Mechanical stress is a fundamental concept in continuum mechanics, describing the internal forces that arise within a material when it undergoes deformation. It's a physical quantity that helps us understand how objects respond to external pressures. For instance, when an elastic band is stretched, it experiences tensile stress, leading to elongation. Conversely, a crumpled sponge is under compressive stress, causing it to shorten. The magnitude of
this stress is directly related to the applied force and inversely proportional to the cross-sectional area over which that force acts. This means a greater force or a smaller area results in higher stress.
Defining and Quantifying Mechanical Stress
Stress is precisely defined as the force acting across a small boundary per unit area of that boundary, considering all possible orientations of the boundary. It is a physical quantity, much like velocity, torque, or energy, and can be quantified and analyzed without needing to explicitly consider the specific nature of the material or the exact physical causes behind it. In the framework of continuum mechanics, stress is considered a macroscopic concept. This implies that the particles involved in its definition and analysis are small enough to be treated as homogeneous in their composition and state, yet large enough to disregard quantum effects and the intricate movements of individual molecules. Therefore, the force between two particles is effectively an average of a vast number of atomic forces between their constituent molecules. Physical quantities such as mass, velocity, and forces acting throughout the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed across them.
Quantitatively, stress is often expressed by the Cauchy traction vector, denoted as T. This vector represents the traction force F between adjacent parts of a material across an imaginary separating surface S, divided by the area of S. In a fluid at rest, this force is perpendicular to the surface, which is the familiar concept of pressure. However, in a solid or a flowing viscous liquid, the force F may not be perpendicular to the surface S. Consequently, the stress across a surface must be treated as a vector quantity, not merely a scalar number. Furthermore, both the direction and magnitude of this stress generally depend on the orientation of the surface S. To fully describe the stress state of a material, a tensor, known as the Cauchy stress tensor, is used. This tensor is a linear function that connects the normal vector 'n' of a surface S to the traction vector T across that surface. When referenced to any chosen coordinate system, the Cauchy stress tensor can be represented as a symmetric 3x3 matrix of real numbers. It's important to note that even within a homogeneous body, the stress tensor can vary from one location to another and may change over time, making stress within a material generally a time-varying tensor field.
Normal, Shear, and Isotropic Stress Types
The stress vector T that one particle applies on another across a surface S can have any direction relative to that surface. This vector can be broken down into two primary components: normal stress and shear stress. Normal stress is the component perpendicular to the surface, representing either compression or tension. If the normal unit vector 'n' of the surface is fixed, the normal component can be expressed as a single number, the dot product T · n. A positive value indicates that one part is "pulling" on the other (tensile stress), while a negative value signifies "pushing" (compressive stress). The second component, shear stress, is parallel to the surface and is represented by the vector T − (T · n)n.
Mechanical stress is measured in units that are equivalent to pressure. The standard international (SI) unit is pascals (Pa), which is newtons per square meter (N/m²). In the Imperial system, pounds per square inch (psi) is used. Given that mechanical stresses can easily exceed a million pascals, megapascals (MPa) is a commonly employed unit for convenience.
Several simple types of stress are frequently encountered in engineering design. Uniaxial normal stress describes a situation where a straight rod with uniform material and cross-section, is subjected to tension or compression along its axis by opposite forces. In equilibrium, the stress (σ) throughout the bar across any horizontal surface is simply the force (F) divided by the cross-sectional area (A), or σ = F/A. This is called tensile stress if stretching or compressive stress if pushing. This analysis often assumes uniform stress distribution, though in practice, F/A might represent an average, known as engineering or nominal stress. Saint-Venant's principle suggests that stress is uniformly distributed far from the ends of a long bar. Normal stress also appears in bending, where an elastic bar's outer part experiences tensile stress and the inner part compression, and as hoop stress in pressurized cylindrical pipes.
Simple shear stress occurs when a layer of elastic material is pulled in opposite directions by forces parallel to the layer, or when a metal bar is cut by a scissors-like tool. Here, the stress (τ) is also calculated as F/A, but it is directed parallel to the cross-section. This average shear stress is often sufficient for practical purposes. Shear stress is also significant in shafts subjected to torques and in the web of I-beams under bending loads.
Finally, isotropic normal stress describes a state where a material body experiences equal compression or tension in all directions. This is typical for a liquid or gas at rest, or a cube of elastic material uniformly pressed or pulled on all six faces. In such cases, the stress across any internal surface is equal in magnitude and always perpendicular to the surface, regardless of its orientation. If compressive, this is known as hydrostatic pressure. While gases cannot withstand tensile stresses, some liquids can endure significant isotropic tensile stress under specific conditions. Parts with rotational symmetry, like wheels, axles, pipes, and pillars, often exhibit stress patterns with rotational or cylindrical symmetry, allowing for simplified analysis.
