Adair R. Aguiar

A constitutive model for a linearly elastic peridynamic body

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.

Self-intersection in elasticity

Abstract This study represents a contribution to the local behavior of the solutions of the governing equations of elastostatics in the vicinity of corners. It contains an asymptotic investigation – within both the linear and two special nonlinear theories of plane strain – of the deformation field near a corner point that separates a free and a fixed part of the boundary. Our computations confirm that the asymptotic solutions represent the local behaviors of equilibrium states very close to the corner. In this neighborhood the linear and nonlinear theories predict very different behavior. Away from the corner the deformation gradient becomes small and the asymptotic solution in the nonlinear theory becomes – expectedly – not valid. However, in an intermediate region our numerical results obtained from both the linear and nonlinear theories show striking similarities and predict a novel behavior of the free surface.

A SINGULAR PROBLEM IN INCOMPRESSIBLE NONLINEAR ELASTOSTATICS

This paper represents a contribution to the numerical treatment of problems in incompressible elasticity theory for large deformations. We are especially concerned about the solution of plane problems with corners. A review of the literature on these problems indicates that the behavior of the solution in the vicinity of a corner is given little attention. We investigate the solution of the compressed bonded block problem corresponding to the compression of an incompressible elastic block of rectangular cross-section and innite transverse length between two opposing bonded rigid surfaces, with the two remaining lateral faces traction-free. We are especially interested in the behavior at a corner where a bonded end is adjacent to a free lateral side. We employ a nite element method based on a reduced and selective integration technique with penalization to construct a numerical solution for this problem. Our computational method converges everywhere except in a small neighborhood of the corner. We appeal to an elementary ap rioriinequality concerning the angle of shear to show that the numerical calculations in this neighborhood are inaccurate and need a more rened study. Based on the inequality, we oer a conjecture concerning the local shape of the deformed free lateral surface at the corner.

Effective Electromechanical Properties of 622 Piezoelectric Medium With Unidirectional Cylindrical Holes

The asymptotic homogenization method (AHM) yields a two-scale procedure to obtain the effective properties of a composite material containing a periodic distribution of unidirectional circular cylindrical holes in a linear transversely isotropic piezoelectric matrix. The matrix material belongs to the symmetry crystal class 622. The holes are centered in a periodic array of cells of square cross sections and the periodicity is the same in two perpendicular directions. The composite state is antiplane shear piezoelectric, that is, a coupled state of out-of-plane shear deformation and in-plane electric field. Local problems that arise from the two-scale analysis using the AHM are solved by means of a complex variable method. For this, the solutions are expanded in power series of Weierstrass elliptic functions, which contain coefficients that are determined from the solutions of infinite systems of linear algebraic equations. Truncating the infinite systems up to a finite, but otherwise arbitrary, order of approximation, we obtain analytical formulas for effective elastic, piezoelectric, and dielectric properties, which depend on both the volume fraction of the holes and an electromechanical coupling factor of the matrix. Numerical results obtained from these formulas are compared with results obtained by the Mori‐Tanaka approach. The results could be useful in bone mechanics. [DOI: 10.1115/1.4023475]

Boundary Layer Effects in a Finite Linearly Elastic Peridynamic Bar

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 Youngs 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.

On the Number of Invariants in the Strain Energy Density of an Anisotropic Nonlinear Elastic Material with Two Material Symmetry Directions

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.

Syzygy entre os Invariantes da Viscoelasticidade Não-Linear Isotrópica.

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.