Download 7. Finite Deformation

ES240
Solid Mechanics
Fall 2011
7. Finite Deformation
References
G.A. Holzapfel, Nonlinear Solid Mechanics, Wiley, 2000.
T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures,
Wiley, 2000.
L.E. Malvern, Introduction to the Mechanics of Continuous Medium, Prentice-Hall, 1969.
C. Trusdell and W. Noll, The Non-linear Field Theories of Mechanics, 3rd edition, Springer,
2004.
C. Trusdell and R.A. Toupin, The Classical Field Theories, in Encyclopedia of Physics, Volume
III/1, Pringer-Verlag, Berlin, 226-793 (1960).
L.R.G. Treloar, The Physics of Rubber Elasticity, 3rd ed. 1975. Reissued in 2005.
Be wise, linearize. Following the advice of George Carrier, we have been mostly looking at
Hookean materials and infinitesimal deformation. We have
mixed the 3 ingredients in solid mechanics (deformation
geometry, momentum balance, material law) without fussing
over subtleties. The results are fascinating and useful. Now we
wish to go nonlinear, hopefully also with wisdom. We will
refine the ingredients by considering non-Hookean materials
and finite deformation. The two refinements need not be mixed.
For example, a viscoelastic material is non-Hookean, but
deformation of such a material can be infinitesimal. In this brief
introduction to finite deformation, we will outline some of the
fundamental considerations, and describe a few illustrative phenomena.
Finite deformation. When a structure deforms, Newton’s law holds true in every deformed state.
We have often violated this law. For example, in analyzing a truss, we have balanced forces as if
the truss did not deform.
You might think that a structure suffering a small strain, say less than 1%, entitles you to neglect
the change in geometry when you balance forces. A counter example is familiar to you. Upon
buckling, the strains in a column are indeed small, but you must enforce equilibrium in the
deflected state of the column. Mechanics of deformation is a tricky business. We proceed with
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caution. The essential point is this: we must
enforce Newton’s law in every deformed state,
and justify any simplification on this basis.
Non-Hookean materials. Moving nonlinear and
inelastic will take us in many directions. For
example, when deformation is large, the force
may vary nonlinearly with the elongation. As
another example, we’ve already looked at timedependent behavior of materials, such as
viscoelasticity. we also have daily experience of metals. After elastic deformation, upon
unloading, a metal recovers its shape. After plastic deformation, upon unloading, the metal does
not fully recover its shape. If we apply an axial force to a metal bar, and measure its length, the
force-length relation is linear for elastic deformation, and is nonlinear for plastic deformation.
During unloading, the metal bar deforms elastically. After plastic loading and elastic unloading,
the force-length relation is not a one-to-one relation, but is history-dependent. Of course, a
viscoelastic material is also history-dependent.
To analyze finite, history-dependent deformation of a structure, a general approach is to evolve
the state incrementally, and enforce Newton’s law in every state.
A rod under axial load. We also proceed with our subject incrementally, beginning with a rod
in incremental states of uniaxial stress. Initially, the rod is unstressed, and has cross-sectional
area A and length L . The rod is then subject to an axial force P, and deforms to cross-sectional
area a and length l. We next examine the 3 ingredients in solid mechanics.
Strain measures. Any state can be used as a reference state. For example, we can take the
initial, unstressed state as the reference state. Define the engineering strain by the elongation
divided by the reference length:
e
elongation
lL
.

length in the reference state
L
Another strain measure is defined as follows. Deform the material from a current length l by a
small amount to l  dl . Define the increment in the strain, d , as the increment in the length of
the rod divided by the current length of the rod, namely,
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d 
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increment in length
dl
 .
length in the current state
l
This equation defines the increment of natural strain. Integrating from L to l, we obtain that
l
L
  log   .
Yet one more strain measure, the Lagrange strain, is defined as
l 2  L2
.

