Variable stiffness boundary condition to determine effective toughness of heterogeneous materials
Variable stiffness boundary condition to determine effective toughness of heterogeneous materials
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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.
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.
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The arbitrary domain containing any source of heterogeneity, denoted as Ωd, has a length of Lx and a width of Ly. The domain Ωvsd is located at the top and bottom of Ωd and has a length of Lx and a width of Lvsd. The stiffness of the variable stiffness domain, denoted as Evsd, is a value that varies steadily along x. An initial crack of length a0 is subjected to loading.
The imposed displacement profile at the remote boundary and the effective displacement at the boundary of the main domain are shown schematically. A contour consisting of straight segments surrounds the crack tip, from Γ1 to Γ5.
Schematic of the toughness-heterogeneous domain Ωd. The domain is partitioned into three vertical regions: the central red region has a higher fracture toughness (Gc₂ = 0.15), while the two gray regions have a lower fracture toughness (Gc₁ = 0.1). The dashed line indicates the projected crack path along the symmetry plane.
The J-integral for the toughness-heterogeneous domain with Gc₁ = 0.1 and Gc₂ = 0.15.
The 3D-printed sample consists of a main domain in the middle, variable stiffness domains (VSDs) above and below the main domain, and grip sections held by the fixture. The VSD strip modulus profiles range from 0° to 90°, and the crack exhibits stable propagation.
The 3D-printed sample consists of a main domain in the middle, variable stiffness domains (VSDs) positioned above and below the main domain, and grip sections held by the fixture. The VSD strip modulus profiles range from 0° to 27°. In the sample with this lower nonlinear profile, the crack grew stably for a shorter distance before it began to jump and bifurcate, causing significant material loss.
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