On the existence of elastic minimizers for initially stressed materials

Abstract

A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of the divergencefree symmetric tensors satisfying the boundary condition in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed accordingly to the symmetry group of the material at fixed initial stresses. These results allow to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials.

Surface tension controls the onset of gyrification in brain organoids

“…These stresses are called residual [Hoger, 1985] and they are created when differential growth in solid matter develops microstructural misfits. These geometrical incompatibilities are restored by elastic distortions of the body, generating mechanical stress [Rodriguez et al, 1994, Goriely, 2017, Riccobelli et al, 2019.…”

The Föppl–von Kármán equations of elastic plates with initial stress

“…Enforcing the compatibility condition (2.4) for the initial stress, we also find that p τ = r . Such an energy guarantees that the material properties are not affected by the initial stress distribution, for details see [45].…”

Activation of a muscle as a mapping of stress–strain curves

“…Summarizing, the active strain approach corresponds to a remodelling [8,29]: there is only a morphological change in the relaxed configuration and the properties of the material are conserved. In the previous sections, we have proved that the active strain cannot fit the experimental data of the uniaxial extension of a skeletal muscle.…”

“…The active strain has been employed in several models (e.g. [6,31,21,25]) due to its robust mathematical properties and the clear physical interpretation: the muscle contraction corresponds to a geometrical remodelling of the microstructure of the body [8,29].…”