2L2
This definition is hard to motivate in 1D. But if you take the view that any increasing function of
l / L is a suitable measure of strain, then no motivation is really needed.
Indeed, even the ratio l / L itself has a name: the stretch is defined as the length of the rod in the
current state divided by the length of the rod in the reference state:

length in current state
l
 .
length in reference state L
There seems to be no lack of human ingenuity to form a dimensionless quantity out of two
lengths L and l. Needless to say, all these strain measures contain the same information. For
example, every one of the measures defined above is an increasing function of stretch:
e    1,   log  ,  
2  1
2
.
Because they are all one-to-one functions, any one measure can be taken to be “basic” and then
used to express all the rest. For example, we can express all measures in terms of the
engineering strain:
  e  1,   log 1  e ,   e 2  e
1
2
Thus, when we call e the engineering strain, we do not mean that e is unscientific or crude or
unnatural strain. We just need a name. When the strain is small, namely, e  1 , the three
measures are approximately equal,     e .
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Later on, we will provide motivations for some of these definitions, but these motivations are
probably elaborate ways to express preferences of individual people.
Stress measures. Work done by a force. When dealing with finite deformation, we must be
specific about the area used in defining the stress. Define the nominal stress, s, as the force in
the current state divided by the area in the reference state:
s
force in the current state
P
 .
area in the reference state A
When the rod elongates from length l to length l  dl , the force P does work Pdl . Recall that
P  sA and dl  Lde , so that the work done by the force is
Pdl  ALsde .
Since AL is the volume of the rod in the reference state, we note that
sde 
increment of work in the current state
.
volume in the reference state
We say that the nominal stress and the engineering strain are work-conjugate. Also note that
de  d . Consequently, the stretch is also work-conjugate to the nominal stress.
Define the true stress,  , as the force in the current state divided by the area in the current state,
namely,

force in the current state P
 .
area in the current state
a
Recall that dl  ld . The work done by the force is
Pdl  ald .
Since al is the current volume of the bar, we note that
d 
increment of work in the current state
.
volume in the current state
That is, the true stress is work-conjugate to the natural strain.
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Given a measure of strain, we can define its work-conjugate stress. For example, consider the
Lagrange strain,  . Subject to an increment in the strain, d , the force acting on the element
does the work. Denote
S 
increment of work in the curent state
.
volume in the reference state
This expression defines a new measure of stress, S.
This new stress measure does not have a “simpler” interpretation than its status as the work
conjugate to the Lagrange strain. Indeed, if we are liberal about the definition of strain measures,
without being obsessive about “motivating” each measure, we may as well take a liberal view to
call the work conjugate of each strain a stress measure, and name the stress measure after a
mechanician who can no longer protest. You can easily invent and name other stress measures,
but the above stress measures have already got names:
  : true stress or the Cauchy stress.
 s : nominal stress or the first Piola-Kirchhoff stress.
 S : the second Piola-Kirchhoff stress.
Recall the relations among the measures of strain:
e    1,   log  ,  
2  1
2
We obtain the relations among their increments:
de  d , d 
d

