iMechanica - Comments for "Principal stress"
https://imechanica.org/node/15329
Comments for "Principal stress"enRecipe to make new tensors
https://imechanica.org/comment/25335#comment-25335
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<p><em>In reply to <a href="https://imechanica.org/node/15329">Principal stress</a></em></p>
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Most tensors are formed from other tensors.
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A scalar is a tensor of order 0. Examples include mass, energy, and entropy. These tensors are difficult to make. We find them in nature.
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A vector is a tensor of order 1. Examples include displacement, force, electric field.
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Start with tensors you know, you can make new tensors by recipes like these:
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<ul><li>A linear map that maps tensor 1 to tensor 2 is a new tensor.</li>
<li>A<a href="http://en.wikipedia.org/wiki/Bilinear_map"> bilinear map</a> that maps tensor 1 and tensor 2 to tensor 3 is a new tensor. </li>
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You can invent your own recipes, so long as you start with known tensors and use linear maps.
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<strong>Stress</strong>. By force balance, stress is a linear map that maps an area vector to the force acting on the area. See my <a href="http://imechanica.org/node/538">notes on finite elasticity</a>. The area vector is a tensor of order 1, and the force is a tensor of order 1. Consequently, the stress is a tensor of order 2.
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<strong>Piezoelectricity</strong>. When a stress is applied on a <a href="http://en.wikipedia.org/wiki/Piezoelectric">piezoelectric material</a>, an electric field arises. This phenomenon is modeled by a linear map that maps the stress to the electric field. The stress is a tensor of order 2, and the electric field is a tensor of order 1. Thus, the map is a tensor of order 3.
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Please take a look at the recent <a href="http://imechanica.org/node/15843">thread on linear algebra and tensors</a>.
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</ul>Sun, 29 Dec 2013 17:42:00 +0000Zhigang Suocomment 25335 at https://imechanica.orggood lecture notes
https://imechanica.org/comment/24984#comment-24984
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<p><em>In reply to <a href="https://imechanica.org/node/15329">Principal stress</a></em></p>
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good lecture notes
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it is difficult for me to understand the third rank tensor.
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The send rank tensor is stress, so what is the third rank tensor.
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</ul>Tue, 24 Sep 2013 15:31:08 +0000Hua Wangcomment 24984 at https://imechanica.org