Me 530 assignment 2
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1. The displacement field for a body is given. The displaced position of a point originally at (1, 2, 3) is calculated. 2. The strain matrix and strain in a given direction are calculated for a given displacement field. 3. The strains are calculated at a point for a given displacement field in several directions.
![Assignment-2
ME 530 Advance Mechanics of Solids 01.09.2015
1. The displacement field for a body is given by 2 2 2 3 4
[( 2) (3 4 ) (2 4 ) ]10u x y i x y j x z k −
= + + + + + + .
What is the displaced position of a point originally at (1, 2, 3)?
2. If the displacement field for a body is given by xu kxy= , yu kxy= , 2 ( )zu k x y z= + , where k is
a constant small enough to ensure applicability of the small deformation theory,
(a) write down the strain matrix
(b) what is the strain in the direction
1
3
x y zn n n= = = ?
3. The displacement field for a body is given by 2
( 2 )xu k x z= + , 2
(4 2 )yu k x y z= + + , 2
4zu kz=
k is a very small constant. What are the strains at (2, 2, 3) in directions
(a)
1 1
0 , ,
2 2
x y zn n n= = =
(b) 1 , 0x y zn n n= = =
(c) 0 . 6 , 0 , 0 . 8x y zn n n= = =
4. The displacement field for a body is given by 2
( 2 )xu k x z= + , 2
(4 2 )yu k x y z= + + , 2
4zu kz=
with k = 0.001, determine the change in angle between two line segments PQ and PR at P (2,
2, 3) having direction cosines
deformation as
(a) PQ: 1 1 1
1
0 ,
2
x y zn n n= = =
PR: 2 2 21 , 0x y zn n n= = =
(b) PQ: 1 1 1
1
0 ,
2
x y zn n n= = =
PR: 2 2 20 .6 1, 0 , 0 .8x y zn n n= = =
5. State the conditions under which the following is a possible system of strains:
2 2 4 4
( )xx a b x y x yε = + + + + , 0yzγ =
2 2 4 4
( )yy x y x yε α β= + + + + , 0zxγ =
2 2 2
( )xy A Bxy x y cγ = + + − , 0zzε =](https://image.slidesharecdn.com/me530assignment2-150902115609-lva1-app6892/85/Me-530-assignment-2-1-320.jpg)
![6. A displacement field given as i ij ju Xλ= where ijλ form a set of constants, is called as
affine deformation. If
0.2 0.05 0.1
0.03 0.1 0.02
0.003 0.2 0.03
ijλ
− −
= −
−
,what is the displacement of a point
whose position vector from the fixed point in the undeformed geometry is ˆˆ ˆ 3r i j k= − +
7. Show that for affine deformations (see the problem 6):
(a) Plane sections remain plane during deformation.
(b) Straight lines remain straight lines during deformation.
8. Suppose that we have two successive small deformations represented by the displacement
gradient tensors at point P:
(1)
0.02 0.01 0
0 0.01 0.02
0 0 0.02
i
j p
u
x
∂
= − ∂ −
and
(2)
0.01 0.015 0.02
0 0 0.01
0 0.03 0.04
i
j p
u
x
−
∂
= − ∂ −
. What is the total change
at point P of vector ∆s given by 3ˆˆ ˆ(6 10 2 ) 10s i j k −
∆ = + + × ?
9. For the following small displacement field 2 2 2 3ˆˆ ˆ[(3 ) (3 ) 2 ] 10u x y i y z j z k −
= + + + + × ft, what are
strain xzε and zzε at (2, 1, 3)? What is the rotation at this point about the y axis?
10. A body element undergoes a small rotation ω given as follows:
ˆˆ ˆ0.0002 0.0005 0.0003i j kω = + − rad. What is the rotation tensor ijΩ at this point?
11. Given the small displacement field 3ˆˆ ˆ[(6 5 ) ( 6 3 ) ( 5 3 ) ] 10u y z i x z j x y k −
= + + − + + − − × ft, show
that this field is that of pure rigid-body rotation. What is the rotation vector ω for the body?
