The 15 governing equations of elasticity form a system of partial differential equations that possess infinitely many solutions. To find the specific solution that corresponds to a particular physical problem, we must impose boundary conditions, which specify the physical constraints on the surface of the elastic body.
For any given point on the surface of the body, one of two types of conditions must be prescribed. Let the body occupy a domain \Omega with a boundary surface \Gamma.
1. Displacement Boundary Conditions
This condition prescribes the displacement of the points on the surface. It is used to model parts of a body that are fixed, clamped, or forced to move in a specific way.
If a portion of the boundary, denoted \Gamma_u, has its displacement specified, the condition is written as: u_i(x,y,z,t) = u_i^* where u_i^* are the known components of the prescribed displacement vector on that surface. A common example is a cantilever beam’s fixed end, where u_i^* = 0.
2. Traction Boundary Conditions
This condition prescribes the forces acting on the surface. These forces are described by the traction vector, \mathbf{t}, which is the force per unit area. It is used to model surfaces subjected to pressures, distributed loads, or contact forces.
The traction vector is related to the internal stress state at the surface by Cauchy’s stress relation: t_i = n_j \sigma_{ji} or \mathbf{t}=\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}. where n_j are the components of the outward-pointing unit normal vector to the surface and summation over the repeated index j is implied.
If a portion of the boundary, denoted \Gamma_\sigma, has its tractions specified, the condition is written as: t_i(x,y,z,t) = n_j \sigma_{ji}= t_i^* or \mathbf{t}=\hat{\mathbf{n}}\cdot\boldsymbol{\sigma}=\mathbf{t}^*. where t_i^* are the known components of the prescribed traction vector. A “free surface,” which has no forces acting on it, is a common and important example where t_i^* = 0.
Mixed Boundary Conditions
In most engineering problems, the boundary conditions are of a mixed type. Displacements are prescribed on one part of the surface (\Gamma_u), while tractions are prescribed on the remaining part (\Gamma_\sigma). The entire boundary must be covered, and these two regions must be disjoint: \Gamma = \Gamma_u \cup \Gamma_\sigma \quad \text{and} \quad \Gamma_u \cap \Gamma_\sigma = \emptyset Together, the governing field equations and a complete set of boundary conditions form a well-posed problem, ensuring a unique solution exists.
