Material Characterization

Everyday experience indicates that pulling identical bars of steel and rubber by the same axial force and hence axial stress results in different elongations of the two bars. In mechanics, this difference in the materials of the two bars is represented through the relationship between the components of the stress and the components of the strain. Writing each of these as a column matrix, i.e,
$$\begin{align}&\{\sigma\}^T = [\sigma_{11}\ \ \sigma_{22}\ \ \sigma_{33}\ \ \sigm... ...{11}\ \ e_{22}\ \ e_{33}\ \ 2e_{23}\ \ 2e_{31}\ \ 2e_{12}],\tag{6.2} \end{align}$$
we have

$$\{\sigma\} = [C]\{e\}\ {\rm or}\ \sigma_\alpha = C_{\alpha\beta}e_\beta ,\ \alpha,\beta = 1,2,\ldots , 6\tag{6.3}$$ (6.3)

where C, a $6\times 6$ matrix, characterizes the material of the body, and is generally called the elasticity matrix and its components elasticities or elastic constants for the material of the body. Note that shear strains have been multiplied by 2 in eqn. (6.2); definitions (6.1) and (6.2) ensure that $\sigma_{ij}e_{ij} = \sigma_\alpha e_\alpha$. Equation (6.3), i.e., the relation between the stresses and strains, is called the constitutive relation for the material of the body. It is tacitly assumed in (6.3) that the body is stress free in the reference configuration from which strain e is measured. A stress-free configuration is called a natural state of the body. Note that equation (6.3) gives ${\mbox{\boldmath {$\sigma$ }}} = \mathbf{0}$ whenever e = . A material whose stress-strain response can be represented by (6.3) is called a linear elastic material, and (6.3) is Hooke's law. The matrix C is symmetric; thus there are 21 elastic constants for a general anisotropic material. For a material with the maximum symmetry, i.e., an isotropic material, there are 2 independent elastic constants, the Young's modulus E and Poisson's ratio $\nu$. For an isotropic material, the material properties are the same in every direction. Said differently, the components of the matrix C have the same values with respect to every rectangular Cartesian coordinate system, and
$$\begin{align}&[\mathbf{C}] = \left[\begin{array}{cccccc}\lambda + 2\mu & \lambda... ...(1 + \nu )},\ \lambda = \frac{E\nu}{(1 + \nu )(1 - 2\nu )}.\tag{6.5} \end{align}$$
The elastic constant $\mu$ equals the shear modulus of the material. Annealed metals and metallic alloys, and most fluids are generally assumed to be isotropic.

A material is said to be transversely isotropic about an axis a if the components of the matrix C are invariant (i.e. are unchanged) with respect to rotations of axes about the vector a, and the direction of a is called the axis of transverse isotropy. For rectangular Cartesian coordinate axes with x₃-axis coinciding with the vector a, the components of the matrix Cwill be same no matter how x and y-axes are chosen. For a transversely isotropic material with the axis of transverse isotropy along the x₃-axis,
$$\begin{align}&[\mathbf{C}] = \left[\begin{array}{cccccc}C_{11} & C_{12} & C_{13}... ...&A = 1 + \nu_{12}, B = 1 - \nu_{12} - 2(E_1/E_3)\nu^2_{31}.\nonumber \end{align}$$
The ratio E₁/E₃ is a measure of the degree of anisotropy. Here
$$\begin{align}&E_1 = \mbox{Young's modulus in}\ x_1\ \mbox{direction},\nonumber\\... ..._{13} = E_1/2(1 + \nu_{13}),\ \nu_{13} = (E_1/E_3)\nu_{31}.\nonumber \end{align}$$
Thus there are 5 independent material parameters, C₁₁,C₁₂,C₁₃,C₃₃ and G₁₃. Materials having a laminated structure are usually modeled as transversely isotropic.

A material (e.g. wood) that has three mutually perpendicular planes of elastic symmetry is called orthotropic. We choose co-ordinate axes so that the coordinate planes coincide with the planes of elastic symmetry. The material properties, i.e., values of components of the matrix C will be unchanged if the direction of the co-ordinate axes were reversed one at a time. For an orthotropic material, the elasticity matrix C has the following form.

$$[\mathbf{C}]= \left[\begin{array}{cccccc} C_{11} & C_{12} & C... ...} & 0\\ 0 & 0 & 0 & 0 & 0 & C_{66}\end{array}\right]\tag{6.8}$$ (6.8)

Thus there are 9 independent material parameters.