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Dependence of the stress vector on surface orientation
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[QUOTE="pasmith, post: 6570499, member: 415692"] The stress vector is a force per unit area, not a force. The resulting force depends not just on the size of the area, but also its normal. The force due to stress experienced by a finite volume [itex]V[/itex] of material with boundary [itex]\partial V[/itex] is [itex]\int_{\partial V} T_{ij} n_j\,dS.[/itex] This is unique once you define the volume. But why should we choose one volume rather than another? The equivalent of Newton's Seond Law for the medium is essentially "density times acceleration equals force per unit volume". By the divergence theorem we have [tex]\int_{\partial V} T_{ij} n_j\,dS = \int_V \frac{\partial T_{ij}}{\partial x_j}\,dV[/tex] for an arbitrary volume [itex]V[/itex], so at each point in the medium the force per unit volume due to stress is the divergence of the stress tensor, [itex]\frac{\partial T_{ij}}{\partial x_j}[/itex]. This is uniquely defined at each point. [/QUOTE]
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Dependence of the stress vector on surface orientation
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