Stress–strain analysis


Stress–strain 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. Strain is the ratio of change in length to the original length, when a given body is subjected to some external force.
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.

Scope

General principles

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 the 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, or cause the particles in the affected part to accelerate. In a solid object, all particles must move substantially in concert to maintain the object's overall shape. It follows that any force applied to one part of a solid object must give rise to internal reaction forces that propagate from particle to particle throughout an extended part of the system. With very rare exceptions, internal forces are due to very short range intermolecular interactions, and are therefore manifested as surface contact forces between adjacent particles — that is, as stress.

Fundamental problem

The fundamental problem in stress analysis is to determine the distribution of internal stresses throughout the system, given the external forces that are acting on it. In principle, that means determining, implicitly or explicitly, the Cauchy stress tensor at every point.
The external forces may be body forces, that act throughout the volume of a material; or concentrated loads, that are imagined to act over a two-dimensional area, or along a line, or at single point. The same net external force will have a different effect on the local stress depending on whether it is concentrated or spread out.

Types of structures

In civil engineering applications, one typically considers structures to be in static equilibrium: that is, are either unchanging with time, or are changing slowly enough for viscous stresses to be unimportant. In mechanical and aerospace engineering, however, stress analysis must often be performed on parts that are far from equilibrium, such as vibrating plates or rapidly spinning wheels and axles. In those cases, the equations of motion must include terms that account for the acceleration of the particles. In structural design applications, one usually tries to ensure the stresses are everywhere well below the yield strength of the material. In the case of dynamic loads, the material fatigue must also be taken into account. However, these concerns lie outside the scope of stress analysis proper, being covered in materials science under the names strength of materials, fatigue analysis, stress corrosion, creep modeling, and other.

Experimental methods

Stress analysis can be performed experimentally by applying forces to a test element or structure and then determining the resulting stress using sensors. In this case the process would more properly be known as testing. Experimental methods may be used in cases where mathematical approaches are cumbersome or inaccurate. Special equipment appropriate to the experimental method is used to apply the static or dynamic loading.
There are a number of experimental methods that may be used:
  • Tensile testing is a fundamental materials science test in which a sample is subjected to uniaxial tension until failure. The results from the test are commonly used to select a material for an application, for quality control, or to predict how a material will react under other types of forces. Properties that are directly measured via a tensile test are the ultimate tensile strength, maximum elongation and reduction in cross-section area. From these measurements, properties such as Young's modulus, Poisson's ratio, yield strength, and the strain-hardening characteristics of the sample can be determined.
  • Strain gauges can be used to experimentally determine the deformation of a physical part. A commonly used type of strain gauge is a thin flat resistor that is affixed to the surface of a part, and which measures the strain in a given direction. From the measurement of strain on a surface in three directions the stress state that developed in the part can be calculated.
  • Neutron diffraction is a technique that can be used to determine the subsurface strain in a part.
  • The photoelastic method relies on the fact that some materials exhibit birefringence on the application of stress, and the magnitude of the refractive indices at each point in the material is directly related to the state of stress at that point. The stresses in a structure can be determined by making a model of the structure from such a photoelastic material.
  • Dynamic mechanical analysis is a technique used to study and characterize viscoelastic materials, particularly polymers. The viscoelastic property of a polymer is studied by dynamic mechanical analysis where a sinusoidal force is applied to a material and the resulting displacement is measured. For a perfectly elastic solid, the resulting strains and the stresses will be perfectly in phase. For a purely viscous fluid, there will be a 90 degree phase lag of strain with respect to stress. Viscoelastic polymers have the characteristics in between where some phase lag will occur during DMA tests.

    Mathematical methods

While experimental techniques are widely used, most stress analysis is done by mathematical methods, especially during design.

Differential formulation

The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies and the Euler-Cauchy stress principle, together with the appropriate constitutive equations.
These laws yield a system of partial differential equations that relate the stress tensor field to the strain tensor field as unknown functions to be determined. Solving for either then allows one to solve for the other through another set of equations called constitutive equations. Both the stress and strain tensor fields will normally be continuous within each part of the system and that part can be regarded as a continuous medium with smoothly varying constitutive equations.
The external body forces will appear as the independent term in the differential equations, while the concentrated forces appear as boundary conditions. An external surface force, such as ambient pressure or friction, can be incorporated as an imposed value of the stress tensor across that surface. External forces that are specified as line loads or point loads introduce singularities in the stress field, and may be introduced by assuming that they are spread over small volume or surface area. The basic stress analysis problem is therefore a boundary-value problem.

Elastic and linear cases

A system is said to be elastic if any deformations caused by applied forces will spontaneously and completely disappear once the applied forces are removed. The calculation of the stresses that develop within such systems is based on the theory of elasticity and infinitesimal strain theory. When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved
Engineered structures are usually designed so that the maximum expected stresses are well within the realm of linear elastic behavior for the material from which the structure will be built. That is, the deformations caused by internal stresses are linearly related to the applied loads. In this case the differential equations that define the stress tensor are also linear. Linear equations are much better understood than non-linear ones; for one thing, their solution will also be a linear function of the applied forces. For small enough applied loads, even non-linear systems can usually be assumed to be linear.