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mudzm20 - Muddy Card Responses Lecture M20 There were a lot...

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Muddy Card Responses Lecture M20 12/1/2003 There were a lot of different matrices presented today. How can I keep them all straight. I certainly don’t expect you to memorize all of this. You should be aware that they all stem from the definitions of the engineering elastic constants – which are all defined in terms of simple loading (i.e one component of stress only). This means that the compliance form (known stress leading to resulting strain) is the most straightforward to follow directly, and I would expect you to be able to explain where this comes from, if not actually remember the terms in it. Ê 1 - n - n 0 0 0 ˆ Ê e x ˆ Á E E E ˜ Ê s x ˆ Á ˜ Á - n 1 - n 0 0 0 ˜ Á ˜ Á e y E E Á s y ˜ ˜ Á - E n - n 1 ˜ Á e z ˜ Á E 0 0 0 ˜ Á s z ˜ E E = Á ˜ Á 1 Á g zy 0 0 0 0 0 ˜ Á Á t zy ˜ ˜ Á G ˜ ˜ Á g zx ˜ Á 0 0 0 0 1 0 ˜ Á t zx ˜ Á ˜ Á ˜ Ë g xy ¯ Á Á 0 G 1 ˜ ˜ Ë t xy ¯ Ë 0 0 0 0 G ¯ I do expect you to be aware of the fact that anisotropic materials exist and to have some physical understanding of why there are different elastic properties for a composite material in particular. Connection between tensor and engineering terms was confusing. Fair enough. The most straightforward route to understanding this is by understanding what the terms in the compliance matrix mean (known stress to unknown strain – given by matrix in previous mud response). These connect directly to the definitions of the Engineering elastic constants, and how they are defined. Once you understand this, it is a small step to convert the engineering notation stress and strain to tensor stress and strain. What exactly are l and m in the 6x6 matrix? They are just convenient groupings of E and n . All we have done is invert the 6x6 compliance matrix (see previous two questions) to allow us to go from known strain to unknown stress. E m = = G 2 1 + n ) ( n E l = ( 1 + n )( 1 - 2 n ) What exactly are l and m ? Starting to understand tensors…. See previous response regarding l and m ? I am glad that you are gaining confidence with tensors. They are not
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really that frightening once you become familiar with them - they are just a convenient way of representing several simultaneous linear equations. What exactly are l and m ? See above. How are are l and m calculated? See above What physically is G? It is the shear modulus. It links shear stress to shear strain, defined in engineering notation. I am wondering if the matrices make understanding the concepts any easier, because they have so many zeros. I guess this depends on one’s point of view. I agree that there are a lot of terms, and if you had to memorize them it might be confusing. However, the nice aspect to using a matrix presentation is that you can see quite readily how different terms link stress to strain in different directions – and that you can relate that back to your physical understanding of the engineering elastic constants.
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