- #1

baw

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In every book I checked, the energy (per unit mass) of elastic deformation is derived as follows:

## \int \sigma_1 d \epsilon_1 = \frac{\sigma_1 \epsilon_1}{2} ##

and then, authors (e.g. Timoshenko & Goodier) sum up such terms and substitute ##\epsilon ## from generalised Hooke's law i.e.

## \epsilon_1=\frac{1}{E} (\sigma_1 -\nu \sigma_2 -\nu \sigma_3) ##

## \epsilon_2=\frac{1}{E} (\sigma_2 -\nu \sigma_1 -\nu \sigma_3) ##

## \epsilon_3=\frac{1}{E} (\sigma_3 -\nu \sigma_2 -\nu \sigma_1) ##

obtaining:

##V=\frac{1}{2E} (\sigma_1^2 +\sigma_2^2+\sigma_3^2 )-\frac{\nu}{E}(\sigma_1 \sigma_2+\sigma_2 \sigma_3 + \sigma_1 \sigma_3) ##

but... is it correct to substitute generalised Hooke's law after the integration? The formula is obtained as if simple ##\sigma = E \epsilon ## was used. As in the attached figure, it looks like they assume that ##\sigma_x ## has no term independent on ##\epsilon_x ##, despite that Hooke's law can be transformed to:

## \sigma_1=\frac{(\nu -1)E}{(\nu +1)(2 \nu-1)} \epsilon_1 - \frac{\nu E}{(\nu+1)(2\nu -1)}(\epsilon_2+\epsilon_3) ##

##\sigma_2=(...) ##

##\sigma_3=(...) ##

where this term is present. Shouldn't we integrate the above formula? Could someone please, explain me why it is correct?

## \int \sigma_1 d \epsilon_1 = \frac{\sigma_1 \epsilon_1}{2} ##

and then, authors (e.g. Timoshenko & Goodier) sum up such terms and substitute ##\epsilon ## from generalised Hooke's law i.e.

## \epsilon_1=\frac{1}{E} (\sigma_1 -\nu \sigma_2 -\nu \sigma_3) ##

## \epsilon_2=\frac{1}{E} (\sigma_2 -\nu \sigma_1 -\nu \sigma_3) ##

## \epsilon_3=\frac{1}{E} (\sigma_3 -\nu \sigma_2 -\nu \sigma_1) ##

obtaining:

##V=\frac{1}{2E} (\sigma_1^2 +\sigma_2^2+\sigma_3^2 )-\frac{\nu}{E}(\sigma_1 \sigma_2+\sigma_2 \sigma_3 + \sigma_1 \sigma_3) ##

but... is it correct to substitute generalised Hooke's law after the integration? The formula is obtained as if simple ##\sigma = E \epsilon ## was used. As in the attached figure, it looks like they assume that ##\sigma_x ## has no term independent on ##\epsilon_x ##, despite that Hooke's law can be transformed to:

## \sigma_1=\frac{(\nu -1)E}{(\nu +1)(2 \nu-1)} \epsilon_1 - \frac{\nu E}{(\nu+1)(2\nu -1)}(\epsilon_2+\epsilon_3) ##

##\sigma_2=(...) ##

##\sigma_3=(...) ##

where this term is present. Shouldn't we integrate the above formula? Could someone please, explain me why it is correct?