Variable stiffness boundary condition to determine effective toughness of heterogeneous materials

We present a computational framework that combines a variable stiffness boundary condition (VSBC) with a phase-field model to evaluate the effective fracture toughness of heterogeneous materials. It is built on the so-called surfing boundary condition (SBC) that applies nonuniform displacement at the remote boundary to stabilize crack-propagation in a heterogeneous medium. Unlike SBC, VSBC is easily implementable in traditional universal testing machines and commercial software packages. The VSBC passively translates a simple, uniform remote displacement into a non-uniform load via an engineered stiffness gradient, enabling the stable, natural propagation of cracks along energetically favorable paths. The framework is validated on homogeneous materials, where the calculated $J$-integral precisely matches the prescribed fracture toughness. When applied to heterogeneous domains, the VSBC method successfully quantifies the increase in effective toughness due to stiffness and toughness contrasts and captures the critical transition from crack penetration to deflection. Experimental validation using 3D-printed samples confirms the model’s predictive capability. The VSBC framework provides a robust easily accessible tool for investigating fracture in complex materials and guide the design of advanced, fracture-resistant composites.