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I need answers for these 5 questions that i postedScreen Shot 2019-10-02</h1><p class='abPRemoveTitle'> at 2.44.07 PM.pngScreen Shot 2019-10-02 at 2.43.51 PM.pngScreen Shot 2019-10-02 at 2.43.40 PM.png

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The first 3 problems are the same three problems of the sample MT#1 posted. 1) 2) (Strain) (l6pt) Start with a given coordinate axes x1 — x2 — x3 , the strain tensor at a given point P in a elastic solid is
0.004 0.001 0 4 1 0 [SI-j]: ? 0.006 ? = ? 6 ? x10‘3, find,a)—d) 4pt each:
? 0.004 0.001 ? 4 1 the eigenvalues and corresponding normalized eigenvectors, using Cardan’s
formula
the 3 principal strains 81 &gt; 82 &gt; £3 , the orientation of the principal axis of £1 (w.r.t. x1 — x2 — x3 ), the octahedral normal so“ and shearing yo“ strains in the octahedral plane. if, without given [80.] , only 31, Em and you are known, can we find 32 and 33?
How? (14pt) (Strain Compatibility, Stress vs Strain) In a region with a Cartesian coordinate system (19,1:2 ,x3) , the small strain components are given by _ 2 _ 2 _ 2
611—351 922—352 333—35 312 = xlxz 323 = x2x3 631 = x1x3 a) (Spt) If E = 32,000ksi and v=0.25, , find the Lamé constants x1 and y , and the corresponding components of stress.
b) (5pt) Determine if the strains compatible in the region by checking ALL six
compatible equations.

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3) (10pt) (Tensor)
Given two vectors SE and j? (w.r.t. x1 — x2 — x3 ). In a rotated coordinate system 3:; — x; — x; , the same two vectors take the form i't' and j&quot; . Find out if the scalar dot an, iv, . products 33- 33 and x - y 1n the two coordinate systems are the same. If they are the
same, prove it. If not, give a counter—example.

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4)
L09: (20pt)
You learned in your sophomore year the Generalised Hooke's Law as
1
Ex
E
(0, - V(0, to2))
Kxx 3
2G
Xx 2
E
(0) - V(ox to2) )
1
E yz
2G
2 yz
E
(0, - V(0 * + 0 ,))
1
E zx
2G
.T zx
a) (10pt) Write this in matrix form (a 6 X 6 matrix), using the X - X2 - X3
coordinate system, expressing the strains in terms of the stresses.
b) (10t) Express the normal stresses -11 , 22, 33 in terms of the normal strains
Ell, E22 and 833 , and similarly the shear stresses in terms of the shear strains. I
n other words, find the inverse relations.
5)
L10: (10pt)
Express the shear modulus G (or u) in terms of E and v, Young's Modulus and Poisson's

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