12. The following state of strain exists at a point in a body
0.01 0.02 0
0.02 0.03 0.01
0 0.01 0
ijε
−
= − −
−
In a direction ˆν having the direction cosines 1 0.6ν = , 2 0ν = , 3 0.8ν = , what is ννε ?](https://image.slidesharecdn.com/me530assignment2-150902115609-lva1-app6892/85/Me-530-assignment-2-2-320.jpg)
Me 530 assignment 2
- 1. Assignment-2 ME 530 Advance Mechanics of Solids 01.09.2015 1. The displacement field for a body is given by 2 2 2 3 4 [( 2) (3 4 ) (2 4 ) ]10u x y i x y j x z k − = + + + + + + . What is the displaced position of a point originally at (1, 2, 3)? 2. If the displacement field for a body is given by xu kxy= , yu kxy= , 2 ( )zu k x y z= + , where k is a constant small enough to ensure applicability of the small deformation theory, (a) write down the strain matrix (b) what is the strain in the direction 1 3 x y zn n n= = = ? 3. The displacement field for a body is given by 2 ( 2 )xu k x z= + , 2 (4 2 )yu k x y z= + + , 2 4zu kz= k is a very small constant. What are the strains at (2, 2, 3) in directions (a) 1 1 0 , , 2 2 x y zn n n= = = (b) 1 , 0x y zn n n= = = (c) 0 . 6 , 0 , 0 . 8x y zn n n= = = 4. The displacement field for a body is given by 2 ( 2 )xu k x z= + , 2 (4 2 )yu k x y z= + + , 2 4zu kz= with k = 0.001, determine the change in angle between two line segments PQ and PR at P (2, 2, 3) having direction cosines deformation as (a) PQ: 1 1 1 1 0 , 2 x y zn n n= = = PR: 2 2 21 , 0x y zn n n= = = (b) PQ: 1 1 1 1 0 , 2 x y zn n n= = = PR: 2 2 20 .6 1, 0 , 0 .8x y zn n n= = = 5. State the conditions under which the following is a possible system of strains: 2 2 4 4 ( )xx a b x y x yε = + + + + , 0yzγ = 2 2 4 4 ( )yy x y x yε α β= + + + + , 0zxγ = 2 2 2 ( )xy A Bxy x y cγ = + + − , 0zzε =
- 2. 6. A displacement field given as i ij ju Xλ= where ijλ form a set of constants, is called as affine deformation. If 0.2 0.05 0.1 0.03 0.1 0.02 0.003 0.2 0.03 ijλ − − = − − ,what is the displacement of a point whose position vector from the fixed point in the undeformed geometry is ˆˆ ˆ 3r i j k= − + 7. Show that for affine deformations (see the problem 6): (a) Plane sections remain plane during deformation. (b) Straight lines remain straight lines during deformation. 8. Suppose that we have two successive small deformations represented by the displacement gradient tensors at point P: (1) 0.02 0.01 0 0 0.01 0.02 0 0 0.02 i j p u x ∂ = − ∂ − and (2) 0.01 0.015 0.02 0 0 0.01 0 0.03 0.04 i j p u x − ∂ = − ∂ − . What is the total change at point P of vector ∆s given by 3ˆˆ ˆ(6 10 2 ) 10s i j k − ∆ = + + × ? 9. For the following small displacement field 2 2 2 3ˆˆ ˆ[(3 ) (3 ) 2 ] 10u x y i y z j z k − = + + + + × ft, what are strain xzε and zzε at (2, 1, 3)? What is the rotation at this point about the y axis? 10. A body element undergoes a small rotation ω given as follows: ˆˆ ˆ0.0002 0.0005 0.0003i j kω = + − rad. What is the rotation tensor ijΩ at this point? 11. Given the small displacement field 3ˆˆ ˆ[(6 5 ) ( 6 3 ) ( 5 3 ) ] 10u y z i x z j x y k − = + + − + + − − × ft, show that this field is that of pure rigid-body rotation. What is the rotation vector ω for the body? 12. The following state of strain exists at a point in a body 0.01 0.02 0 0.02 0.03 0.01 0 0.01 0 ijε − = − − − In a direction ˆν having the direction cosines 1 0.6ν = , 2 0ν = , 3 0.8ν = , what is ννε ?