Solid Mechanics

Solid 机械 provides the theoretical 框架 for understanding how solid objects respond to stimuli. A central concept is the relationship between stress (internal forces per unit area) and strain (relative deformation). For many materials under small loads, this relationship is linear and described by Hooke’s Law, [latex]\sigma = E \epsilon[/latex], where [latex]\sigma[/latex] is stress, [latex]\epsilon[/latex] is strain, and [latex]E[/latex] is the Young’s modulus, a measure of stiffness. In three dimensions, these quantities are represented by tensors, the stress tensor and the strain tensor, which capture the state of stress and deformation at any point within the body.

The field is broadly divided into statics, which deals with bodies at rest or in equilibrium, and dynamics, which studies bodies in motion and includes phenomena like vibrations and wave propagation. When loads exceed a material’s elastic limit, it enters the plastic regime, where permanent deformation occurs. Solid mechanics provides theories to predict the onset of this yielding, using criteria like the von Mises or Tresca yield criteria. Furthermore, fracture mechanics, a subfield, analyzes the behavior of materials containing cracks. It aims to predict crack growth and prevent catastrophic failure in structures. These principles are applied computationally using methods like the Finite Element Method (FEM) to solve complex real-world engineering problems that would be intractable to solve analytically.