The Governing Equations of Elasticity

The goal of any problem in solid mechanics is to determine the distribution of displacements, strains, and stresses throughout a body subjected to external forces. This requires a set of governing equations that are based on three core physical principles: the balance of forces (equilibrium), the geometry of deformation (kinematics), and the material’s response (constitutive law).

For a three-dimensional elastic body, we must solve for a total of 15 unknown field quantities at every point within the body.

The 15 Unknowns

The 15 unknowns can be grouped into three categories:

1. Displacement Vector (3 unknowns): These describe how each point in the body moves.
   • , ,
2. Strain Tensor (6 independent unknowns): These describe the deformation (stretching and shearing) of the material. The strain tensor is symmetric (), so it has 6 unique components.
   • Normal Strains:
   • Shear Strains:
3. Stress Tensor (6 independent unknowns): These describe the internal forces acting on infinitesimal surfaces within the material. Due to the balance of moments, the stress tensor is also symmetric (), giving it 6 unique components.
   • Normal Stresses:
   • Shear Stresses:

Total Unknowns = 3 (Displacements) + 6 (Strains) + 6 (Stresses) = 15 Quantities.

To solve for these 15 unknowns, we need an equal number of independent equations, which are provided by the fundamental laws of continuum mechanics.

The 15 Governing Equations

The 15 equations are derived from three fundamental principles:

1. Equilibrium Equations (3 Equations)

Considering infinitesimal element within the body, it follows from Newton’s second law () that the stress components satisfy the following equations of motion: In the above equations, is the displacement field and is the body force per unit mass.

The equilibrium equations are often written in compact index notation as where means differentiation with respect to , summation over the repeated index is implied (Einstein notation), and a double dot is used to denote second derivative with respect to time.

Another form of writing the above equation is as follows

2. Kinematic (Strain-Displacement) Equations (6 Equations) These are geometric relationships that define the components of the strain tensor in terms of the derivatives of the displacement vector. They are valid for the assumption of small deformations. In a compact form: or 3. Constitutive Equations / Generalized Hooke’s Law (6 Equations) These equations describe the material’s intrinsic behavior by relating stress to strain. In general: * Notice that summation over the repeated indices and is implied. That is, For a linear, isotropic elastic material, this relationship is defined by two material constants, typically Young’s Modulus () and Poisson’s Ratio (). or in a compact way or #### Summary

The theory of linearized elasticity is a closed and complete mathematical framework. We have established a system of 15 unknowns and a corresponding set of 15 independent equations:

• 3 Equilibrium Equations
• 6 Kinematic Equations
• 6 Constitutive Equations

This equality ensures that the problem is mathematically well-posed. By applying appropriate boundary conditions (i.e., specifying the forces or displacements on the surface of the body), a unique solution for the stress, strain, and displacement fields can be determined.