Understanding the behavior of materials when loaded is a vital part of creating safe and reliable structures. Of the various stresses a given structural element may be subjected to, tensile stress is perhaps the most concerning. – Not that materials are necessarily always that much stronger in compression and shear, given an equal force per unit area.
But it doesn’t take a genius to see that beams are weaker to bending than columns are to buckling. This is due to the direction and position of the primary loads and not because of the material’s tensile properties. This becomes apparent when both beam and column are of the same material, length, and cross section, yet the beam will exhibit major deflection long before the column comes even near buckling.
Due to this phenomenon, a beam must be carefully designed, with its second moment of area and material make-up ultimately deciding its resistance to deformation and failure. Rather than only tension, a horizontal structural member will more likely experience both tension on the lower face, and compression on the upper face, as it bends downward when loaded.
Generally speaking, there are two kinds of material-types, ductile and brittle.
By using a stress-strain curve one can graph the behavior of these materials starting from its original state to its point of rupture. Ductile materials will exhibit both an engineering or apparent stress-strain curve as well as a ‘true’ or actual stress-strain curve.
The difference is that the engineering curve bases its stress on the initial cross section of the material in question, whereas the true curve takes instantaneous ratios as the cross sectional area decreases due to Poisson’s contractions. So with the engineering curve, there’s a jump in the rate of strain as the cross section decreases, whereas the true curve will remain proportional to the cross section at all times.
Under normal test conditions, the true stress-strain curve is difficult to determine without continually monitoring the cross section. This is why the engineering curve is commonly used to make quick analysis of materials, even though the true curve is a better representation. As long as you don’t get the two mixed up, they both have their individual pros and cons.
Strain-hardening and Necking
Ductile materials also exhibit a phenomenon called strain-hardening or work-hardening when its yield point has been crossed and it enters plastic deformation. As the stress increases due to the strain-hardening, it will eventually reach its ultimate strength, at which point it will begin necking.
Necking is when the material’s cross section begins decreasing rapidly in a localized point, as opposed to a uniform decrease across the entire member. Once a member begins necking, rupture will soon follow. Necking is strictly a property of ductile materials, as opposed to brittle materials which will fracture before displaying any major cross-sectional reduction.
Brittle materials do not exhibit the discrepancy between true stress-strain and engineering stress-strain, as they have neither yield point nor strain-hardening properties. What this means is that the stress-strain curve of brittle materials will be linear, and extend in a linear fashion right up to their point of rupture. There is no yield point where the stress decreases, and no necking after ultimate strength – it will simply rupture while still deforming elastically.
An interesting result of the fracture of brittle materials is that the reconnection of the two broken parts form the same shape as the original. This is possible because there is no necking prior to the fracture, unlike ductile materials which often demonstrate large areas of plastic deformation and necking prior to fracture. This can be seen as a condition which defines ductile and brittle materials.
The above graph represents the difference in energy-absorption between brittle and ductile materials. As you can see, the shaded areas under the curves equal the total amount of energy absorbed by the two materials. Brittle materials don’t deform plastically, leaving no choice but fracture. In contrast, ductile materials can absorb large amounts of energy. These two properties must be carefully balanced to meet the needs of various applications.