A free boundary problem for an elastic material

In this paper we investigate the dynamics of an elastic material, for example, a spring with some weight. Such dynamics is usually represented by the ordinary differential equation for the length of the spring or the partial differential equation with the linear strain on a fixed domain. The main purpose of this paper is to propose a new free boundary problem with a nonlinear strain as a mathematical model for an elastic material. Also, we establish the wellposedness for initial boundary value problem with the nonlinear strain on the cylindrical domain.

1. Introduction. The following ordinary differential equation is well-known as a mathematical model for a spring with some weight: where = (t) is the length of the spring for t > 0, m is the mass of the weight, κ is a positive constant from Hooke's law and L is the original length of the spring. When we adopt this model, we suppose that the dynamics of the spring is independent of changes of some variables in space so that the unknown function does not depend on the space variable x. However, in order to deal with a one-dimensional material made of a shape memory alloy we usually consider a mathematical model including parameters depending on the space variable. Moreover, we must consider a free boundary problem, because the length of the material may change. These facts are the motivations of the present paper. From now on the mathematical model will be given.
is the position of the point at t = 0 (see Figure 1). Also, we denote by ε(t, x) the strain at (t, x). Then, easily, we obtain

Figure 1
If u x = 1, then ∂X ∂x = 0. This shows that the different points have moved to the same place and it is impossible, physically. Usually, we suppose that u x is sufficiently small so that by using Taylor's expansion we regard the definition of the strain as ε = u x .
Next, we calculate the velocity v as follows: . Furthermore, in our problem the density ρ is not a constant function. Then, by mass conservation law we have Here, we note the momentum balance law: where f is the internal force. Thus we obtain where ρ 0 (x) = ρ(0, x). Therefore, Hooke's law σ = κε implies that Moreover, let (t), (t) > 0, be the length of the one-dimensional elastic material and fix the material at x = 0. This leads to the homogeneous Dirichlet boundary u(t, 0) = 0. On the other hand, on the free boundary x = (t) it holds that u(t, (t)) = (t)−X(t, (t)) = (t)− 0 and m (t) = mg−σ(t, (t)), where 0 = (0).
From the above argument we get the following system: where , and u 0 and v 0 are initial functions, 0 is an initial value of the velocity of the free boundary.
Here, we clarify the connection between the ordinary differential equation (1) and the above system (3), briefly. The connection means that the neglect of the mass of the spring, ρ = 0, leads to (1). In fact, first we have According to the boundary condition for u we obtain u(t, x) = (t)− 0 (t) . By substituting this function u into the free boundary condition m (t) = mg − κ ux (1). It is very difficult to solve the above system (3), directly. Then as the first step of this research we consider the following initial boundary value problem (P1) on the cylindrical domain Q(T ) = (0, T ) × (0, 1), T > 0: where γ and µ are positive constants. Since our final goal of this research is to investigate free boundary problem for shape memory alloys and the equation u tt + γu xxxx − µu txx = f + σ x is well known as one of mathematical models for shape memory alloys (cf. [4]), we adopt (4) as the approximation of (2). In Section 2 we give a precise definition of a solution of (P1) and show a theorem concerned with the existence and the uniqueness of a solution.
The other purpose of the present paper is to propose the following free boundary problem (P2) with the linear strain: The problem is to find = (t) and In author's forthcoming paper [2] we shall discuss the free boundary problem (P2).
First, we give a definition of a solution of (P1) as follows: Definition 1. Let u be a function on Q(T ). We call that u is a weak solution of (P1) on [0, T ], T > 0, if the conditions (S1) ∼ (S4) hold. 1)) and (4) holds a.e. on Q(T ), then we say that u is a strong solution of (P1) on [0, T ].
The next theorem guarantees the well-posedness of (P1). 1). Then there exists a positive constant c independent of T such that if We can prove the uniqueness under the more general assumptions.
Then, since β λ is Lipschitz continuous, we see that