[Paper Review] Quasistatic crack growth in finite elasticity

Gianni Dal Maso, Gilles A. Francfort+1 more|ArXiv.org|Jan 16, 2004

Composite Material MechanicsEngineering23 references16 citations

TL;DR

This paper establishes the existence of quasistatic crack growth in n-dimensional finite elasticity using a variational model based on energy minimization, where crack paths emerge naturally from energy balance. The key contribution is a rigorous existence result for irreversible evolution of minimum energy configurations under time-dependent boundary conditions and quasiconvex bulk energy, valid for arbitrary n ≥ 1 and general heterogeneous, anisotropic materials.

ABSTRACT

In this paper, we prove a new existence result for a variational model of crack growth in brittle materials proposed in [15]. We consider the case of $n$-dimensional finite elasticity, for an arbitrary $n\ge1$, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, both depending on time.

Motivation & Objective

• To establish a rigorous mathematical framework for quasistatic crack growth in brittle materials governed by energy balance.
• To extend the variational model of crack propagation to n-dimensional finite elasticity with general quasiconvex bulk energy.
• To prove the existence of an irreversible quasistatic evolution of minimum energy configurations under time-dependent boundary conditions and applied loads.
• To handle general heterogeneous and anisotropic materials by allowing crack surface energy density to depend on position and normal direction.
• To develop a functional setting in GSBV spaces that accommodates discontinuous deformations with jump sets contained in evolving cracks.

Proposed method

• Formulate the problem in the space of functions of generalized bounded variation (GSBV) to model deformations with discontinuities on codimension-one sets.
• Define the total energy as the sum of elastic energy, crack surface energy, and negative work done by applied loads, with crack surface energy given by an integral over the crack with anisotropic, heterogeneous density κ.
• Use quasiconvexity of the stored energy function W(x,ξ) to ensure lower semicontinuity of the elastic energy functional.
• Model time-dependent boundary conditions and applied loads via conservative forces with work functionals F(t,x,u) and G(t,x,u) satisfying specific regularity and growth conditions.
• Define admissible configurations as pairs (u, Γ) where u has jump set S(u) ⊆ Γ and satisfies time-dependent Dirichlet traces on ∂DΩ \ Γ.
• Define minimum energy configurations as those minimizing the total energy over all larger cracks and compatible deformations, ensuring energy balance via Griffith’s criterion.

Experimental results

Research questions

• RQ1 Can a variational model of quasistatic crack growth be rigorously established in n-dimensional finite elasticity with general quasiconvex energy and time-dependent loads?
• RQ2 How can the crack path be determined intrinsically through energy minimization without prior prescription?
• RQ3 What functional analytic framework is suitable for handling discontinuous deformations and evolving cracks in finite elasticity?
• RQ4 Under what conditions on the energy density and applied loads does an irreversible quasistatic evolution of minimum energy configurations exist?
• RQ5 How can the model accommodate heterogeneous and anisotropic materials through position- and orientation-dependent crack surface energy?

Key findings

• The paper proves the existence of an irreversible quasistatic evolution of minimum energy configurations in n-dimensional finite elasticity for arbitrary n ≥ 1.
• The existence result holds under general assumptions: quasiconvexity of the bulk energy W(x,ξ), and suitable growth and regularity conditions on the applied loads and crack surface energy κ.
• The crack path is not prescribed but emerges as part of the solution, with the crack set Γ(t) evolving in time as a rectifiable set of finite (n−1)-Hausdorff measure.
• The model allows for heterogeneous and anisotropic materials through a crack surface energy functional depending on position x and normal νΓ(x).
• The functional setting in GSBV(Ω;ℝⁿ) ensures that deformations are well-defined almost everywhere and that jump sets are contained within the evolving crack.
• The existence result is robust under time-dependent boundary conditions and conservative body and surface forces, with the work done by loads modeled via integrals over the domain and boundary with appropriate regularity.