Kinetic relations and local energy balance for LEFM from a nonlocal peridynamic model
- PDF / 864,438 Bytes
- 15 Pages / 547.087 x 737.008 pts Page_size
- 4 Downloads / 182 Views
ORIGINAL PAPER
Kinetic relations and local energy balance for LEFM from a nonlocal peridynamic model Prashant K. Jha
· Robert P. Lipton
Received: 23 March 2020 / Accepted: 24 August 2020 © Springer Nature B.V. 2020
Abstract A simple nonlocal field theory of peridynamic type is applied to model brittle fracture. The kinetic relation for the crack tip velocity given by Linear Elastic Fracture Mechanics (LEFM) is recovered directly from the nonlocal dynamics, this is seen both theoretically and in simulations. An explicit formula for the change of internal energy inside a neighborhood enclosing the crack tip is found for the nonlocal model and applied to LEFM. Keywords Fracture · Peridynamics · LEFM · Fracture toughness · Stress intensity · Local power balance
1 Introduction The fracture of solids can be viewed as a collective interaction across length scales. Application of sufficient stress or strain to a brittle material breaks atomistic bonds leading to fracture at macroscopic scales. This material is based upon work supported by the U. S. Army Research Laboratory and the U. S. Army Research Office under contract/grant number W911NF1610456. P. K. Jha Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA e-mail: [email protected] R. P. Lipton (B) Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA e-mail: [email protected]
The appeal of a nonlocal fracture theory like peridynamics Silling (2000), Silling et al. (2007) is that fracture is captured as an emergent phenomenon. At the same time such a theory needs to recover the established theory of dynamic fracture mechanics described in Freund (1990), Ravi-Chandar (2004), Anderson (2005), Slepian (2002) as a limiting case. Motivated by these observations we consider a nonlocal peridynamic model (cohesive dynamics) proposed in Lipton (2014, 2016). The length scale of nonlocal interaction between any material point and its neighbors is called the horizon. Here the force strain relation between two points is linear elastic for small strains, softens under sufficiently large strain and ultimately becomes zero, see Fig. 2. In this nonlocal model displacement gradients can become steep and localize onto thin regions, see Jha and Lipton (2020). This model is used to show the kinetic relation for the velocity of the crack tip given by LEFM Freund (1990) follows in the limit of vanishing horizon. In this paper the kinetic relation of LEFM is recovered from the nonlocal model in two different ways. The first approach to recovering the kinetic relation is to note that the same equation of motion applies everywhere in the body for the nonlocal model. We use this to show that local power balance is given by the stationarity in time of the internal energy of a small domain containing the crack tip. The change in internal energy is shown to be the difference between the elastic energy flowing into the crack and the kinetic energy
123
P. K. Jha, R
Data Loading...
