Any material would deform under an applied load. If all the deformation may recover upon unloading, this type of deformation is known as elastic deformation

Figure 5.2 (a) A schematic of a tensile tester. Adapted from Ref. [1]. (b) An Instron universal tester that can perform tensile test among many other tests. Courtesy: Instron.

as we defined earlier. However, if the load is large enough, plastic deformation sets in so that the material undergoes a permanent deformation, that is, upon unload­ing only the elastic portion of the deformation recovers but the rest remains.

Even though the raw data obtained from the tension test are load and elongation, they need to be converted into stress and strain, respectively, to have meaningful use in engineering considerations. First, let us talk about the engineering stress — strain curve that is of prime importance in many load-bearing applications. Engi­neering stress and strain are determined based on the original dimensions of the specimens. Engineering stress (ae), also known as conventional or nominal stress,

Figure 5.3 Design of an ASTM standard round tensile specimen.

186 I 5 Properties of Materials thus is given by P

Se = —, (5.5)

where P is the instantaneous load and A0 is the initial diameter of the gauge sec­tion. Engineering strain is defined as the ratio of the elongation of the gauge length (d = L — L0) to the original gauge length (L0) as shown in the equation below:

_ d _ L — L0 L0 L0

where L is the instantaneous gauge length. Engineering strain is generally expressed in terms of percent elongation. As the engineering stress-strain curves are obtained based on the original dimensions (constant), the engineering stress — strain curve and load-elongation curve have the similar shape. A schematic engi­neering stress-strain curve is shown in Figure 5.4.

Let us talk about the general shape of the engineering stress-strain curve.

a) In the initial linear portion of the curve, stress is proportional to strain. This is the elastic deformation (instantaneously recoverable) regime where Hooke’s law is applicable. The modulus of elasticity or Young’s modulus (Eq. (5.3)) can be determined from the slope of the straight portion on the stress-strain curve. Young’s modulus is determined by the interatomic forces that are difficult to change significantly without changing the basic nature of the materials. Hence, this is the most structure-insensitive mechanical property. However, any signifi­cant change in the crystal structure (such as material undergoing polymorphic transformation) would also change the elastic modulus. It is only affected to a small extent by alloying, cold working, or heat treatment. Most aluminum alloys have Young’s moduli close to 72 GPa. Modulus of elasticity does decrease with increasing temperature as the interatomic forces become weaker.

Strain to fracture

Uniform strain

Offset yield strength

Tensie strength

fracture stress

Elongation (%)

Figure 5.4 A typical engineering stress-strain curve.

b) When the deformation proceeds past a point, it becomes nonlinear, and the point at which this linearity ends is known as the proportional limit and the stress at which plastic deformation or yielding starts is known as yield stress or yield strength that depends on the sensitivity of the strain measurement. A majority of materials show a gradual transition from the elastic to the plastic regime and it becomes difficult to determine exactly what the yield stress actu­ally is. That is why yield strength is generally taken at an offset strain of 0.2%, as shown in Figure 5.4. In cases, where there is no straight portion in the stress — strain curve (such as for soft copper and gray cast iron), the yield strength is defined as the stress that generates a total strain of0.5%.

Another shape of stress-strain curve that is commonly observed in some spe­cific materials represents discontinuous yielding. Figure 5.5 shows such an engineering stress-strain curve. Some metals/alloys (in particular, low carbon steels) exhibit a localized, heterogeneous type of transition from elastic to plastic deformation. As shown in the figure, there is a sharp stress drop as the stress decreases almost immediately after the elastic regime. This maximum point on the linear stress-strain curve is known as “upper yield point.” Following this, the stress remains more or less constant, and this is called “lower yield point.”

Yield point phenomenon occurs because the dislocation movement gets impeded by interstitial atoms such as carbon, nitrogen, and so on forming sol­ute atmospheres around the dislocations. However, at higher stress, the disloca­tions break away from the solutes and thus requires less stress to move. That is why a sharp drop in load or stress is observed. The elongation that occurs at constant load or stress at the lower yield point is called “yield point elongation.” During yield point elongation, a type of deformation bands known as Liiders bands (sometimes known as Hartmann lines or stretcher strains) are formed across this regime. This particular phenomenon is called Piobert effect. After Liiders bands cover the whole gauge length of the specimen, the usual strain hardening regime sets in, as shown in Figure 5.5. However, temperature of test­ing could change the behavior drastically.

