Question

A wire of cross-sectional area $A$ and initial length $x_0$ is stretched. The normal stress $\sigma$ acting in the wire varies linearly with strain, $\epsilon$, where $$\epsilon =(x-x_0)/x_0$$ and $x$ is the length of the wire. Assuming the cross-sectional area remains constant, derive an expression for the work done on the wire as a function of strain.

Short answer

The work done on the wire is $W=\frac{1}{2}EA{x}_0{\epsilon }^2$.

Step-by-step solution

Define the Strain

The strain, $\epsilon$, is defined as the change in length divided by the original length. Mathematically, it is given by: $$\epsilon =\frac{x-x_0}{x_0}$$.

Express the Length

Rearrange the strain equation to express the length of the wire $x$ in terms of the strain $\epsilon$ and the initial length $x_0$: $$x=x_0(1+\epsilon )$$.

Define the Normal Stress

Given that the normal stress $\sigma$ varies linearly with strain $\epsilon$, we can write: $$\sigma =E\epsilon$$, where $E$ is the Young's modulus of the material.

Relate Force to Stress

The force $F$ can be related to the stress $\sigma$ and the cross-sectional area $A$: $$F=\sigma A=EA\epsilon$$.

Determine Work Done

Work done $W$ on the wire is the integral of force over the change in length. As force varies with length, we express it in terms of length: $$dW=Fdx$$.

Integrate to Find Work

Substitute the expression for force $F$ into the integral: $$W={\int }_{x_0}^{x}EA\epsilon dx$$. Substitute $dx=x_{0}d\epsilon$: $$W=EA\text{[}{\int }_0^{\epsilon }{x}_{0}d\epsilon$$.\]

Solve the Integral

Evaluate the integral: $$W=EA{x}_0\text{[}\frac{{\epsilon }^2}{2}$$. Therefore, the work done on the wire as a function of strain is given by: $$W=\frac{1}{2}EA{x}_0{\epsilon }^2$$.\]

Key concepts

Normal Stress-Strain Relationship

In the context of material mechanics, the relationship between normal stress and strain is fundamental. Normal stress, represented by $\sigma$, is the internal force per unit area, acting perpendicular to the cross-sectional area of a material. Strain, denoted as $\epsilon$, measures the deformation of the material—specifically, the change in length relative to the original length. For a given wire, the strain can be expressed as: $$\epsilon =\frac{x-x_0}{x_0}$$ Here, $x$ is the current length and $x_0$ is the initial length. This linear relationship implies that as the wire stretches, the strain increases proportionally to the length change. Understanding this relationship helps in predicting how a material will react under various loads, which is crucial for applications in engineering and material science.

Young's Modulus

Young's modulus, represented by $E$, is a measure of a material's stiffness and is a critical parameter in stress-strain relationships. It describes how much a material will deform under a given amount of stress. Mathematically, Young's modulus is defined as the ratio of stress to strain: $$E=\frac{\sigma }{\epsilon }$$ In the given problem, since stress varies linearly with strain, we have: $$\sigma =E\epsilon$$ This equation signifies that for materials following Hooke's Law (linear elasticity), the stress increases in direct proportion to the strain. Young's modulus is a constant that characterizes the elasticity of the material. Higher values indicate a stiffer material, which deforms less under the same amount of stress.

Mechanical Work in Elastic Deformation

When a material like a wire is stretched, mechanical work is done on it. This work results in elastic deformation, which the wire can recover from when the load is removed. The mechanical work done, $W$, can be found by integrating the force applied over the distance the material stretches. Given the force $F=EA\epsilon$ and displacement $dx$, the expression for infinitesimal work is: $$dW=Fdx$$ Substituting the force and integrating over the strain yields: $$W=EA{\int }_0^{\epsilon }{x}_{0}d\epsilon =EA{x}_0\frac{{\epsilon }^2}{2}$$ Therefore, the work done on the wire as a function of strain is: $$W=\frac{1}{2}EA{x}_0{\epsilon }^2$$ This equation illustrates how the work done is stored as potential energy in the material, which can be fully recovered upon unloading, assuming the material remains within its elastic limit.

Linear Elasticity

Linear elasticity describes the behavior of materials that deform linearly under applied loads and return to their original shape upon unloading. The relationship between stress and strain is governed by Hooke's Law, stating that stress is directly proportional to strain within the elastic limit of the material. For a wire, this is expressed as: $$\sigma =E\epsilon$$ This linear relationship remains valid only up to the yield point of the material, beyond which permanent deformation occurs. Understanding linear elasticity is crucial when designing structures and components to ensure they operate safely within the elastic range, thereby avoiding permanent deformation or failure. This concept helps engineers predict and control the deformation behavior of materials under various loading conditions.