Journal of Elasticity最新文献

The representation theory of tensor functions is a powerful mathematical tool for constitutive modeling of anisotropic materials. A major limitation of the traditional theory is that many point groups require fourth- or sixth-order structural tensors, which significantly impedes practical engineering applications. Recent advances have introduced a reformulated representation theory that enables the modeling of anisotropic materials using only lower-order structural tensors (i.e., second-order or lower). Building upon the reformulated theory, this work establishes the representations of tensor functions for three-dimensional centrosymmetric point groups. For each point group, we propose a lower-order structural tensor set and derive the representations of tensor functions explicitly. For scalar-valued and second-order symmetric tensor-valued functions, our theory is indeed applicable to all three-dimensional point groups because their representations are determined by the corresponding centrosymmetric groups. The representation theory presented here is broadly applicable for constitutive modeling of anisotropic materials.

Materials are known to behave in strange and novel ways in the neighborhood of critical points. The softening of various material moduli is commonly reported, and the smooth change of homogeneous states into complex multiphase microstructures is possible. For elastic solids, the analysis of this behavior is complicated because the full notion of the stress and deformation gradient tensor fields, including shearing, must be considered rather than simply the classical effects associated with pressure, specific volume, and temperature. Here, we are concerned with sequences of static (equilibrium) coexistent phases, induced by thermal and mechanical loading, and the asymptotic limits and relations between various thermodynamic fields for nonlinear elastic solids in the neighborhood of a critical point. (For fluids and gasses, see the works of Fisher (J. Math. Phys. 5:944–962, 1964), Fisher (Rep. Prog. Phys. 30:615–730, 1967), Griffiths (J. Chem. Phys. 43:1958–1968, 1965), and the monograph of Rowlinson and Swinton (Liquids and Liquid Mixtures. Butterworths Monographs in Chemistry. London, 1982).) A generalized form of the famous Rushbrooke inequality from physical chemistry is obtained.

We solve the inverse problems in both three-dimensional and two-dimensional elasticity associated with the design of harmonic ellipsoidal and elliptical isotropic elastic inhomogeneities with spring-type imperfect interface. The first invariant of the stress tensor in the infinite isotropic elastic matrix subjected to uniform remote normal stresses remains unchanged after the introduction of the harmonic inhomogeneity. The jump in normal displacement across the interface is proportional, in terms of the corresponding three-variable or the two-variable imperfect interface function, to the normal traction; the interface does not sustain any shear traction. The three-dimensional inverse problem is solved via the use of a Newtonian potential and its two-dimensional counterpart via Muskhelishvili’s complex variable method. In order to achieve the harmonic condition, the two ratios of the remote normal stresses for the three-dimensional problem and the single ratio of the remote normal stresses for the two-dimensional problem are uniquely determined for given geometric and material parameters.

We derive an effective model for a periodic chain of linearly-elastic springs, achieving second-order accuracy in the scale separation parameter (varepsilon ll 1). The chain has finite length and is made up of springs connecting both nearest- and next-nearest-neighbors: it serves as a one-dimensional prototype for higher-order periodic homogenization problems with boundaries. This type of problem has been approached by inserting two-scale expansions into the equations of equilibrium in the bulk and by matching them with boundary-layer solutions. We explore an alternative method operating at the energy level, bypassing the cumbersome matching procedure. We start from an ansatz of the microscopic displacement accounting for both boundary layers and for small-scale fluctuations in the bulk, and insert it into the discrete energy. This yields a continuous energy functional depending on the macroscopic displacement (u), in the form of a series expansion in powers of (varepsilon ). We call it a pseudo-energy (Phi _{varepsilon } [u]) as it is not positive when truncated at order (varepsilon ^{2}). The boundary terms in the pseudo-energy account for boundary layers in an effective way. By making the pseudo-energy stationary order by order in (varepsilon ), we derive the homogenized equations of equilibrium along with effective boundary conditions. We provide quantitative validation showing that the effective model is correct to second order. We point out the special form of the effective higher-order tractions, which has been overlooked in the strain-gradient theories proposed so far.

