3.2. Elastic Stiffness Constant and Elastic Compliance

The stress components are specified as σij, where i is the direction of stress and j is the direction perpendicular to the plane upon which the stress is acting. The simple stress tensor is shown in Figure 2. Meanwhile, σxy is a stress in the x direction acting on the plane perpendicular to the y axis. Taking into account Figure 2, it reveals that there are nine stress components, however, since the presumption is that none are rotating, (σij= σji), the stress components are reduced to six: three axial, namely σxx, σyy and σzz, and three shear, namely σxy, σyz and σxz [18].

Elaborates of stress.

According to the Equation (1) (Hooke’s law), the stress is proportional to the strain for small displacements. In the generalized form, this proportionality principle is extended to the six stresses and strains [19].

Nevertheless, Hooke’s law can be written such as:

σxx= C11εxx + C12εyy + C13εzz + C14εyz + C15εzx + C16εxy

σyy= C21εxx + C22εyy + C23εzz + C24εyz + C25εzx + C26εxy

σzz= C31εxx + C32εyy + C33εzz + C34εyz + C35εzx + C36εxy

σyz= C41εxx + C42εyy + C43εzz + C44εyz + C45εzx+ C46εxy

σzx= C51εxx + C52εyy + C53εzz + C54εyz + C55εzx + C56εxy

σxy= C61εxx + C62εyy + C63εzz + C64εyz + C65εzx + C66εxy

The constants of proportionality Cij are introduced as the elastic constants. Since for a small displacement the elastic energy density is a quadratic function of the strains, and the elastic constants are given by the second partial derivatives of the energy density, it can be shown that Cij= Cji. Meanwhile, the 36 elastic constants result in the matrix form and they can be decreased to 6 diagonal components and 15 off-diagonal elaborates. In addition, it is possible for each system of crystal structures that the number of matrices can be decreased. According to Equation (2) in matrix form, these elastic constants can be written as [20,21]

The specified form of equation 1 can be written in decreased tensor notation and the form is the so-called Voigt notation (Equation (3)).

i and j equal 1 to 6, and i, j, which goes from 1 through 6, is related to xx, yy, zz, yz, zx and xy, respectively.

Furthermore, in the practical subjects, it is preferred to take strains into account in terms of the stresses. According to Equation (3), Equation (4) can be written

Here, Sij is introduced as the elastic compliance constants, so the exact formula can be written as the below:

ε1= S11σ1 + S12σ2 + S13σ3 + S14σ4 + S15σ5 + S16σ6

ε2= S21σ1 + S22σ2 + S23σ3 + S24σ4 + S25σ5 + S26σ6

ε3= S31σ1 + S32σ2 + S33σ3 + S34σ4 + S35σ5 + S36σ6

ε4= S41σ1 + S42σ2 + S43σ3 + S44σ4 + S45σ5 + S46σ6

ε5= S51σ1 + S52σ2 + S53σ3 + S54σ4 + S55σ5 + S56σ6

ε6= S61σ1 + S62σ2 + S63σ3 + S64σ4 + S65σ5 + S66σ6

There are seven conventional unit cells of Bravais lattices consisting of triclinic, monoclinic, orthorhombic, tetragonal, cubic, trigonal and hexagonal. Moreover, each lattice has special elastic stiffness constant values and elastic compliance values. The triclinic matrix has 21 elastic constants and it is the conventional wisdom. However, Landau and Lifshitz suggested that this number should be decreased by 3 to 18 due to the choice of the Cartesian orthogonal axes x, y and z, and it is arbitrary and could be interchanged, therefore decreasing the number by three [22].

Triclinic matrix |C11C12C13C14C15C16C21C22C23C24C25C26C31C32C33C34C35C36C41C42C43C44C45C46C51C52C53C54C55C56C61C62C63C64C65C66|

Monoclinic has 13 elastic constants and due to the suitable selection of the coordinate axes, these should be decreased to 12.

Monoclinic matrix |C11C12C130C150C21C22C230C250C31C32C330C350000C440C46C51C52C530C550000C640C66|

Orthorhombic elastic constants are obtained from nine components.

Orthorhombic matrix |C11C12C13000C21C22C23000C31C32C33000000C44000000C55000000C66|

Trigonal has two types of elastic constants: the first one is for the symmetry groups that have seven constants and the second one is for the suitable selection of coordinates, and these should be decreased to six constants.

Trigonal matrix (1) |C11C12C13C14−C250C12C11C13−C14C250C13C13C33000C14−C140C440C25−C25C2500C44C14000C25C1412(C11−C12)|

Trigonal matrix (2) |C11C12C13C1400C12C11C13−C1400C13C13C33000C14−C140C440C25−C25C2500C44C140000C1412(C11−C12)|

The tetragonal system has two types of elastic constants as well: the first one is for the symmetry groups that have seven constants and the second one is for the suitable selection of coordinates, and these should be decreased to six constants [21,23].

Tetragonal matrix (1) |C11C12C1300C16C12C11C1300−C16C13C13C33000000C44000000C440C16−C16000C66|

Tetragonal matrix (2) |C11C12C13000C12C11C13000C13C13C33000000C44000000C44000000C66|

The hexagonal system has five elastic constants.

Hexagonal matrix |C11C12C13000C12C11C13000C13C13C33000000C44000000C4400000012(C11−C12)|

The cubic system has three elastic constants.

Cubic matrix |C11C12C13000C12C11C13000C13C13C33000000C44000000C44000000C44|

For conventional systems consisting of cubic and hexagonal structures, the relationships between Cij and Sij are presented in Equations (5)–(12), respectively.

For cubic [24,25]:

For hexagonal [24,26]:

Mostly, elastic constant values of materials are tabulated in the literature [25,26]. Furthermore, the Young’s modulus of each plane (E_hkl) can be expressed for cubic and hexagonal crystals respectively as Equations (13) and (14).

For cubic [27]:

For hexagonal [28,29]: