me601_asst5.pdf

ME 601 Homework Assignment 5 – Due Date: 26 th August 2015 1. For a linear isotropic material prove that a) !! = (3 + 2) !! and b) !" = ! !! !" − ! !!!!! !! !" , Where = !" !!! (!!!!) . 2. A rectangular steel plate of thickness 4mm as shown in fig is subjected to uniform biaxial stress field assuming all fields are uniform, determine the changes in the dimensions of the plate under this loading.(E=207GPa) 3. An elastic layer is sandwiched between two perfectly rigid plates, to which it is bonded. The layer is compressed between the plates, the compressive stress being ! . Supposing that attachment to plates prevents lateral strains ! & ! completely, find the apparent Young’s modulus (i.e. ! / ! ) in terms of E & . 4. (a) From generalized Hooke’s law for linear isotropic elastic solids 2 ij kk ij ij e e σ λ δ μ = + , show that, 1 ( ) 2 3 2 ij ij kk ij e λ σ σδ μ λ μ = − + (b) Show that Hooke’s law for an isotropic material may be expressed in terms of spherical and deviatoric tensors by the two relations ˆ ˆ 3 , 3 ij ij ij ij ke ke σ σ = = % % 5. (a) If the elastic constants E, k, and m are required to be positive, show that Poisson’s ratio must satisfy the inequality 1 0.5 υ − < < . For most real materials it has been found that 0 0.5 υ < < .Show that this more restrictive inequality in this problem implies that 0 λ > . (b) Under the condition that E is positive and bounded, determine the elastic moduli, , and k λ μ for the special cases of Poisson’s ratio: 0,1/4,1/2 υ = 6. Show for an isotropic elastic medium that (a) 1 2( ) 1 3 2 λ μ υ λ μ + = + + , (b) 1 2 υ λ υ λ μ = − + , (c) 2 3 1 2 1 K μυ υ υ υ = − + 7. For the general threedimensional thermoelastic problem with no body forces, explicitly develop the BeltramiMichell compatibility equations, , , , , 1 1 ( ) 0 1 1 1 ij kk kk ij ij ij kk E T T α υ σ σ δ υ υ υ + + + + = + + −

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ME 601 -‐ Homework Assignment 5 – Due Date: 26th August 2015

1. For a linear isotropic material prove that a) 𝜎!! = (3𝜆 + 2𝜇)𝜀!! and

b) 𝜀!" =!!!

𝜎!" −!

!!!!!𝜎!!𝛿!" , Where 𝜆 = !"

!!! (!!!!) .

2. A rectangular steel plate of thickness 4mm as shown in fig is subjected to uniform biaxial stress field assuming all fields are uniform, determine the changes in the dimensions of the plate under this loading.(E=207GPa)

3. An elastic layer is sandwiched between two perfectly rigid plates, to which it is bonded.

The layer is compressed between the plates, the compressive stress being𝜎!. Supposing that attachment to plates prevents lateral strains 𝜀! & 𝜀! completely, find the apparent Young’s modulus (i.e. 𝜎!/𝜀! ) in terms of E & 𝜗.

4. (a) From generalized Hooke’s law for linear isotropic elastic solids 2ij kk ij ije eσ λ δ µ= + ,

show that, 1 ( )2 3 2ij ij kk ije λ

σ σ δµ λ µ

= −+

(b) Show that Hooke’s law for an isotropic material may be expressed in terms of spherical and deviatoric tensors by the two relations ˆ ˆ3 , 3ij ij ij ijke keσ σ= =% %

5. (a) If the elastic constants E, k, and m are required to be positive, show that Poisson’s ratio must satisfy the inequality 1 0.5υ− < < . For most real materials it has been found that 0 0.5υ< < .Show that this more restrictive inequality in this problem implies that

0λ > . (b) Under the condition that E is positive and bounded, determine the elastic moduli, , and kλ µ for the special cases of Poisson’s ratio: 0,1/ 4,1/ 2υ =

6. Show for an isotropic elastic medium that

(a) 1 2( )1 3 2

λ µυ λ µ

+ +, (b)

1 2υ λυ λ µ=

− +, (c) 2 3

1 2 1Kµυ υ

υ υ=

− +

7. For the general three-‐dimensional thermo-‐elastic problem with no body forces, explicitly develop the Beltrami-‐Michell compatibility equations,

, , , ,1 1( ) 01 1 1ij kk kk ij ij ij kk

E T Tα υσ σ δ