# Strain_gages - Stress Strain and Strain Gages Author John M Cimbala Penn State University Latest revision 26 October 2007 Introduction Stress and

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Stress, Strain, and Strain Gages Author: John M. Cimbala, Penn State University Latest revision: 26 October 2007 Introduction Stress and strain are important aspects of Mechanical Engineering, especially in structural design. In this learning module, we discuss stress and strain and their relationship, and how to measure them. Definitions Stress o When a material is loaded with a force, the stress at some location in the material is defined as the applied force per unit of cross-sectional area . o For example, consider a wire or cylinder, anchored at the top, and hanging down. Some force F (for example, from a hanging weight) pulls at the bottom, as sketched, where A is the original cross-sectional area of the wire, and L is the original wire length. o In this situation, the material experiences a stress, called an axial stress , denoted by the subscript a , and defined as a F A σ = . o Notice that the dimensions of stress are the same as those of pressure – force per unit area. Strain o In the above simple example, the wire stretches vertically as a result of the force. Strain is defined as the ratio of increase in length to original length . o Specifically, when force is applied to the wire, its length L increases by a small increment δ L , while its cross-sectional area A decreases, as sketched. o In the axial direction (the direction of the applied force), axial strain ε a is defined as a L L = . o The dimensions of strain are unity – strain is a nondimensional quantity . Hooke’s law o It turns out that for elastic materials, stress is linearly proportional to strain . o Mathematically, this is expressed by Hooke’s law , which states a E a = , where E = Young’s modulus , also called the modulus of elasticity . o Young’s modulus is assumed to be constant for a given material. o Hooke’s law breaks down when the strain gets too high. On a typical stress-strain diagram, Hooke’s law applies only in the elastic stress region , in which the loading is reversible . Beyond the elastic limit (or proportional limit ), the material starts to behave irreversibly in the plastic deformation region , in which the stress vs. strain curve deviates from linear, and Hooke’s law no longer holds, as sketched. o In this learning module, only the elastic stress region is considered. Wire resistance The electrical resistance R of a wire of length L and cross-sectional area A is given by L R A ρ = , where is the resistivity of the wire material. (Do not confuse with density, for which the same symbol is used.) The electrical resistance of the wire changes with strain: o As strain increases, the wire length L increases, which increases R . o As strain increases, the wire cross-sectional area A decreases, which increases R . o For most materials, as strain increases, the wire resistivity also increases, which further increases R .

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## This note was uploaded on 07/23/2008 for the course ME 345 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.

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Strain_gages - Stress Strain and Strain Gages Author John M Cimbala Penn State University Latest revision 26 October 2007 Introduction Stress and

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