1. 2 STRESS-STRAIN CURVES 13 nine constants required to describe the elastic response of an orthorhombic crystal five for hexagonal, and only three for cubic crystals. For the latter, the elastic com- pliance matrix reduces to 000 12S12S11000 5=00 0s4400 0000 00000 It can be shown for the case of cubic crystals that the modulus of elasticity in any given direction may be given by Eq. 1-14 in terms of these three independent elastic constants and the direction cosines of the crystallographic direction under study 少=-261-51)-54B2+B3+1(14) where l1, I2, l3 direction cosines. Note that the elastic modulus for a given cubic material depends only on the magnitude of the direction cosines, with values for the principal crystallographic directions in the cubic lattice being given in Table 1. 2. For example,the modulus in the(100)direction is given by 1/s1l, since 2lg=0.By omparison,244=3(the maximum value)in the(111)direction so that 1/E=S11 AI(s11 -S12)-h2S44] Depending on whether (51 -S12)is larger or smaller than 12544, the modulus of elasticity may be greatest in either the (111)or(100) rection( see Problem 1.9). By comparison, the modulus in the (110) direction is in good agreement with the average value of E for a polycrystalline sample of the same material(see Example 1.2 and Problem 1.10). The elastic constants for several materials are given in Table 1.3, and their relative elastic anisotropy is tabulated in Table 1. 4. Note the large anisotropy exhibited by many of these crystals as compared with the isotropic behavior of tungsten for which 5u -S12)=h2s44. Owing to this equality, the modulus of elasticity in tungsten is independent of the direction cosines(Eq. 1-14) EXAMPLE 1.2:99056820029466002485922593 Compute the modulus of elasticity for tungsten and iron in the(110)direction.From Tables 1.2 and 1.3 we obtain the necessary information regarding elastic compliance TABlE 1.2 Direction Cosines for rincipal Directions in Cubic Lattice Direction 100) (111
14 CHAPTER 1 TENSILE RESPONSE OF MATERIALS TABLE 1.3 Stiffiness and Compliance Constants for Selected Crystals (100Pa) Aluminum 1082 057 1684 12.14 1860 l5.70 4.20233 23.70 14.10 11 0.80 63 1.16 155 0096 0648 1247 odium chloride 4.87 1272.29 Spinel(MgAl2O小27.9 0208 Titanium carbide 51.3 178 021 0.561 Tungsten 19.8 1514026 1079 722 4.1220 3.55540 Cobalt 30.70165010.3035817.53 032132 Titanium 8 0.180.692.15 Zinc 16.10 25016103.8 Data adapted from H B. Huntington, Solid State Physics, Vol 7, Academic, New York, 1958, p. 213 ege, Elastic, Piezoelectric and Related Constants of Crystals, Springer-Verlag, Berlin values and direction cosines. The modulus of elasticity in the(110) direction is then determined from Eq. 1-14 =0.26-2{026-(-0.07)]-(066(4) =026-(0)(4) TABLE 1. 4a Elastic Anisotropy of Selected Materials Degree of Anisotropy 岛岛[ .203 s 170.0 2.133 0.877 4292 1000 384.6
1. 2 STRESS-STRAIN CURVES 15 TABLE 1.4b Elastic Anisotropy of Selected Materials Aluminum 92 3.203 97 2.87 169 2.72 2.133 Titanium carbide Tungsten 1.000 55.8 Therefore E10=3846GPa which is the same value given in Table 1. 4 for Elll and E100- For iron E10=0.80-20.80-(-0.28)-(0:6)(4) E10=210.5GPa Note that E111>E1o>E1oo and that enlo is in good agreement with the average value of E for a polycrystalline sample (Table 1.1) For the case of tungsten and any other isotropic material, then, 5120z ∈a=S120x+sn2y+St Yxy= S44xy= 2(511-512)tx SiT 2(51-S2)ry If we compare Eqs. 1-15 with 1-10, the elastic constants sy may be described in terms of the familiar strength of materials elastic constants. Therefore (1-16a) 2(511-12) 1
16 CHAPTER 1 TENSILE RESPONSE OF MATERIALS Finally, from Eq. 1-16, G=2-51)=2+ E 2(1+v) For the case of hexagonal crystals, the matrix in Eq. 1-11 reduces to E=1(1-2+3y+(23+1(1-13 where 11, L2, I3 are direction cosines for directions in the hexagonal unit cell. from Eq. 1-17 note that in hexagonal crystals e depends only on the direction cosine l3, which lies normal to the basal plane. Consequently, the modulus of elasticity in hexagonal crystals is isotropic everywhere in the basal plane. 1.2.1.2 Resiliency The resilience of a material is a measure of the amount of energy that can be absorbed under elastic loading conditions and which is released completely when the loads are removed. From this definition, resilience may be measured from the area under the curve in Fig. 1.1 where omax =maximum stress for elastic conditions Emax elastic strain limit Should an engineering design require a material that allows only for elastic response with large energy absorption(such as in the case of a mechanical spring), the appro- priate material to choose would be one possessing a high yield strength but low modulus of elasticity 1.2.2 Elastic-Homogeneous Plastic Response: Type II When a material has the capacity for plastic deformation--irreversible flow-the stress-strain curve often assumes the shape of Curve I ( Fig. 1.5). Here we see the same elastic region at small strains but now find a smooth parabolic portion of the curve, which is associated with homogeneous plastic deformation processes, such as
1. 2 STRESS-STRAIN CURVES 17 FIGURE 1.5 Type I stress-strain behavior revealing elastic behavior followed by a region of homogeneous plastic deformation Data are plotted on the basis of engineering and true stress-strain definitions. the irreversible movement of dislocations in metals, ceramics, and crystalline poly mers, and a number of possible deformation mechanisms in amorphous polymers. That the curve continues to rise to a maximum stress level reflects an increasing resistance on the part of the material to further plastic deformation--a process known s strain hardening. The portion of the true stress-strain curve(from the onset of ielding to the maximum load)may be described empirically by the relationship generally attributed to Hollomon where g= true stress ∈= true plastic strain n= strain-hardening coefficient K material constant, defined as the true stress at a true strain of 1.0 However, Bulfinger' proposed a similar parabolic relationship between stress and strain almost 200 years earlier The magnitude of the strain ning coefficien reflects the ability of the material to resist further deformation In the limit, n may equal to unity, which represents ideally elastic behavior, or equal to zero, which represents an ideally plastic material. Selected values of strain-hardening coefficients for some engineering metal alloys are given in Table 1.5. Note that n values are sensitive to thermomechanical treatment; they are generally larger for materials in the annealed condition and smaller in the cold-worked state. Such data may be derived by plotting true stress and associated true strain values on log-log paper. If Eq. 1-20 was absolutely correct, a straight line should result with a slope equal to n. However, this is found not always to be the case and reflects the