Stress (mechanics)

Lore & Background

Stress, in continuum mechanics, is a physical quantity representing the forces that arise within a material during deformation. It is defined as the force acting across a small boundary per unit area, for all possible orientations of that boundary. The dimension of stress is force per area, with SI units of newtons per square meter (N/m²), also called the pascal (Pa). Stress is typically denoted by the lowercase Greek letter sigma (σ). Greater force or a smaller cross-sectional area results in higher stress. For instance, a stretched elastic band experiences tensile stress and may elongate, while a crumpled sponge undergoes compressive stress and shortens. Stress expresses the internal forces that neighboring particles of a continuous material exert on each other; for example, particles in a vertical bar supporting an overhead weight push on those below, while particles in a pressurized liquid are pushed by surrounding particles and container walls. These macroscopic forces are the net result of numerous intermolecular forces and collisions. Strain measures the relative deformation of the material. In solids, any deformation generates an internal elastic stress that tends to restore the original shape, analogous to a spring. In liquids and gases, only volume-changing deformations produce persistent elastic stress, though gradual deformations also generate viscous stress. Significant stress can exist even without deformation, as in prestressed concrete or tempered glass, and can arise from temperature changes, chemical composition shifts, or electromagnetic fields, without net external forces. The relationship between stress, strain, and strain rate can be complex, though a linear approximation often suffices for small quantities. Stress exceeding a material’s strength limits may cause permanent deformation, fracture, or changes in crystal structure.

Reader's Guide

Augustin-Louis Cauchy gave the first rigorous and general mathematical model of a deformed elastic body by introducing the notions of stress and strain. He observed that the force across an imaginary surface was a linear function of its normal vector and must be a symmetric function. Stress is defined as the force across a small boundary per unit area for all orientations, expressed by the Cauchy traction vector. The stress state must be described by a tensor, which can be represented as a symmetric matrix of 3×3 real numbers. Stress may exist even without deformation, as in prestressed concrete and tempered glass, and can be imposed by changes in temperature, chemical composition, or electromagnetic fields. The relation between stress, strain, and strain rate can be complicated, though a linear approximation may be adequate for sufficiently small quantities. Stress exceeding certain strength limits results in permanent deformation such as plastic flow, fracture, or cavitation.

Did You Know?

• Stress has dimension of force per area, with SI units of newtons per square meter (N/m²) or pascal (Pa).
• Stress may exist even when deformation is negligible or non-existent, as in prestressed concrete and tempered glass.
• The Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers.
• Stress in liquids started with Newton, who provided a differential formula for friction forces (shear stress) in parallel laminar flow.