, d  d .
Consequently, the three measures of stress are related as

V 2
 S , s  S .
v
All stress measures are linear in force applied to the cross-sectional area, but differ by geometric
factors.
Material laws. For a metal undergoing large, plastic deformation, the stress-strain curve
(without unloading) is often fit to a power law in terms of the true stress and the natural strain:
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    K N
where K and  are parameters to fit experimental data. Some representative values: N = 0.150.25 for aluminum, N = 0.3-0.35 for copper, N = 0.45-0.55 for stainless steel. K has the
dimension of stress; it represents the true stress at strain   1. Representative values for K are
100 MPa – 1GPa. At large deformation, volumetric strain is negligible compared to tensile
strain. Consequently, the material is often taken to be incompressible.
Rubbers are often assumed to obey the neo-Hookean law (more details later). For a rubber rod in
uniaxial states of stress, the stress-strain data are fit to
     2  1 .
Recall that   1  e . For small strains, e  1 , the above reduces to
  3e .
Thus, we interpret 3 as Young’s modulus and  the shear modulus. Rubbers are nearly
incompressible, so that Poisson’s ratio is taken to be ½. Representative values for  are 1 MPa
– 100 MPa.
Are these alternatives necessary? Now we have described the 3 ingredients for a rod under
uniaxial tension. Even in this simplest setting, for each ingredient we have given several
alternative descriptions. Some alternatives are necessary; for example, metals and rubbers
behave differently. But the difference in their force-displacement relations does not justify us to
use different stress and strain measures to describe different materials. In fact, to see the
difference in material behavior, we would like to use the same stress and strain measures for both
materials. For example, we can use the natural strain to describe the stress-strain relation for
rubbers:
    exp 2   exp   .
This change of variable immediately brings out a key insight: in tension, the stress in rubbers
rises more steeply than in metals. We will return to this insight shortly.
Are these alternative stress and strain measures necessary? I have my own thoughts, but you
should form your own opinions. The question perhaps boil down to something no more profound
than asking, “Is it necessary to know many alternative roads to Boston Common?” Whatever
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your opinions are, however, it may alleviate some of your pains in studying the subject by
knowing that textbooks of nonlinear continuum mechanics are full of equivalent alternatives at
every turn. These alternatives often hide behind forests of notation and verbiage, and may offer
some tantalizing sights. You will just have to look beyond them for matters of consequence.
Exercise. Use the 3 ingredients outlined about to obtain the force-deflection relation for the truss
sketched in the beginning of the notes. Assuming all three members of the truss are made of
rubber bands, and that deformation is large.
Necking in a bar. Considère condition. Let us try to apply the newly refined 3 ingredients to a
specific phenomenon: necking. Subject to a tensile force, a metal bar first elongates uniformly
and then, at some strain, a small part of the rod starts to thin down preferentially, forming a neck.
By contrast, a rubber band under tension usually does not form a neck. We would like to
interpret these observations. To do so we must be explicitly specify the measures of stress and
strain.
Here is a summary of the 3 ingredients, using a specific set of alternatives:
 Force balance: P  a
 Material law: For a metal bar under uniaxial tension, the true stress relates to the natural
strain as      .
 Deformation geometry:   log l / L . We will assume that the volume of the rod is
constant during deformation, AL  al , or a  A exp    .
Put the three ingredients together, and we obtain the force as a function of strain:
P  a  A  exp   .
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Plot P as a function of . In plotting the figure, I’ve set   K N , with N = 0.5. Observe the two
competing factors: material hardening and geometric softening. As the bar elongates, the material
hardens, as reflected by the hardening exponent in the stress-strain relation      . At the
same time, the elongation reduces the cross-sectional area, an effect known as geometric
softening. For small deformation, P  0 as   0 ; material hardening prevails, and the force
increases as the bar elongates. For large deformation, so long as the stress-strain relation   
increases slower than exp   , P  0 as    , geometric softening prevails, and the force
drops as the bar elongates.
To determine the peak force, note that
dP
 

