Continuum mechanics

Continuum mechanics models matter as continuously distributed, using fields such as displacement, velocity, and stress to describe motion and deformation.

Continuum hypothesis: The assumption that matter can be treated as continuously distributed rather than as discrete particles. It justifies describing material properties at every point despite matter’s molecular structure.
Balance of linear momentum: The local equation equating a material’s acceleration to the divergence of stress and applied body forces. It links stress fields and external forces to the acceleration of matter.
Elasticity: The study of reversible deformation in materials under applied forces. Elasticity provides the standard continuum model for recoverable solid deformation.
Molecular dynamics: A computational method that simulates interacting atoms or molecules to predict their motion. It represents matter discretely, unlike continuum mechanics’ smoothly varying fields.
Deformation gradient: A tensor that maps infinitesimal material line elements from a reference configuration to a deformed configuration. It captures local changes in length, angle, and orientation during deformation.
Constitutive equation: A material-specific relation connecting quantities such as stress, strain, temperature, and their rates. It supplies the response law that balance equations alone cannot determine.
Fluid mechanics: The study of fluids in motion and at rest, including their forces and stresses. Fluid mechanics applies continuum fields to liquids and gases.
Finite element method: A numerical method that approximates field equations by dividing a domain into smaller elements. It is a widely used way to solve continuum mechanics problems approximately.
Cauchy stress tensor: A tensor describing internal force per unit area acting across planes through a point in a material. It expresses the internal forces that balance loads and drive deformation.
Kinematics of continua: The description of material motion and deformation using mappings, velocity fields, and strain measures. Kinematics defines deformation independently of the forces that produce it.

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