Upper yield

Yield elongation

Lower yield

Stress

•Liiders

Wr band

Unyielded metal

Elongation

Figure 5.5 Typical discontinuous stress-strain curve with distinct yield point phenomenon. From Ref.[2].

c) After proportional limit or yielding, the material enters a strain hardening regime where stress increases with increase in strain until it reaches a stress where non­uniform plastic deformation (or necking starts). This stress is known as tensile strength or ultimate tensile stress (UTS) corresponding to the maximum load in the load-elongation curve. So, basically UTS is given by the maximum load divided by the original cross-sectional area of the specimen. UTS in itself is not a property of fundamental significance. But this has long been used in the design of materials with a suitable safety factor (~2). Nowadays yield strength rather than UTS is used for the purpose of designing. But still it can serve as a good quality control indicator and in specifications of the products. However, since UTS is easy to determine and is quite a reproducible property, it is still used in practice. For brittle materials, UTS is considered a valid design criterion.

d) Ductility as determined by the tension testing is a subjective property; however, it does have great significance: (i) Metal deformation processing needs a mate­rial to be ductile without fracturing prematurely. (ii) A designer is interested to know whether a material will fail in service in a catastrophic manner or not. That information may come from ductility. (iii) If a material is impure or has undergone faulty processing, ductility can serve as a reliable indicator even though there is no direct relationship between the ductility measurement and service performance.

Generally, two measures of ductility as obtained from a tension test are used — total fracture strain (ef) and the reduction of area at fracture (q). These properties are obtained after fracture during tension test using the following equations:

where Lf and Af are the final gauge length and cross-sectional area, respectively. Generally total fracture strain is composed of two plastic strain components (prenecking deformation that is uniform in nature, and postnecking deforma­tion that is nonuniform in nature). Both these properties are expressed in per­centage. We note from Eq. (5.7) that the elongation to fracture depends on the gauge length (L0). That is why it is customary to mention the gauge length ofthe tension specimen while reporting the elongation values; 2 in. gauge lengths are generally used. On the other hand, percentage reduction in area is the most structure-sensitive property that can be measured in a tension test even though it is difficult to measure very accurately or in situ during testing.

e) Fracture stress or breaking stress (Eq. (5.9)) are often used to define the engi­neering stress at which the specimen fractures. However, the parameter does not have much significance as necking complicates its real value.

Pf

A ’

where Pf is the load at fracture.

f)We discuss two more properties that are of importance — resilience and tough­ness. From dictionary meaning, they would be considered synonyms. But in the context of material properties, they are bit different. Resilience is the ability of a material to absorb energy when deformed elastically. Modulus of resilience (U0) is used as its measure and is given by the area under the stress-strain curve up to yielding:

U0 = ^. 0 2E

where se0 is the yield strength and E is the modulus ofelasticity.

Toughness is the ability of a material to absorb energy in the plastic range. As we will see later, there are two more types of toughness that are often used: frac­ture toughness and impact toughness. Hence, the toughness one obtains from stress-strain curves is known as tensile toughness. The area under the stress — strain curve generally indicates the amount of work done per unit volume on the material without causing its rupture. Toughness is generally described as a parameter that takes into account both strength and ductility. There are empiri­cal relations that express toughness; however, as they are based on original dimensions, they do not represent the true behavior in the plastic regime.

Even though we use the parameters obtained from engineering stress-strain curves for engineering designs, they do not represent the fundamental deforma­tion behavior of the material as it is entirely based on the original dimensions of the tensile specimen. In the engineering stress-strain curve, the stress falls off after the maximum load due to the gradual load drop and calculation based on the original cross-sectional area of the specimen. But in reality, the stress does not fall off after maximum load. Actually, the strain hardening effect (the stress in fact increases) remains in effect until the fracture, as shown in Figure 5.6, but the cross-sectional area of the specimen decreases more compared to the load drop, thus increasing the stress. This happens because the true stress (st) is based on the

instantaneous cross-sectional area (A) and is expressed as P

St = A = Se(e + 1). (5.12)

Before necking, it is better to calculate the true stress from the engineering stress and engineering strain based on the constancy of volume and a homogeneous dis­tribution of strain across the gauge length. However, after necking, these assump­tions are hardly valid and the true stress should then be calculated by using the actual load and cross-sectional area in the postnecking regime. True strain, on the other hand, is generally calculated from the expression as given below:

However, the second equality of the equation does not hold valid after necking. That is why beyond the maximum load, true strain should be calculated based on actual area or diameter of the specimen measurement following the relation given below:

For the true stress-strain curves, some parameters such as true stress at maxi­mum load, true fracture stress, true fracture strain, true uniform strain, and so on can be calculated using appropriate relations.

