A constitutive model for a linearly elastic peridynamic body

Abstract

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.

On the Determination of a Peridynamic Constant in a Linear Constitutive Model

“…Preliminary results of this work are detailed in Aguiar and Fosdick [1]. Let B ⊂ E 3 be the undistorted reference configuration of a peridynamic body at time t = 0 relative to a fixed frame F and let y := χ(x, t) be the position of the particle x ∈ B at time t ≥ 0 relative to the frame F in E 3 .…”

“…With the definitions (3) through (5), Aguiar and Fosdick [1] propose the following reduced quadratic form for the free energy function of an isotropic simple elastic material,…”

“…In the absence of residual force vector state, the linearized force response function state is given by (Aguiar and Fosdick [1])…”

“…In Sect. 3 we write the expression of the quadratic free energy function proposed by Aguiar and Fosdick [1] in terms of the radial and non-radial components of h and show that it reduces to the energy density function proposed by Silling et al [4] near the natural state of a peridynamic body if two peridynamic constants are zero and a certain weighting function has a multiplicative decomposition. In Sect.…”

“…In this work we want to find the peridynamic constant α 33 , which is one of the two constants that were left undetermined in Aguiar and Fosdick [1]. For this, we decompose the difference displacement quotient state, h, at a point in terms of radial and non-radial components and set the radial component, ϕ, equal to zero.…”