CEE 213—Deformable Solids
Contents.
CEE 213—Deformable Solids
Hooke’s Law
Linear elastic constitutive equations

E

Keith D. Hjelmstad
The Mechanics Project
Arizona State University
© Keith D. Hjelmstad
1.
2.
3.
4.
5.
6.
7.
8.
9.
Constitutive models.
Dilatation.
Stress and strain in three dimensions.
Stress in terms of strain.
Strain in terms of stress.
How does the matrix equation work?
Plane stress.
Plane strain.
Summary.
CEE 213—Deformable Solids
Hooke’s Law
Constitutive models. To model the behavior of materials it
turns out that a very useful approach is to write relationships
between stress at a point and strain at a point. At this
fundamental level we can successfully write “universal” laws
that govern the mechanical behavior of certain classes of
materials. In general terms we write either (1) stress is a
function of strain or (2) strain is a function of stress.
Constitutive Models

E

S  f ( E)

G

Isotropic means that the
material looks the same in all
directions. A counterexample is
wood that has an oriented grain
structure. Steel is generally
view as isotropic.
Linear simply means loading
and unloading happens along a
straight line (as above).
© Keith D. Hjelmstad
E  g (S)
A constitutive model is a relationship between stress and
strain. This is the third leg of the mechanics stool—force is
related to stress through equilibrium and displacement is
related to strain through kinematics.
One such law, which will be our focus here is Hooke’s Law,
which models isotropic, linear elasticity.
CEE 213—Deformable Solids
Hooke’ Law
Dilatation. If the deformation of a solid body involves pure
expansion in all directions but no shearing, we call the
deformation dilatation. Considering the sketch at left, we can
see that a change in volume involves a change in length of
each side. Because the normal strains, xx, yy, and zz, give the
ratio of the change in length per original length in those
directions we can write the volume of the deformed cuboid as
Dilatation
e2
e3
e1
a  xx
c  zz
b  yy
V  a 1   xx  b 1   yy  c 1   zz 
 abc    xx   yy   zz  abc  higher order terms
 Vo    xx   yy   zz Vo
 1  e Vo
where we have defined
e  tr(ε)   xx   yy   zz
Dilatation is pure expansion of
a body. Change in volume per
original volume is the sum of
the normal strains.
© Keith D. Hjelmstad
Note that the notation tr(E) stands for “add up the diagonal
terms” of the tensor (matrix) E. Rearranging the above
equation shows that change in volume per original volume is
V
e
Vo
CEE 213—Deformable Solids
Hooke’s Law
Stress and strain in three dimensions
e2
e3
 yy
 yx
 yz
 zy
 zz
 zx
e1
 xy
 xz
 xx
To satisfy balance of moments the
stress tensor must be symmetric.
We take the strain tensor to be
symmetric, too. Hence,
© Keith D. Hjelmstad
 xy   xy
 xy   xy
 zx   xz
 zy   yz
 zx   xz
 zy   yz
Stress and strain in three dimensions. If the state of stress
and strain is fully three dimensional then we write the stress
and strain tensors as
 xx  xy  xz 
S   yx  yy  yz 


 zx  zx  zz 
Stress
 xx

E   yx
 zx

Strain
 xy  xz 

 yy  yz 
 zy  zz 
The convention for the components are shown in the sketch. If
you look across a row of the stress matrix those quantities are
the components of the traction vector on one face (the face
associated with the axis of the normal vector). For a general
state of stress or strain any or all of these components can be
non-zero.
Among the interesting invariants the “trace” (the sum of the
diagonal components) is important to Hooke’s Law. We
designate these invariants as
tr(E)  e   xx   yy   zz
tr(S)  p   xx   yy   zz
CEE 213—Deformable Solids
Hooke’s Law
Stress in terms of strain. Hooke’s Law is a relationship
between stress and strain. The general relationship can be
written in the form of stress as a function of strain as
Stress in terms of strain
e2
 yy
 yx
 yz
 zy
 zz
 zx
 xz
e3
S
e1
where
 xy
 xx
E
E
tr(E) I 
E
(1   )(1  2 )
1 
tr(E)  e   xx   yy   zz
and where E is Young’s modulus and  is Poisson’s Ratio.
These constants are material properties, which can be
determined by laboratory tests (or you can often look them up
in a handbook). The specific form of this relationship is
important because it includes all of the special cases (i.e., E is
the constant of proportionality between axial stress and axial
strain in a uniaxial tension test and shear stress is proportional
to shear strain in torsion).
Poisson’s effect is where a body in tension contracts laterally.
As shown below. Hooke’s law incorporates this behavior.
1
 xx
 xx
 yy
1
 xx
© Keith D. Hjelmstad
CEE 213—Deformable Solids
Hooke’s Law
Strain in terms of stress. We can put Hooke’s Law into a
form that gives strain as a function of stress. The easiest way
to do that is to take the “trace” of both sides of the previous
equation to get an expression for tr(E) in terms of tr(S). Then
substitute back and rearrange terms to get:
Strain in terms of stress
e2
e3
 yy
 yx
 yz
 zy
 zx
 zz
e1
 xy
 xz

tr(S) tr(I ) 
E
1  2

tr(S)
E
 xx
1 
tr(S)
E
Thus,
© Keith D. Hjelmstad
E
tr(E)
1  2
E
tr(S) I 
1 
S
E
tr(S)   xx   yy   zz
Again, the equations involve E (Young’s modulus) and 
(Poisson’s ratio). It is important to note that these two forms of
Hooke’s Law are completely equivalent. They are simply an
algebraic manipulation of the equations. To get back to the
form on the previous page compute tr(S) and substitute back
in…
 E
 1 
tr(E)  I 
E 
S
E  1  2
E

