Introduction:

The study of the elastic behavior of solids is extremely significant in the basic and technical researches. In technology, it would state us regarding the strength of the materials. In basic research, it is of interest because of the insight it gives in to the nature of binding forces in solids. They are as well of significance for the thermal properties of solids.

Definition:

Elasticity is basically the study of the capability of crystals to incorporate changes or adapt to new conditions easily.

Analysis of elastic strains and stresses:

The local elastic strain of a body might be specified through six numbers. If α, β, γ are the angles between the unit cell axes a, b, c, the strain might be specified through the changes Δα, Δβ, Δγ; Δa, Δb, Δc resultant from the deformation. This is an excellent physical specification of strain, however for non-orthogonal axes it leads to the arithmetical complications. The strain might be specified in terms of the six components e_xx, e_yy, e_zz, e_xy, e_yz, e_zx which are stated below. We assume that three orthogonal axes f, g, h of unit length are embedded securely in the unstrained solid, as described in the figure below. We assume that after a small uniform deformation has taken place the axes that we now label, f', g', h' are distorted in orientation and in length, in such a way that having the similar atom as origin we might represent.

f' = (1 + ε_xx)f + ε_xyg + ε_xzh;

g' = ε_yxf + (1 + ε_yy)g + ε_yzh;

h' = ε_zxf + ε_zyg + (1 + ε_zz)h;

The fractional changes of length of f, g and h. axes are ε_xx, ε_yy, ε_zz correspondingly, to the first order. We state the strain components e_xx, e_yy, e_zz through the relations:

e_xx = ε_xx; e_yy = ε_yy; e_zz = ε_zz;

The strain components e_xy, e_yz, e_zx might be stated as the changes in angle between the axes, in such a way that to the first order:

e_xy = f'.g' = ε_yx + ε_xy;

e_yz = g'.h' = ε_zy + ε_yz;

e_zx = h'.f' = ε_zx + ε_xz;

This completes the statement of the six strain components. A deformation is uniform when the values of the strain components are independent of the choice of origin.

We note that just rotating the axes doesn't change the angle between them, therefore for a pure rotation ε_yx = -ε_xy; ε_zy = -ε_yz; ε_zx = -ε_zx.  Whenever we exclude pure rotations, we might without additional loss of generality take ε_yx = ε_xy; ε_zy = ε_yz; ε_zx = ε_zx. So that in terms of the strain components we encompass:

f' - f = e_xx + 1/2 e_xyg + 1/2 e_zxh;

g' - g = 1/2 e_xyf + e_yyg + 1/2 e_yzh;

h' - h = 1/2 e_zxf + 1/2 e_yzg + e_zzh;

We suppose beneath a deformation that is substantially uniform close to the origin a particle originally at the position:

r = xf + yg + zh

Subsequent to deformation the particle is at:

r' = xf' + yg' + zh'

In such a way that the displacement is represented by:

ρ = r' - r = x(f' - f) + y(g' - g) + z(h' - h)

If we represent the displacement as:

ρ = uf + vg + wh

The expressions for the strain components are as follows:

e_xx = ∂u/∂x; e_yy = ∂v/∂y; e_zz = ∂w/∂z;

e_xy = ∂v/∂x + ∂u/∂y; e_yz = ∂w/∂y + ∂v/∂z; e_zx = ∂u/∂z + ∂w/∂x;

We have represented derivatives for application to non-uniform strain. The above expressions are often employed in the literature to state the strain components. Occasionally statements of e_xy, e_yz and e_zx are given that differ by a factor 1/2 from those given here. For a uniform deformation the displacement 'ρ' consists of the components.

u = e_xxx + 1/2 e_xyy + 1/2 e_zxz;

v = 1/2 e_xyx + e_yyy + 1/2 e_yzz;

w = 1/2 e_zxx + 1/2 e_yzy + e_zzz;

Dilation:

The fractional increment of volume caused through a deformation is termed as the dilation. The unit cube of edges f, g and h, after deformation consists of a volume:

V' = f'.g' X h' ≈ 1 + e_xx + e_yy + e_zz

Here squares and products of the strain components are neglected. Therefore the dilation is:

δ = ΔV/V' = e_xx + e_yy + e_zz

Shearing strain:

We might interpret the strain components of the kind:

e_xy = ∂v/∂x + ∂u/∂y

as build up of two simple shears. In one of the shears, planes of the material normal to the x-axis slide in the y-direction; in the other shear, planes normal to y slide in the x-direction.

Stress Components:

The force acting on the unit area in the solid is stated as the stress. There are nine stress components: X_x, X_y, X_z, Y_x, Y_y, Y_z, Z_x, Z_y, Z_z. The capital letter points out the direction of the force, and the subscript points out the normal to the plane to which the force is applied. Therefore the stress component X_x symbolizes a force applied in the x-direction to a unit area of a plane whose normal lies in the x-direction; the stress component X_y symbolizes a force applied in the x-direction to a unit area of a plane whose normal lies in the y-direction.

The number of independent stress components is decreased to six via applying to an elementary cube as shown in the figure above. The condition that the angular acceleration disappear and therefore that the total torque should be zero. It follows that:

Y_z = Z_y, Z_x = X_z, X_y = Y_x

And the independent stress components might be taken as X_x, Y_y, Z_z, Y_z, Z_x, X_y. The stress components encompass the dimensions of force per unit area or energy per unit volume that the strain components are dimensionless.

