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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.

Although photobiomodulation therapy (PBM) has been applied clinically for the treatment of pain and inflammation, wound healing, sports and soft tissue injuries, as well as to repair injured spinal cords and peripheral nerves, it remains unclear which molecular substrates (receptor) are implicated in the cellular mechanisms of PBM. Here, we reported that PBM (660 nm, 30 mW, 0.06 cm

De um modo geral, estruturas biologicas sao anisotropicas, heterogeneas e estao sujeitas a grandes deformacoes. Muito embora seja comum a utilizacao da teoria de elasticidade para modelar o comportamento destas estruturas, sabe-se que o comportamento de, por exemplo, tendoes e ligamentos, e melhor modelado no contexto da teoria de viscoelasticidade. Neste trabalho considera-se uma classe de materiais viscoelasticos compressiveis e isotropicos do tipo diferencial de primeira ordem que satisfaz o principio da invariancia sob mudanca de observador. Neste caso, a funcao resposta mecanica que fornece a tensao de Cauchy depende do tensor deformacao de Cauchy-Green a esquerda B e da parte simetrica do gradiente de velocidade D. Utilizando a teoria de representacao de funcoes isotropicas, mostra-se que esta funcao resposta e dada em termos de produtos dos tensores B e D e de coeficientes multiplicando estes produtos que dependem de dez invariantes dos mesmos. Mostramos que somente nove dos dez invariantes sao independentes e que existe um syzygy entre os mesmos. Consequentemente, quaisquer outros conjuntos de dez invariantes, determinados de forma unica deste conjunto, tem somente nove invariantes independentes. Esta investigacao e relevante em procedimentos experimentais empregados na caracterizacao de relacoes constitutivas de tecidos biologicos.

This study concerns the local behavior of the solutions of the governing equations of non-linear elastostatics in the vicinity of the corner of a wedge-shaped region of angle α ∈ ( 0 , 2 π ] . It contains an asymptotic investigation in the plane strain regime – using a subclass of non-linear harmonic elastic solids that have a proper asymptotic constitutive structure in the vicinity of the corner – of the deformation field near the corner point that separates a free from an adjoining fixed segment of the boundary. It is well-known that, as the corner point is approached, the singular field behavior predicted by the classical linear theory of elasticity yields oscillatory deformations that are not injective. This anomalous behavior is related to a spurious self-intersection anomaly. Using a proper constitutive structure within a subclass of non-linear harmonic materials in the vicinity of the corner, we obtain an asymptotic expansion of the deformation field for which the foregoing anomalous behavior is avoided. The conditions on this field which guarantee injectivity lead to an unexpected behavior of the deformed free surface that is physically possible. In the particular case of α = π , the second-order expansion of the deformation field agrees with its counterpart found elsewhere in the literature. For another particular case, concerning α = π / 2 , the second-order expansion is an improvement over its counterpart found in the literature.

Constitutive modeling within peridynamic theory considers the collective deformation at each time of all the material within a δ-neighborhood of any point of a peridynamic body. The assignment of the parameter δ, called the horizon, is treated as a material property. The difference displacement quotient field in this neighborhood, rather than the extension scalar field, is used to generate a three-dimensional state-based linearly elastic peridynamic theory. This yields an enhanced interpretation of the kinematics between bonds that includes both length and relative angle changes. A free energy function for a linearly elastic isotropic peridynamic material that contains four material constants is proposed as a model, and it is used to obtain the force vector state and the associated modulus state for this material. These states are analogous to, respectively, the stress field and the fourth-order elasticity tensor in classical linear theory. In the limit of small horizon, we find that only three of the four peridynamic material constants are related to the classical elastic coefficients of an isotropic linear elastic material, with one of the three constants being arbitrary. The fourth peridynamic material constant, which accounts for the coupling effect of both bond length and relative angle change, has no effect on the limit, but remains a part of the peridynamic model. The determination of the two undetermined constants is the subject of future investigation. Peridynamic models proposed elsewhere in the literature depend on the deformation state through its dilatational and deviatoric parts and contain only two peridynamic material constants, in analogy to the classical linear elasticity theory. Observe from above that our model depends on both length and relative angle changes, as in classical linear theory, but, otherwise, is not limited to having only two material constants. In addition, our model corresponds to a nonordinary material, which represents a substantial break with classical models.

