3.3 Mechanical Strength
3.3.1 Classic Mechanics
In a typical engineering design, a mechanical component is subject to a stress below the elastic limit. Therefore, Hook’s law applies in most engineering analyses and the material will linearly extend along the direction where an external load is applied. The material will elastically recover to the original dimensions after the removal of the applied load. When the applied stress is beyond the elastic limit, a permanent plastic deformation will remain.
The average strain is often referred to as an engineering or normal strain.
Consider a uniform tensile test specimen that is subject to an axial static tensile load and extended to the final length Lf ; the engineering strain, εe , is defined by Equation 3.5 as the ratio of the change in length, ∆L, to the origi- nal length, Lo:
εe o
f o
o
L L
L L
= ∆ = L−
(3.5) The nominal normal stress in the axial direction, σe , also referred to as the engineering stress, is defined by Equation 3.6:
σe o
P
= A (3.6)
where Ao is the original cross-sectional area and P is the total load. In the International System of Units (SI/Systeme International), the unit for stress is Newton per square meter, N/m2, or pascal, Pa. The stress is expressed as pound per square inch, psi, or 1,000 pounds per square inch, ksi, in the U.S. customary system. One pascal only represents a very small stress, 1 Pa = 0.000145 psi. The SI stress is therefore usually expressed in MPa = 106 N/m2 = 145 psi. The strain is dimensionless.
To the first degree of approximation, the engineering strain is linearly pro- portional to the engineering stress following Hook’s law when the stress is small. The proportionality constant is Young’s modulus, or the modulus of elasticity, E, as defined by Equation 3.7. The value of Young’s modulus is relatively independent of the manufacturing process, but it heavily depends on alloy composition. Both cast and extruded 2024 aluminum alloy (with a nominal composition of Al–4.5% Cu–1.5% Mg) show similar Young’s modulus values. However, the addition of 1% lithium to the 2024 aluminum alloy can increase its Young’s modulus by 8 to 9%. In reverse engineering, a comparison between the values of two Young’s moduli can be used as a barometer to verify the equivalency of some part characteristics, such as the elastic instability of a slender column due to buckling. However, most of the mechanical properties, such as yield strength, usually are a function of the manufacturing process and independent of Young’s modulus.
σ ε
e e
=E (3.7)
A typical engineering stress-strain curve in tension for ductile metals is illustrated in Figure 3.5. A proportional limit exists just below the elastic limit. The linearity between stress and strain as stated in Hook’s law starts to deviate beyond the proportional limit. When the stress reaches a critical
value, the material becomes unstable and continues to yield with perma- nent deformation at the same level of stress. A distinct yield point sometimes does not exist, particularly for brittle materials. In contrast to ultimate tensile strength that is well defined universally, yield strength has more than one definition. For engineering purposes, the yield strength is usually defined as the stress that will produce a small amount of permanent deformation, for example, 0.2%, the so-called 0.2% offset yield strength. In reverse engineer- ing applications, the engineer should verify that the same definition of yield strength applies to all data before the comparison. A higher stress is required to further deform the alloy beyond the yielding point due to strain harden- ing, until the maximum stress is reached. The ratio of the maximum load and the original cross-sectional area is defined as ultimate tensile strength.
The cross-sectional area of a ductile alloy usually begins to decrease rapidly beyond the maximum load. As a result, the total load required to further deform the specimen is decreased until the specimen fails at the fracture stress, as shown in Figure 3.5.
The yield strength depends on material composition as well as its micro- structure. The grain size has a profound effect on yield strength. Equation 3.8 is the mathematical formula of the Hall-Petch equation. It is an empirical relationship between yield strength and grain size and is based on the pio- neering work of Eric Hall (1951) and Norman Petch (1953). This is a functional relationship applicable to most polycrystalline alloys with grain size ranging from 1 mm to 1 µm. When the grain size is in this range, the impediment of dislocation movement is the determining factor of yield strength. The smaller the grain size is, the more grain boundaries there will be, and the more difficult dislocations can move from one grain to another—therefore, the material is stronger.
σy=σo+kd−1 2 (3.8)
Fracture strength Ultimate tensile
strength
Proportional Elastic
limit
Yield strength
Strain
Stress
FIgurE 3.5
Schematic of an engineering stress-strain curve.
where σy is the yield strength, σo and k are constant material parameters, and d is the average grain size.
In contrast to engineering stress, the true stress actually imposed on to a ten- sile specimen increases continuously as the true cross-sectional area, A, shrinks during the test. The true stress, σt , is defined by Equation 3.9 as force per unit true area at that instant, where P and A are force and area, respectively:
σt= P
A (3.9)
Similar to the true stress, the true or natural strain, εt , at a point is a local strain calculated against the actual length at the point of interest and at that instant. It is mathematically defined by Equation 3.10:
εt L
L f
o
dL L
L
o L
=
∫
f =ln (3.10)where L is specimen length at the moment, Lf is final specimen length, and Lo is original specimen length.
For very small elastic strains, the true and engineering strains are virtually the same value. However, the true strain more truthfully reflects the large plas- tic deformation. For instance, the engineering tension strain is εe = 2LL = 100%, while the true tension strain is εt = ln(2LL) = 69.3% when a specimen doubles its original length. A 69.3% true compressive strain [εt = ln(0.5LL) = –69.3%]
implies a reduction in length by half that represents an opposite but simi- lar deformation in tension at a 69.3% strain. In other words, in terms of true strain, 69.3% in tension means doubling the length, and 69.3% in compression means reducing the length by half. However, in terms of engineering strain, 100% in tension means doubling the length, while a 100% compressive strain (εe = ΔLL = –LL = –100%) implies a complete depression of the specimen from its original length to virtually zero length and represents a very different mag- nitude in deformation. Nonetheless, the simplicity in measurement of engi- neering stress and strain has made them overwhelmingly adopted in most engineering practices, as well as in reverse engineering applications, with only a few exceptions, such as true stress creep test.
Elastic deformation might also result in an angular or shape change, as shown in Figure 3.6. The angular change in a right angle is known as shear strain. The right angle at A was reduced by a small amount θ due to the application of a shear stress. The shear strain is the ratio between the dis- placement and the height, as defined by Equation 3.11, where τ, γ, a, and h are shear stress, shear strain, displacement, and height, respectively:
γ= =a θ θ≈
h tan (3.11)
For small θ values in radians, tanθ ≈ θ, and shear strains are often referred to as angles of rotation. Finally, it is worth noting that the tensile properties are func- tions of specimen size, loading rate, and testing environment, such as tempera- ture. The accuracy of comparative analysis based on tensile properties in reverse engineering directly relies on careful verification of these test parameters.