8 CHAPTER 1 TENSILE RESPONSE OF MATERIALS TABLE 1.1b Elastic Properties of Engineering Materials" Material at68° (10 psi) Metals 10 225 3.8 30.6 11,8 6.5 289 110 0312 Niobium 15.2 0397 Silver Tantalum 26.9 10.0 0342 tanium 59.6 Vanadium 185 0365 Aluminum oxide(fully dense) 140 class (heavy flint 11.6 4.6 0.2 Polyethylene(high density) 00580.19 0350.49 0.160.39 Quartz(fused 106 0.I70 ilicon carbide 022 aG. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants, 14th ed, Longman, London, 1973,p.3I separation, dF/dx (i.e, dE/dx), then describes the stiffness or relative resistance to eparation of the two particles. As such, dF/dx is analogous to the youngs modulus quantity given in Eq. 1-7. a simple analysis of bonding forces shows that the elastic ess is proportional to 1/x0. Examples of the strong dependence of elastic stiffness on xo for alkali metals are shown in Fig. 1.2c From the above discussion, it follows that values of E for metals and ceramics should decrease with increasing temperature. This is related to the fact that the distance of atom or ion separation increases with temperature (i.e, materials expand when heated). Note the dotted line in Fig. 1.2a, which corresponds to the locus of values of xo at temperatures above absolute zero. The loss of material stiffness with increasing temperature is gradual, with only a small percent decrease occurring for a 100%C (180F)temperature change(Fig. 1. 2d). Since E depends on the strength of interatomic forces that vary with the type of bonding found in a given material, it relatively insensitive to changes in microstructure. As a result, while heat treatment and minor alloying additions may cause the strength of a steel alloy to change from
1.2 STRESS-STRAIN CURVES 9 dFdx Slope - -4 A2O3+60 interatomic distances, A FIGURE 1.2 Dependence of elastic stiffness on interatomic spacing:(a) Potential en- ergy versus interatomic spacing;(b) Force versus interatomic spacing;(c) Elastic stiff ness of alkali metals versus interatomic spacing. (From J. J. Gilman, Micromechanics of Flow in Solids, McGraw-Hill, New York, 1969, with permission.); (d) variation of Young 's modulus with temperature in selected metals and ceramics. From K M. Ralls, T. H. Courtney, and J. Wulff, introduction to Materials Science and Engineering, wiley, 1976, with permissi 210 to 2400 MPa, the modulus of elasticity of both materials remains relatively unchanged-about 200 to 210 GPa. It is found that if the loads are removed from the tensile sample before the point of fracture, the corresponding strain will retrace itself along the same linear plot back to zero. The reversible nature of strain in this portion of the a-e curve is a basic element of elastic strains in any material, whether it is capable of much larger tot strain or not. When a material is characterized by such a stress-strain curve and exhibits no plastic deformation, there is great concem for its ability to resist brittle (ow energy)premature fracture. This point is treated extensively in Chapters Typical materials that behave in this manner include glasses, rocks, many ceramics, heavily cross-linked polymers, and some metals at low temperature. Although these materials are not suitable for engineering applications involving tensile loading, they
10 CHAPTER 1 TENSILE RESPONSE OF MATERLALS may be used with considerable success in situations involving compression loads for which the material exhibits much greater resistance to fracture. It is not uncommon to find the compressive strength of a brittle solid to be several times greater than the tensile value. Concrete is an excellent example of an industrial material used exten- ively in compression but not in tension. When tensile loads avoidable, the concrete is reinforced by the addition of steel bars that assume the tensile stresses Before proceeding with additional discussion of Hooke's law, it should be noted that elastomers also exhibit elastic, though nonlinear, behavior in nature. A brief discussion of the response of this material is found in Section 1. 2.5. Anelastic defor- mation(time-dependent-reversible strain) is discussed in Section 6.3 1.2.1.1 Generalized Hookes law Hooke's law can be generalized to account for multiaxial loading conditions as well as material anisotropy. Regarding the former, readers should recall from their studies of the strength of materials that a stress in one direction(say the Y direction) will cause not only a strain in the y direction but in the X and Z directions as well. Hence (1-8a) E (1-8b) where ory stress acting normal to y plane and in Y direction Erm Eyn,Ez= corresponding strains in orthogonal directions v= Poisson's ratio[= E= modulus of elasticity From Fig. 1.3, typical normal and shear strain components may be given by au Yry- tano tanB dy au with the other normal and shear strains defined in similar fashion when multiaxial stresses are applied, the total strain in any given direction is the sum of all strains resulting from each normal and shear stress component. For the case of an isotropic
1. 2 STRESS-STRAIN CURVES 1 1 y dy dx FIGURE 1.3 Distortion of the Z face of a cubical element. The dashed lines indicate the unstrained position of the cube. +可a) v(σx+ Ye G where Ti= stress acting on I plane and in J direction G= shear modulus The situation is complicated greatly when the material is anisotropic wherein the elastic constants vary as a function of crystallographic orientation. Since this is the ase for practically all crystalline solids, it is important to consider the general loading
12 CHAPTER 1 TENSILE RESPONSE OF MATERIALS FIGURE 1. 4 Stress components acting on a volume element. conditon shown in Fig. 1.4. We see that there are three normal and six shear stress components. However, since yx Tx Ty= Ty, and T =Te(so as to avoid rotation of the cube), only six independent stress components remain that determine the strains of the body. The strains in each direction may be given by Exx=S11O+ $120yy $130z S14T SISTx t SITry eyy=S2rOx $220yy $230x S24T SxsTzx $26try ea=530x+5320y+830z+539Tx+s3zx+536T(1-11) Exy=541Ux $420xy S430x+ S44Tyz SasTax S46Txy Ez= SiOx Ss20yy Ss30z Ss4T t SssTx ss6Try Exx=$610xx $620yy t $6302 S64Ty S6sTxx $6sTry where Sy elastic compliances. Solving for stresses instead, we have Ox=CulEx C12Eyy C13Ex t C14Yy C1sYx c1yy Oyy=C21E cz2eyy Cz3Ez C24Yyz C2sYzx C26Yxy Oz- C31Exx C32eyy C33E= C3 Yy C35Yax C36lxy (1-12) Ty=C41Ex C42Eyy Ca3ezx+ Ca4'Yy CasA c46Yay Tax= Cs1Ex Cs2ey Cs3e= Cs4yy CssYzx Cs6Yay Txy = C61Ex C62Eyy C636= C64yyz C65Yxx c66Yry where cy= elastic stiffnesses The reversibility of elastic strains leads to the fact that sy=Si and cy=gi,which reduces the number of independent matrial constants from 36 to 21. As a result of symmetry considerations, the number of independent constants decreases further, with