[Paper Review] A 3-dimensional singular kernel problem in viscoelasticity: an existence result

Sandra Carillo|arXiv (Cornell University)|Aug 7, 2018

Elasticity and Material Modeling2 citations

TL;DR

This paper establishes the existence of a weak solution to a three-dimensional initial boundary value problem in viscoelasticity with a singular kernel, where the relaxation modulus is unbounded at t=0 and its time derivative is not integrable. By constructing a sequence of regularized approximations and leveraging weak convergence and a crucial lemma on kernel convergence, the authors prove the existence of a solution under relaxed regularity conditions compared to classical viscoelastic models.

ABSTRACT

Materials with memory, namely those materials whose mechanical and/or thermodynamical behaviour depends on time not only via the present time, but also through its past history, are considered. Specifically, a three dimensional viscoelastic body is studied. Its mechanical behaviour is described via an integro-differential equation, whose kernel represents the relaxation modulus, characteristic of the viscoelastic material under investigation. According to the classical model, to guarantee the thermodynamical compatibility of the model itself, such a kernel satisfies regularity conditions which include the integrability of its time derivative. To adapt the model to a wider class of materials, this condition is relaxed; that is, conversely to what is generally assumed, no integrability condition is imposed on the time derivative of the relaxation modulus. Hence, the case of a relaxation modulus which is unbounded at the initial time t = 0, is considered, so that a singular kernel integro-differential equation, is studied. In this framework, the existence of a weak solution is proved in the case of a three dimensional singular kernel initial boundary value problem.

Motivation & Objective

• To address the lack of existence results for viscoelastic models with singular kernels, particularly when the time derivative of the relaxation modulus is not integrable.
• To generalize classical viscoelastic models by removing the standard integrability condition on the derivative of the relaxation modulus.
• To establish the existence of a weak solution for a three-dimensional initial boundary value problem in viscoelasticity under weaker regularity assumptions.
• To extend previous one-dimensional results to the three-dimensional setting, enhancing applicability to real materials such as polymers and biological tissues.
• To provide a mathematical foundation for modeling materials with memory where the relaxation response is singular at initial time.

Proposed method

• Formulate the viscoelastic problem using a linear integro-differential equation with a relaxation modulus K(t), where K'(t) is not in L¹(0,T), leading to a singular kernel.
• Introduce a regularized approximation of the relaxation modulus by shifting the kernel: Kεh(s) = ∫₀^{s} G(εh + τ)dτ, with G ∈ L¹(0,T) but Ẇ ∉ L¹.
• Define a weak formulation of the problem using test functions in H¹₀(Ω×(0,T)), leading to an integral identity involving the stress and strain history.
• Construct a sequence of approximate solutions {uεh} to the regularized problem and prove their convergence: weak in H¹(0,T;H¹₀(Ω)) and strong in L²(Ω×(0,T)).
• Prove a key lemma (Lemma 2) showing that the difference between the regularized and original kernel integrals converges to zero in the L² sense.
• Use the convergence of the approximated solutions and the kernel convergence lemma to pass to the limit in the weak formulation, thereby proving existence of a weak solution to the singular problem.

Experimental results

Research questions

• RQ1 Can the existence of a weak solution be established for a 3D viscoelastic initial boundary value problem when the relaxation modulus has a non-integrable derivative?
• RQ2 How can a singular kernel model, where the relaxation modulus is unbounded at t=0, be mathematically formulated and analyzed?
• RQ3 What approximation strategy can be used to handle the singularity in the kernel while preserving the physical and mathematical structure of the problem?
• RQ4 To what extent can the classical existence theory for viscoelasticity be extended to models with less regular relaxation functions?
• RQ5 What is the role of the homogeneous boundary conditions and the regularity of initial data in ensuring convergence of the approximated solutions?

Key findings

• The paper proves the existence of a weak solution to a 3D initial boundary value problem in viscoelasticity with a singular kernel, where the relaxation modulus G satisfies G ∈ L¹(0,T) but Ẇ ∉ L¹(0,T).
• A sequence of regularized problems is constructed by introducing a shift εh in the kernel, ensuring the approximated kernels are smooth and integrable.
• The approximated solutions {uεh} converge weakly in H¹(0,T;H¹₀(Ω)) and strongly in L²(Ω×(0,T)) as εh → 0, guaranteeing the existence of a limit function u.
• Lemma 2 establishes that the difference between the regularized and original kernel integrals converges to zero in the L² sense, which is essential for passing to the limit.
• The weak formulation of the original problem is recovered in the limit, confirming that the limit function u satisfies the integral equation in the weak sense.
• The result generalizes previous 1D existence results and extends the analytical framework to 3D, enabling modeling of materials with memory that exhibit singular relaxation behavior.