Respuesta :
T=σ(x)n(x), where σ is the stress tensor and n is the outward unit normal to the surface at x, gives the stress vector t operating on a surface at a point x.
What do you mean by stress tensor?
The stress tensor, also known as the Cauchy stress tensor σ, genuine stress tensor, or simply the stress tensor in continuum mechanics, is made up of nine elements σ[tex]_{ij}[/tex] that collectively characterize the state of stress at a point inside a material that is in a deformed condition, location, or configuration. The stress tensor and traction vector have the SI unit, which corresponds to the stress scalar, N/m2. A unit vector has no dimensions.
Continued Answer:
The stress tensor σ and unit normal n are constant for all x on the plane, as you have already noticed. The tension vector t is therefore given by
t=σn= [tex]\left[\begin{array}{ccc}1&1&0\\1&1&1\\0&1&1\end{array}\right] \left[\begin{array}{ccc}\frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\\frac{1}{\sqrt{3}} \end{array}\right] = \left[\begin{array}{ccc}\frac{2}{\sqrt{3}} \\\frac{2}{\sqrt{3}}\\\frac{2}{\sqrt{3}}\end{array}\right][/tex]
Since n has magnitude 1, the component of t that is perpendicular to the plane, c, can be calculated by
[tex]c = (t . n)n = \frac{7}{3}n[/tex]
The angle θ between the stress vector and the plane's normal is finally given by:
θ = [tex]cosx^{-1} (\frac{t . n}{|t| . |n|}) = cos^{-1}(\frac{7/3}{\sqrt{17}/\sqrt{3}})[/tex]
To learn more about stress tensor, use the link given
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