ES 240
Solid Mechanics
Z. Suo
Homework
Due Friday, 20 October (except for Problem 16)
16. Recommend a textbook that you think will help students in this course
(Please post this entry on iMechanica by Wednesday, 25 October)
Even though the lectures and homework problems are reasonably self-contained, you
should always get familiar with a few textbooks on the subject. You have multiple resources to
help you locate textbooks: the books reserved in the library for the course, Amazon.com, and
word of mouth. Ultimately, you have to decide which books are good for you and others with
similar tastes. Please select one textbook, and post an entry in iMechanica. Please include the
following items.
 Title of the post. The title of the book and the names of authors (Example: Theory of


Elasticity by S.P. Timoshenko and J.N. Goodier).
Tags for the post. Use the following free tags: ES 240, solid mechanics, Fall 2006,
students, textbooks.
If there are already helpful reviews of the book on Amazon.com, please make a hyperlink
in your post to the web page of the book on Amazon.
Outline the content of the book.
Why do you think that this book complements this course?


Anything particular that makes you like this book and recommend it to others?
Feel free to write anything else about the book and its relation to you and to the course.


17. Yield conditions under the biaxial stress states
A metal has the yield strength  Y under the uniaxial stress state. Load a sheet of this metal
into a biaxial stress state, 1 in the x-direction and  2 in the y-direction. The two stress
components can have different magnitudes, and can be either tensile or compressive. All the
other stress components are zero.
(a) Obtain the yield condition according to the Mises law. Represent the yield condition on
the plane spanned by 1 and  2 .
(b) For this part of the problem, assume that the uniaxial yield strength is Y 100 MPa ,
and the applied loads are such that the two stress components are related as 1  22 ,
with 1 being compressive and  2 being tensile. According to the Mises law, at what
stresses 1 and  2 will the sheet yield?
18. Design a rotating disk to avert plastic deformation
The problem of the stresses in disks spinning at high speed is of vital importance in the
design of many machines. We now model a solid disk in the polar coordinates r, , and
calculate the stresses in the disk caused by the centrifugal force. The nonzero stress components
are  r and   . Recall the relations between the radial displacement u and the strains:
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ES 240
Solid Mechanics
u
r
   , r 
Z. Suo
du
.
dr
(a) Apply Newton’s second law to a differential element in the polar coordinates, and show that
d r  r   

   2 r ,
dr
r
where  is the angular velocity, and  the density.
(b) Assume the disk deforms elastically. Formulate a boundary value problem and determine the
stress field in the disk.
(c) If you want to design a steel disk to rotate at 3600 rotations per minute (rpm), what is the
2
largest radius that the disk can have? (   7800kg/m , E  210 GPa,  = 0.3,  Y  500MPa ).
19. A half space of an elastic material subject to a periodic traction on the surface.
An elastic material occupies the half space z  0 . The surface of the material, z  0 , is
subject to a traction in shear,  xz x, y, z    0 sin kx , where  0 is the amplitude and k the wave
number. Determine the stress field and displacement field inside the elastic material.
20. Orthotropy rescaling
When I was a graduate student, I noticed a relation, which I called orthotropy rescaling,
and published a paper (http://www.deas.harvard.edu/suo/papers/006.pdf). The basic idea is
outlined in this problem. For orthotropic materials, Hooke's law is
0
0   x 
  x   s11 s12 s13 0
  y   s21 s22 s23 0
0
0   y 
  z  s
0
0
0   z 
31 s32 s33
   
 
0
0
0 s44 0
0   yz
yz
  
 
 zx
0
0
0
0 s55 0  zx
  
 

0
0
0
0
0
s
 xy  
66   xy 
where sij are compliances.
i) Consider plane stress problems in the (x,y)-plane. Prove that the Airy stress function
(x,y) is governed by
 4
 4
 4

2




0
x 4
x 2 y 2
y 4
where
s11
2s  s
,   12 66
s22
2 s11 s22
ii) Argue that the governing equation is of the same form for plane strain problems but sij
in i) should be replaced by
 
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ES 240
Solid Mechanics
s'ij  sij 
Z. Suo
si3 s j3
s33
iii) For traction-prescribed plane problems on simply-connected domains, argue that the
in-plane stresses depend on elastic constants only through  and .
iv) Consider edge cracks (slits with no thickness) in an infinite strip subjected to a tensile
load (Fig. 12). Show that stress component x is independent of . (Hint. Examine the
governing equation and boundary condition in coordinates (, y), where  = 1/4x.)


y
x
Strip with edge cracks.
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