Crack studies are essential for predicting the failure and lifespan of materials under various loading, enabling the development of more resilient structures and enhancing safety and reliability. This knowledge aids in the design of components that can withstand higher loads and harsh environments, minimizing catastrophic failures. Numerical methods such as FEM, XFEM, CZM, and non-local damage models enable detailed simulation of crack initiation, propagation, and interaction, offering crucial insights for designing resilient materials and structures under various loading conditions. And more recently, the phase-field model has proven itself to be a superior method for modeling crack propagation in brittle fracture. By using a continuous development of field variables across an interfacial zone, the phase-field model blurs boundaries, offering a realistic representation of crack growth. This method, inspired by the Ginzburg-Landau equation, utilizes an order parameter to represent cracks, facilitating accurate simulations of gradient crack evolution through the incorporation of the damage gradient and a crack width parameter.

In this work, we conduct an in-depth examination of phase-field modeling for brittle fracture, focusing on five established methods for elastic strain energy decomposition: strain spectral, stress spectral, strain volumetric/deviatoric, and the orthogonal decomposition of the stiffness tensor. Additionally, we introduce a novel sixth method involving the weakest cleavage plane, formulated using the compliance tensor. We evaluate these methods under tension/compression and pure shear testing to highlight their respective advantages and limitations across different loading scenarios.

This study aims to honor past research while advancing phase-field modeling methodologies, ultimately enhancing our understanding of material degradation and fracture mechanics. By comparing these decomposition methods, we provide valuable insights into their performance and applicability, contributing to the development of more accurate and reliable models for predicting material failure.