Stress-Strain Relationship
Most polycrystalline materials have within their elastic range an almost constant relationship between stress and strain. Experiments by English scientist Robert Hooke led to the formation of Hooke's Law, which states that in the elastic range of a material, strain is proportional to stress. The ratio of stress to strain, or the gradient of the stress-strain graph, is called the Young's Modulus.
Elastic Moduli
The elastic moduli relevant to polycrystalline materials are Young's Modulus of Elasticity, the Shear Modulus of Elasticity, and the Bulk Modulus of Elasticity.
Young's Modulus
Young's Modulus of Elasticity is the elastic modulus for tensile and compressive stress and is usually assessed by tensile tests. A separate saVRee article discusses Young's Modulus of Elasticity in greater detail.
Shear Modulus
The Shear Modulus of Elasticity is derived from the torsion of a cylindrical test piece. Its symbol is G.
Bulk Modulus
The Bulk Modulus of Elasticity is the elastic response to hydrostatic pressure and equilateral tension or the volumetric response to hydrostatic pressure and equilateral tension. It is also the property of a material that determines the elastic response to the application of stress.
Tensile (Load) Tests and Stress-Strain Curves
To determine the load-carrying ability and the amount of deformation before fracture, a sample of material is commonly tested by a Tensile Test. This test consists of applying a gradually increasing force of tension at one end of a sample length of the material. The other end is anchored in a rigid support so that the sample is slowly pulled apart. The testing machine is equipped with a device to indicate, and possibly record, the magnitude of the force throughout the test. Simultaneous measurements are made of the increasing length of a selected portion at the middle of the specimen, called the gage length. The measurements of both load and elongation are ordinarily discontinued shortly after plastic deformation begins; however, the maximum load reached is always recorded. Fracture point is the point where the material fractures due to plastic deformation. After the specimen has been pulled apart and removed from the machine, the fractured ends are fitted together and measurements are made of the now extended gage length and of the average diameter of the minimum cross section. The average diameter of the minimum cross section is measured only if the specimen used is cylindrical.
The tabulated results at the end of the test consist of the following.
a. Designation of the material under test.
b. Original cross section dimensions of the specimen within the gage length.
c. Original gage length.
d. A series of frequent readings identifying the load and the corresponding gage length dimension.
e. Final average diameter of the minimum cross section.
f. Final gage length.
g. Description of the appearance of the fracture surfaces (for example, cup-cone, wolf's ear, diagonal, start).
A graph of the results is made from the tabulated data. Some testing machines are equipped with an autographic attachment that draws the graph during the test (the operator need not record any load or elongation readings except the maximum for each). The coordinate axes of the graph are strain for the x-axis or scale of abscissae, and stress for the y-axis or scale of ordinates. The ordinate for each point plotted on the graph is found by dividing each of the tabulated loads by the original cross-sectional area of the sample; the corresponding abscissa of each point is found by dividing the increase in gage length by the original gage length. These two calculations are made as follows.
Stress and strain, as computed here, are sometimes called "engineering stress and strain." They are not true stress and strain, which can be computed on the basis of the area and the gage length that exist for each increment of load and deformation. For example, true strain is the natural log of the elongation (ln (L/Lo)), and true stress is P/A, where A is area and P is pressure. The latter values are usually used for scientific investigations, but the engineering values are useful for determining the load-carrying values of a material. Below the elastic limit, engineering stress and true stress are almost identical.
The graphic results, or stress-strain diagram, of a typical tension test for structural steel is shown in the image below. The ratio of stress to strain, or the gradient of the stress-strain graph, is called the Modulus of Elasticity or Elastic Modulus. The slope of the portion of the curve where stress is proportional to strain (between Points 1 and 2) is referred to as Young's Modulus and Hooke's Law applies.
Typical Ductile Material Stress-Strain Curve
The following observations are illustrated in the image above:
- Hooke's Law applies between Points 1 and 2.
- Hooke's Law becomes questionable between Points 2 and 3 and strain increases more rapidly.
- The area between Points 1 and 2 is called the elastic region. If stress is removed, the material will return to its original length.
- Point 2 is the proportional limit (PL) or elastic limit, and Point 3 is the yield strength (YS) or yield point.
- The area between Points 2 and 5 is known as the plastic region because the material will not return to its original length.
- Point 4 is the point of ultimate strength and Point 5 is the fracture point at which failure of the material occurs.
The image above shows ductile material where the strength is small, and the plastic region is great. The material will bear more strain (deformation) before fracture.
The image below is a stress-strain curve typical of a brittle material where the plastic region is small, and the strength of the material is high.
Typical Brittle Material Stress-Strain Curve
The tensile test supplies three descriptive facts about a material. These are: the stress at which observable plastic deformation or "yielding" begins; the ultimate tensile strength or maximum intensity of load that can be carried in tension; and the percent elongation or strain (the amount the material will stretch) and the accompanying percent reduction of the cross-sectional area caused by stretching. The rupture or fracture point can also be determined.
Summary
The important information in this section is summarised below.
Stress-Strain Relationship Summary
- Bulk Modulus
The Bulk Modulus of Elasticity is the elastic response to hydrostatic pressure and equilateral tension, or the volumetric response to hydrostatic pressure and equilateral tension. It is also the property of a material that determines the elastic response to the application of stress.
- Fracture point is the point where the material fractures due to plastic deformation.
- Ductile material will deform (elongate) more than brittle material. The stress-strain curves discussed in this article for ductile and brittle demonstrated how each material would react to stress and strain.
- With reference to the previously seen graphs, Hooke's Law applies between Points 1 and 2, the elastic region is between Points 1 and 2, and the Plastic region is between Points 2 and 5.