# Last edited on 2013-02-06 21:06:27 by stolfilocal [[File:Plastic_Protractor_Polarized_05375.jpg|240px|thumb|right|Built-in stress in a plastic [[protractor]] revealed by its [[photoelasticity|effect on polarized light]].]] {{Infobox Physical quantity |bgcolour={default} |name=Stress |image=[[File:Stress in a continuum.svg|400px]] |caption=Figure 1.1 shows stress in a loaded deformable material body assumed as a continuum. |unit=[[pascal (unit)|pascals]] (Pa) |symbols=σ |derivations=σ = [[force|F]] / [[area|A]] }} {{Continuum mechanics|cTopic=[[Solid mechanics]]}} In [[continuum mechanics]], '''stress''' is a [[physical quantity]] that expresses the internal [[force]]s that neighboring [[particle]]s of a [[continuous medium|continuous material]] exert on each other. For example, when a [[solid]] vertical bar is supporting a [[weight]], each particle in the bar pulls on the particles immediately above and below it. When a [[liquid]] is under [[pressure]], each particle gets pushed inwards by all the surrounding particles, and, in [[reaction force|reaction]], pushes them outwards. These forces are actually the average of a very large number of [[intermolecular force]]s and [[atomic collision|collision]]s between the [[molecule]]s on both sides of S. Stress inside a body may arise by various mechanisms, such as reaction to external forces applied to the bulk material (like [[gravity]]) or to its surface (like [[contact force]]s, external pressure, or [[friction]]). Significant stress may exist even when deformation is negligible (as is often done when studying the flow of water) or non-existent. It may also be created directly, for example by [[thermal expansion|changes in temperature]] or [[chemistry|chemical]] composition, or by external [[electromagnetic field]]s (as in [[piezoelectricity|piezoelectric]] and [[magnetostriction|magnetostrictive]] materials). Quantitatively, stress is defined as the force F between adjacent parts of the material across an imaginary separating surface S, divided by the area A of that surface. In a [[fluid]] at rest the force F is perpendicular to S, and is the [[hydrostatic pressure|pressure]]. In a [[solid]], or in a [[flow]] of viscous [[liquid]], the stress may not be perpendicular to S; hence the stress across S is a vector quantity, not a scalar. Moreover, its direction and magnitude generally depend on the orientation of S. Thus the stress state of the material is a [[tensor]], called the [[Cauchy stress tensor|(Cauchy) stress tensor]]; which is a [[linear map|linear function]] that relates the [[surface normal|normal vector]] of a surface S to the stress across S. With respect to any chosen [[Cartesian coordinates|coordinate system]], the Cauchy stress tensor can be represented as a [[symmetric matrix|symmetric]] [[matrix (mathematics)|matrix]] of 3x3 real numbers. Even within a homogeneous body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying [[tensor field]]. The relation between stress, deformation, and the [[strain rate tensor|rate of change of deformation]] can be quite complex, although a [[linear elasticity|linear approximation]] may be adequate in practice if the quantities are small enough. In some branches of [[engineering]], the term "stress" is occasionally used in a looser sense as a synonym of "internal force". For example, in the analysis of [[truss]]es, it may refer to the total traction or compression force acting on a beam, rather than the force divided by the area of its [[cross-section]]. ---> Compression, traction and shearing stress. ==Units of measure== Each element of the Cauchy stress tensor \sigma is multiplied by one coordinate of the surface normal to yield a component of the stress across a surface. Since the surface normal coordinates are [[dimensionless]] numbers, the elements of \sigma too are measured in pressure units. ---------------------------------------------------------------------- == Definition == Consider an imaginary flat [[surface element]] S separating two particles P and Q, with [[surface normal|normal vector]] u, directed from P to Q. Let F be the [[macroscopic]] force that P exerts on Q across S. The stress across S is the force F divided by the area A of S. Being a force per unit area, the stress on a surface element is [[dimensional analysis|dimensionally]] like the The direction and magnitude of this vector generally depend on the orientation of the surface element S. Thus, the stress state of the particle P as a whole is neither a scalar nor a single a vector, but a [[function (mathematics)|function]] \sigma that relates the outwards-pointing [[surface normal|normal vector]] u of a surface element S to the stress \sigma(u) on S. By the laws of [[physics]], this function must be a [[linear map|linear function]] between the two vectors; that is a (second-order) [[tensor]], called the [[Cauchy stress tensor|(Cauchy) stress tensor]]. With respect to any chosen coordinate system, the stress tensor can be represented as a [[symmetric matrix|symmetric]] [[matrix (mathematics)|matrix]] of 3x3 real numbers. Even within a homogeneous body, the Cauchy stress tensor may vary from place to place, and may change over time; it is therefore a (time-varying) [[tensor field]]. Any [[deformation (mechanics)|strain (deformation)]] of a solid material generates an internal [[elastic stress]] that tends to restore the material to its original (undeformed) state. For fluids (liquids and [[gas]]es), only deformations that change the volume generate persistent elastic stress. However, if the deformation is gradually changing with time, even in fluids there will usually be some [[viscous stress]], opposing that change. On the other hand, stress may exist with negligible or no deformation: water, for example, can be assumed to be incompressible, yet its stress (hydrostatic pressure) can hardly be neglected. The relation between stress, deformation, and