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Usage of Taylor's formula on stress analysis

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  1. Feb 16, 2017 #1
    Moderator Note: Thread moved from forum Atomic, Solid State, Comp. Physic, so no homework template shown.

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    What function do they use Taylor's formula on? And can you show how they derive it? I know how one derives Taylor formula. Thanks! The text i taken from this page:http://ingforum.haninge.kth.se/armin/fluid/exer/deriv_navier_stokes.pdf
     
    Last edited by a moderator: Feb 17, 2017
  2. jcsd
  3. Feb 17, 2017 #2
    It isn't clear what you are asking. What exactly do you mean by Taylor's formula? Are you talking about the use of a Taylor series in stress to derive the stress-equilibrium equation?
     
  4. Feb 17, 2017 #3
    They use Taylors formula on the functions ##F_1##, ##F_2##, ##F_3##, ##F_4##, ##F_5## and ##F_6##. Where do they get the formulas from and show how the Taylors formulas are used on the functions. I only wonder about the mathematical relations not about any understanding of stress-equiibriums.
     
    Last edited: Feb 17, 2017
  5. Feb 17, 2017 #4
    They are making use of a truncated Taylor series expansion as follows:$$f\left(x+\frac{\Delta x}{2}\right)=f(x)+f'(x)\frac{\Delta x}{2}$$
    $$f\left(x-\frac{\Delta x}{2}\right)=f(x)-f'(x)\frac{\Delta x}{2}$$

    Assuming that the stress tensor varies with spatial position, they are finding the components of the forces exerted by the surrounding material on the 2 faces of the little cube that are perpendicular to the x axis. These two faces are located at ##x+\Delta x/2## and at ##x-\Delta x/2##, while the center of the cube is at x. So F1 is the normal component of force acting on the face at ##x-\Delta x/2##, and F2 is the normal component of force acting on the face at ##x+\Delta x/2##. They are doing all this so that they can do a differential force balance on the cube.
     
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