[Paper Review] A quantitative geometric rigidity result in SBD

Manuel Friedrich, Bernd Schmidt|arXiv (Cornell University)|Mar 23, 2015

Numerical methods in engineering32 references8 citations

TL;DR

This paper establishes a quantitative geometric rigidity estimate for special functions of bounded deformation (SBD) in two dimensions, generalizing classical rigidity results to brittle materials with discontinuities. It proves that any deformation can be decomposed into a piecewise rigid motion plus a displacement field, with control over both bulk and surface energy contributions, yielding a piecewise Korn-Poincaré inequality in the geometrically linear setting.

ABSTRACT

We present a quantitative geometric rigidity estimate for special functions of bounded deformation in a planar setting generalizing the result by Friesecke, James, Müller obtained in nonlinear elasticity theory and the piecewise rigidity result by Chambolle, Giacomini, Ponsiglione for brittle materials which do not store elastic energy. We show that for each deformation there is an associated triple consisting of a partition of the domain, a corresponding piecewise rigid motion being constant on each connected component of the cracked body and a displacement field measuring the distance of the deformation from the piecewise rigid motion. We also present a related estimate in the geometrically linear setting which can be interpreted as a `piecewise Korn-Poincaré inequality'.

Motivation & Objective

• To extend geometric rigidity estimates from Sobolev spaces to functions of bounded deformation (SBD) with jump discontinuities.
• To model brittle fracture by characterizing deformations as piecewise rigid motions with controlled displacement fields.
• To derive a quantitative estimate linking the deviation of the deformation gradient from rotations to a global displacement field measuring distance from piecewise rigid motion.
• To establish a geometrically linear version of the rigidity result, interpreted as a 'piecewise Korn-Poincaré inequality'.
• To control both bulk and surface energy contributions in the presence of cracks, enabling application to variational fracture models.

Proposed method

• Introduce a triple decomposition: a partition of the domain, a piecewise rigid motion constant on each connected component, and a displacement field measuring deviation from rigidity.
• Use a localization and modification technique to extend functions across cracks while preserving $H^1$-regularity on modified sets.
• Apply a Korn-Poincaré inequality in localized regions to control the $L^2$-norm of the displacement field.
• Construct a modified deformation $\hat{y}$ on a slightly enlarged domain $\Omega_\rho$ to handle jump sets and ensure regularity.
• Employ a covering argument with dyadic cubes to control the $L^p$-norms of the gradient difference $\nabla \hat{y} - R_j$ on non-regular parts.
• Use a partition of unity and approximation via piecewise rigid motions to derive uniform estimates in the $SBD$-setting.

Experimental results

Research questions

• RQ1 Can a quantitative geometric rigidity estimate be established for functions in SBD, which allow for jump discontinuities?
• RQ2 How can the deviation of a deformation from piecewise rigid motion be quantitatively controlled in the presence of cracks?
• RQ3 What is the relationship between the bulk energy (via $\nabla u$) and surface energy (via $J_u$) in a variational fracture model?
• RQ4 Can a piecewise Korn-Poincaré inequality be derived in the geometrically linear setting for SBD functions?
• RQ5 How can the rigidity estimate be extended to handle small sets and irregular crack geometries?

Key findings

• For any $u \in SBD(\Omega)$, there exists a partition of $\Omega$ into sets $P_j$, a piecewise rigid motion $R_j x + c_j$ on each $P_j$, and a displacement field $u - (R_j x + c_j)$ such that the $L^2$-norm of the displacement is controlled by the $L^2$-norm of the strain and the total variation of the jump set.
• The $L^2$-norm of the displacement field on each component $P_j$ is bounded by $C(\rho) \varepsilon$, where $\varepsilon$ controls the distance of $\nabla u$ from $SO(2)$.
• The $L^4$-norm of the gradient difference $\nabla \hat{y} - R_j$ on the modified domain is bounded by $C(\rho)\varepsilon$, ensuring regularity in the approximation.
• The total perimeter of the jump set $J_{\hat{y}}$ satisfies $\mathcal{H}^1(J_{\hat{y}}) \leq \mathcal{H}^1(J_y) + C_1 \rho$, showing control over surface energy.
• In the geometrically linear setting, the displacement field satisfies a piecewise Korn-Poincaré inequality with a constant depending on $\rho$ and $\varepsilon$.
• The result holds uniformly up to a set of small measure, with error terms controlled by $\varepsilon^{1-\eta}$ and $\rho$.