Elastic Compliance and Stiffness Constants:

Hooke's law defines that for the small deformations the strain is proportional to the stress, in such a way that the strain components are linear functions of the stress components:

e_xx = s₁₁X_x + s₁₂Y_y + s₁₃Z_z + s₁₄Y_z + s₁₅Z_x + s₁₆X_y;

e_yy = s₂₁X_x + s₂₂Y_y + s₂₃Z_z + s₂₄Y_z + s₂₅Z_x + s₂₆X_y;

e_zz = s₃₁X_x + s₃₂Y_y + s₃₃Z_z + s₃₄Y_z + s₃₅Z_x + s₃₆X_y;

e_yz = s₄₁X_x + s₄₂Y_y + s₄₃Z_z + s₄₄Y_z + s₄₅Z_x + s₄₆X_y;

ezx = s₅₁X_x + s₅₂Y_y + s₅₃Z_z + s₅₄Y_z + s₅₅Z_x + s₅₆X_y;

e_xy = s₆₁X_x + s₆₂Y_y + s₆₃Y_z + s₆₄Y_z + s₆₅Z_x + s₆₆X_y;

On the contrary, the stress components are linear functions of the strain components:

X_x = c₁₁e_xx + c₁₂e_yy + c₁₃e_zz + c₁₄e_yz + c₁₅e_zx + c₁₆e_xy;

Y_y = c₂₁e_xx + s₂₂e_yy + c₂₃e_zz + c₂₄e_yz + c₂₅e_zx + c₂₆e_xy;

Z_z = c₃₁e_xx + c₃₂e_yy + c₃₃e_zz + c₃₄e_yz + c₃₅e_zx + c₃₆e_xy;

Y_z = c₄₁e_xx + c₄₂e_yy + c₄₃e_zz + c₄₄e_yz + c₄₅e_zx + c₄₆e_xy;

Z_x = c₅₁e_xx + c₅₂e_yy + c₅₃e_zz + c₅₄e_yz + c₅₅e_zx + c₅₆e_xy;

X_y = c₆₁e_xx + c₆₂e_yy + c₆₃e_zz + c₆₄e_yz + c₆₅e_zx + c₆₆e_xy;

The quantities s₁₁.... s₁₂ are termed as the elastic constants or elastic compliance constants; the quantities c₁₁.... c₁₁ are termed as the elastic stiffness constants or moduli of elasticity. Other names are as well current. The S's and C's encompass the dimension of area per unit force or volume per unit energy and force per unit area or energy per unit volume correspondingly.

Energy Density:

We compute the increment of work 'δW' done via the stress system in straining a small cube of side 'L', with the origin at one corner of the cube and the coordinate axes parallel to the cube edges. We encompass:

δW = F.δρ

Here 'F' is the applied force and

δρ = fδu + gδv + hδw

is the displacement. If X, Y, Z represent the components of 'F' per unit area, then

δw = L²(Xδu + Yδv + Zδw)

We note that the displacement of the three cube faces having the origin is zero, in such a way that the forces all act at a distance 'L' from the origin. Now by the statement of the strain components:

δu = L(δe_xx + 1/2 δe_xy + 1/2 δe_zx) and so on, in such a way that:

δW = L³(X_xδe_xx + Y_yδe_yy + Z_zzδe_zz + Y_zδe_yz + Z_zxδe_zx + X_yδe_xy)

The increment δU of the elastic energy per unit volume is represented by:

δU = X_xδe_xx + Y_yδe_yy + Z_zδe_zz + Y_zδe_yz + Z_xδe_zx + X_yδe_xy

We have δU/δe_xx = X_x and δU/δe_yy = Y_y and on further differentiation:

δX_x/δe_yy = δY_y/δe_xx

From the above, we get the relation:

c₁₂ = c₂₁

And in general we encompass:

c_ij = c_ji

Providing fifteen relations between the thirty non-diagonal terms of the matrix of the Cs. The thirty-six elastic stiffness constants are in this way decreased to twenty-one coefficients. Identical relations hold among the elastic compliances. The matrix of the Cs or S's is thus symmetrical.

Cubic crystal:

The number of independent elastic stiffness constants is generally decreased if the crystal has symmetry elements, and in the significant case of cubic crystals there are just three independent stiffness constants, as we now represent. We assume that the coordinate axes are selected parallel to the cube edges. From the equation above which represents stress components are linear functions of the strain components, we have:

c₁₄ = c₁₅ = c₁₆ = c₂₄ = c₂₅ = c₂₆ = c₃₄ = c₃₅ = c₃₆ = 0

As the stress should not be modified by reversing the direction of one of the other coordinate axes. As the axes are equivalent, we as well have:

c₁₁ = c₂₂ = c₃₃,

And c₁₂ = c₁₃ = c₂₁ = c₂₃ = c₃₁ = c₃₂,

We also have:

c₄₄ = c₁₅ = c₆₆

by equivalence of the axes, and the other constants all disappear due to the reason of their behavior on reversing the direction of one or other axis. The array of values of the elastic stiffness constant is thus decreased for a cubic crystal to the matrix illustrated below:

This is readily observed that for a cubic crystal:

U = (1/2) c₁₁(e²_xx + e²_yy + e²_zz) + c₁₂(e_yye_zz + e_zze_xx + e_xxe_yy) + (1/2) c₄₄(e²_yz + e²_zx + e²_xy)

satisfies the equation for the elastic energy density function.

For illustration, δU/δe_yy = c₁₁e_yy + c₁₂e_zz + c₁₂e_xx = Y_y

By using the equation of cubic crystal, for cubic crystals the compliance and stiffness constants are associated by:

C₁₁ = [(s₁₁ + s₁₂)/(s₁₁ - s₁₂)(s₁₁ + 2s₁₂)];

C₁₂ = [(-s₁₂)/(s₁₁ - s₁₂)(s₁₁ - 2s₁₂)]

C₁₄ = 1/s₄₄

A general evaluation of elastic constant data and of relationships between different coefficients for the crystal classes has been illustrated through Hearmon in the year 1946).

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