Understanding Elastic Constants and Their Relationships

In mechanics of materials, we study how solid bodies deform under stress. Elastic constants are properties of a material that describe its stiffness or resistance to deformation under applied forces in the elastic range. These constants relate stress and strain. The question asks about the relationship between three fundamental elastic constants: Modulus of Rigidity, Modulus of Elasticity, and Poisson's ratio.

Key Elastic Constants Defined

• Modulus of Elasticity (E): Also known as Young's Modulus, it measures the tensile or compressive stiffness of a material. It is defined as the ratio of uniaxial stress to uniaxial strain in the elastic range. Mathematically, \(E = \frac{\sigma}{\epsilon}\), where \(\sigma\) is the stress and \(\epsilon\) is the strain.
• Modulus of Rigidity (C or G): Also known as Shear Modulus, it measures the stiffness of a material against shear deformation. It is defined as the ratio of shear stress to shear strain in the elastic range. Mathematically, \(C = \frac{\tau}{\gamma}\), where \(\tau\) is the shear stress and \(\gamma\) is the shear strain.
• Poisson's Ratio (\(\nu\)): This is a dimensionless ratio that describes the ratio of transverse strain to axial strain when a material is stretched or compressed uniaxially. Mathematically, \(\nu = - \frac{\epsilon_{\text{transverse}}}{\epsilon_{\text{axial}}}\). The negative sign indicates that materials typically contract transversally when stretched axially (positive axial strain leads to negative transverse strain).

Relationship Between Elastic Constants

These elastic constants are not independent of each other. For isotropic materials (materials whose properties are the same in all directions), there are specific relationships that connect E, C, \(\nu\), and the Bulk Modulus (K). The question focuses on the relationship between E, C, and \(\nu\).

The widely accepted and experimentally verified relationship between Modulus of Elasticity (E), Modulus of Rigidity (C), and Poisson's ratio (\(\nu\)) for an isotropic material is given by the formula:

\[ E = 2C(1 + \nu) \]

This formula directly relates the three constants mentioned in the question.

Finding the Correct Expression

The options provided express C in terms of E and \(\nu\). We can rearrange the formula \(E = 2C(1 + \nu)\) to solve for C:

Divide both sides by \(2(1 + \nu)\):

\[ C = \frac{E}{2(1 + \nu)} \]

Now let's compare this derived expression with the given options:

1. \(E = (1 + \nu)\) - This option is incorrect as it doesn't include C and is dimensionally inconsistent.
2. \(C = E(2 + \nu)\) - This option is incorrect. The standard relationship is different.
3. \(C = \frac{E}{2(1+v)}\) - This option matches the derived relationship.
4. \(C = \frac{E}{1+2v}\) - This option is incorrect. The denominator should be \(2(1 + \nu)\).

Therefore, the exact relationship expressing Modulus of Rigidity (C) in terms of Modulus of Elasticity (E) and Poisson's ratio (\(\nu\)) is \(C = \frac{E}{2(1+v)}\).

Revision Table: Elastic Constant Formulas

Additional Information on Elastic Constants

Besides E, C, and \(\nu\), the Bulk Modulus (K) is another important elastic constant. It measures a material's resistance to uniform compression or volume change. It is defined as the ratio of volumetric stress to volumetric strain.

For any isotropic elastic material, only two of the four elastic constants (E, C, K, \(\nu\)) are independent. If you know any two, you can determine the other two using the relationships summarized in the tables above.

It's important to note that these relationships apply specifically to isotropic materials that follow Hooke's Law (linear elastic behavior). Anisotropic materials, like wood or composite materials, have properties that vary with direction, and their stress-strain relationships are more complex, requiring more than two independent elastic constants to describe their behavior.