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Additional file 1: Figure S1 (a) Time course of mibefradil mediated reversal of mechanical hypersensitivity in mice with a chronic constriction injury of the sciatic nerve. Statistical analyses were performed by two-way ANOVA followed by Tukey's test. Asterisks denote a significant difference of **P < 0.01 and ***P < 0.001 when compared with the control group (n = 5–7). Hashtags denote P < 0.001 for comparison with the sham-operated group. (b) Mibefradil at 10 mg/kg inhibits pain responses in CCI operated wild type mice, but not in Cav3.2 null mice *P < 0.05 and ***P < 0.001 (n = 5–6).

We consider the equilibrium problem, with no body force, of a cylindrically orthotropic disk subject to a prescribed displacement along its boundary. In classical linear elasticity, the solution of this problem predicts material overlapping, which is not physically realistic. One way to prevent this anomalous behavior is to consider the minimization of the total potential energy of classical linear elasticity subject to the local injectivity constraint. In the context of this constrained minimization theory, bifurcation occurs from a radially symmetric solution to a secondary solution. In this work we present analytical and computational results indicating that this secondary solution is rotationally symmetric. Initially, we assume that the solution is rotationally symmetric, solve the Euler-Lagrange equations of the corresponding minimization problem in the region where the local injectivity constraint is not active, and obtain a solution that depends on constants of integration that are determined numerically. In the region where the constraint is active, we determine a nonlinear relation between the radial and tangential displacements, which contains a constant of integration that is also determined numerically. Still assuming rotational symmetry, we use an interior penalty formulation together with a standard finite element method to obtain sequences of numerical solutions that converge to a limit function that is in very good agreement with our analytical results in the non active region. To confirm these findings, we also search for an asymmetric solution numerically. In this case, there is no a priori assumption on symmetry and we only obtain either the radially or the rotationally symmetric solution. In all the cases investigated numerically, we have to introduce a small perturbation to obtain the latter solution. The rotationally symmetric solution presents a novel behavior that is not reported in the literature. The tangential displacement is linear near the center of the disk and the corresponding angle of rotation has the value $\pi $ at this center and decreases as we move away from it. Finally, we investigate numerically the influence of both the shear modulus and the boundary condition on the existence of the rotationally symmetric solution. For a given mesh, there is a maximum value of the shear modulus above which and a minimum value of the boundary condition below which this solution is not possible. This research is of interest in the investigation of solids having stiffer response in the radial direction than in the tangential direction, such as in the case of carbon fibers with radial microstructure and certain types of wood.

We investigate the properties of an isotropic linear elastic peridynamic material in the context of a three-dimensional state-based peridynamic theory, which considers both length and relative angle changes, and is based on a free energy function proposed in previous work that contains four material constants. To this end, we consider a class of equilibrium problems in mechanics to show that, in interior points of the body where deformations are smooth, the corresponding solutions in classical linear elasticity are also equilibrium solutions in peridynamics. More generally, we show that the equations of equilibrium are satisfied even when two of the four peridynamic constants are arbitrary. Pure torsion of a cylindrical shaft and pure bending of a cylindrical beam are particular cases of this class of problems and are used together with a correspondence argument proposed elsewhere to determine these two constants in terms of the elasticity constants of an isotropic material from the classical linear elasticity. One of the constants has a singularity in the Poisson ratio, which needs further investigation. Two additional experiments concerning bending of cylindrical beam by terminal load and anti-plane shear of a hollow cylinder, which do not belong to the previous class of problems, are used to validate these results.

The paper focuses on the problem of evaluating analytically the electric and elastic fields in a piezoelectric material of the symmetry class 6 containing a penny-shaped crack loaded by both normal and tangential tractions and by electric charges. A solution of this kind has been developed for symmetry class 6 mm by Karapetian et al. (Arch. Appl. Mech. 70: 201–229, 2000). To the best of our knowledge, it has never been derived for a material of class 6. The research is motivated by needs in evaluation of damage accumulated in bone tissue, which possesses such a symmetry. The problem is solved by the method of potential functions developed by Fabrikant (Applications of Potential Theory in Mechanics: Selection of New Results. Kluwer, Dordrecht,1989). Closed form solutions are obtained for the full electric and elastic fields. The approach is first verified by comparison with the results of Karapetian et al. (Arch. Appl. Mech. 70: 201–229, 2000) for the limiting case of piezoceramic of class 6 mm (PZT-6B) and then applied to calculation of the electroelastic fields associated with a penny-shaped crack in cortical bone (material of class 6).

The evaluation of the effective properties of nonhomogeneous solids using analytical methods is, in general, based on the assumption that these solids have infinite dimensions. Here, we investigate the influence of both the number of holes and the boundary layer of a solid with finite dimensions on the determination of these properties. We use the Asymptotic Homogenization Method (AHM) to determine the effective shear modulus of an elastic solid with infinite dimensions containing a uniform and periodic distribution of circular cylindrical holes arranged on a hexagonal lattice. We also use the Finite Element Method (FEM) to determine this modulus in the case of a solid with finite dimensions containing the same uniform distribution of cylindrical holes away from its boundary. Near the boundary, we consider a layer of material with no holes, which is usually left in the fabrication process of samples. Both solids have the same elastic properties and are subjected to similar anti-plane shear loadings. For the finite medium, we study two sequences of domains discretized by the FEM, which are called the Fixed Layer Sequence (FLS) and the Fixed Domain Sequence (FDS). For the FLS, the layer thickness is kept fixed and both the dimensions of the domain and the number of holes vary. For the FDS, the dimensions of the domain are kept fixed and both the number of holes and the layer thickness vary. Results obtained from numerical simulations are then used to generate graphs of the effective shear modulus versus void volume fraction. It is observed that, in the FLS case, the shear modulus obtained from the numerical simulations converges to the analytical solution obtained via AHM. It is also observed that, in the FDS case, the shear modulus obtained from the numerical simulations converges to a limit function, which is close to the analytical solution obtained via AHM. For comparison purposes, we have also calculated the effective shear modulus of porous elastic solids containing a square array of circular cylindrical holes. We then show graphs of this modulus versus void volume fraction for both hexagonal and square arrangements that are very close to each other up to void volume fraction of 0.5.

