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zm12_m13

# zm12_m13 - Symmetry of Stress Tensor Consider moment...

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Symmetry of Stress Tensor Consider moment equilibrium of differential element: Taking moments about x 1 axis (i.e point C): È Â M 1 = 0 : 2 s 23 d x 3 d x 1 ) 2 Area of È ( d x 2 ˘ - 2 s 32 ( d x 2 d x 1 ) d x 3 ˘ = 0 Î ˚ Î 2 ˚ Moment fi s 23 = face arm s 32 Thus, in general s mn = s nm Stress tensor is symmetric. Six independent components of the stress tensor. s 11 s 12 Ê= s 21 ˆ s 22 s 23 Á Á = s 32 ˜ ˜ s 33 s 31 Ë = s 13 ¯ Note a positive (tensile) component of stress acts on a face with a positive normal in a positive direction. Thus a stress acting on a negative normal face, in a negative direction is also positive. If the stresses do not vary over the infinitesimal element, s mn acts on opposite faces, in opposite directions but with equal magnitude.

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But what happens if stress varies with position? Stress Equations of Equilibrium Consider equilibrium in x 1 direction. Let stresses vary across cube. Also allow there to be a body force (per unit volume) f 1 acting on the cube in the x1 direction: x 3 s 11 d x 2 ∂s 31 d x 3 s 31 + x 3 ∂s 21 d x 2 s 21 + x 2 d x 3 x 2 d x 1 s 31 x 1 ∂s 11 d x 1 f 1 s 11 + x 1 f n is a “body force”, e.g. due to the weight of the element ( r g), centrifugal acceleration ( r r w 2 ), electromagnetic fields etc. Taking equilibrium of forces in the x 1 direction gives: Ê ˆ ˆ Á ∂s 11 d x 1 ˜ ( d x 2 d x 3 ) - s 11 ( d x 2 d x 3 ) + Á Ê s 21 + ∂s 21 d x 2 ˜ ( d x 1 d x 3 ) - s 21 ( d x 1 d x 3 ) Ë s 11 + x 1 ¯ Ë x 2 ¯ ˆ + Ê Á s 31 + ∂s 31 d x 3 ˜ ( d x 1 d x 2 ) - s 31 ( d x 1 d x 2 ) + f 1 ( d x 1 d x 2 d x 3 ) = 0 Ë x 3 ¯
which simplifies to: ∂s 11 + ∂s 21 + ∂s 31 + f 1 = 0 x 1 x 2 x 3 similarly, equilibrium in the x 2 and x 3 directions yields: ∂s 23 ∂s 12 + ∂s 22 + ∂s 32 + f 2 = 0 and

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