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What I do not understand is while balancing the torque, why the $\sigma_{11}$ is added to the product of $dx_{1}$ and the derivative of $\sigma_{11}$?

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The authors are applying a Taylor series expansion and making some assumptions based on the use of an infinitesimal element. Briefly, they define $\sigma_{11}(0)$, i.e., the normal leftward-pointing tensile stress at $x_1=0$, and then they expand $\sigma_{11}(dx_1)$, i.e., the normal rightward-pointing tensile stress at $x_1=dx_1$, as

$$\sigma_{11}(dx_1)=\sigma_{11}(0+dx_1)=\sigma_{11}(0)+\sigma^\prime_{11}(0)\,dx_1+\frac{1}{2!}\sigma^{\prime\prime}_{11}(0)\,(dx_1)^2+\frac{1}{3!}\sigma^{\prime\prime\prime}_{11}(0)\,(dx_1)^3+\cdots$$

and drop all but the first and second terms as negligible, as $dx_1$ is so small that $(dx_1)^2$ and so on are insignificant. Similarly, a single uniform $\frac{\partial \sigma_{11}}{\partial x_1} \approx\sigma^\prime_{11}(0)$ is assumed to apply across the entire infinitesimal element.

This is a very widely used technique to derive constitutive equations because the net rightward normal force can be easily expressed as $\sigma_{11}+\frac{\partial \sigma_{11}}{\partial x_1}dx_1-\sigma_{11}=\frac{\partial \sigma_{11}}{\partial x_1}dx_1$. The same approach is applied for the other directions.

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  • $\begingroup$ Thank you ma'am/sir. The stress and displacement here is related to hook's law I suppose. And that's why it gives the credibility to assume stress as a function of displacement. Am I right? $\endgroup$
    – papun
    Nov 21, 2020 at 5:09
  • $\begingroup$ Stress can be a function of displacement, but here, stress is treated as a function of $x_1$, which is a location, not a displacement. $\endgroup$ Nov 22, 2020 at 0:27
  • $\begingroup$ Firstly, thanks a lot for your time. I have some more queries. Is there any mandate for considering stress as a function of location? How stress behaves with the material can be seen by measuring the location of different parts of the body. Is this the reason why stress is considered as a function of length/ location? Can you please suggest any reading material for in-depth knowledge about this thing. $\endgroup$
    – papun
    Nov 22, 2020 at 11:16
  • $\begingroup$ Start with a mechanics of materials textbook such as Beer & Johnston or Hibbeler. $\endgroup$ Nov 22, 2020 at 16:06
  • $\begingroup$ Okay. Thank you $\endgroup$
    – papun
    Nov 24, 2020 at 3:35

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