CHAPTER 1 TENSILE RESPONSE OF MATERIALS The tensile test is the experimental test method most widely employed to characterize the mechanical properties of materials. From any complete test record, one can obtain important information concerning the material,s elastic properties, the character and extent of plastic deformation, yield and tensile strengths, and toughness. That so much information can be obtained from one test justifies its extensive use in engineering materials research. To provide a framework for the varied response to tensile loading in load-bearing materials, several stress-strain plots reflecting different deformation characteristics will be introduced in this chapter. 1.1 DEFINITION OF STRESS AND STRAIN Before discussing engineering material stress-strain response, it is appropriate to define the terms, stress and strain. This may be done in two generally accepted forms The first definitions, used extensively in engineering practice, are Ceng engineenng stress initial cross-sectional area A (1-1a) Feng engineering strain change in length_12-Lo where lf=final gage length Lo initial gage length Alternatively, stress and strain may be defined by True uue stress-instantaneous cross-sectional area A (1-2ax) ere=true strain In final length= (1-2b) nitial length The fundamental distinction concerming the definitions for true stress and strain is recognition of the interrelation between gage length and diameter changes associated
4 CHAPTER 1 TENSILE RESPONSE OF MATER with plastic deformation. That is, since plastic deformation is a constant-volume process such that any extension of the original gage length would produce a corresponding contraction of the gage diameter. For example, if a 25-mm(1-in. *-long sample were to extend uniformly by 2.5 mm owing to a tensile load P, the real or mue stress would have to be higher than that computed by the engineering stress formulation. Since I2/,=1.1 from Eq. 1-3 A/A2= 1. 1, so that A2= A /l 1. The true stress is then shown to be (l) By combining Eqs. 1-1b and 1-2b, true and engineering strains may be related by ∈ue=ln(∈an+1) The need to define true strain as in Eq. 1-4 stems from the fact that the actual strain it any given time depends on the instantaneous gage length Li. Consequently, a fixed Al displacement will result in a decreasing amount of incremental strain, since the length at any given time, i, will ing thermore, it should be possible to define the strain given to a rod by considering the total change in length of the rod as having taken place in either one step or any number of discrete steps. Stated mathematically, E,En=Er. As a simple example, take the case of a wire drawn in two steps with an intermediate annealing treatment. On the basis of engineering strain, the two deformation strains would be(1-LoyLo and(2-/l. Adding these two increments does not yield a final strain of( 2 Lo)/Lo. On the other hand, a summation of true strains does lead to the correct result Therefore + EXAMPLE 1 1 A 25-cm(10-in. t-long rod with a diameter of 0. 25 cm is loaded with a 4500-newton (1012-1b) *weight. If the diameter decreases to 0.22 cm, compute the following t from inches to millimeters, multiply by 25. 4. f To convert from inches to centimeters, multiply by 2.54. To convert from pounds to newtons, multiply by 4.448
1.1 DEFINITION OF STRESS AND STRAIN 5 (a) The final length of the rod Since A141= A22(from Eg. 1-3) (025)2 (022)2 l2=32.3 The true stress and true strain at this load P 4500 (m14)22×10-3)2 omue=1185MPa(172,000psi)* 32.3 ∈mue=0.256or25.6% (c) The engineering stress and strain at this load (25×10 917M 32.3-25 0292or292% The use of true strains offers an additional convenience when considering the constan volume plastic deformation process in that ex E+Ez=0. In contrast, we find a less convenient relationship, (1+Ex)(1+ey)(1 +e=1, for the case of engineering strains To convert from psi to pascals, multiply by 6.895 x 10
6 CHAPTER 1 TENSILE RESPONSE OF MATERLALS 1.2 STRESS-STRAIN CURVES 1.2.1 Elastic Response: Type Over 300 years ago Robert Hooke reported in his classic paper"Of Spring"the following observation Take a wire string of 20 or 30 or 40 feet long and fasten the upper part..to a nail, and to the other end fasten a scale to receive the weights. Then with a pair of compasses (measure the distance [from the bottom of the scale [tol the ground or floor beneath. Then put . weights into the . scale and measure the several stretching s of the said string and set them down. Then compare the severo ral stretchings of the... string and you will find that they will always bear the same proportions one to the other that the weights do that made them. This observation may be described mathematically by the equation for an elastic (1-6) F= applied force I associated displacement proportionality factor often referred to as the spring constant When the force acts on a cross-sectional area a and the displacement x related to some reference gage length L, Eq. 1-6 may be rewritten as (1-7) where FlA stress E= proportionality constant(often referred to as Youngs modulus or the odulus of elasticity) Equation 1-7--called Hooke's law-describes a material condition where stresses and strains are proportional to one another, leading to a stress-strain response shown in Fig. 1.l. A wide range of values of the modulus of elasticity for many materials is shown in Table 1.1. The major reason for these large property variations is related to differences in the strength of the interatomic forces between adjacent atoms or ions. To illustrate this fact, let us consider how the potential energy E between two adjacent particles changes with their distance of separation x(Fig. 1. 2a). The equilibrium distance of particle separation xo, corresponding to a minimum in potential energy, is ssociated with a balance of the energies of repulsion and attraction between two djacent atoms or ions. The form of this relationship is given by E /x+ B/x, where -a/xm and B/x correspond to the energies of attraction and repulsion, respectively, and n>m. At xo, the force(F= dE/dx) acting on the particles is equal to zero(Fig. 1.2b). The first derivative of the force with respect to distance of particle
1.2 STRESS-STRAIN CURVES 7 FIGURE 1.1 Type I stress-strain behav- or revealing completely elastic material response TABLE 1.1a Elastic Properties of Engineering Materials Material at 20C (GPa) Metals Aluminum 703 26,1 Cadmium 0.300 Chromium 279.1 1154 0210 2114 0312 Niobium 115.7 0321 411.0 1276 Aluminum oxide(fully dense) Diamond 二 12-29 Polyethylene(high densi 04-13 243.4 二 aG. w.C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants, 14th ed, Longman, London, 1973.p.31