Problem 2 An isotropic elastic body (E = 207 GPa and (mu =...

Problem 2 An isotropic elastic body (E = 207 GPa and (mu = 79.2) GPa) has a uniform state of stress given by: (sigma = egin{bmatrix} 100 & 40 & 60 \ 40 & -200 & 0 \ 60 & 0 & 200 end{bmatrix}) MPa. (a) What are the strain components? (b) What is the total change of volume for a five-centimeter cube of the material?

Transcript

00:01 Hello students in this question given as a stress matrix 100 40 60 40 minus 200 0 60 0 200 mega pascal and this is equal to the this sigma xx tau xy tau xz and tau yx sigma yy tau yz tau zx tau zy and sigma zz in this question given as the e is equals to 207 mega pascal and mu is equal to 79 .2 mega pascal.

01:17 Now in the first part of the question, we have to calculate the matrix for the strain.

01:24 So for linear strain, we will calculate for first of all the linear strain which is equals to the sigma xx by e and sigma epsilon y is equals to sigma yy upon e and this is equals to sigma zz upon so this is equal to the sigma xx is equals to 100 mega pascal 10 to the power 6 upon e is equals to 207 giga pascal.

01:55 Now, this will be equal to the 0 .8 0 .483 10 to the power minus 3 next this will be equal to the minus 200 into 10 to the power 6 upon 207 10 to the power 9.

02:15 This is equal to the minus 0 .97 10 to the power minus 3 next this is equals to 200 into 10 to the power 6 upon 207 10 to the power 9 is equal to 0 .98 into 10 to the power minus 3 next we will calculate the shear strain.

02:43 So this is equal to gamma xy upon 2 is equal to gamma y upon x will be same because this is equals to tau xy upon mu and the tau xy is equals to tau yx by mu and this tau xy and tau yx which is same that is 40 into 10 to the power 6 upon 79 .2 10 to the power 9.

03:19 So after calculation, this is equal to 0 .5 1 10 to the power minus 3 similarly, we will calculate the next value sigma gamma zx by 2 gamma xz by 2 is equals to tau zx by mu is equals to tau zx z by mu is equal to 60 into 10 to the power 6 upon 79 .2 into 10 to the power 9 is equal to 0 .76 into 10 to the power 3 similarly, this value will be equal to the 0 now the epsilon now the strain matrix will be equal to the epsilon x gamma xy gamma xz by 2 gamma.

04:37 Now, we will substitute all the values which is 0 .4830 .51 0 .76 0 .51 minus 0 .97 0 0 .76 0 0 .98 into 10 to the power minus a in the next part of the question.

05:08 We have to calculate the change in volume...

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