Tensile testing

In the classic form of tensile testing, both ends of a material specimen are clamped between a pair of jaws. The lower jaw or holder is heldfixed in place. As shown in

Figure 2.2, the upper jaw is attached to the moving crosshead of a tensile tester via a load cell, which measures the applied force. The material specimens tested can be bar shaped with straight sides and uniform rectangular or round cross-sections. Alternatively, the test specimens can have varying cross-sections and be shaped into “dog bone” (rectangular cross-section) or “dumbbell” shapes (circular cross- section), as seen in Figure 2.3. An extensiometer is used to measure the change in the dimension of the test sample along the axis of loading. Using this test, information on force versus deformation is collected and used to develop a stress– strain plot. Shown inEqs. (2.1)and(2.2)are the formulae for stress and strain, where stress has units of force per unit area, and strain is dimensionless and has no units:

stressð Þ ¼σ force

original cross-sectional area, ð2:1Þ

strainðεÞ ¼ change in length

original length : ð2:2Þ

The gage section of the test sample is measured to determine both its cross- sectional area and the length (Figure 2.4). For straight-edge samples, the gage section is the portion of the sample between the upper and lower jaws. For the shaped dog bone and dumbbell samples, the gage section is the mid-section, which has a uniform but reduced cross-section.

F F (b) F F (c) F F (a) Figure 2.1

It is imperative to understand that force and stress are not one and the same.

A force of the same magnitude can generate very different stress levels based on the area over which it is applied.

Example: a force applied on an apple through the flat side of a knife blade (large area) may cause a small depression but when the same force is applied through the edge of the blade (small area), it will cut the fruit because of the very high stress generated.

In the seventeenth century, Robert Hooke showed that when a solid material is subjected to a tensile force, its deformation is proportional to the load applied. This is known as Hooke’s law and is represented by the linear region of the material’s stress–strain curve. An example of a stress–strain curve is shown in

Figure 2.2

Figure 2.5. In the linear region of the stress–strain curve, the material is said to be elastic and behaves like a spring. The slope of this linear region of the curve is known as the elastic modulus, E.Equation (2.3)defines the relationship between elastic modulus, stress, and strain:

σ ¼ Εε: ð2:3Þ

The proportional limit is the point on the stress–strain curve corresponding to the highest stress at which the stress is linearly proportional to the strain.

Gage Length (a) (b)( ) Gage Length Figure 2.3

Standard types of specimens for testing the mechanical properties of a material using a tensile testing machine: (a) dog bone and (b) dumbbell.

F F A Δᐍ ᐍo F A Δᐍ s = ᐍo e = Figure 2.4

The yield stress,σy, is the stress which causes the onset of permanent deformation in the material, and this stress is sometimes defined as the yield strength or the elastic limit of a material. Permanent deformation implies deformation that persists even when all loading is removed from the material. The point on the stress–strain curve corresponding to the yield stress is known as the yield point. The proportional limit and the yield stress can be the same value, but the propor- tional limit often precedes the yield stress. If the stress–strain curve is not linear or a clear yield point is not obvious, then alternate means are used to determine the yield stress. In such circumstances, a 0.2% strain technique is generally used to determine the yield point. In this method, a line parallel to the linear part of the stress–strain plot is drawn to intersect the strain axis at 0.2% strain. The point where this line intersects the stress–strain curve is designated as the yield point.

Under elastic conditions, the material reverts to its original dimension when unloaded. On the other hand, if the stress is increased beyond the elastic limit, the material may either undergo failure or become permanently or plastic- ally deformed. Plastic deformation is made possible by the movement of large numbers of atoms or as in the case of polymers, movement of molecular chains. Brittle materials, such as ceramics, do not exhibit plastic deformation or yield points. Other mechanical properties can also be obtained from the stress– strain curve: the maximum stress reached prior to fracture is defined as

Figure 2.5

A stress–strain curve showing the yield stress (σy), the ultimate strength (σu), and the elastic

modulus (_{E) given by slope of the linear portion. The shaded area under the curve represents the} toughness of the material.

the ultimate tensile strength; the area under the curve up to the failure point is a measure of the work required to cause fracture and is a measure of the toughness of the material.

For many ductile metals the stress levels continue to increase after the yield stress is reached and plastic deformation is initiated. This is due to a phenomenon known as strain hardening, which is the resistance to further deformation. This resistance is caused by the decreased mobility of atomic planes within the material due to the interaction of multiple dislocations (imperfections in the lattice of atoms in a material). As a result, the stress continues to increase until it reaches the ultimate strength, beyond which the material deforms rapidly until it fractures. It is interesting to note that materials with very few dislocations, such as single crystals, or materials with large number of dislocations tend to have high strength.

The stress value calculated for most engineering design considerations is called the engineering stress and is defined as σ ¼ F/A, where A represents the original cross-sectional area of the sample.

In reality the cross-sectional area changes with loading as the sample deforms.

The true stress in the sample can be calculated by using the instantan- eous area.

In general, the elastic modulus of polymeric materials is low and they tend to undergo large plastic deformations. The linear region of their stress–strain plot is usually short or may be non-existent. In some polymers, the slope of the stress–strain plot may increase after initial loading due to the alignment of molecular chains. Most polymers have relatively low ultimate strength but deform significantly before fracture and possess high toughness (Figure 2.6). Metals have higher elastic modulus, a clear linear region and often a well-defined yield point, beyond which they undergo plastic deformation. They have high ultimate strength. Ceramic materials have very high elastic modulus values and very little deformation. They usually have high strength but are brittle with very low toughness.

As a material is loaded, it undergoes deformation along the axis of the load. However, since its volume does not change, the longitudinal deformation necessi- tates a corresponding transverse deformation. For example, if a cylindrical specimen is loaded in tension and undergoes an increase in length, it will also exhibit a simultaneous decrease in diameter in order to maintain a constant volume. This is known as the Poisson Effect. The ratio of the transverse strain

divided by the longitudinal strain is called the Poisson’s ratio, as denoted by ν (nu). For most isotropic simple materials, the Poisson’s ratio lies in the range 0.2–0.5. Steels, when loaded within their elastic limits (before yield), usually exhibit aν value of approximately 0.3. Rubber has a ν value of nearly 0.5, whereas cork has very little transverse deformation and has aν value close to 0. A negative value of Poisson’s ratio indicates a material which, when stretched in one direc- tion, becomes thicker in the perpendicular direction. Such materials are called auxetic and are usually composed of macroscopic elements with hinge-like struc- tures. Some polymer foams behave in this manner.