Stress–strain curve

During testing of a material sample, the stress–strain curve is a graphical representation of the relationship between stress, derived from measuring the load applied on the sample, and strain, derived from measuring the deformation of the sample, i.e. elongation, compression, or distortion. The nature of the curve varies from material to material. The following diagrams illustrate the stress–strain behaviour of typical materials in terms of the engineering stress and engineering strain where the stress and strain are calculated based on the original dimensions of the sample and not the instantaneous values.

Ductile materials

1. Ultimate Strength
2. Yield Strength
3. Rupture
4. Strain hardening region
5. Necking region.]

Steel generally exhibits a very linear stress–strain relationship up to a well defined yield point (figure 1). The linear portion of the curve is the elastic region and the slope is the modulus of elasticity or Young's Modulus. After the yield point the curve typically decreases slightly due to dislocations escaping from Cottrell atmospheres. As deformation continues the stress increases due to strain hardening until it reaches the ultimate strength. Until this point the cross-sectional area decreases uniformly due to Poisson contractions.

However, beyond this point a "neck" forms where the local cross-sectional area decreases more quickly than the rest of the sample resulting in an increase in the true stress. On an engineering stress–strain curve this is seen as a decrease in the stress. Conversely, if the curve is plotted in terms of "true stress" and "true strain" the stress will continue to rise until failure. Eventually the neck becomes unstable and the specimen ruptures (fractures).

Less ductile materials such as aluminum and medium to high carbon steels do not have a well-defined yield point. For these materials the yield strength is typically determined by the "offset yield method", by which a line is drawn parallel to the linear elastic portion of the curve and intersecting the abscissa at some arbitrary value (most commonly 0.2%). The intersection of this line and the stress–strain curve is reported as the yield point.

Brittle materials

Brittle materials such as concrete and carbon fiber do not have a yield point, and do not strain-harden which means that the ultimate strength and breaking strength are the same. A most unusual stress-strain curve is shown in the figure. Typical brittle materials like glass do not show any plastic deformation but fail while the deformation is elastic. One of the characteristics of a brittle failure is that the two broken parts can be reassembled to produce the same shape as the original component as there will not be a neck formation like in the case of Ductile materials. A typical stress strain curve for a brittle material will be linear. Testing of several identical specimens will result in different failure stresses. The curve shown below would be typical of a brittle polymer tested at very slow strain rates at a temperature above its glass transition temperature. Some engineering ceramics show a small amount of ductile behaviour at stresses just below that causing failure but the initial part of the curve is a linear.

In brittle materials such as rock, concrete, cast iron, or soil, tensile strength is negligible compared to the compressive strength and it is assumed zero for many engineering applications. Glass fibers have a tensile strength stronger than steel, but bulk glass usually does not. This is due to the Stress Intensity Factor associated with defects in the material. As the size of the sample gets larger, the size of defects also grows. In general, the tensile strength of a rope is always less than the tensile strength of its individual fibers.

Properties

The area underneath the stress–strain curve is the toughness of the material—the energy the material can absorb prior to rupture.

The resilience of the material is the triangular area underneath the elastic region of the curve. By Definition Resilience is the property of a material to absorb energy when it is deformed elastically and then, upon unloading to have this energy recovered.


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