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Mathematical analysis of stresses in solids
Stress–strain analysis (or stress analysis) is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. In simple terms we can define stress as the force of resistance per unit area, offered by a body against deformation. Stress is the ratio of force over area (S = R/A, where S is the stress, R is the internal resisting force and A is the cross sectional area). Strain is the ratio of change in length to the original length, when a given body is subjected to some external force (Strain= change in length÷the original length). Stress analysis is a primary task for civil, mechanical and aerospace engineers involved in the design of structures of all sizes, such as tunnels, bridges and dams, aircraft and rocket bodies, mechanical parts, and even plastic cutlery and staples. Stress analysis is also used in the maintenance of such structures, and to investigate the causes of structural failures. Typically, the starting point for stress analysis are a geometrical description of the structure, the properties of the materials used for its parts, how the parts are joined, and the maximum or typical forces that are expected to be applied to the structure. The output data is typically a quantitative description of how the applied forces spread throughout the structure, resulting in stresses, strains and the deflections of the entire structure and each component of that structure. The analysis may consider forces that vary with time, such as engine vibrations or the load of moving vehicles. In that case, the stresses and deformations will also be functions of time and space. In engineering, stress analysis is often a tool rather than a goal in itself; the ultimate goal being the design of structures and artifacts that can withstand a specified load, using the minimum amount of material or that satisfies some other optimality criterion. Stress analysis may be performed through classical mathematical techniques, analytic mathematical modelling or computational simulation, experimental testing, or a combination of methods. The term stress analysis is used throughout this article for the sake of brevity, but it should be understood that the strains, and deflections of structures are of equal importance and in fact, an analysis of a structure may begin with the calculation of deflections or strains and end with calculation of the stresses.

Stress analysis is specifically concerned with solid objects. The study of stresses in liquids and gases is the subject of fluid mechanics. Stress analysis adopts the macroscopic view of materials characteristic of continuum mechanics, namely that all properties of materials are homogeneous at small enough scales. Thus, even the smallest particle considered in stress analysis still contains an enormous number of atoms, and its properties are averages of the properties of those atoms. In stress analysis one normally disregards the physical causes of forces or the precise nature of the materials. Instead, one assumes that the stresses are related to strain of the material by known constitutive equations. By Newton's laws of motion, any external forces that act on a system must be balanced by internal reaction forces, and they co exist with the stress fields that balance the external forces. In linear elasticity, their presence is required to satisfy the strain/displacement compatibility requirements and in limit analysis their presence is required to maximise the load carrying capacity of the structure or component. Such built in stress may occur due to many physical causes, either during manufacture (in processes like extrusion, casting or cold working), or after the fact (for example because of uneven heating, or changes in moisture content or chemical composition). However, if the system can be assumed to behave in a linear fashion with respect to the loading and response of the system, then effect of preload can be accounted for by adding the results of a preloaded structure and the same non preloaded structure. If linearity cannot be assumed, however, any built in stress may affect the distribution of internal forces induced by applied loads (for example, by changing the effective stiffness of the material) or even cause an unexpected material failure. For these reasons, a number of techniques have been developed to avoid or reduce built in stress, such as annealing of cold worked glass and metal parts, expansion joints in buildings, and roller joints for bridges.

Stress analysis is simplified when the physical dimensions and the distribution of loads allow the structure to be treated as one or two dimensional. In the analysis of a bridge, its three dimensional structure may be idealized as a single planar structure, if all forces are acting in the plane of the trusses of the bridge. Further, each member of the truss structure might then be treated a uni dimensional members with the forces acting along the axis of each member. In which case, the differential equations reduce to a finite set of equations with finitely many unknowns. If the stress distribution can be assumed to be uniform (or predictable, or unimportant) in one direction, then one may use the assumption of plane stress and plane strain behavior and the equations that describe the stress field are then a function of two coordinates only, instead of three. Even under the assumption of linear elastic behavior of the material, the relation between the stress and strain tensors is generally expressed by a fourth order stiffness tensor with 21 independent coefficients (a symmetric 6 × 6 stiffness matrix). This complexity may be required for general anisotropic materials, but for many common materials it can be simplified. For orthotropic materials such as wood, whose stiffness is symmetric with respect to each of three orthogonal planes, nine coefficients suffice to express the stress–strain relationship. For isotropic materials, these coefficients reduce to only two. One may be able to determine a priori that, in some parts of the system, the stress will be of a certain type, such as uniaxial tension or compression, simple shear, isotropic compression or tension, torsion, bending, etc. In those parts, the stress field may then be represented by fewer than six numbers, and possibly just one.

In any case, for two or three dimensional domains one must solve a system of partial differential equations with specified boundary conditions. Analytical (closed form) solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. For more complicated problems one must generally resort to numerical approximations such as the finite element method, the finite difference method, and the boundary element method.

The evaluation of loads and stresses within structures is directed to finding the load transfer path. Loads will be transferred by physical contact between the various component parts and within structures. The load transfer may be identified visually or by simple logic for simple structures. For more complex structures more complex methods, such as theoretical solid mechanics or numerical methods may be required. Numerical methods include direct stiffness method which is also referred to as the finite element method. The object is to determine the critical stresses in each part, and compare them to the strength of the material (see strength of materials). For parts that have broken in service, a forensic engineering or failure analysis is performed to identify weakness, where broken parts are analysed for the cause or causes of failure. The method seeks to identify the weakest component in the load path. If this is the part which actually failed, then it may corroborate independent evidence of the failure. If not, then another explanation has to be sought, such as a defective part with a lower tensile strength than it should for example.

Mohr's circle, Lame's stress ellipsoid (together with the stress director surface), and Cauchy's stress quadric are two dimensional graphical representations of the state of stress at a point. They allow for the graphical determination of the magnitude of the stress tensor at a given point for all planes passing through that point. Mohr's circle is the most common graphical method. Mohr's circle, named after Christian Otto Mohr, is the locus of points that represent the state of stress on individual planes at all their orientations. The abscissa, , and ordinate, , of each point on the circle are the normal stress and shear stress components, respectively, acting on a particular cut plane with a unit vector with components .

The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints of all stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i. e., the radius vector from the center of the ellipsoid, located at the material point in consideration, to a point on the surface of the ellipsoid is equal to the stress vector on some plane passing through the point. In two dimensions, the surface is represented by an ellipse (Figure coming).

The Cauchy's stress quadric, also called the stress surface, is a surface of the second order that traces the variation of the normal stress vector as the orientation of the planes passing through a given point is changed. The complete state of stress in a body at a particular deformed configuration, i. e., at a particular time during the motion of the body, implies knowing the six independent components of the stress tensor , or the three principal stresses , at each material point in the body at that time. However, numerical analysis and analytical methods allow only for the calculation of the stress tensor at a certain number of discrete material points. To graphically represent in two dimensions this partial picture of the stress field different sets of contour lines can be used: are a system of curves which are at each material point tangent to the principal axes of stress see figure
Isoclinics are curves on which the principal axes make a constant angle with a given fixed reference direction. These curves can also be obtained directly by photoelasticity methods. Slip lines are curves on which the shear stress is a maximum.