Variational damage model: A new paradigm for fractures

Abstract The computational modeling of fracture in solids using damage mechanics faces challenges with complex crack topology. This can be addressed by using a variational framework to reformulate the damage mechanics. In this paper, we propose several mathematically elegant variational damage models (VDMs) for fracture mechanics without explicitly using damage variables. Based on the energy density ϕ , the fracture energy density is formulated as $$\hat{\phi}=\phi(1+\ell \phi/G_{\mathrm{c}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:math> and the damage variable is expressed as s = ϕ /( ϕ + G c / ℓ ), which satisfy $$\hat{\phi}\mid_{\phi \rightarrow\infty}=G_{\mathrm{c}}/\ell$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>ϕ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> <mml:msub> <mml:mo>∣</mml:mo> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>ℓ</mml:mi> </mml:math> and s ∣ ϕ →∞ = 1 as ϕ approaches infinity. These limits demonstrate that the new energy density converges to the Griffith energy release rate at full-damage state. The VDM profoundly modified the energy functional, implicitly incorporating the damage field. As a generalization of previous model, we propose a family of VDMs of varying orders. Additionally, we develop a multi-damage model to account for different types of energy densities, such as elastic thin plate and gradient elasticity. Using this functional, it is straightforward to deduce the governing equation for automatically evolving fractures. These formulations can be employed in conventional finite element method or other numerical methods with minimal modifications. Compared to the phase field method with the same mesh density, a sharper crack interface can be achieved. We demonstrate the capabilities of the proposed variational damage formulations using representative numerical examples.