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It seems that you confused the Jaumann derivative $\overset{o}{{S}}$ (in your notation $\overset{\bigtriangleup}{{S}}$) with the time derivative ${\dot{S}}$ $$\frac{dS}{dt} = {\dot{S}} = \overset{o}{{S}} -{S} \cdot {w} +{w} \cdot {S}$$ See how it is derived in "http://www.continuummechanics.org/cm/corotationalderivative.html". Using the argument that $w^T ... 0 If you are able to make the switch from the conventional scalar representation of material properties to tensor representations, you will be better off. It will open up an entirely new way of looking at things, if you can handle the math. 0 This is the engineering stress-strain graph. In this graph, engineering stress calculate by force/initial area. But, what you talk about is real stress that calculate force/instantaneous area. So, there is no wrong. After ultimate strength, the cross section area decreases and strain continues without force increasing. And engineering stress decreases. ... 3 At least a part of this comes from understanding where the stress-strain curve comes from. Normally from a physics background we think of applying a force to a sample and seeing how it responds. Instead, experimental results like what you show are done differently in materials science - the sample is mounted in the testing machine (Instron for example), and ... 0 No, Young's modulus is an intrinsic property of the material (e.g. steel) that does not depend on its form (e.g. wire diameter). 0 The young's modulus is like the 'spring constant' for a material. It comes from treating the atoms in a material as harmonic oscillators. It is a material property that does not depend on geometry. The young's modulus for both of the materials you mention is equal. Now the stress they feel will be different under the same applied load. 3 Suppose you have some weightless elastic strip and two weights of equal mass$M/2\$. You arrange the weights in two different ways: In the strip on the left the tensions in the two halves of the strip are different because the upper half is stretched by both weights while the lower half is stretched by just a single weight. In the strip on the right the ...