One important parameter that can be obtained from the true stress-true strain curve is the strain hardening exponent (n). This is generally described by Hollo — mon’s equation:

S = Ken,

Figure 5.7 Double logarithmic plot illustrating the method of determining strain hardening exponent and strength coefficient.

where K is the strength coefficient and n is known as strain hardening (or work hardening) exponent or parameter or coefficient. As the above equation is in the form of a power law, the true stress and true strain data when plotted on a double logarithmic scale and fitted to a straight line will yield a slope that is equal to strain hardening exponent, and the strength coefficient (i. e., the true stress at e = 1.0) can be calculated from the extrapolated line, as shown in Figure 5.7. Note that the data used for the calculation of n should not be taken beyond the maximum load (i. e., at or after maximum load). Theoretically, n value can range between 0 (elastic solid, follows Hooke’s law) and 1 (perfectly plastic solid). For most metals/alloys, n values are found to be 0.1-0.5. Note that these aspects are valid for stresses and strains beyond the elasticity (i. e., in the plastic deformation regime). Strain hardening is sensitive to the microstructure, which in turn depends on processing. The rate of strain hardening is obtained by differentiating Eq. (5.15) with respect to e and then replacing Ken by st, the following equation is obtained:

t St

n— .

e

Note that often the data at low strains and high strains tend to deviate from Hollomon’s equation. That is why many other techniques, such as Ludwik’s equa­tion, Ludwigson’s equation, Ramberg-Osgood relation, and so on have been pro­posed over the past years to obtain a better estimate of strain hardening exponents.

One important aspect of strain hardening exponent is that it represents the true uniform strain, and is thus related to the ductility of a material. In order to derive the relation, we need to first discuss about the instability effect that occurs in tension. In tension, necking (i. e., localized deformation) occurs at

the maximum load (generally in a ductile material since brittle materials frac­ture well before reaching that point). The load-carrying capacity of the material increases as the strain increases; however, as noted before, the cross-sectional area decreases as an opposing effect. However, at the onset of necking, the increase in stress due to decrease in cross-sectional area becomes higher than the concomitant increase in the load-carrying ability of the material because of strain hardening. The condition of instability can be written in a differential form as dP = 0 where P is a constant (the maximum load). Now replacing P with (st ■ A) in this instability equation, we obtain

dP — d(at ■ A) — A ■ dat + at ■ dA — 0

or

dA dat

A at

Due to the constancy of volume (V) during plastic deformation, we can also write

dV — 0, V — AL; hence dV — d(AL) —A ■ dL + L ■ dA — 0

or

By comparing Eqs (5.17) and (5.18), we can write

which is valid at the condition oftensile instability (or at maximum load). Now comparing Eqs (5.16) and (5.19), we obtain

n — eu, (5.20)

where eu is the true uniform strain, that is, true strain at the maximum load. Hence, it can be noted that the higher the strain hardening exponent, the greater the uniform elongation, which in turn helps in improving the overall ductility. Note that Considere’s criterion (da/de — a or da/de — a/(1 + e)) can be used to estimate the strain hardening exponent from a graphical plot, this is however out­side the scope of this book. For further details on these relations, readers are referred to the suggested texts enlisted in Bibliography.

5.1.1.1 Effect of Strain Rate on Tensile Properties

The rate at which strain is imposed on a tensile specimen is called strain rate (e — de/dt). The unit is generally expressed in s_1. It is instructive to know the ranges of generic strain rates used in different types of mechanical testing and deformation processing. Generally, quasi-static tension testing involves strain rates in the range of 10_5-10_1 s_1. Generally, tension testing is done by placing the tension specimen in the cross-head fixture and running the test at a constant

cross-head speed. However, it is important to know what the strain rate is in the specimen. Following Nadai’s analysis, the nominal strain rate is expressed in terms of cross-head speed (v) and original gauge length (L0):

v

Lo

However, the true strain rate changes as the gauge length changes:

Therefore, it must be noted that most tensile tests are not conducted at constant true strain rates. However, specific electronic feedback (open loop before necking and closed loop after necking) system can be set up with the tension tester where the cross-head speed is continuously increased as the test progresses in order to maintain the same true strain rate throughout the test. But the test becomes more complicated without much benefit.

The flow stress increases with increasing strain rate. The effect of strain rate becomes more important at elevated temperatures. The following equation shows the relation between the flow stress and true strain rate at a constant strain and temperature:

S = Cem (5.23)

where C is a constant and m is the strain rate sensitivity (SRS). The exponent m can be found out from the slope of the double logarithmic plot of true stress versus true strain rate. The value of strain rate sensitivity is quite low (<0.1) at room tempera­ture, but it increases as the temperature becomes higher with a maximum value of 1 when the deformation is known as viscous flow. Figure 5.8 shows flow stress (at 0.2% strain) versus strain rate on a double logarithmic plot for an annealed 6063 Al-Mg-Si alloy. In superplastic materials, the strain rate sensitivity is higher (0.4-0.6). But these materials require finer grain diameter (<10 pm) and tempera­tures at or above half the melting temperature (in K). Superplastic materials exhibit higher than normal ductility (as rule of thumb more than 200%) and utilize strain rate hardening instead of strain hardening.