Overview
This paper investigates the variational derivation of reduced models pertinent to elastic membranes that experience fracture. The analysis specifically addresses scenarios where constraints are imposed on the determinant of the deformation gradient. Two distinct physical settings are explored: one where the deformation maintains local orientation (defined by $\det \nabla u > 0$) and another where the deformation inherently preserves volume (characterized by $\det \nabla u = 1$). Within these frameworks, the surface energy density is formulated to incorporate a dependence on the jump amplitude. This allows the models to encompass cohesive fracture phenomena that exhibit an activation threshold. A central technical achievement of the research is the development of recovery sequences. These sequences are designed to concurrently adhere to the specified determinant constraint while also optimizing the associated surface energy.
Research Context
The study specifically focuses on elastic membranes, which are foundational structures in mechanics, and their behavior when subjected to fracture. The introduction of constraints on the determinant of the deformation gradient is a key aspect, distinguishing this work within the broader field of fracture mechanics. The problem is approached through a variational framework, a common method for analyzing and modeling physical systems by optimizing an energy functional. The inclusion of cohesive fracture models, which account for the energy required to create new surfaces, is achieved by allowing the surface energy density to vary with the jump amplitude. The consideration of both orientation-preserving and incompressible regimes means the models are intended to be applicable to different material responses and deformation characteristics, reflecting their relevance to varied physical phenomena where local orientation or volume preservation are critical factors.
Approach
The research's approach centers on the variational derivation of reduced models. This involves defining an energy functional and seeking its minimizers under specific conditions. The conditions include constraints on the determinant of the deformation gradient ($\det \nabla u > 0$ for orientation-preserving, and $\det \nabla u = 1$ for incompressible deformations). A crucial element of the methodology is the way surface energy is modeled: its density is defined to depend on the jump amplitude. This dependency is key to integrating cohesive fracture models that feature an activation threshold into the variational framework.
The core technical contribution involves constructing recovery sequences. The design of these sequences is complex, requiring them to satisfy two simultaneous criteria: maintaining the determinant constraint and optimizing the surface energy. This construction relies on a combination of specific mathematical techniques:
- The use of $C^\infty$ diffeomorphisms that converge to the identity. These diffeomorphisms are employed to rotate the normal vector to the jump set. The purpose of this rotation is to minimize the reduced surface energy, suggesting an optimization step within the recovery sequence construction.
- A novel smooth approximation result for $GSBV^p$ functions. This indicates an advancement in approximation theory applied to generalized special functions of bounded variation. $GSBV^p$ spaces are relevant in fracture mechanics for representing functions with jump discontinuities, which accurately describe fractured materials. The new approximation result facilitates the construction of the required recovery sequences by providing a mechanism to smooth these functions while respecting their underlying structure relevant to fracture.
Findings
The primary finding of this research is the successful variational derivation of reduced models for fractured elastic membranes under determinant constraints. The derived models are capable of describing both orientation-preserving deformations (where $\det \nabla u > 0$) and incompressible deformations (where $\det \nabla u = 1$). A key aspect of these models is the incorporation of surface energy densities that are dependent on the jump amplitude, which allows them to account for cohesive fracture models that possess an activation threshold. The central technical outcome is the effective construction of recovery sequences. These sequences were successfully demonstrated to simultaneously satisfy the imposed determinant constraint on the deformation gradient and optimize the surface energy concurrently. This was achieved through the specific use of $C^\infty$ diffeomorphisms converging to the identity, which served to rotate the normal to the jump set for reduced surface energy minimization, and by applying a newly developed smooth approximation result for $GSBV^p$ functions.