Rearrange to get
S
tr(S) 

where the trace of the stress tensor is
Take the trace of both sides…
tr(E)  
E
E
1 

E
E


E
tr(E) I  
tr(E) I 
E 
1  2
1 

 1   1  2 
Which is exactly what we have on the previous page.
CEE 213—Deformable Solids
Hooke’s L aw
How does the matrix equation work?
Stress vs. strain version
of Hooke’s Law
S
How does the matrix equation work? It is a good idea to
take a look under the hood of the constitutive equation. It is a
matrix relationship and each one of the nine elements actually
gives a single scalar relationship. Let’s see how that works…
E
E
tr(E) I 
E
(1   )(1  2 )
1 
 xx  xy  xz 
 e 0 0
E



0 e 0   E




xy
yy
yz

 (1   )(1  2 ) 
 1 
 xz  yz  zz 
0 0 e 


E
 yz 
 yz
1 
 xx 
E
E
 xx   yy   zz  
 xx

(1   )(1  2 )
1 
Normal stress vs. normal strain. Notice how these
equations are couple by the e term. This gives rise to
Poisson’s effect.
© Keith D. Hjelmstad
e  tr(E)   xx   yy   zz
 xx

 xy
 xz

 xy  xz 

 yy  yz 
 yz  zz 
Shear stress vs. shear strain. Notice that
these equations are not coupled because the
e term is not involved (off diagonal). Note
also that the shear modulus is defined as
G
E
2 1   
So we can also write this equation as
 yz  2G  yz  G yz
CEE 213—Deformable Solids
Hooke’s L aw
Plane stress. Plane stress is a condition in which there are no
tractions (and hence no stresses) on two opposing faces (the
ones with normal in the z direction in this case). Just plug in to
find zz in terms of xx and yy .
Plane stress
 yy
 xx  xy

 yy
 xy
 0
0
 xx
 zz 
Hooke’s Law for the special
case of plane stress:
 zz   xz   yz  0
E
  xx    yy 
1  2
E
 yy 
   xx   yy 
1  2
E
 xy 
 xy  2G  xy
1 
 xy
 yy
E
E
 xx   yy   zz  
 zz

(1   )(1  2 )
1 
0

E
E
E

 xx   yy   

(1   )(1  2 )
 (1   )(1  2 ) 1  
0
 E (1   ) 
E
 xx   yy   

  zz
(1   )(1  2 )
(1
)(1
2
)







  zz

Solve for zz
 xx 
© Keith D. Hjelmstad
 xx

 xy
 0
0
 e 0 0
E
0 e 0   E
0 
 (1   )(1  2 ) 
 1 
0 0 e 
0 
tr(E) 
1  2
 xx   yy 
1 
Substitute zz in 3D equations
to get plane stress equations.
 zz  

1 

xx
  yy 
The out-of-plane strain
is not zero!
0
0
0

 zz 
CEE 213—Deformable Solids
Hooke’s L aw
Plane strain. Plane strain is a condition in which there are
no strains on two opposing faces (the ones with normal in the
z direction in this case). Just plug in to find zz in terms of xx
and yy .
Plane strain
 yy
 xx
 xx

 xy
 0
 xy 0 

 yy 0   
0
p
0
E
 0
0 
0
p
0
0
1 
0 

E
p 
 xx  xy 0 

 yy 0 
 xy
 0
0  zz 
 zz
1 
 zz
E

  1  
0    xx   yy     
 zz
E
E
E


 zz  
Hooke’s Law for the special
case of plane strain:
 zz   xz   yz  0
1 
1   xx   yy 
E
1 
 yy 
 xx  1   yy 
E
1 
 xy 
 xy
E
0


E
xx


E
xx
  yy   zz  
1
  yy     zz
E
Solve for zz
 xx 
© Keith D. Hjelmstad
tr(S)  1     xx   yy 
Substitute zz in 3D equations
to get plane stress equations.
 zz     xx   yy 
The out-of-plane stress
is not zero!
CEE 213—Deformable Solids
Hooke’s Law
Summary
Robert Hooke (1635 – 1703)
Summary. Behavior of materials is a huge and important part
of solid mechanics. All sorts of behaviors have been observed
in material testing. Among those models that have been
formulated for deformable solids we find:
1.
Elasticity. Hooke’s law, but also for cases that are not
isotropic. Also, nonlinear elasticity where the body
returns to its original shape upon unloading but the
response is not linear.
2.
Plasticity. Models that capture yielding of material that
prevents the body from returning to its original state upon
unloading.
3.
Visco-elasticity or visco-plasticity. Like the two above
only with time-dependent (or viscous) effects.
and many others.
This set of notes has presented only the simplest case of material behavior (isotropic, linear elasticity). This
model is called Hooke’s Law (named after Robert Hooke a famous contemporary of Isaac Newton). Hooke’s
Law isn’t really a “law” of nature, but rather a model that describes the behavior of certain materials very
well. The model incorporates the important observation that a bar stretched in tension will contract laterally
(Poisson’s Effect) as well as the normal straining caused by normal stresses and shear straining caused by
shear stresses. This model was implicit in the constitutive models used for the bar in axial tension, the bar in
torsion, and the beam in flexure. With Hooke’s Law in three dimensions we can relate multiaxial states of
stress to multiaxial states of strain.
© Keith D. Hjelmstad