Computational Mechanics of Damage and Fracture in Quasi-brittle Materials

Quasi-brittle materials are characterized by progressive damage and crack propagation, leading to softening and irreversible strain after the peak load is reached. Examples include concrete as well as many types of rocks, including shale, granite, and sandstone. In this talk, a field theory is presented for predicting damage and fracture in quasi-brittle materials. The approach taken here is formulated using a nonlocal constitutive law together with a phase field that is nonlocal in space and time. The displacement field inside the material is shown to be uniquely determined by an initial boundary value problem. The fracture set is characterized by the evolving phase field taking the value one inside intact material and zero in the fully damaged material. The theory satisfies energy balance, with positive energy dissipation rate in accordance with the laws of thermodynamics. Notably, these properties are not imposed but follow directly from the evolution equation by multiplying the equation of motion by the velocity and integrating by parts. The field theory uses a peridynamic interpretation of Newton’s 2nd law to evolve the displacement inside the material and there is no separate equation for phase field evolution. Instead, the phase field is part of a history dependent constitutive law dynamically coupled to the displacement field.

 

The model requires a material’s elastic moduli, the strain at the onset of nonlinearity, the ultimate tensile strength, and the fracture toughness.  Here, the characteristic length scale L is derived using geometric measure theory and is proportional to the ratio of fracture toughness to material strength. The formulation delivers a mesh-free method for predicting crack patterns.  The computational method successfully captures the cyclic load–deflection response of crack mouth opening displacement, the structural size-effect related to ultimate load, and fracture nucleating from boundary defects. It provides dynamic results identical to dynamic phase field methods of M.J. Borden, C.V. Verhoosel, M.A. Scott, T.J. Hughes, and C.M. Landis, CMAME  (2012). These results are reported in Coskun, Damircheli, and Lipton, JMPS (2025). 

Robert Lipton is the Nicholson Professor in the Department of Mathematics at Louisiana State University and an Adjunct Professor at the Center for Computation & Technology (CCT).

 

Lipton earned his B.S. in Electrical Engineering from the University of Colorado in 1981 and his M.S. (1984) and Ph.D. (1986) in Mathematics from the Courant Institute of Mathematical Sciences at New York University. Before joining LSU, he served on the faculty at Worcester Polytechnic Institute and as a C.B. Morrey Assistant Professor at the University of California, Berkeley.

 

His research focuses on the multi-scale analysis of heterogeneous media, with applications in photonics, metamaterials, fracture mechanics, and composite materials. He studies wave phenomena arising from sub-wavelength resonances and multiple scattering, non-local models of fracture and interface formation, and the influence of microstructure on macroscopic material behavior and strength.

Lipton is a Fellow of the American Mathematical Society, a Fellow of the Society for Industrial and Applied Mathematics, and a Fellow of the American Association for the Advancement of Science.