Structural materials experiencing complex stresses due to varied external forces may suffer elastic, anelastic and plastic deformations. Elastic strain is an instantaneous and completely recoverable deformation, the extent of which depends on the elastic modulus of the material and, in a simple uni­axial loading case,

Ze = |, [1.2]

E

where a is the stress (load per unit area), E the modulus of elasticity (also known as Young’s modulus) and eE is the instantaneous elastic strain (change in length per unit length). Anelastic strain is time dependent, completely reversible and generally small in magnitude — albeit non-negligible in some cases — as will be discussed in detail in Chapter 3. On the contrary, plas­tic strain is permanent and remains even after removal of the stresses; it is generally time — and rate-dependent. A typical stress vs strain curve under uniaxial loading is shown in Fig. 1.2a8 and the important design parameters are the yield strength, tensile strength, uniform elongation and ductility or total elongation to fracture. The deformation beyond the elastic limit obeys a power relation between the true stress (a) and the true plastic strain (ep):

a = K(£p)n, [1.3]

where K is the strength coefficient and n is the strain-hardening exponent. The area under the stress-strain curve represents the energy to deformation and fracture (referred to as resilience and toughness in the elastic and plastic regime, respectively), and this grades a material as brittle or ductile (Fig. 1.2b). The various mechanical properties of a material are also rate dependent and the flow stress is often characterized by the strain-rate sensitivity (m):

Ir, e = A m. [1.4]

The higher the n value, the higher is the uniform elongation, while a higher m value means a higher total elongation to fracture. The maximum possible value for m is unity which corresponds to viscous flow as seen in fluids, and this is generally noted in metals and ceramics at relatively high tempera­tures and at low strain-rates (or stresses).

Time dependent plastic deformation that occurs under constant load or stress (creep) becomes important above homologous temperature (T/TM > 0.4, where TM is the melting point in absolute temperature). The reader is referred to Chapter 3 for more detail on the underlying creep mechanisms and phenomenological descriptions of the creep rupture life. A typical creep curve is illustrated in Fig. 1.3 and design allowances are limited to the total strain accumulation in the primary and secondary regimes. Thus the strain at any instant of time is given by the sum of instantaneous recoverable elastic

strain, instantaneous plastic strain and time-dependent strain component from primary and secondary creep regimes:

£ = £0 + £ (l-) + £st, [1.5]

where £0 is the instantaneous strain (the majority from elastic deforma­tion), £t is the extent of primary creep strain, r is the rate at which strain decreases with time during primary creep regime and subscript ‘s’ stands for steady-state creep rate. The steady-state creep rate is a unique function of the applied stress and temperature for a given material

£s = Aone-Qc/RT, [1.6]

where Qc is the activation energy for creep, n is the stress exponent, R is the gas constant and T is absolute temperature. The activation energy for creep can generally be matched with that for self diffusion and the above relation­ship can be rewritten as

£s = A’Don, [1.7]

where D stands for appropriate diffusion coefficient and A’ could be grain size dependent (see Chapter 3 for details). In general, lattice diffusion is temperature dependent:

D = ea2vDCVe ~Qm/RT, [1.8a]

6

and

CV=e ~Qv /RT, [1.8b]

where в is the coordination number, a is the atomic jump distance, vD is Debye frequency, CV is vacancy concentration and Qm and QV are the acti­vation energies for migration and formation of a vacancy, respectively. It should be noted that higher stress increases the diffusion and leads to higher creep rate with reduced rupture time.