Elastic deformation of metals

7.2.1 Elastic deformation of metals

It is well known that metals deform both elastically and plastically. Elastic deformation takes place at low stresses and has three main characteristics, namely (1) it is reversible, (2) stress and strain are linearly proportional to each other according to Hooke’s Law

Figure 7.6 Bend test configurations. MoR D modulus of and (3) it is usually small (i.e. <1% elastic strain). rupture, F D applied force, L D outer span, L i

The stress at a point in a body is usually defined span, b D breadth of specimen, d D depth of specimen.

D inner

by considering an infinitesimal cube surrounding that point and the forces applied to the faces of the cube by the surrounding material. These forces may be resolved

known as the modulus of rupture (MoR) and expresses into components parallel to the cube edges and when the maximum tensile stress which develops on the con-

divided by the area of a face give the nine stress vex face of the loaded beam. Strong ceramics, such as

components shown in Figure 7.7. A given component silicon carbide and hot-pressed silicon nitride, have

ij is the force acting in the j-direction per unit very high MoR values. The four-point loading method

area of face normal to the i-direction. Clearly, when is often preferred because it subjects a greater volume

xx ) and area of the beam to stress and is therefore more

which may be either tensile (conventionally positive) searching. MoR values from four-point tests are often

xy ) substantially lower than those from three-point tests

the stress components are shear. These shear stresses on the same material. Similarly, strength values tend

exert couples on the cube and to prevent rotation of the to decrease as the specimen size is increased. To pro-

cube the couples on opposite faces must balance and vide worthwhile data for quality control and design

ji . 1 Thus, stress has only six independent activities, close attention must be paid to strain rate

ij

components.

and environment, and to the size, edge finish and sur- When a body is strained, small elements in that face texture of the specimen. With oxide ceramics and

body are displaced. If the initial position of an element silica glasses, a high strain rate will give an appre- ciably higher flexural strength value than a low strain

1 ij form a second-rank rate, which leads to slow crack growth and delayed

tensor usually written

fracture (Section 10.7). The bend test has also been adapted for use at high

temperatures. In one industrial procedure, specimens

of magnesia (basic) refractory are fed individually

from a magazine into a three-point loading zone at the and is known as the stress tensor.

202 Modern Physical Metallurgy and Materials Engineering

Figure 7.8 Deformation of a square OABC to a parallelogram PQRS involving (i) a rigid body translation

OP allowed for by redefining new axes X 0 Y 0 , (ii) a rigid body rotation allowed for by rotating the axes to X 00 Y 00 , and (iii) a Figure 7.7 Normal and shear stress components .

change of shape involving both tensile and shear strains . is defined by its coordinates ⊲x, y, z⊳ and its final

is simply e D ε xx . However, because of the strains position by ⊲x C u, y C v , z C w⊳ then the displacement

introduced by lateral contraction, ε yy and ε zz D is ⊲u, v , w⊳ . If this displacement is constant for all elements in the body, no strain is involved, only a

of the strain tensor are zero. rigid translation. For a body to be under a condition

At small elastic deformations, the stress is linearly of strain the displacements must vary from element

proportional to the strain. This is Hooke’s law and in to element. A uniform strain is produced when the

its simplest form relates the uniaxial stress to the uni- displacements are linearly proportional to distance. In

axial strain by means of the modulus of elasticity. For one dimension then u D ex where e D du/dx is the

a general situation, it is necessary to write Hooke’s law coefficient of proportionality or nominal tensile strain.

as a linear relationship between six stress components For a three-dimensional uniform strain, each of the

and the six strain components, i.e. three components u, v , w is made a linear function in terms of the initial elemental coordinates, i.e.

wDe zx xCe zy yCe zz z yz Dc 41 ε xx Cc 42 ε yy Cc 43 ε zz Cc 44 yz Cc 45 zx Cc 46 xy The strains e xx

zx Dc 51 D dw/dz are ε xx Cc 52 ε yy Cc 53 ε zz Cc 54 yz Cc 55 zx Cc 56 xy the tensile strains along the x, y and z axes, respec-

D du/dx, e yy Dd v / dy, e zz

xy Dc 61 ε xx Cc 62 ε yy Cc 63 ε zz Cc 64 yz Cc 65 zx Cc 66 xy tively. The strains e xy ,e yz , etc., produce shear strains

and in some cases a rigid body rotation. The rotation The constants c 11 ,c 12 ,...,c ij are called the elastic produces no strain and can be allowed for by rotat-

stiffness constants. 1

ing the reference axes (see Figure 7.8). In general, Taking account of the symmetry of the crystal, many therefore, e ij Dε ij Cω ij with ε ij the strain compo-

of these elastic constants are equal or become zero. nents and ω ij the rotation components. If, however,

Thus in cubic crystals there are only three indepen- the shear strain is defined as the angle of shear, this

dent elastic constants c 11 ,c 12 and c 44 for the three is twice the corresponding shear strain component, i.e.

independent modes of deformation. These include the ij D 2ε . The strain tensor, like the stress tensor, has

application of (1) a hydrostatic stress p to produce a nine components which are usually written as:

dilatation  given by

cube face in the direction of the cube axis defining

44 , and (3) a rotation about a where ε xx etc. are tensile strains and xy , etc. are

shear strains. All the simple types of strain can be

1 D 2 ⊲c 11 12 ⊳ . produced from the strain tensor by setting some of

1 is the elastic anisotropy factor and the components equal to zero. For example, a pure

in elastically isotropic crystals it is unity with 2c 44 D dilatation (i.e. change of volume without change of shape) is obtained when ε xx Dε yy Dε zz and all other

1 Alternatively, the strain may be related to the stress, e.g. components are zero. Another example is a uniaxial

C . . ., in which case the tensile test when the tensile strain along the x-axis

ε x Ds 11 xx Cs 12 yy Cs 13 zz

constants s 11 ,s 12 ,...,s ij are called elastic compliances.

Mechanical behaviour of materials 203

Table 7.1 Elastic constants of cubic crystals . GN/m 2 /

c 11 12 ; the constants are all interrelated with c 11 D

3 ,c 12

3 and c 44 Table 7.1 shows that most metals are far from

isotropic and, in fact, only tungsten is isotropic; the alkali metals and ˇ-compounds are mostly anisotropic.

Generally, 2c 44 > ⊲c 11 12 ⊳ and hence, for most elas-

tically anisotropic metals E is maximum in the h1 1 1i and minimum in the h1 0 0i directions. Molybde- num and niobium are unusual in having the reverse anisotropy when E is greatest along h1 0 0i directions. Most commercial materials are polycrystalline, and consequently they have approximately isotropic prop- erties. For such materials the modulus value is usually independent of the direction of measurement because the value observed is an average for all directions, in the various crystals of the specimen. However, if dur- ing manufacture a preferred orientation of the grains in the polycrystalline specimen occurs, the material will behave, to some extent, like a single crystal and some ‘directionality’ will take place.