Formulation of Boundary Value Problem - Civil Engineering (CE) PDF Download

Overview

Before we endeavor on the formulation of the boundary value problem and discuss strategies to solve it, we recollect and record the basic equations. If u represents the displacement that the particles undergo from a stress free reference conguration to the deformed current conguration with the deformation taking place due to application of the traction on the boundary, then we de ne the linearized strain as,

         (7.1)

where, h = grad(u). The equation (7.1) is called as the strain displacement relation. The constitutive relation that relates the stress to the strain that is to be used in this course is the Hooke's law. The various forms of the same Hooke's law that would be used is recorded:

where σ is the Cauchy stress, λ and μ are Lam� constants and E is the Young's modulus and v is the Poisson's ratio. The conservation of linear momentum equation,

      (7.5)

where ρ is the density, b is the body force per unit mass and a is the acceleration, for our purposes here reduces to:

           (7.6)

since we have ignored the body forces and look at congurations that are in static equilibrium under the action of the applied static traction. That is the displacement is assumed not to depend on time. It is appropriate that we discuss these assumptions in some detail. By ignoring the body forces we are ignoring the stresses that arise in the body due to its own mass. Since, these stresses are expected to be much small in comparison to the stresses induced in the body due to traction acting on its boundary and these stresses
practically do not vary over the surface of the earth, ignoring the body forces is justiable. In the same spirit, all that we require is that the magnitude of acceleration be small, if not zero.

Finally, we document the compatibility conditions; the conditions that ensures the existence of a smooth displacement eld given a strain field:

Formulation of boundary value problem

Formulation of the boundary value problem involves specification of the geometry of the body, the constitutive relation and the boundary conditions. To elaborate, we have to define the region of the Euclidean point space the body occupies in the reference configuration, which is denoted by B. For example, the body might occupy a region that is a unit cube, then using Cartesian coordinates the body is defined as B = {(X, Y, Z)|0 ≤ X ≤ 1, 0 ≤ Y ≤ 1, 0 ≤ Z ≤ 1}. Alternatively, the body might be the annular region between two annular cylinders of radius R_o and R_i and height H, then using cylindrical polar coordinates the body may be defined as B = {(R, Θ, Z)|R_i ≤ R ≤ R_o, 0 ≤ Θ ≤ 2π, 0 ≤ Z ≤ H}. The boundary of the body, denoted as ∂B is a surface that encloses the body. For illustration, in the cube example, the surface is composite of 6 different planes, defined by X = 0, X = 1, Y = 0, Y = 1, Z = 0 and Z = 1 respectively and in the annular cylinder example the surface is composed of 4 different planes, defined by R = R_i , R = R_o, Z = 0, Z = H. In this course, at least, the constitutive relation is known; it is Hooke’s law for isotropic materials. In real life problem, the weakest link in the formulation of the boundary value problem is the constitutive relation. Then finally one needs to prescribe boundary conditions. These are specifications of the displacement or traction on the surface of the body. Depend-ing on what is prescribed on the surface, there are four type of boundary conditions. They are

1. Displacement boundary condition: Here the displacement is specified on the entire boundary of the body. This is also called as Dirichlet boundary condition

2. Traction boundary condition: Here the traction is specified on the entire boundary of the body. This is also called as Neumann boundary condition

3. Mixed boundary condition: Here the displacement is specified on part of the boundary and traction is specified on the remaining part of the boundary. Both traction as well as displacement are not specified over any part of boundary

4. Robin boundary condition: Here both the displacement and the traction are specified on the same part of the boundary.

Once, the geometry of the body, constitutive relations and boundary conditions are prescribed then finding the Cauchy stress and displacement over the entire region of the body such that the displacement is continuous and differentiable over the entire region occupied by the body and the stress computed using this displacement field from the constitutive relation satisfies the equilibrium equations is called as solving the boundary value problem. If for a given geometry of the body, constitutive relations and boundary conditions, there exists only one displacement and stress field as a solution to the boundary value problem then the solution to the boundary value problem is said to be unique.

Now it is appropriate to make a few comments regarding the choice of the independent variable for the boundary value problem. That is, when we say the region occupied by the body, we should have been more specific and said whether this is the region occupied by the body in the reference or current configuration. As described in section 3.4 of chapter 3, the displacement and the stresses can be given an Eulerian or Lagrangian description, i.e.,

                 (7.13)

In this course, to be specific, we follow the Eulerian description of these fields. That is we use the region occupied by the body in the current configuration as the spatial domain of the boundary value problem. However, this domain is not known a priori. Therefore, we approximate the region occupied by the body in the current configuration with the region occupied by the body in the reference configuration. This approximation is justified, since we are interested only in problems where the components of the gradient of the displacement field are small and the magnitude of the displacement is also small. Then, one might ponder as to why we cannot say we are following the Lagrangian description of these fields. In chapter 5 section 5.4.1, we showed that the conservation of linear momentum takes different forms depending on the description of the stress field. The equation used here (7.5) is derived assuming Eulerian or spatial description of the stress field. If we use (5.46) instead of (7.5) then we would have said we are following Lagrangian description of these fields. Following Lagrangian description also, we would obtain the same governing equations if we assume that the components of the gradient of the deformation field are small and the magnitude of the displacement is small; but we have to do a little more algebra than saying upfront that we are following an Eulerian or spatial description.