Chapter 6. Mechanical Properties of Metals 
Mechanical Properties of Metals
Often materials are subject to forces (loads) when they are used. Mechanical engineers calculate those forces and material scientists how materials deform (elongate, compress, twist) or break as a function of applied load, time, temperature, and other conditions. Materials scientists learn about these mechanical properties by testing materials. Results from the tests depend on the size and shape of material to be tested (specimen), how it is held, and the way of performing the test. That is why we use common procedures, or standards, which are published by the ASTM. To compare specimens of different sizes, the load is calculated per unit area, also called normalization to the area. Force divided by area is called stress. In tension and compression tests, the relevant area is that perpendicular to the force. In shear or torsion tests, the area is perpendicular to the axis of rotation.
The unit is the Megapascal = 10^{6} Newtons/m^{2}. There is a change in dimensions, or deformation elongation, DL as a result of a tensile or compressive stress. To enable comparison with specimens of different length, the elongation is also normalized, this time to the length L. This is called strain, e.
The change in dimensions is the reason we use A_{0} to indicate the initial area since it changes during deformation. One could divide force by the actual area, this is called true stress (see Sec. 6.7). For torsional or shear stresses, the deformation is the angle of twist, q (Fig. 6.1) and the shear strain is given by:
Elastic deformation. When the stress is removed, the material returns to the dimension it had before the load was applied. Valid for small strains (except the case of rubbers). Deformation is reversible, non permanent Plastic deformation. When the stress is removed, the material does not return to its previous dimension but there is a permanent, irreversible deformation. In tensile tests, if the deformation is elastic, the stressstrain relationship is called Hooke's law:
That is, E is the slope of the stressstrain curve. E is Young's modulus or modulus of elasticity. In some cases, the relationship is not linear so that E can be defined alternatively as the local slope:
Shear stresses produce strains according to:
where G is the shear modulus. Elastic moduli measure the stiffness of the material. They are related to the second derivative of the interatomic potential, or the first derivative of the force vs. internuclear distance (Fig. 6.6). By examining these curves we can tell which material has a higher modulus. Due to thermal vibrations the elastic modulus decreases with temperature. E is large for ceramics (stronger ionic bond) and small for polymers (weak covalent bond). Since the interatomic distances depend on direction in the crystal, E depends on direction (i.e., it is anisotropic) for single crystals. For randomly oriented policrystals, E is isotropic. Here the behavior is elastic but not the stressstrain curve is not immediately reversible. It takes a while for the strain to return to zero. The effect is normally small for metals but can be significant for polymers. Materials subject to tension shrink laterally. Those subject to compression, bulge. The ratio of lateral and axial strains is called the Poisson's ratio n.
The elastic modulus, shear modulus and Poisson's ratio are related by E = 2G(1+n) Yield point. If the stress is too large, the strain deviates from being proportional to the stress. The point at which this happens is the yield point because there the material yields, deforming permanently (plastically). Yield stress. Hooke's law is not valid beyond the yield point. The stress at the yield point is called yield stress, and is an important measure of the mechanical properties of materials. In practice, the yield stress is chosen as that causing a permanent strain of 0.002 (strain offset, Fig. 6.9.) The yield stress measures the resistance to plastic deformation. The reason for plastic deformation, in normal materials, is not that the atomic bond is stretched beyond repair, but the motion of dislocations, which involves breaking and reforming bonds. Plastic deformation is caused by the motion of dislocations. Tensile strength. When stress continues in the plastic regime, the stressstrain passes through a maximum, called the tensile strength (s_{TS}) , and then falls as the material starts to develop a neck and it finally breaks at the fracture point (Fig. 6.10). Note that it is called strength, not stress, but the units are the same, MPa. For structural applications, the yield stress is usually a more important property than the tensile strength, since once the it is passed, the structure has deformed beyond acceptable limits. Ductility. The ability to deform before braking. It is the opposite of brittleness. Ductility can be given either as percent maximum elongation e_{max} or maximum area reduction.
These are measured after fracture (repositioning the two pieces back together). Resilience. Capacity to absorb energy elastically. The energy per unit volume is the area under the strainstress curve in the elastic region. Toughness. Ability to absorb energy up to fracture. The energy per unit volume is the total area under the strainstress curve. It is measured by an impact test (Ch. 8). When one applies a constant tensile force the material will break after reaching the tensile strength. The material starts necking (the transverse area decreases) but the stress cannot increase beyond s_{TS}. The ratio of the force to the initial area, what we normally do, is called the engineering stress. If the ratio is to the actual area (that changes with stress) one obtains the true stress. If a material is taken beyond the yield point (it is deformed plastically) and the stress is then released, the material ends up with a permanent strain. If the stress is reapplied, the material again responds elastically at the beginning up to a new yield point that is higher than the original yield point (strain hardening, Ch. 7.10). The amount of elastic strain that it will take before reaching the yield point is called elastic strain recovery (Fig. 6. 16). Compressive and shear stresses give similar behavior to tensile stresses, but in the case of compressive stresses there is no maximum in the se curve, since no necking occurs. Hardness is the resistance to plastic deformation (e.g., a local dent or scratch). Thus, it is a measure of plastic deformation, as is the tensile strength, so they are well correlated. Historically, it was measured on an empirically scale, determined by the ability of a material to scratch another, diamond being the hardest and talc the softer. Now we use standard tests, where a ball, or point is pressed into a material and the size of the dent is measured. There are a few different hardness tests: Rockwell, Brinell, Vickers, etc. They are popular because they are easy and nondestructive (except for the small dent). Tests do not produce exactly the same result because of variations in the test equipment, procedures, operator bias, specimen fabrication, etc. But, even if all those parameters are controlled within strict limits, a variation remains in the materials, due to uncontrolled variations during fabrication, non homogenous composition and structure, etc. The measured mechanical properties will show scatter, which is often distributed in a Gaussian curve (bellshaped), that is characterized by the mean value and the standard deviation (width). To take into account variability of properties, designers use, instead of an average value of, say, the tensile strength, the probability that the yield strength is above the minimum value tolerable. This leads to the use of a safety factor N > 1 (typ. 1.2  4). Thus, a working value for the tensile strength would be s_{W }= s_{TS} / N. Important Terms:
