Strain-limiting viscoelasticity
Keywords: elasticity
regularity
global existence
evolutionary problem
viscoelasticity
weak solution
partial differential equations
linear growth
nonlinear
strain-limiting theory

This thesis is concerned with PDEs that have a strain-limiting constitutive relation, originating in the framework of continuum mechanics and implicit constitutive the- ory. The work presented here is broadly split into two halves. In the first half, we consider the balance of linear momentum
u_{tt} = div(T) + f in (0,T)×Ω=:Q, (1)
where Ω ⊂ R^d is a domain under the action of some external body force f and u : Q → R^d denotes the displacement with Cauchy stress tensor T : Q → R^{d×d}. This is supplemented with the strain-limiting constitutive relation
ε(u_t+αu)=F(T):= T / (1 + |T|^a)^{1/a}, (2)
where ε is the symmetric gradient operator and a > 0 is a fixed parameter. The linearised strain ε(u) is assumed to be small. Provided that the strain at the initial time t = 0 is sufficiently small, in fact it follows from (2) that the linearised strain ε(u) is uniformly bounded. This is the motivation for the terminology strain-limiting.
We prove the existence of unique weak solutions to (1), (2) under periodic boundary conditions, Dirichlet boundary conditions and mixed Dirichlet–Neumann boundary conditions. The theory is first developed in the simplest case, namely, the periodic setting. The techniques are built upon so that we can eventually investigate the most challenging case of mixed Dirichlet–Neumann boundary conditions. The main ideas used in the existence proofs are as follows. First, we introduce an elliptic regularisation term into the constitutive relation (2) on the right-hand side, replacing F with a function F_n that is a bijection on R^{d×d} to itself. Existence of a solution to the regularised problem is proven via a Galerkin approximation in space. Then n-uniform estimates are constructed. The main technical difficulty here is that the sequence of approximate stress tensors is, at best, bounded in L^∞(0,T;L^1(Ω)^{d×d}), which has poor compactness properties. We must construct weighted higher regularity estimates that allow the deduction of a pointwise convergence result. However, in the case that the parameter a is small, a higher integrability estimate can in fact be proven. The limit in the regularisation parameter n can then be taken and a weak solution of the strain-limiting problem constructed.
In the second half of the thesis, the focus is on nonlinear dynamic fracture problems. We consider a domain that is experiencing damage and cracking under the action of an external force. As in the previous half of the thesis, there is an unknown displacement u and Cauchy stress tensor T, but now we also deal with an unknown phase-field function v : Q → [0, 1] which approximates an a priori unknown time- dependent crack set to a certain approximation level ε. The function v takes the value 1 ‘away’ from the crack set and the value 0 ‘near’ the crack set. The balance of linear momentum is replaced with
u_{tt} = div ((v^2 + η)T) in Q, (3)
where η is a fixed parameter that prevents degeneracy near the crack set. This is supplemented with a nonlinear constitutive relation, either a strain-limiting relation of the form (2), where the strain is necessarily small, or the relation
ε(u_t + αu) = F_p(T) := |T|^{p−2}T, (4)
where p ∈ (1,∞). First, we consider the problem (3), (4) and develop a general theory of nonlinear dynamic fracture problems of this kind. The existence of weak energy solutions is proven, meaning that they solve (3) weakly, (4) holds pointwise, a minimisation problem which ensures growth in the crack set wherever possible (elim- inating the case of a stationary crack) and the satisfaction of an energy-dissipation equality. Then we combine the techniques developed with those from the first half of the thesis to investigate dynamic fracture problems with a strain-limiting consti- tutive relation. We solve (3) coupled with (2). A partial existence result is proven in the case of mixed Dirichlet–Neumann boundary conditions and a full existence result in the case of fully Dirichlet boundary conditions.