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Correct me if i understood something wrongly.As i understand,you are asking why do we analyze using stress rather than forces? Well,stresses are easier to work with.Suppose we have a certain material with a cubic shape.A force is applied to it and so you make your analysis with the forces.ok,its all good.But when you want to make your analysis with a ...


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Of course, I cannot be sure what you read, so, for what it's worth, Euler's buckling theory (http://en.wikipedia.org/wiki/Buckling#Columns )can also be relevant. It determines the maximum axial load a column can withstand without losing stability. I guess Euler derived the theory for a round column, but there should be formulas for an arbitrary shape of the ...


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What you are looking for is the famous Theorema Egregium by Gauss, which asserts that the Gaussian curvature of a surface is invariant under local isometry. At the same time, the Gaussian curvature of a surface is the product of the principal curvatures. Regarding a slight bend along the middle as a local isometry (of course, this conceptualization breaks ...


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Pressure is perpendicular to the object, it is an external force only. Pressure causes stress inside of the object, so stress is an internal force.


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The following is basically what the Ashcroft/Mermin says about it. The idea is as following: in harmonic approximation a relative displacement $u$ results in an energy $U=- \frac 1 4 (\vec{ u }(\vec R) - \vec{ u }(\vec R ')) \mathbf{D}(\vec{ u }(\vec R ') - \vec{ u }(\vec R)) $ The tensor $\mathbf D$ already has natural symmetry ...


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You should start with the strain energy density $\psi$, then define: $$ C_{ijkl} = \frac{\partial^2 \psi}{\partial \epsilon_{ij}\partial \epsilon_{kl}}, $$ and then define $$ \sigma_{ij} = C_{ijkl} \epsilon_{kl} $$ The remainder of my answer will be about explaining why you have to do it that way. Firstly it is physical, there really is energy associated ...


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Since $\epsilon$ is a symmetric tensor, it has 6 independent component that determine it. Hence use a multi-index $I\in\{(i,j)|1\leq i\leq j\leq 3\}$ to denote them. The strain energy density then becomes (perhaps one has to be careful with "diagonal" terms here in order to get the right coefficients) $$\psi = C_{IJ}\epsilon_I\epsilon_J$$ where summation is ...



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