For theories working with fully implicit constitutive relations we find conditions for strict hyperbolicity for the system of elastodynamic equations in one space dimension. To bypass the fact that stress is only implicitly related to strain and not explicitly, we utilize three partial differential equations to form our system: the momentum equation, the compatibility equation as well as the time differentiated fully implicit constitutive law. This way we built up a system of three quasilinear partial differential equations of first order for the velocity ((u_{t})), the displacement gradient ((u_{x})) (or the strain ((e))) as well as the stress ((sigma )). For this system we find conditions for strict hyperbolicity using the characteristic polynomial. We do that for cases where the implicit constitutive laws are either of the form (f(sigma , u_{x})=0), or (f(sigma , e)=0). The same method applies also to constitutive laws of the form (e=g(sigma )).

Shock wave propagation in nonlinear viscoelastic solids is fundamental to understanding their response to rapid dynamic loading. We analyze shock wave structures within a recent symmetric hyperbolic nonlinear viscoelastic model, proposed by Ruggeri (in Int. J. Non-Linear Mech. 160, 2024) and derived within the framework of Rational Extended Thermodynamics. An explicit expression is obtained for the critical Mach number at which a smooth traveling wave loses its (C^{1}) regularity and a sub-shock forms. For shocks propagating into an undeformed equilibrium state, the critical Mach number depends solely on the linearized material response, in particular on the ratio of the elastic moduli in the Zener constitutive model. The model admits both compressive and tensile (expansive) shock waves. Numerical simulations based on a Mooney–Rivlin elastic potential coupled with a quadratic viscous energy confirm the theoretical predictions for shock propagation in vulcanized rubber.

Having been studied for a long time, simple and pure shear deformation models are well known in elasticity. For small deformations, these two shear models differ only by a rigid rotation. On the contrary, for large deformations, the two models do not differ only by a rigid rotation. Therefore, in the latter situation, one cannot expect both models to fit the same constitutive properties for a prescribed form of the stored energy function. The kinematic and static differences between the two shear models, as well as their inadequacies for simulating experimental evidences, are discussed in this paper. To overcome these critical issues, a study was developed, which led quite naturally to the definition of a new shear deformation model, here called purely angular shear, based on the direct extension of the linearized pure shear model. The new model, characterized by a simple and immediate physical meaning, is particularly suitable for matching experimental tests.

A new formulation is proposed for linearized elastic solids, which can be used for the analysis of boundary value problems. This formulation is based on considering both the displacement field and the stress tensor as main variables for the problem, solving in parallel the equation of motion and the constitutive equation (expressing the linearized strain as a function of the stresses) to find such unknown variables. Some boundary value problems are solved using separation of variables for the fully dynamic case for isotropic bodies. The application of the above method is briefly considered for anisotropic bodies, and also for a relatively new class of constitutive equation, wherein the linearized strain is a nonlinear function of the stress.

We apply the formalism of analytical mechanics for constrained systems to reformulate the equilibrium theory of uniaxial nematic elastomers, allowing for constitutive dependence on the gradient (boldsymbol{G}) of the director (boldsymbol{n}). In this setting, inextensibility is enforced by requiring that (|boldsymbol{n}|^{2}=1) and that (boldsymbol{G}^{top }boldsymbol{n}=boldsymbol{0}). Starting from these constraints, and using the principle of virtual work within a thermomechanically consistent framework, we derive boundary-value problems for determining equilibrium configurations. We show that the original formulation yields an underdetermined system for the Lagrange multiplier fields unless ancillary gauge conditions are imposed. To resolve this indeterminacy, we introduce two effective Lagrange multiplier fields: one defined in the interior of the referential region and the other on that portion of the boundary where the director traction is prescribed.