Higher-Order Peridynamic Material Correspondence Models for Elasticity
- PDF / 1,068,659 Bytes
- 27 Pages / 439.37 x 666.142 pts Page_size
- 87 Downloads / 220 Views
Higher-Order Peridynamic Material Correspondence Models for Elasticity Hailong Chen1
· WaiLam Chan1
Received: 23 April 2020 © Springer Nature B.V. 2020
Abstract Higher-order peridynamic material correspondence model can be developed based on the formulation of higher-order deformation gradient and constitutive correspondence with generalized continuum theories. In this paper, we present formulations of higherorder peridynamic material correspondence models adopting the material constitutive relations from the strain gradient theories. Similar to the formulation of the first-order deformation gradient, the weighted least squares technique is employed to construct the secondorder and the third-order deformation gradients. Force density states are then derived as the Fréchet derivatives of the free energy density with respect to the deformation states. Connections to the second-order and the third-order strain gradient elasticity theories are established by realizing the relationships between the energy conjugate stresses of the higher-order deformation gradients in peridynamics and the stress measures in strain gradient theories. In addition to the horizon, length-scale parameters from strain gradient theories are explicitly incorporated into the higher-order peridynamic material correspondence models, which enables application of peridynamics theory to materials at micron and sub-micron scales where length-scale effects are significant. Keywords Peridynamics · Material correspondence model · Higher-order deformation gradient · Length-scale effect · Size dependence · Strain gradient theory Mathematics Subject Classification (2010) 74A05 · 74A20 · 74A70
1 Introduction The classical continuum mechanics theories were originally developed based on the assumption that matter is continuously distributed throughout the body. These theories provide rea-
B H. Chen
[email protected] W. Chan [email protected]
1
Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506, USA
H. Chen, W. Chan
sonable basis for analyzing the behavior of continuous materials at macro-scale where material underlying microstructure can be neglected. Although these theories have been widely applied to study large and small scales problems, experiments such as those in Ref. [1–8] have repeatedly shown that the material behavior displays strong size dependence at the micron or sub-micron scales. Classical continuum mechanics theories, however, cannot explain this size dependence observed at these scales because their constitutive models possess no internal material lengths. In addition, the classical continuum mechanics theories break down at spatial discontinuities such as crack tips due to their mathematical setups. The inability to represent geometric singularities can be traced back to the kinematic requirement of a sufficiently smooth and differentiable deformation field appearing in the governing partial differential equations. In order to circumvent these limitations of the classical continuum mechanics theories,
Data Loading...
