Hybrid lattice particle modeling: Theoretical considerations for a 2D elastic spring network for dynamic fracture simulations G. Wang a, * , A. Al-Ostaz a , A.H.-D. Cheng a , P.R. Mantena b a Department of Civil Engineering, University of Mississippi, Oxford, MS 38677-1848, United States b Department of Mechanical Engineering, University of Mississippi, Oxford, MS 38677-1848, United States article info Article history: Received 30 May 2008 Received in revised form 1 July 2008 Accepted 23 July 2008 Available online 7 September 2008 PACS: 68.45.kg 61.43.Bn 40.30.My 46.30.Nz 83.20._d 83.80.Nb Keywords: Particle modeling Lattice model Dynamic fracture Impact Constitutive relations abstract A new hybrid lattice particle modeling (HLPM) scheme is proposed. The particle–particle interaction is derived from lattice modeling (LM) theory, whereas the computational scheme follows particle modeling (PM) technique. The newly proposed HLPM considers different particle interaction schemes, involving not only particles in the nearest neighborhood, but also the second nearest neighborhood. Different mesh structures with triangular or rectangular unit cells can be used. The current paper is concerned with the mathematical derivations of elastic interaction between contiguous particles in 2D lattice networks, accounting for different types of linkage mechanism and different shapes of lattice. Axial (a) and com- bined axial-angular (a  b) models are considered. Derivations are based on the equivalence of strain energy stored in a unit cell with its associated continuum structure in the case of in-plane elasticity. Con- ventional PM technique was restricted to a fixed Poisson’s ratio and had a strong bias in crack propaga- tion direction, as a result of the geometry of the adopted lattice network. The current HLPM is free from the above-mentioned deficiencies and can be applied to a wide range of impact and dynamic fracture fail- ure problems. Although the current analysis is based on the linear elastic spring model, inelastic consid- erations can be easily implemented, as HLPM has the same force interaction scheme as PM, based on the Lennard–Jones potential. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction Since its introduction in the 1980s by Greenspan [1], particle modeling (PM) has found good success in a number of applications [2–10]. PM is a numerical technique similar to the molecular dy- namic (MD) simulation; but rather than simulating actual atoms, it is based on lumped mass particles distributed on a grid to allow macro scale modeling. The PM utilizes an equivalent Lennard– Jones potential to model the nonlinear constitutive law at the continuum, macroscopic level. The mass has inertia that obeys Newton’s second law of motion. It is a Lagrangian model that keeps track of particle location and velocity. Despite its success, the PM has a few deficiencies. One major shortcoming is that in the modeling of solid, the potential type for- mulation allows only one elastic constant to be modeled, which is typically selected as the bulk modulus, or Young’s modulus. The second elastic constant present in an isotropic material, say, Poisson’s ratio, becomes the property of the grid system used, such as the triangular or rectangular networks. The use of spring network model, or lattice model (LM), to mod- el elastic solid has more than 60 years of history [11]. Literature re- view on the successive development of these models can be found in [12–15]. Lattice modeling technique has been widely applied in the computation of effective elastic moduli and simulation of brit- tle intergranular fracture (BIF) in ceramics, intermetallics, and refractory metals [16–21]. Lattice model, when applied in its original form, also has the deficiency that only one elastic constant can be modeled. For example, on a 2D equilateral triangular lattice, the equivalent Poisson’s ratio is fixed to the value of 1/3. This situation can be remedied by the use of the more advanced lattice models. For example, Ostoja–Starzewski [14,15] has manipulated several types of spring systems, including central (a), angular (b) and the mixed (a  b) interactions, coupled with different lattice net- works, triangular, rectangular, etc. that allows the modeling of two elastic constants of isotropic materials, as well as the poten- tial for applying to anisotropic materials. Other deficiencies of the conventional LM include the fact that it is of an Eulerian model. It does not allow the particles to move around and even be frag- mented from the main lattice. It is also basically a static model, and does not model the dynamic fracture process. In Table 1, 0927-0256/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2008.07.032 * Corresponding author. Tel.: +662 915 5369; fax: +662 915 5523. E-mail address: gewang@olemiss.edu (G. Wang). Computational Materials Science 44 (2009) 1126–1134 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci