Final answer:
The resultant stress, normal stress, and shear stress on a plane making equal angles with the x, y, z axes can be calculated using the stress tensor matrix. The stress invariants and principal stresses can also be determined, along with the absolute maximum shear stress.
Step-by-step explanation:
To determine the given quantities, we can use the stress tensor matrix.
a) The resultant stress S can be calculated using the formula:
S = sqrt(x^2 + y^2 + z^2 + 2xy + 2xz + 2yz) = sqrt(1^2 + 3^2 + 5^2 + 2(4) + 2(6) + 2(8)) = sqrt(1 + 9 + 25 + 8 + 12 + 16) = sqrt(71) ≈ 8.43MPa
b) To find the normal stress σn and shear stress τnt, we need to calculate the average stress on a plane making equal angles with the x, y, z axes. The average stress is given by:
σavg = (x + y + z) / 3 = (1 + 3 + 5) / 3 = 3MPa
The normal stress σn is equal to the average stress, so σn = σavg = 3MPa. The shear stress τnt can be calculated using the formula:
τnt = sqrt((x - σavg)^2 + (y - σavg)^2 + (z - σavg)^2 + 2(xy - τxy) + 2(xz - τxz) + 2(yz - τyz)) = sqrt((1 - 3)^2 + (3 - 3)^2 + (5 - 3)^2 + 2(4 - 6) + 2(6 - 6) + 2(8 - 8)) = sqrt(4 + 0 + 4 + 2(2) + 2(0) + 2(0)) = sqrt(12) ≈ 3.464MPa
c) The stress invariants can be calculated using the formulas:
I1 = x + y + z = 1 + 3 + 5 = 9MPa
I2 = xy + xz + yz = 4 + 6 + 8 = 18MPa
I3 = xyz + 2τxyz - τyz(x^2 + y^2 + z^2) - σyz(x + y + z) + Στ^2 = 1(2) - 8(1^2 + 3^2 + 5^2) - 3(1 + 3 + 5) + (4^2 + 6^2 + 8^2) = -41MPa
The principal stresses can be determined using the formulas:
λ^3 - I1λ^2 + I2λ - I3 = 0
By solving this cubic equation, we can find the values of λ, which are the principal stresses.
d) The absolute maximum shear stress can be calculated using the formula:
τmax = sqrt(((x - y)/2)^2 + ((y - z)/2)^2 + ((z - x)/2)^2 + τxy^2 + τxz^2 + τyz^2) = sqrt(((1 - 3)/2)^2 + ((3 - 5)/2)^2 + ((5 - 1)/2)^2 + 4^2 + 6^2 + 8^2) = sqrt((-1/2)^2 + (-1/2)^2 + (2/2)^2 + 16 + 36 + 64) = sqrt(1/4 + 1/4 + 1 + 16 + 36 + 64) = sqrt(117.5) ≈ 10.83MPa