Elastic deformation

6.2 Elastic deformation

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, and (3) it is usually small (i.e. <1% elastic strain).

The stress at a point in a body is usually defined 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 into components parallel to the cube edges and when divided by the area of a face give the nine stress components shown in Figure 6.6. A given component σ ij is the force acting in the j-direction per unit area of face normal to the i-direction. Clearly, when i = j we have normal stress components (e.g. σ xx ) which may be either tensile (conventionally positive) or compressive (negative), and when i

xy or τ xy ) the stress components are shear. These shear stresses exert couples on the cube and to prevent rotation of the cube the couples on opposite faces must balance,

and hence σ ij =σ ji . 2 Thus, stress has only six independent components.

2 The nine components of stress σ ij form a second-rank tensor usually written as:

σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz

and is known as the stress tensor.

Mechanical properties I 295

Figure 6.6 Normal and shear stress components.

When a body is strained, small elements in that body are displaced. If the initial position of an element is defined by its coordinates (x, y, z) and its final position by (x + u, y + v, z + w), then the displacement is (u, v, w). If this displacement is constant for all elements in the body, no strain is involved, only a rigid translation. For a body to be under a condition of strain the displacements must vary from element to element. A uniform strain is produced when the displacements are lin- early proportional to distance. In one dimension then u = ex, where e = du/dx is the coefficient of proportionality or nominal tensile strain. For a three-dimensional uniform strain, each of the three components u, v, w is made a linear function in terms of the initial elemental coordinates, i.e.

u =e xx x +e xy y +e xz z v =e yx x +e yy y +e yz z w =e zx x +e zy y +e zz z.

The strains e xx = du/dx, e yy = dv/dy, e zz = dw/dz are the tensile strains along the x-, y- and z-axes, respectively. The strains e xy ,e yz , etc. produce shear strains and in some cases a rigid-body rotation.

The rotation produces no strain and can be allowed for by rotating the reference axes (see Figure 6.7). In general, therefore, e ij =ε ij +ω ij , with ε ij the strain components and ω ij the rotation components. If, however, the shear strain is defined as the angle of shear, this is twice the corresponding shear strain component, i.e. γ = 2ε ij . The strain tensor, like the stress tensor, has nine components, which are usually written as:

where ε xx etc. are tensile strains and γ xy , etc. are shear strains. All the simple types of strain can be produced from the strain tensor by setting some of the components equal to zero. For example, a pure dilatation (i.e. change of volume without change of shape) is obtained when ε xx =ε yy =ε zz and all other components are zero. Another example is a uniaxial tensile test when the tensile strain along the

296 Physical Metallurgy and Advanced Materials

Figure 6.7 Deformation of a square OABC to a parallelogram PQRS involving: (i) a rigid-body translation OP allowed for by redefining new axes X ′ Y ′ , (ii) a rigid-body rotation allowed for by rotating the axes to X ′′ Y ′′ , and (iii) a change of shape involving both tensile and shear strains.

x-axis is simply e =ε xx . However, because of the strains introduced by lateral contraction, ε yy = −νe and ε zz = −νe, where ν is Poisson’s ratio; all other components of the strain tensor are zero.

At small elastic deformations, the stress is linearly proportional to the strain. This is Hooke’s law and in its simplest form relates the uniaxial stress to the uniaxial strain by means of the modulus of elasticity. For a general situation, it is necessary to write Hooke’s law as a linear relationship between six stress components and the six strain components, i.e.

σ xx =c 11 ε xx +c 12 ε yy +c 13 ε zz +c 14 γ yz +c 15 γ zx +c 16 γ xy σ yy =c 21 ε xx +c 22 ε yy +c 23 ε zz +c 24 γ yz +c 25 γ zx +c 26 γ xy σ zz =c 31 ε xx +c 32 ε yy +c 33 ε zz +c 34 γ yz +c 35 γ zx +c 36 γ xy τ yz =c 41 ε xx +c 42 ε yy +c 43 ε zz +c 44 γ yz +c 45 γ zx +c 46 γ xy τ zx =c 51 ε xx +c 52 ε yy +c 53 ε zz +c 54 γ yz +c 55 γ zx +c 56 γ xy τ xy =c 61 ε xx +c 62 ε yy +c 63 ε zz +c 64 γ yz +c 65 γ zx +c 66 γ xy .

The constants c 11 ,c 12 ,...,c ij are called the elastic stiffness constants. 3

Taking account of the symmetry of the crystal, many of these elastic constants are equal or become zero. Thus, in cubic crystals there are only three independent elastic constants c 11 ,c 12 and c 44 for the three independent modes of deformation. These include the application of (1) a hydrostatic stress p

1 p =− (c

3 11 + 2c 12

where κ is the bulk modulus, (2) a shear stress on a cube face in the direction of the cube axis defining the shear modulus μ =c 44 , and (3) a rotation about a cubic axis defining a shear modulus μ 1 = (c 11 −c 12 )/2. The ratio μ/μ 1 is the elastic anisotropy factor and in elastically isotropic crystals it is unity with 2c 44 =c 11 −c 12 ; the constants are all interrelated, with c 11 = κ + 4μ/3, c 12 = κ − 2μ/3 and c 44 = μ.

3 Alternatively, the strain may be related to the stress, e.g. ε x =s 11 σ xx +s 12 σ yy +s 13 σ zz + . . . , in which case the constants s 11 ,s 12 ,…,s ij are called elastic compliances.

Mechanical properties I 297 Table 6.1 Elastic constants of cubic crystals (GN m −2 ).

Table 6.1 shows that most metals are far from isotropic and, in fact, only tungsten is isotropic; the alkali metals and β-compounds are mostly anisotropic. Generally, 2c 44 > (c 11 –c 12 ) and, hence, for most elastically anisotropic metals E is maximum in the directions. Molybdenum and niobium are unusual in having the reverse anisotropy when E is greatest along approximately isotropic properties. 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 during 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.