Linear Elastic Systems and Strain Energy

Introduction:
In linear elastic systems, the relationship between stress and strain is linear within the elastic limit. This means that the strain energy stored in the system can be calculated using a displacement function. The type of displacement function determines the form of the strain energy equation.

Explanation:
The displacement function for strain energy can be determined by analyzing the behavior of the system under different types of loading conditions. The most common loading condition is uniaxial tension, where a linear relationship between stress and strain is observed. In this case, the displacement function is quadratic.

Quadratic Displacement Function:
For linear elastic systems, the quadratic displacement function is expressed as:
u(x) = ax^2 + bx + c
where u(x) represents the displacement at a given position x, and a, b, and c are constants.

Strain Energy Equation:
The strain energy equation for a linear elastic system with a quadratic displacement function can be derived using the principle of virtual work. The strain energy U in the system is given by:
U = (1/2) ∫[σ(x) * ε(x)] dx
where σ(x) represents the stress at a given position x, and ε(x) represents the strain at that position.

Derivation:
By substituting Hooke's Law (σ = E * ε) into the strain energy equation, we get:
U = (1/2E) ∫[E * ε(x)^2] dx
Since ε(x) can be expressed in terms of displacement u(x) as ε(x) = du/dx, we have:
U = (1/2E) ∫[E * (du/dx)^2] dx
Simplifying the equation, we get:
U = (1/2E) ∫[E * (a^2) * x^2 + 2abx + b^2] dx
U = (1/2E) [E * (a^2) * (x^3/3) + ab * (x^2) + b^2x] + constant

Conclusion:
The strain energy equation for linear elastic systems with a quadratic displacement function is a quadratic equation. This is because the displacement function represents a parabolic shape, which is characteristic of a quadratic equation. Therefore, the correct answer is option 'B' - quadratic.