There are three basic types of deformation: elastic, plastic, and creep deformation. Elastic deformation is discussed next, and this leads to some rough theoretical estimates of strength for solids.
Elastic Deformation
Elastic deformation is associated with stretching, but not breaking, the chemical bonds between the atoms in a solid. If an external stress is applied to a material, the distance between the atoms changes by a small amount that depends on the material and the details of its structure and bonding. These distance changes, when accumulated over a piece of material of macroscopic size, are called elastic deformations. If the atoms in a solid were very far apart, there would be no forces between them.
As the distance x between atoms is decreased, they begin to attract one another according to the type of bonding that applies to the particular case. This is illustrated by the upper curve in Fig. 1.
A repulsive force also acts that is associated with resistance to overlapping of the electron shells of the two atoms. This repulsive force is smaller than the attractive force at relatively large distances, but it increases more rapidly, becoming larger at short distances. The total force is thus attractive at large distances, repulsive at short distances, and zero at one particular distance xe, which is the equilibrium atomic spacing. This is also the point of minimum potential energy.
Elastic deformations of engineering interest usually represent only a small perturbation about the equilibrium spacing, typically less than 1% strain. The slope of the total force curve over this small region is approximately constant.
Let us express force on a unit area basis as stress, σ = P/A, where A is the cross-sectional area of material per atom. Also, note that strain is the ratio of the change in x to the equilibrium distance xe.
Since the elastic modulus E is the slope of the stress–strain relationship, we have
Hence, E is fixed by the slope of the total force curve at x = xe, which is illustrated in Fig. 1.
Trends in Elastic Modulus Values
Strong primary chemical bonds are resistant to stretching and so result in a high value of E. For example, the strong covalent bonds in diamond yield a value around E = 1000 GPa, whereas the weaker metallic bonds in metals give values generally within a factor of three of E = 100 GPa.
In polymers, E is determined by the combination of covalent bonding along the carbon chains and the much weaker secondary bonding between chains. At relatively low temperatures, many polymers exist in a glassy or crystalline state.
The modulus is then on the order of E = 3 GPa, but it varies considerably above and below this level, depending on the chain-molecule structure and other details. If the temperature is increased, thermal activation provides increased free volume between chain molecules, permitting motion of increased lengths of chain.
A point is reached where large scale motions become possible, causing the elastic modulus to decrease, often dramatically. This trend is shown for polystyrene in Fig. 2.
The temperature where the rapid decrease in E occurs varies for different polymers and is called the glass transition temperature, Tg. Melting does not occur until the polymer reaches a somewhat higher temperature, Tm, provided that chemical decomposition does not occur first.
Above Tg, the elastic modulus may be as low as E = 1 MPa. Viscous flow is now prevented only by tangling of the long chain molecules and by the secondary bonds in any crystalline regions of the polymer.
A polymer has a leathery or rubbery character above its Tg, as do vulcanized natural rubber and synthetic rubbers at room temperature. For single crystals, E varies with the direction relative to the crystal structure; that is, crystals are more resistant to elastic deformation in some directions than in others.
But in a polycrystalline aggregate of randomly oriented grains, an averaging effect occurs, so that E is the same in all directions. This latter situation is at least approximated for most engineering metals and ceramics.
Theoretical Strength
A value for the theoretical cohesive strength of a solid can be obtained by using solid-state physics to estimate the tensile stress necessary to break primary chemical bonds, which is the stress σb corresponding to the peak value of force in Fig. 1.
These values are on the order of σb = E/10 for various materials. Hence, for diamond, σb ≈ 100 GPa, and for a typical metal, σb ≈ 10 GPa. Rather than the bonds being simply pulled apart in tension, another possibility is shear failure. A simple calculation can be done to obtain an estimate of the theoretical shear strength.
Consider two planes of atoms being forced to move slowly past one another, as in Fig. 3. The shear stress τ required first increases rapidly with displacement x, then decreases and passes through zero as the atoms pass opposite one another at the unstable equilibrium position x = b/2. The stress changes direction beyond this as the atoms try to snap into a second stable configuration at x = b. A reasonable estimate is a sinusoidal variation
where τb is the maximum value as τ varies with x; hence, it is the theoretical shear strength. The initial slope of the stress–strain relationship must be the shear modulus, G, in a manner analogous to E for the tension case previously discussed. Noting that the shear strain for small values of displacement is γ = x/h, we have
Obtaining dτ/dx from Equation 1 and substituting its value at x = 0 gives τb:
τb = Gb/2πh
The ratio b/h varies with the crystal structure and is generally around 0.5 to 1, so this estimate is on the order of G/10.
σb = 2τb = Gb/πh
In a tension test, the maximum shear stress occurs on a plane 45◦ to the direction of uniaxial stress and is half as large. Thus, a theoretical estimate of shear failure in a tension test is
Since G is in the range E/2 to E/3, this estimate gives a value similar to the previously mentioned σb = E/10 estimate based on the tensile breaking of bonds.
Theoretical tensile strengths around σb = E/10 are larger than the actual strengths of solids by a large amount, typically by a factor of 10 to 100. This discrepancy is thought to be due mainly to the imperfections present in most crystals, which decrease the strength. However, small whiskers can be made that are nearly perfect single crystals.
Also, thin fibers and wires may have a crystal structure such that strong chemical bonds are aligned with the length direction. Tensile strengths in such cases are indeed much higher than for larger and more imperfect samples of material. Strengths in the range from E/100 to E/20, corresponding to one-tenth to one-half of the theoretical strength, have been achieved in this way, lending credence to the estimates. Some representative data are given in following Table.
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