zm17_20 - Block 3-Materials and Elasticity Lecture M17...

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Block 3 -Materials and Elasticity Lecture M17: Engineering Elastic Constants There are three purposes to this block of lectures: 1. To complete our quick journey through continuum mechanics, to provide you with a continuum version of a constitutive law - at least for linear elastic materials s pq = E ? e mn Elasticity Where does it come from? 2. Increasingly, materials are designed along with the structure, you need insight into what contributes to material properties. What you can control. What you cannot. This will also allow us to understand the limits of the model of linear elasticity for a material. 3. To allow you to select quantitatively materials for applications as part of the design process. The lectures associated with objectives 2 and 3 will closely follow Ashby and Jones chapters 1-7. This is an excellent reference and will not be supplemented by web-posted notes. The notes for the lectures associated with objective 1 are reproduced here. Engineering Elastic Properties of Materials In order to understand how we link stress and strain we need to understand that there are two points of view to this matter. There is the experimental point of view that some properties (behaviors) are easier to measure than others, and there is the mathematical point of view that some representations of physical phenomena are mathematically easier to handle than others. In the present case, engineering elastic constants are derived from an experimental point of view, whereas the stress and strain tensors, are mathematically useful. Ultimately we need to resolve these two points of view.
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Young’s modulus and Poisson’s ratio From the truss and strain laboratories you are now familiar with at least two elastic constants. If we apply a uniaxial tensile stress s L to a constant cross-section rod of material, we will obtain a biaxial state of strain, consisting of an axial tensile strain e L and a transverse strain T . The axial strain will be tensile for a tensile applied stress, and the transverse strain will usually be compressive. We can measure the strains using resistance strain gauges.
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For many materials, over some range of applied stress, the applied stress and the resulting strains will follow a linear relationship. This observation is the basis for the definition of the engineering elastic constants. The Young’s modulus, E, is defined as the constant of proportionality between a uniaxial applied stress and the resulting axial strain, i.e: s L = E e L Note. This only applies for a uniaxial applied stress, and the component of strain in the direction of the applied stress. We can also define the Poisson’s ratio, n , as the ratio of the transverse strain to the axial strain. Since for the vast majority of materials the transverse strain is compressive for a tensile applied stress, the Poisson’s ratio is defined as the negative of this ratio, to give a positive quantity. I.e: T = - L A similar process, of performing experiments in which a well-defined stress state is
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This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.

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zm17_20 - Block 3-Materials and Elasticity Lecture M17...

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