the [[strain rate tensor|rate of change of deformation]] can be quite complex, although a [[linear elasticity|linear approximation]] may be adequate in practice if the quantities are small enough. Stress that exceeds certain [[strength of materials|strength limits]] of the material will result in permanent deformation (such as [[plasticity|plastic flow]],[[fracture]], [[cavitation]]) or even change of chemical state. ---------------------------------------------------------------------- Quantitatively, stress is defined by considering an imaginary flat [[surface element]] S in the solid. Let F be the force that the material on one side of S applies directly on the material on the other side of S, divided by the area A of S. In a [[fluid]] at rest the force F is perpendicular to S, and is the [[hydrostatic pressure|pressure]]. In a [[solid]], or in a [[flow]] of viscous [[liquid]], the stress may not be perpendicular to S; hence the stress across S is a vector quantity, not a scalar. Moreover, its direction and magnitude generally depend on the orientation of S. Thus the stress state of the material is a [[tensor]], called the [[Cauchy stress tensor|(Cauchy) stress tensor]]; which is a [[linear map|linear function]] \sigma that relates the [[surface normal|normal vector]] u of a surface element S to the stress \sigma(u) across S. With respect to any chosen [[Cartesian coordinates|coordinate system]], the Cauchy stress tensor can be represented as a [[symmetric matrix|symmetric]] [[matrix (mathematics)|matrix]] of 3x3 real numbers. Even within a homogeneous body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying [[tensor field]]. ---------------------------------------------------------------------- . In fact, the stress on the boundary of a particle P may be directed into P, away from P, or parallel to its boundary, depending on which surface element S we consider. Thus, the stress state of the particle P as a whole is neither a scalar nor a single a vector, but a [[function (mathematics)|function]] \sigma that relates the outwards-pointing [[surface normal|normal vector]] u of a surface element S to the stress \sigma(u) on S. By the laws of [[physics]], this function must be a [[linear map|linear function]] between the two vectors; that is a (second-order) [[tensor]], called the [[Cauchy stress tensor|(Cauchy) stress tensor]]. With respect to any chosen coordinate system, the stress tensor can be represented as a [[symmetric matrix|symmetric]] [[matrix (mathematics)|matrix]] of 3x3 real numbers. Even within a homogeneous body, the Cauchy stress tensor may vary from place to place, and may change over time; it is therefore a (time-varying) [[tensor field]]. ---------------------------------------------------------------------- Being a force per unit area, the stress on a surface element is [[dimensional analysis|dimensionally]] like the [[hydrostatic pressure|pressure]] within a [[fluid]]. However, in many materials—including solids and flowing [[viscosity|viscous]] [[fluid]]s—the force F may not be perpendicular to S. Therefore, the stress F/A across a surface element is a [[vector]] quantity, not a simple [[scalar (mathematics)]] like pressure. Moreover, the ---------------------------------------------------------------------- a [[function (mathematics)|function]] \sigma that relates the outwards-pointing [[surface normal|normal vector]] u of a surface element S centered on p to the stress \sigma(u) on S. By the laws of [[physics]], this function must be a [[linear map|linear function]] between the two vectors; that is a (second-order) [[tensor]], called the [[Cauchy stress tensor|(Cauchy) stress tensor]] of the material at that point. With respect to any chosen coordinate system, the stress tensor can be represented as a [[symmetric matrix|symmetric]] [[matrix (mathematics)|matrix]] of 3x3 real numbers. Even within a homogeneous body, the Cauchy stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is a [[tensor field]], often varying with time. ---------------------------------------------------------------------- Like other concepts of continuum mechanics, stress is a [[macroscopic]] concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore [[quantum mechanics|quantum]] effects and the detailed motions of molecules (and, depending on the context, other microscopic features like the grains of a [[metal]] rod or the [[fiber]]s of [[wood]]). Thus the stress across a surface element S is actually the average of a very large number of forces and collisions between the molecules on both sides of S. ---------------------------------------------------------------------- Any symmetric 3×3 matrix has three [[eigenvector]]s with real [[eigenvalue]]s. Therefore, at any instant and any point in the medium, there are three mutually orthogonal directions (unit-length vectors) e_1, e_2, and e_3 such that the stress across a surface element S<\math> perpendicular to each e_1 is aligned with e_1 and therefore perpendicular to S<\math>. That is, \mu\u_i =\lambda_i u_i<\math> for some real numbers \lambda_1, \lambda_2, and \lambda_3. Thus the stress state around a point is always a combination of normal stresses (compression or tension) along three mutually different directions. If we use e_1, e_2, and e_3 as coordinate axes, the stress tensor matrix becomes diagonal; that is, \epsilon_{i i} = \lambda_i \epsilon_{i j} = 0 if i\neq j. If the three eigenvalues \lambda_1, \lambda_2, and \lambda_3 are all distinct, the directions are unique except that each e_i can be negated, and they can be picked in any order. If two of them are equal, the two corresponding eigenvectors can be rotated about the third. If all three are equal, the stress across any surface element S<\math> centered at p is perpendicular to S<\math>, and its magnitude does not depend on direction: it is an isotropic compression (like hydrostatic pressure) or tension.