Variable-order fracture mechanics and its application to dynamic fracture

Abstract

This study presents the formulation, the numerical solution, and the validation of a theoretical framework based on the concept of variable-order mechanics and capable of modeling dynamic fracture in brittle and quasi-brittle solids. More specifically, the reformulation of the elastodynamic problem via variable and fractional-order operators enables a unique and extremely powerful approach to model nucleation and propagation of cracks in solids under dynamic loading. The resulting dynamic fracture formulation is fully evolutionary, hence enabling the analysis of complex crack patterns without requiring any a priori assumption on the damage location and the growth path, and without using any algorithm to numerically track the evolving crack surface. The evolutionary nature of the variable-order formalism also prevents the need for additional partial differential equations to predict the evolution of the damage field, hence suggesting a conspicuous reduction in complexity and computational cost. Remarkably, the variable-order formulation is naturally capable of capturing extremely detailed features characteristic of dynamic crack propagation such as crack surface roughening as well as single and multiple branching. The accuracy and robustness of the proposed variable-order formulation are validated by comparing the results of direct numerical simulations with experimental data of typical benchmark problems available in the literature.

An Enhanced Lemaitre Model and Fracture Map for Cr5 Alloy Steel during High-Temperature Forming Process

“…The Lemaitre damage model is a damage model that clearly expresses the damage of materials and considers more parameters and variables, and can be directly applied to guide the production and manufacture of products in industrial production. The traditional Lemaitre damage model considers the material to be damaged or destroyed when the absolute damage value D suffered by the material during processing is greater than or equal to the critical damage value D c of the material itself, and its mathematical formula is expressed as follows [ 33 ]: where dD represents the absolute damage value increment; represents the stress triaxiality function; represents the stress triaxiality; E represents the elastic modulus of the material; S represents the anti-damage factor; represents the equivalent plastic strain; represents the equivalent plastic strain increment; v is the Poisson’s ratio of the material; σ m represents the mean stress; σ represents the equivalent stress experienced by the material.…”

Fractional Modeling in Action: A Survey of Nonlocal Models for Subsurface Transport, Turbulent Flows, and Anomalous Materials

“…Finally, variable-order operators also proved to be useful mathematical tools to determine the onset of fracture. Patnaik and Semperlotti [175] employed a variable fractional-order activation function for damage, where the sharp power-law activation threshold induced by the fractional operator was successfully employed to determine crack propagation and branching of brittle materials. We refer the reader to the recent review works on the use of variable-order [176] and distributed-order [69] fractional models in viscoelasticity and structural mechanics.…”

“…In practice, FPDEs appear within tractable mathematical models for anomalous transport, ranging from complex fluids to non-Newtonian rheology and the design of aging materials [241,118,147,107,108], but also in modeling transport phenomena when rates of change in the quantity of interest depend on space or time. In this context, FPDEs with "variable orders" can be exploited in diverse physical and biological applications [176,175,263] to capture transitions between different transport regimes. Moreover, even classical long-standing issues such as monotonicity, anisotropy, and multi-fractal scaling laws in turbulence can be reformulated and reinterpreted in the context of fractional calculus and probability theory.…”

A New Updated Reference Lagrangian Smooth Particle Hydrodynamics Framework for Large Strain Solid Dynamics and its Extension to Dynamic Fracture

“…Two versions of this experiment are widely studied in the literature and are known either as the Kalthoff-Winkler experiment (double notched plate) or the Zhou-Rosakis-Ravichandran experiment (single notched plate) [117]. Many authors have studied the problem in a variety of computational fracture mechanics approaches [1,69,114,[117][118][119][120][121][122][123][124][125][126]. The model presented herein consists of a double notched plate and follows the characteristics presented in [69,114], i.e., a two-dimensional (plane strain) rectangular plate with a length of L = 0.200m, and width W = 0.100m, the main characteristics of the model are illustrated in figure 9.5.…”

“…This example illustrates the prediction of dynamic crack propagation in a beam with an offset edge crack subjected to an impact load. Experimental data for this classic benchmark problem is available in [127], many authors have studied the same problem in computational mechanics ( [1,119,122,124,125,[128][129][130]). The model presented herein consists of a two-dimensional (plane strain) rectangular beam with a length of L = 0.2286m and width W = 0.0762m, the main characteristics of the model are illustrated in figure 9.8.…”