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# lec10-2 - Tensors in continuum mechanics When we apply...

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Tensors in continuum mechanics When we apply forces on a deformable body (stress) we get a deformation (strain) If the stresses are fairly small, the strains will be small For small stress/strain, the relationship between stress and strain is linear (Just like Hooke’s law F = - kx ) The stress and strain tensors are rank 2 Stress tensor: σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz The first index refers to the surface where the force is applied, and the second represents the force direction

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Strain tensor We can also characterize the deformation by a tensor, in this case the strain tensor xx xy xz yx yy yz zx zy zz To get the meaning of the strain tensor, define u ( x , y , z ), v ( x , y , z ) and w ( x , y , z ) which are the components of the displacements of the an element of material originally at x , y , and z (in the unstrained state) Separately, the u , v , and w are scalar fields, and the relevant quantity is the gradient of them xy = 1 2 u y + v x
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