An existence result for accretive growth in elastic solids (Q6633037)

The authors consider a continuous function \(\theta :\mathbb{R} ^{d}\rightarrow \lbrack 0,1)\), the sets\ \(\Omega (0)=\int\{x\in \mathbb{R} ^{d}:\theta (x)=0\}\) and \(\Omega (t)=\{x\in \mathbb{R}^{d}:\theta (x)<t\}\) for all \(t>0\), and the equilibrium of a growing object written in its quasi-static approximation as: \(-\nabla \cdot \sigma (x,t)=f(x,t)\), for \( x\in \Omega (t)\), \(t\in \lbrack 0,T]\).\ Here, \(f\in C^{0}([0,T];L^{2}( \mathbb{R}^{d};\mathbb{R}^{d}))\) are the applied forces, \(\sigma \) the stress obeying the constitutive relation: \(\sigma (x,t)=\mathbb{C} (\varepsilon (u(x,t))-\alpha (x))\), for \(x\in \Omega (t)\), \(t\in \lbrack 0,T] \), where \(u\) the displacement, \(\mathbb{C}\) the symmetric and positive definite elasticity tensor, \(\varepsilon \) the strain, and \(\alpha :\Omega (T)\rightarrow \mathbb{R}^{d\times d}\) the backstrain which has been accumulated during growth.\ In the present study, the authors choose this backstrain as: \(\alpha (x)=(K\overline{\varepsilon })(u)(x,\theta (x))\), for \(x\in \Omega (T)\), \(\overline{\varepsilon }(u)(\cdot ,t)\) being the trivial extension of \(\varepsilon (u)(\cdot ,t)\) to the whole \(\mathbb{R}^{d}\) and \(K \) a space-time convolution operator of the form \((K\overline{\varepsilon } (u))(x,t)=\int_{0}^{t}\int_{\mathbb{R}^{d}}k(t-s)\phi (x-y)\overline{ \varepsilon }(u)(y,s)dyds\), for given time- and space-kernels \(k\in W^{1,1}(0,T)\) and \(\phi \in H^{1}(\mathbb{R}^{d})\) with compact support, respectively. They impose the boundary condition \(u(x,t)=0\) on \(\omega \times \lbrack 0,T]\), for a nonempty, open, and connected docking set \( \omega \subset \subset \Omega (0)\), satisfying \(\operatorname{dist}(\omega ;\partial \Omega (0))>0\). They finally assume that a point \(x(t)\in \partial \Omega (t)\) follows the normal accretion law: \(\overset{.}{x}(t)=\gamma \eta (x(t))\), where \(\eta (x(t))\) indicates the outward normal to \(\partial \Omega (t)\) at \(x(t)\) and \(\gamma \in C^{0,1}(\mathbb{R}^{d}\times \mathbb{R}^{d\times d}; \mathbb{R})\) is the growth rate. The evolution of \(\Omega (t)\) is determined by solving the generalized eikonal equation: \(\gamma (x,\alpha (x))\left\vert \nabla \theta (x)\right\vert =1\), with the initial condition \( \theta =0\) on \(\Omega (0)\). The authors assume that \(\Omega (0)\neq \varnothing \) is a John domain, that is there exists a specific point \( x_{0}\in \Omega (0)\) and a John constant \(C_{J}\in (0,1]\) such that for all points \(x\in \Omega (0)\) one can find an arc-length parametrized curve \(\rho :[0,L]\rightarrow \Omega (0)\) with \(\rho (0)=x\), \(\rho (L)=x_{0}\), and \(\operatorname{dist}(\rho (s);\partial \Omega (0))\geq C_{J}s\) for all \(s\in \lbrack 0,L]\). They introduce a weak formulation of this problem that includes an equilibrium system and a growth system, and the associated notions of viscosity sub-, super- and solution. The first main result proves that if \( \theta :\mathbb{R}^{d}\rightarrow \lbrack 0,1)\) is given, so that the corresponding set-valued map \(t\in \lbrack 0,T]\rightarrow \Omega (t)\) takes values in \(\Theta =\{\Omega \subset \subset D:\Omega \) is a John domain with respect to the point \(x_{0}\in \omega \), with John constant \(C_{J}\), and \(\operatorname{dist}(\omega ;\partial \Omega )\geq \rho _{0}>0\), there exists a unique measurable function \(u:Q=\cup _{t\in (0,T)}\Omega (t)\times \{t\}\rightarrow \mathbb{R}^{d}\), with \(u(\cdot ,t)\in H_{\omega }^{1}(\Omega (t);\mathbb{R} ^{d})\) for a.e. \(t\in (0,T)\) and \(t\rightarrow \left\Vert u(\cdot ,t)\right\Vert _{H^{1}}\in L^{1}(0,T)\), that solves the equilibrium system. For the proof, the authors first establish a uniform Korn inequality in the present context. They use a contraction argument in an appropriate function space to establish a solution to the equilibrium problem on \((0,T_{0})\) for some \(T_{0}\in (0,T]\) small. They finally proceed on intervals \( (0,jT_{0})\) adapting the estimates required for the application of a contraction argument. The second main result proves that for \(T>0\) small enough, there exist a Lipschitz continuous function \(\theta :\mathbb{R} ^{d}\rightarrow \lbrack 0,1)\) and a measurable function \(u:Q\rightarrow \mathbb{R}^{d}\), with \(u(\cdot ,t)\in H^{1}(\Omega (t);\mathbb{R}^{d})\), for almost every \(t\in (0,T)\), that solves the complete weak formulation of the coupled equilibrium and growth problem. The proof of this existence result for \(T>0\) small is based on a Schauder fixed-point argument on the function \( \alpha \) in an appropriate space.