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If you have learned about the concept of pressure in Fluid Mechanics, then you already know what stress is. When you apply a force to something, that force is distributed over a surface of influence and the amount of stress is the applied force per unit of that surface, just like pressure. That is why if I poke a nail against your skin you will feel terrible but if I poke that same nail with the same force on a piece of wood on top of your skin you won't feel a thing! The force divided by the surface of the nail's tip applies much pressure comparatively that it surpasses the strength of the skin and goes through it while in the latter case, the wood piece distributes the force and stress is negligible that we can bear easily. The same story is true for walking on a slab of iced lake or crouching on it.
The stress is usually shown as lbf/in2, ksi, and Pa and there are two types of them that are more important for our purpose here: Normal Stress (σ), where force is normal to the surface and Shear stress (τ), where force is parallel to the surface.
Strain is the deformation per unit. For example, linear strain (ϵ or normal, longitudinal, axial strain) is the change in length per unit of length (ϵ = δ / L), which obviously is a unitless quantity. The reason strain is used instead of deformation is to have a standard unit of stress influence. For example, assume two steel cylinders with the same cross section but different length are being pulled with the same force F. Obviously the stress applied to them are the same according to σ = F/A, but the change in their length will not be the same and that is because their initial length are not the same. So in order to have a better understanding of the effect of stress we normalize the change in length and name it strain. Now, we must have the same strain for both cylinders because they are both the same material and the same force is applied to them. Shear strain, γ is another form of strain that shows the angular deformation that results from shear stress.
As long as a material behaves elastically, there is a linear relationship between stress and strain that can be represented by Hook's Law: in the case of normal stress, stress is proportional to strain by the Modulus of Elasticity, E, or Young's Modulus (σ = Eϵ). If we press an elastic material (say a sponge) there will be axial deformation and lateral deformation: while it shrinks, it also somehow expands laterally. The ratio of lateral strain to axial strain is called Poisson's Ratio, ν. Note that this is a property of material that is used for the case of axial loading which varies between 0 to 0.5.
In case of pure shear stress, we can still apply Hook's law and say τ = Gγ, in which G is called the Shear Modulus, or Modulus of Rigidity. The following relationship exists between the moduli of elesticity and rigidity: E = 2G(1- ν). It is nice to know that these moduli and Poisson's ratio plus another modulus (bulk modulus) are called Elastic Constants, and as long as any two of them are known, the other two can be calculated. Here is a page (accessed 9/12/2015) for a class I used to teach, which is a nice reference on elastic constants. It must be known that Hook's law can also be used for 3D stress-strain relationships. In the page above (accessed 9/13/2015) in the details section, all the relationships in 3D are also explained.
If a member with length L is only axially loaded, its deformation δ can be found from Hook's law: δ = Lϵ = L(σ/E) = PL/AE. Now here we had the cross sectional area as constant, but if the cross section changes through an axial member, then the different deformation is added together to get the total deformation: δ = pΣ(L/AE). The Σ changes into integral if one of the variables next to it continuously changes through the length.
Elastic energy or the internal work is the energy per unit of volume stored in the deformed object, which is the area underneath the stress-strain curve. Next, we have the biaxial loading (plane stress) and triaxial loading, which are the case for most of the real structures that undergo different types of loading from different angles. Although, most of the problems can be solved in plane stress as in the third direction stresses are usually zero or negligible. If the normal and shear stress is known for a set of orthogonal planes in an element, then using transformation of axes, the stresses for any other plane within the element can be determined. Here is a nice pdf handout (accessed 9/13/2015) about this subject from the university of Colorado at Boulder.
Another theory is the maximum shear stress theory, which is good to predict yielding points (or failure) for ductile material. The concept is similar to the first theory mentioned above. For a lot more details on failure theories, see this page (accessed 9/13/2015). Another theory is the distortion energy theory, which is similar to the concept of strain energy. This theory is good in predicting the tensile and shear failure for ductile material under static biaxial loading, in which, the von Mises' stress or the effective stress is calculated using the principal stresses and the failure criterion is that it must exceed the ultimate strength.
Next, we talk about other types of stresses that happen in torsion, or as a result of temperature, etc.
In order to be able to predict when solid materials fail under external loading, failure theories were proposed. We need to have an idea about some of these theories here. For example, the maximum normal stress theory works good for brittle material under static biaxial loading. Failure occurs when the largest principal stresses exceed the strength of material either in compression or tension. Brittle material have a higher strength in compression, while ductile material have higher strength in tension. The failure theories make sure that the applied stresses does not reach the ultimate strength. For this purpose, we need a safety factor (SF), which is the ultimate strength Su divided by actual stress. Therefore, given the safety factor, material fails if σ > Su/SF.
The plane where the shear stress is zero can be found for any point in a loaded object. The normal stresses for such a plane are called principal stresses, which are the maximum and minimum normal stresses at that point. The equations of axes transformation can be used to find these extremum stresses in addition to the extremum shear stress, which result in finding the principal planes as well. Same page (accessed 9/13/2015) referred above provides ample information on this subject. All of the above can be graphically demonstrated by Mohr's Circle.
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