Existence of solutions for time-dependent Signorini-type problems in linearised viscoelasticity

Technical Summary: Existence of Solutions for Time-Dependent Signorini-Type Problems in Linearised Viscoelasticity

Problem Formulation
This paper addresses the mathematical modeling of a three-dimensional linearly viscoelastic body constrained to remain within a prescribed half-space. The body is subject to time-dependent applied body forces and is modeled using linearised elasticity theory with isotropic, homogeneous material properties. The confinement is governed by a Signorini-type condition, which requires that the deformed configuration of the body does not cross the boundary of the half-space defined by a unit normal vector $q$.

The core challenge lies in formulating a well-posed initial-boundary value problem for a hyperbolic variational inequality that incorporates memory-dependent constitutive laws (viscosity) and unilateral contact constraints. A specific focus of the work is the treatment of initial conditions where the body may already be in contact with the obstacle at $t=0$, a scenario often excluded in prior literature due to model limitations. The governing problem, denoted as $P(\Omega)$, seeks a displacement field $u$ satisfying specific regularity requirements and a set of hyperbolic variational inequalities involving inertia, elasticity, viscosity, and external forces.

Methodology
The existence of solutions is established through a two-layer approximation strategy:

1. Penalization: The unilateral constraint is relaxed by introducing a penalty parameter $\kappa > 0$. This transforms the variational inequality into a sequence of penalized variational equations, $P_\kappa(\Omega)$. The penalty term represents the energy cost of violating the confinement condition and is formulated using a non-linear operator involving the negative part of the gap function.
2. Galerkin Discretization: The penalized problem is discretized in space using the Galerkin method. By projecting the problem onto finite-dimensional subspaces spanned by an orthogonal basis, the partial differential equations are reduced to a system of ordinary differential equations (ODEs). The existence of solutions for these ODE systems is guaranteed by standard theorems due to the Lipschitz continuity of the non-linear penalty term.

Key Analytical Tools and Results
The passage to the limit as the discretization dimension $m \to \infty$ and the penalty parameter $\kappa \to 0$ relies on several critical analytical steps:

• Energy Estimates: Uniform bounds are derived for the approximate solutions in appropriate Sobolev-Bochner spaces ($L^\infty(0, T; V(\Omega))$, $L^\infty(0, T; L^2(\Omega))$, etc.). These estimates utilize Korn's inequality, the positive-definiteness of the elasticity and viscosity tensors, and the Gronwall inequality.
• Compactness of the Trace Operator: A pivotal contribution of the paper is Lemma 4.2, which establishes the compactness of the time-dependent version of the trace operator. This result is essential for handling the boundary terms arising from the contact condition and is proven using the Aubin-Lions-Simon theorem.
• Monotonicity and Convergence: The monotonicity and Lipschitz continuity of the penalty operator allow for the application of Minty-Browder type arguments. This ensures that the limit of the penalized solutions satisfies the original variational inequality and that the constraint is strictly enforced in the limit.
• Initial Conditions: The methodology explicitly handles initial velocity data $u_1$ and allows for initial displacement $u_0$ to be in contact with the obstacle ($u_0 \in U(\Omega)$), overcoming limitations found in previous approaches that required non-contact initial states or imposed terminal conditions.

Main Theorems

• Theorem 4.4: Under the assumption that the applied body force $f$ belongs to $H^1(0, T; L^2(\Omega))$, the paper proves the existence of at least one solution to the time-dependent Signorini problem $P(\Omega)$. The solution possesses the regularity $u \in L^\infty(0, T; U(\Omega))$ and $\dot{u} \in C^0([0, T]; L^2(\Omega)) \cap L^2(0, T; V(\Omega))$.
• Theorem 4.5: For materials satisfying a specific "smallness condition" relating the mass density, elasticity, and viscosity constants (implying elastic dominance over viscous effects), the paper demonstrates asymptotic stability. Specifically, if the external force is removed after a time $T_0$, the body returns to its reference configuration at an exponential rate of decay. This aligns with the theoretical behavior of Kelvin-Voigt materials.

Significance and Novelty
The paper claims several distinct contributions to the literature on viscoelastic obstacle problems:

1. Novel Solution Concept: It introduces a solution concept for time-dependent Signorini problems in linearised viscoelasticity that aligns with classical elliptic obstacle problems and accommodates initial contact with the obstacle.
2. Methodological Advancement: Unlike previous works that relied on vector-valued measures or imposed terminal conditions, this approach uses a penalization method combined with a novel compactness result for the trace operator.
3. General Applicability: The methodology applies to vectorial magnitude problems in three dimensions and does not require the additional regularity assumptions or specific initial configurations (non-contact) often found in related studies.
4. Asymptotic Analysis: The paper provides a rigorous proof of exponential decay for the solution when external forces are lifted, confirming the physical intuition of energy dissipation in Kelvin-Voigt viscoelastic bodies under confinement.

The work synthesizes ideas from variational inequalities, geometric confinement frameworks, and evolutionary equation theory to resolve existence questions for a class of problems previously considered open in the context of three-dimensional time-dependent viscoelasticity.