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Chapter 10: Problem 1

Prove that the sum of the normal strains in perpendicular directions is constant.

Short Answer

Expert verified
The sum of the normal strains in perpendicular directions is constant if the sum of the corresponding normal stresses is constant, as indicated by the relationship derived from Hooke's law: \( \epsilon_x + \epsilon_y = \frac{1 - \nu}{E} (\sigma_x + \sigma_y) \).

Step by step solution

01

Understand Normal Strain

Normal Strain, denoted as \( \epsilon \), is defined as the change in length per unit length. It is a measure to quantify the deformation of a material body. Mathematically, it's given as: \( \epsilon = \frac{\Delta L}{L} \) where \( \Delta L \) is the change in length and \( L \) is the original length.
02

Define Normal Strains in Perpendicular directions

Consider a two-dimensional material element subjected to normal stresses in perpendicular directions. The normal strains corresponding to these stresses are \( \epsilon_x\) and \( \epsilon_y\), where the subscript refers to the direction.
03

Apply Hooke's Law

According to Hooke's law, for an isotropic material, \( \epsilon_x = \frac{1 - \nu}{E} \sigma_x \) and \( \epsilon_y = \frac{1 - \nu}{E} \sigma_y \) where \( \sigma_x\) and \( \sigma_y\) are the normal stresses, \( \nu \) is Poisson's ratio and \( E \) is Young's modulus. Adding these two: \( \epsilon_x + \epsilon_y = \frac{1 - \nu}{E} (\sigma_x + \sigma_y) \) Thus, we see that the sum of normal strains in perpendicular directions depends upon the sum of corresponding normal stresses. If \( \sigma_x + \sigma_y \) remains constant, the sum of normal strains in perpendicular directions remain constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Hooke's Law
Hooke's Law is fundamental in understanding how materials deform under stress. It provides a simple relationship between the stress applied to a material and the resulting strain (deformation).
In the context of normal strain in materials, Hooke's Law can be represented as:
\[\text{Stress} = \text{Young's modulus} \times \text{Strain}\]
where the stress is the force acting on the material's cross-sectional area, and strain is the deformation experienced by the material as a proportion of its original length. In mathematical symbols, Hooke's Law is often notated as:
\[\text{Stress} = E \times \text{Strain}\]
Here, the constant of proportionality is known as Young's modulus (E), which is unique for each material and dictates how much it will stretch or compress under a given load.
This relationship is linear until a material reaches what's known as the elastic limit 鈥� the point beyond which it will not return to its original shape. Understanding this concept is crucial for engineers and designers when predicting how materials will behave when forces are applied, ensuring safety and functionality in their applications.
Exploring Poisson's Ratio
Poisson's ratio (\(u\)) plays a key role in understanding the multidirectional deformation of materials. When a material is stretched in one direction, it not only elongates but also tends to contract in the directions perpendicular to the stretch.
This phenomenon is quantified by Poisson's ratio, which is the negative ratio of transverse strain to axial strain. For most common materials, Poisson's ratio is positive, meaning that they get thinner in cross-section when stretched. Mathematically, Poisson's ratio can be expressed as:
\[u = -\frac{\text{Transverse Strain}}{\text{Axial Strain}}\]
It is a dimensionless number and usually falls in the range from 0 to 0.5 for most materials, where a value of 0.5 corresponds to a perfectly incompressible material. Understanding this property is essential for engineers, as it impacts how materials will contract or expand in applications where multi-axial stresses occur, such as in pressurized tanks or bridges supporting weight.
Young's Modulus and Material Behavior
Young's modulus is a measure of the stiffness of a material. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in the elastic region of a material's stress-strain curve.
Technically, it is the ratio of tensile stress to tensile strain and is represented by the symbol E. In equations:
\[E = \frac{\text{Stress}}{\text{Strain}}\]
The value of Young's modulus is crucial when it comes to material selection in engineering and construction. A high Young's modulus indicates that a material is rigid and requires a great deal of force to deform. Conversely, a low Young's modulus implies that the material is flexible.
Materials with different Young's moduli will react differently under the same applied stress, which is why knowing this value is essential for predicting material behavior under load. Apart from helping in material selection, it is also used to determine the load a structure can sustain and how much it will deflect when a load is applied.

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Most popular questions from this chapter

For the case of plane stress, show that Hooke's law can be written as \\[\sigma_{x}=\frac{E}{\left(1-\nu^{2}\right)}\left(\boldsymbol{\epsilon}_{x}+\nu \boldsymbol{\epsilon}_{y}\right), \quad \sigma_{y}=\frac{E}{\left(1-\nu^{2}\right)}\left(\boldsymbol{\epsilon}_{y}+\nu \boldsymbol{\epsilon}_{x}\right)\\]

The gas tank is made from \(A-36\) steel and has an inner diameter of \(1.50 \mathrm{m}\). If the tank is designed to withstand a pressure of \(5 \mathrm{MPa}\), determine the required minimum wall thickness to the nearest millimeter using (a) the maximum-shear-stress theory, and (b) maximum- distortion-energy theory. Apply a factor of safety of 1.5 against yielding.

The \(60^{\circ}\) strain rosette is mounted on a beam. The following readings are obtained for each gauge: \(\epsilon_{a}=600\left(10^{-6}\right), \quad \epsilon_{b}=-700\left(10^{-6}\right), \quad\) and \(\quad \epsilon_{c}=350\left(10^{-6}\right)\). Determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and average normal strain. In each case show the deformed element due to these strains.

The \(60^{\circ}\) strain rosette is attached to point \(A\) on the surface of the support. Due to the loading the strain gauges give a reading of \(\epsilon_{a}=300\left(10^{-6}\right), \epsilon_{b}=-150\left(10^{-6}\right),\) and \(\epsilon_{c}=-450\left(10^{-6}\right) .\) Use Mohr's circle and determine (a) the in-plane principal strains and (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of each element that has these states of strain with respect to the \(x\) axis.

The state of strain at the point on the bracket has components \(\epsilon_{x}=350\left(10^{-6}\right), \epsilon_{y}=-860\left(10^{-6}\right), \gamma_{x y}=250\left(10^{-6}\right)\). Use the strain-transformation equations to determine the equivalent in-plane strains on an element oriented at an angle of \(\theta=45^{\circ}\) clockwise from the original position. Sketch the deformed element within the \(x-y\) plane due to these strains.
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