting effect. Another example of such an effect arises when a soft material specimen is being laterally sheared or rotated between two platens. The necessity to apply a compressive lateral normal force in order to maintain the relative distance between the platens is also often referred to as a Poynting-type effect. Both effects are inherently nonlinear phenomena. In recent years, a large body of experimental and theoretical work on the Poynting effect has been carried out. In particular, a reverse Poynting effect has been investigated where the cylinder contracts under torsion unless a tensile axial force is applied or in the case of the lateral shear problem, the platens tend to draw together laterally unless a tensile lateral normal force is applied. The purpose of the present article is to review recent research findings on both of these effects for soft materials.

The material correspondence formulation plays an essential role in connecting the Classical Continuum Mechanics and Peridynamics. In this paper, we analyze the material correspondence formulation in both the generalized two-parameter bond-based and state-based Peridynamics with a particular emphasis on the material symmetry principles. We discover a “trade-off balance” law between material symmetry and Poisson’s ratio in Peridynamics. Specifically, in the generalized two-parameter bond-based Peridynamics, the Poisson’s ratio limitation is eliminated, but the symmetry of the homogenized fourth-order material tensor in this model differs from that in Classical Continuum Mechanics. This asymmetry in the material tensor leads to energy incompatibility between the bond-based Peridynamics and Classical Continuum Mechanics. Furthermore, it can be proved that this incompatible energy has an upper bound and approaches zero as the characteristic length of the non-local interaction domain vanishes. In the case of the state-based Peridynamics, the symmetry of material tensors aligns with Classical Continuum Mechanics. However, the material correspondence formulation imposes a lower bound constraint on the Poisson’s ratio for the state-based Peridynamics. Inspired by this ‘trade-off balance’ law in Peridynamics, we propose a novel continuum model that maintains symmetry consistency. The proposed model integrates local and non-local energy into a single energy functional. By employing the Hamilton’s variational principle, we derive the governing equations with exact force boundary conditions. Unlike Peridynamics, the proposed model exerts the force boundary on the outer surface of the solids. We demonstrate that the proposed model is asymptotically compatible with Classical Continuum Mechanics. Wave dispersion analysis shows that the proposed model does not exhibit zero-energy mode oscillations.

Herein we take a first step towards merging nonlinear elasticity with the two non-relativistic Galilean-covariant limits of electromagnetism, namely the electric limit and the magnetic limit, the results of which we call Galilean electroelasticity and Galilean magnetoelasticity, respectively. Using the first law of thermodynamics for dynamical adiabatic processes, we derive, for systems (with zero free-charge and free-current densities) which undergo such processes, the internal energy density function and its associated constitutive equations in Galilean electroelasticity and magnetoelasticity, respectively. Each of the two internal energy density functions (per unit reference volume) thus obtained agrees with one of the two total energy density functions introduced by Dorfmann and Ogden in their work on electro-elastostatics and magneto-elastostatics, respectively. For linear polarizable and magnetizable dielectrics, Galilean-invariant expressions of the Maxwell stress are obtained for the electric limit and for the magnetic limit, respectively.

Linear elasticity comprises the fundamental branch of continuum mechanics that is extensively used in modern structural analysis and engineering design. In view of this concept, the displacement field provides a measure of how solid materials deform and become internally stressed due to prescribed loading conditions, a fact which is associated with linear relationships between the components of strain and stress, respectively. The mathematical characteristics of these dyadic fields are combined within the Hooke’s law via the stiffness tetratic tensor, which embodies either the isotropic or the anisotropic behavior, exhibited by materials with linear properties. In fact, Hooke’s law is incorporated into the general law of Newton that actually defines the principal spatial and temporal second-order non-homogeneous partial differential equation for the displacement. In this study, we construct handy closed-form solutions for Newton’s law in the Cartesian regime, implying time-independence and considering the case of absence of body forces. Towards this direction, our aim is twofold, in the sense that an efficient analytical technique is introduced that generates homogeneous polynomial solutions of the displacement field for both the typical isotropic and the cubic-type anisotropic structure in the invariant Cartesian geometry. The reliability of the presented methodology is verified by reducing the results for each polynomial degree from the anisotropic to the isotropic eigenspace, in terms of a simple transformation, while we demonstrate our theory with an important application, wherein the effect of a prescribed force on an isotropic half-space to the neighboring half-space of cubic anisotropy is examined.