Question

The stress matrix at a point relative to the xyz coordinate system is given by [25 [0] = | 10 115 10 0 0 15 1 0 | MPa –20] The axes of a new coordinate system XYZ is defined by three vectors X = 2i-2j+k, Y = -i-j, Z= i-j-4k, where i, j, k are the unit vectors along the x, y and z directions. Determine the stress matrix at the point in the XYZ coordinate system. Verify if the stress invariants in both coordinate systems are the same.

Please help with the following problem and please show all work, thank you.

Answer 1

the directional vector for X-axis is given by

2 -2.j+k (22 + 22 + 12) دهم تا ان +

the directional vector for Y-axis is given by

-i-j ↓(1 + 12 + 02) - |

the directional vector for Y-axis is given by

i-j-4k W(12 + 12 + 42) - 7

therefore the transformation matrix, T is given by

C-1-.... -- ej«- ----

given the stress tensor as

[0] = 25 10 15 10 0 0 15 0 -20

therefore the stress tensor/ matrix at the given point is given by

Lo] = [T] [0] [7] = 25 10 10 0 | 15 0 1510 0| -20|| 13

5.286 -14.714 -0.6066 -14.714 24.1421 -4.1074 -0.6066 - 4.1074 MPa -24.4281

Stress Invariants

for [o], 11 = 011 + 022 +033 = 25 +0 - 20 = 5M Pa

0 0 .25 15 25 022 023 032 033 -825M Pa 12= 011 013 011 012 031 033021 022 + +

25 10 13 = det(o) = 100 15 0 15 0 -20 = 2000

for [o]

I = σι1 + σ22 + σ33 = 5.286 + 24.1421 – 24.4281 = 5MPa.

I2 = 1-14.714 24.1421 - -825MPα 2 σ22 σ23 | σι1 σ13.σ11 σ12 σ32 σ33 |σ31 σ33 |σ21 σ22 | 24.1421 -4.1074T 5.286 -0.6066 -4.1074 -24.4281-0.6066 -24.4281 |-0.6066 -4.1074

13 = det(o') = 5.286 -14.714 -14.714 24.1421 -0.6066 -4.1074 -0.6066 -4.1074 -24.4281 = 2000

therefore the stress invariants are the same for both the coordinate systems.