Necessary and sufficient conditions for strong ellipticity of the equilibrium equations governing finite plane deformations are established for the class of orthotropic and compressible hyperelastic materials. The conditions are then specialized to the case of an orthotropic material having the symmetry axes parallel to the principal directions of deformation. This case is relevant in experimental protocols employed in the characterization of constitutive relations of rubber-like materials and biological tissues. The classical conditions for infinitesimal linear elasticity and isotropic finite elasticity are recovered as particular cases.

We determine the minimum number of independent invariants that are needed to characterize completely the strain energy density of a compressible hyperelastic solid having two distinct material symmetry directions. We use a theory of representation of isotropic functions to express this energy density in terms of eighteen invariants, from which we extract ten invariants to analyze two cases of material symmetry. In the case of orthogonal directions, we recover the classical result of seven invariants and offer a justification for the choice of invariants found in the literature. If the directions are not orthogonal, we find that the minimum number is also seven and correct a mistake in a formula found in the literature. An energy density of this type is used to model, on the macroscopic scale, engineering materials, such as fiber-reinforced composites, and biological tissues, such as bones.

This work concerns the behavior of a two-phase periodic laminate composed of homogeneous, isotropic, and hyperelastic layers in equilibrium in the absence of body force and subjected to a finite deformation on its boundary. The effective behavior of unbounded laminates is studied elsewhere using the tangent second-order homogenization method. In the case of a laminate composed of penalty compressible Neo-Hookean phases and subjected to pure shear deformation on its boundary, analytical results indicate that a combination of loading conditions, material properties, and phase concentrations may lead to a reduction of the overall stiffness of the laminate, which may be related to the loss of overall strong ellipticity of the corresponding effective medium, in spite of the fact that the layers are strongly elliptic. Here, the Asymptotic Homogenization Method (AHM) and the Finite Element Method (FEM) are used to verify and extend these results for laminates subjected to similar boundary conditions and compressible Neo-Hookean phases. Using FEM, the rotation angles of the layers at the center of the laminate are calculated and compared to the rotation angles obtained analytically. For certain loading conditions and material properties, the rotation angles are very close to each other up to a critical shear deformation predicted analytically, after which, the angle obtained via FEM changes abruptly from the angle obtained analytically, suggesting a bifurcation-like behavior. This critical deformation depends strongly on the heterogeneity contrast ratio between the phases. For a contrast ratio close to one, no such behavior is observed. As we increase this ratio, the critical deformation becomes smaller and tends to the identity deformation as the contrast ratio tends to infinity. This work may be of interest to practitioners doing numerical simulation of problems involving composites and to researchers interested in material instabilities of effective nonlinear elastic media.

A methodology is proposed to find either implicit or explicit relations between invariants in a minimal integrity basis for n symmetric secondorder tensors defined on a three-dimensional euclidean space. The implicit relations are called syzygies. In particular, the methodology i) allows the construction of a set of 6 n − 3 independent invariants, ii) yields explicit non-polynomial expressions for certain invariants in terms of the remaining invariants in the integrity basis, and iii) allows the construction of syzygies. The results of this investigation are important in the modeling of biological structures, which, in general, are non-homogeneous and made of anisotropic viscoelastic materials that are subjected to large deformations.

The peridynamic theory is an extension of the classical continuum mechanics theory. The peridynamic governing equations involve integrals of interaction forces between near particles separated by finite distances. These forces depend upon the relative displacements between material points within a body. On the other hand, the classical governing equations involve the divergence of a tensor field, which depends upon the spatial derivatives of displacements. Thus, the peridynamic governing equations are valid not only in the interior of a body, but also on its boundary, which may include a Griffith crack, and on interfaces between two bodies with different mechanical properties. Near the boundary, the solution of a peridynamic problem may be very different from the classical solution. In this work, we investigate the behavior of the displacement field of a unidimensional linearly elastic bar of length L near its ends in the context of the peridynamic theory. The bar is in equilibrium without body force, is fixed at one end, and is subjected to an imposed displacement at the other end. The bar has micromodulus C, which is related to the Young's modulus E in the classical theory and is given by different expressions found in the literature. We find that, depending on the expression of C, the displacement field may be singular near the ends, which is in contrast to the linear behavior of the displacement field observed in the classical linear elasticity. In spite of the above, we show that the peridynamic displacement field converges to its classical counterpart as a length scale, called peridynamic horizon, tends to zero.

This work is an extension of previous investigation concerning a free energy function for an isotropic, linearly elastic peridynamic material that depends quadratically on infinitesimal normal and shear strain states. The free energy function contains four peridynamic material constants, from which three constants are related to the classical elasticity coefficients of an isotropic linear elastic material, with one of the three constants being arbitrary. To determine this arbitrary constant, the difference displacement quotient state at a point is decomposed in terms of radial and non-radial components. If the radial component is zero, the quadratic free energy function reduces to an integral expression that multiplies the arbitrary constant. This result together with a correspondence argument is used next to find a general expression for this constant. A simple experiment in mechanics is then used to evaluate this constant in terms of the classical shear modulus and the horizon \(\delta\). The correspondence argument can also be used to find a general expression for the fourth peridynamic constant that appears in the quadratic free energy function.