Multiple cracking model in a 3D GraFEA framework

In this work, a thermodynamically consistent three-dimensional (3D) small strain-based theory to describe the deformation and fracture in quasi-brittle and brittle elastic solids is presented. The description of fracture at a material point resembles the microplane fracture approach developed by Bažant et al. (J Eng Mech 126(9):944–953, 2000, J Eng Mech 122(3): 245–254, 1996), but the present theory has the following novel features: (a) a probabilistic description of fracture propagation is used, developing evolution equations for the probability of a microcrack occurring at a given location and (b) a kinematical approach to modeling crack opening and closing. The new 3-D constitutive theory, in which elements were recently proposed by Srinivasa et al. (Mech Adv Mater Struct 80(27–30):2099–2108, 2020), has been computationally implemented within a Graph-based Finite-Element Analysis (GraFEA) framework developed by Reddy and colleagues (Khodabakhshi et al. in Meccanica 51:3129–3147, 2016, Acta Mech 230:3593–3612, 2019), and it has also been implemented into the dynamics-based Abaqus/Explicit (Reference manuals. Simulia-Dassault Systémes, 2020) finite element program through a vectorized user–material subroutine interface. Our computational approach for fracture modeling is intra-element-based, which is central to the GraFEA approach rather than inter-element fracture, as is done in cohesive zone-based numerical methods, together with selective non-locality where the non-locality is only for probability evolution motivated by population dynamic models that allows us to perform efficient implementation of the code without special elements or other numerical artifacts. Several homogeneous deformation cases for fracture in cementitious and brittle elastic materials were modeled, and the response obtained from the constitutive theory and its finite element implementation are qualitatively similar to that obtained in the literature. In particular, we show that our computational procedure is able to model crack closure in solids in a robust, relatively simple and elegant manner instead of relying on a previously developed method of decomposing the stored energy into “positive” and “negative” portions (Amor et al. in J Mech Phys Solids 57(8):1209–1229, 2009, Miehe et al. in Int J Numer Meth Eng 83:1273–1311, 2010).