Understanding Composite Modulus under Isostress Conditions

This question asks us to determine the modulus of elasticity of a composite material made of E-glass fibres and epoxy resin when it is subjected to stress under isostress conditions. A composite material combines two or more constituent materials with significantly different physical or chemical properties, remaining separate and distinct at the macroscopic or microscopic level within the finished structure.

The modulus of elasticity (Young's modulus) is a measure of the stiffness of an elastic material. It is defined as the ratio of stress ($\sigma$) to strain ($\epsilon$) in a material under tensile or compressive stress:

\[ E = \frac{\sigma}{\epsilon} \]

For composite materials, the overall modulus depends on the properties of the constituent materials (fibre and matrix), their volume fractions, and how they are oriented relative to the applied stress. Two common simplified models are used: the isostrain model (rule of mixtures) and the isostress model (inverse rule of mixtures or Reuss model).

Isostress Condition in Composites

Under isostress conditions, it is assumed that the stress is uniform throughout the composite material, meaning the stress experienced by the fibre is the same as the stress experienced by the matrix and the overall composite. However, the strain is not uniform. This condition is often considered when the load is applied perpendicular to the direction of continuous fibres or when the fibres are short and randomly oriented.

For a composite material under isostress conditions, the reciprocal of the composite modulus ($E_c$) is the weighted average of the reciprocals of the moduli of the constituents, weighted by their volume fractions. The formula for the modulus of elasticity under isostress conditions is:

\[ \frac{1}{E_c} = \frac{V_f}{E_f} + \frac{V_m}{E_m} \]

Where:

• \( E_c \) is the modulus of elasticity of the composite
• \( V_f \) is the volume fraction of the fibre
• \( E_f \) is the modulus of elasticity of the fibre
• \( V_m \) is the volume fraction of the matrix
• \( E_m \) is the modulus of elasticity of the matrix

We are given the following values:

• Modulus of E-glass fibre (\( E_f \)) = 72 GPa
• Modulus of epoxy resin matrix (\( E_m \)) = 3 GPa
• Volume fraction of E-glass fibre (\( V_f \)) = 60% = 0.60
• Volume fraction of epoxy resin matrix (\( V_m \)) = 40% = 0.40

Note that \( V_f + V_m = 0.60 + 0.40 = 1.00 \), as expected.

Calculating the Composite Modulus

Now, we can plug the given values into the isostress formula:

\[ \frac{1}{E_c} = \frac{0.60}{72 \text{ GPa}} + \frac{0.40}{3 \text{ GPa}} \]

First, let's calculate the terms on the right side:

\[ \frac{0.60}{72} = \frac{60}{7200} = \frac{1}{120} \]

\[ \frac{0.40}{3} = \frac{4}{30} = \frac{2}{15} \]

Alternatively, to find a common denominator for 120 and 3:

\[ \frac{0.60}{72} = \frac{60}{7200} \]

\[ \frac{0.40}{3} = \frac{0.40 \times 24}{3 \times 24} = \frac{9.6}{72} \]

Let's use the simpler fractions:

\[ \frac{1}{E_c} = \frac{1}{120} + \frac{2}{15} \]

To add these fractions, we need a common denominator, which is 120:

\[ \frac{2}{15} = \frac{2 \times 8}{15 \times 8} = \frac{16}{120} \]

So, the equation becomes:

\[ \frac{1}{E_c} = \frac{1}{120} + \frac{16}{120} \]

\[ \frac{1}{E_c} = \frac{1 + 16}{120} = \frac{17}{120} \]

Now, we find \( E_c \) by taking the reciprocal:

\[ E_c = \frac{120}{17} \text{ GPa} \]

Calculating the numerical value:

\[ E_c \approx 7.0588 \text{ GPa} \]

The question asks for the modulus of elasticity to the nearest unit magnitude. Rounding 7.0588 GPa to the nearest whole number gives 7 GPa.

Comparing with Options

Let's compare our calculated value with the given options:

Our calculated value of approximately 7.06 GPa, when rounded to the nearest unit, is 7 GPa, which corresponds to Option 4.

Revision Table: Composite Modulus Calculation

Additional Information: Composite Properties

Composite materials are designed to combine the desirable properties of different materials. Fibres often provide strength and stiffness, while the matrix binds the fibres together and transfers stress. The overall properties of the composite depend heavily on the properties of the constituents, their arrangement, and the interface between them.

Two primary ideal conditions are often considered for estimating composite properties based on constituent properties and volume fractions:

1. Isostrain Condition (Rule of Mixtures): This condition assumes that the strain is uniform throughout the composite, fibre, and matrix. This typically applies when stress is applied parallel to continuous fibres. The composite modulus under isostrain is given by: \[ E_c = V_f E_f + V_m E_m \] This model often provides an upper bound for the composite modulus.
2. Isostress Condition (Inverse Rule of Mixtures or Reuss Model): This condition assumes that the stress is uniform throughout the composite, fibre, and matrix. This typically applies when stress is applied perpendicular to continuous fibres or for composites with short, randomly oriented fibres. The composite modulus under isostress is given by: \[ \frac{1}{E_c} = \frac{V_f}{E_f} + \frac{V_m}{E_m} \] This model often provides a lower bound for the composite modulus.

In this specific question, the problem explicitly states "isostress conditions", guiding us to use the second formula. The calculated value of 7 GPa falls between the matrix modulus (3 GPa) and the fibre modulus (72 GPa), which is expected for a composite.