Where did I go wrong in calculating the traction fields?

[SP90]

Homework Statement

Question (specifically part b):

Solution:

The Attempt at a Solution

My issue is that I don't agree with [itex]\Gamma^{0}_{3}[/itex] and [itex]\Gamma^{1}_{3}[/itex], surely if they're given by [itex][n][/itex] they should be given by [itex]\mp(0,0,\rho g x_{3})[/itex]. The answers they have don't make sense, because traction fields should satisfy [itex]t(-n,x) =-t(n,x)[/itex] (i.e. every reaction has an equal and opposite reaction).

Can someone point out to me where I'm going wrong?

 

[mark726]

Dear fellow scientist,

I understand your concerns regarding the given values for \Gamma^{0}_{3} and \Gamma^{1}_{3}. It is important to note that the values given in the solution are based on the assumption that the traction fields are proportional to the normal vector n. This is a simplification that is commonly used in mechanics problems. However, in reality, traction fields can have more complex dependencies on the normal vector and other parameters.

In this case, the given values for \Gamma^{0}_{3} and \Gamma^{1}_{3} may not seem intuitive, but they are consistent with the given assumption. It is also possible that there are other factors at play that are not explicitly mentioned in the problem, which could affect the values of these terms.

I suggest further exploring the problem and considering other possible factors that could influence the traction fields. If you are still unsure, it may be helpful to consult with a colleague or mentor to gain a different perspective.

I hope this helps address your concerns. Good luck with your analysis!