The neurotoxin 1-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP) induces motor and nonmotor dysfunctions resembling Parkinson's disease (PD); however, studies investigating the effects of 1-methyl-4-phenylpyridinium (MPP

A unidirectional fiber composite is considered here, the fibers of which are empty cylindrical holes periodically distributed in a transversely isotropic piezoelectric matrix. The empty-fiber cross-section is circular and the periodicity is the same in two directions at an angle π/2 or π/3. Closed-form formulae for all electromechanical effective properties of these 3–1 longitudinally periodic porous piezoelectric materials are presented. The derivation of such expressions is based on the asymptotic homogenization method as a limit of the effective properties of two-phase transversely isotropic parallel fiber-reinforced composites when the fibers properties tend to zero. The plane effective coefficients satisfy the corresponding Schulgasser–Benveniste–Dvorak universal type of relations. A new relation among the antiplane effective constants from the solutions of two antiplane strains and potential local problems is found. This relation is valid for arbitrary shapes of the empty-fiber cross-sections. Based on such a relation, and using recent numerical results for isotropic conductive composites, the antiplane effective properties are computed for different geometrical shapes of the empty-fiber cross-section. Comparisons with other analytical and numerical theories are presented.

There are problems in the classical linear theory of elasticity whose closed form solutions, while satisfying the governing equations of equilibrium together with well-posed boundary conditions, predict the existence of regions, often quite small, inside the body where material overlaps. Of course, material overlapping is not physically realistic, and one possible way to prevent it combines linear theory with the requirement that the deformation field be injective. A formulation of minimization problems in classical linear elasticity proposed by Fosdick and Royer [3] imposes this requirement through a Lagrange multiplier technique. An existence theorem for minimizers of plane problems is also presented. In general, however, it is not certain that such minimizers exist. Here, the Euler–Lagrange equations corresponding to a family of three-dimensional problems is investigated. In classical linear elasticity, these problems do not have bounded solutions inside a body of anisotropic material for a range of material parameters. For another range of parameters, bounded solutions do exist but yield stresses that are infinite at a point inside the body. In addition, these solutions are not injective in a region surrounding this point, yielding unrealistic behavior such as overlapping of material. Applying the formulation of Fosdick and Royer on this family of problems, it is shown that both the displacements and the constitutive part of the stresses are bounded for all values of the material parameters and that the injectivity constraint is preserved. In addition, a penalty functional formulation of the constrained elastic problems is proposed, which allows to devise a numerical approach to compute the solutions of these problems. The approach consists of finding the displacement field that minimizes an augmented potential energy functional. This augmented functional is composed of the potential energy of linear elasticity theory and of a penalty functional divided by a penalty parameter. A sequence of solutions is then constructed, parameterized by the penalty parameter, that converges to a function that satisfies the first variation conditions for a minimizer of the constrained minimization problem when this parameter tends to infinity. This approach has the advantages of being mathematically appealling and computationally simple to implement.

Results concerning the behavior of interfaces in the presence of multiple scattering are obtained here for the first time. To characterize interface effects, we consider an elastic slab of finite thickness containing a random distribution of circular cylindrical cavities. Using elasticity theory, and taking first-order multiple scattering effects between cavities into account, we obtain two new results. First, inside the slab, within one cavity-diameter from the two slab boundaries, the coherent displacement is strongly inhomogeneous (with singular first and second derivatives). Second, inside the slab, away from the preceding singular regions, the coherent displacement is governed by a complex-valued wavenumber (which is independent of slab thickness). Exhaustive analytical work, together with time-consuming finely-tuned numerical work, has been performed, which has allowed us to evaluate the complex-valued wavenumber. We plot the corresponding speed and attenuation as functions of frequency for various cavity densities. This work will help researchers improve the precision of ultrasound technologies in materials engineering or biomaterials engineering.