 A exp   
 .
d
 

Consequently, the force P peaks when the true stress equals the tangent modulus:
d  
    .
d
This equation, known as the Considère condition, determines the strain at which the force peaks.
For the power-law material, the force peaks at the critical strain
c  N .
When the metal bar is loaded beyond this critical strain, deformation becomes nonuniform, with
a segment of the bar elongates at a higher strain than the rest of the bar. That is, a neck forms in
the bar. For an analysis of the nonuniform deformation, see Needleman (1972, A numerical
study of necking in circular cylindrical bar, Journal of the Mechanics and Physics of Solids, 20,
111). ABAQUS can be used to study the necking process.
For a rubber band, assume the material is Neo-Hookean:
   2  1    exp 2   exp   .
Thus, at a large tensile strain, the true stress increases exponentially with the natural strain, so
that the force P  A  exp    always increases with the strain. The rubber band will not
form a neck under uniaxial tension.
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Hyperelastic materials. We next explore nonlinear stress-strain relations under multiaxial
states of stress. Of course, the only way to really know such relations is to run tests for a given
history of state of stress, but this would be too time-consuming and quickly become impractical.
We’ll have to reduce the number of tests by some approximations. The art of making such
compromise between accuracy and labor is known as formulating constitutive laws. As an
example, here we attempt to describe this art for rubbers.
A rubber rod, length L and cross-sectional area A in the unstressed state, is stretched by force P to
length l and cross-sectional area a. When the rod extends from length l to length l  dl , the
force does work Pdl . Recall that P  sA and dl  Ld , so that the work is
Pdl  ALsd .
Since AL is the volume of the rod in the undeformed state, we note that
sd 
increment of work in the current state
.
volume of the reference state
We say that the nominal stress is work-conjugate to the stretch. Define the nominal energy
density by
W 
energy in the current state
.
volume of the reference state
Assuming that the work done by the force is fully stored as energy in the rod, we obtain that
dW  sd .
Note that W should be considered the Helmholtz free energy. We’ll only consider the isothermal
conditions, so that that we’ll drop the dependence on temperature.
We can measure experimentally the nominal stress as a function of stretch, s  , and then
integrate the curve s  to obtain W   , known as the strain-energy function. Alternatively, we
can obtain an expression of W   from some theoretical considerations, and then obtain s  by
s   
dW  
.
d
For an isotropic material, we can represent a material particle by a rectangular block cut in the
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orientation of the three principal stresses. Let the block be stretched in the three directions by 1,
 2 and  3, and the corresponding nominal stresses be s1, s2 and s3. The complete stress-strain
relations involve three functions s1 1 , 2 , 3  , s 2 1 , 2 , 3  and s3 1 , 2 , 3  . In general, we
can run tests to determine these functions. However, as we indicated above, running test alone
would be too time-consuming. Instead, we will formulate a constitutive law on the basis of the
assumption that work done by the forces is all stored in the block as energy.
When the stretches increase by d1 , d2 and d3 , following the same line of reasoning as given
above, we conclude that the free energy changes by
dW  s1d1  s2 d2  s3 d3 .
Now the strain energy is a function of the three stretches, W 1 , 2 , 3  . Once the function
W 1 , 2 , 3  is determined by experiment, say, the stresses are calculated from the partial
derivatives:
s1 
W 1 , 2 , 3 
W 1 , 2 , 3 
W 1 , 2 , 3 
.
, s2 
, s3 
1
2
3
An elastic material whose stress-strain relation is derivable from a strain-energy function is
known as a hyperelastic material. An elastic material whose stress-strain relation is not
derivable from a strain-energy function is called a hypoelastic material.
A rectangular block of a material, lengths L1 , L2 and L3 in the undeformed state, is subject to
forces P1 , P2 and P3 on its faces, and is deformed into a block of lengths l1 , l 2 and l 3 . The
nominal stress on one face is
s1 
P1
.
L2 L3
The true stress on the same face is
1 
P1
.
l 2 l3
Consequently, the true stress relates to the nominal stress by
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1 
s1
 2 3
Fall 2011
.
The true stress is derivable from the strain-energy function:
1 
W 1 , 2 , 3 
.
2 3 1
The other two true stress components can be similarly obtained.
Incompressible, isotropic, hyperelastic material. When a material undergoes large
deformation, the amount of volumetric deformation is often small compared to the overall
deformation. Consequently, we may neglect the volumetric deformation, and assume that the
material is incompressible.
A block of a material, of lengths L1 , L2 and L3 in the undeformed state, is deformed into a
rectangle of lengths l1 , l 2 and l 3 . If the material is incompressible, the volume of the block must
remain unchanged, namely, L1 L2 L3  l1l 2 l3 , or
123  1 .
The incompressibility places a constraint among the three stretches: they cannot vary
independently. We may regard 1 and  2 as independent variables, so that
3  12 1 .
Consequently, the strain-energy density is a function of two independent variables, W 1 , 2  –
only biaxial tests need be done to fully characterize the material.
Inserting the constraint 3  12 
that
1



into the expression dW  s1d1  s2 d2  s3 d3 , we obtain

dW  s1  12 21 s3 d1  s 2  22 11 s3 d2 .
Once the function W 1 , 2  is determined, the stress-stretch relation is given by differentiation:
s1  12 21 s3 
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W 1 , 2 
W 1 , 2 
, s 2  2 2 11 s3 
.
1
2
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These relations, together with the incompressibility condition 123  1 , replace Hooke’s law
and serve as the stress-strain relations for incompressible, isotropic and hyperelastic materials.
For incompressible materials, the true stresses relate to the nominal stresses as
 1  1 s1 ,  2  2 s 2 ,  3  3 s3 .
Consequently, the stress-strain relations become
 1   3  1
W 1 , 2 
,
1
 2   3  2
W 1 , 2 
.
2
Stress-strain relations for rubbers. The function W 1 , 2  can be determined by subjecting a
sheet of rubber under biaxial stress states. The form of the function is sometimes inspired by
theoretical considerations. Here are three often-used forms.
The Neo-Hookean materials are materials whose energy density is expressed by
W 


2
2
1

 22  32  3 .
Inserting the constraint 3  12  , we obtain the stress-stretch relations:
1
 1   3   12  32  ,
 2   3   22  32  .
A neo-Hookean material is characterized by a single elastic constant,  . The constant may be
determined experimentally. For example, consider a rod in a state of uniaxial stress:
 1   , 2   3  0 .
Let the stretch along the axis of loading be 1   . Incompressibility dictates that the stretches in
the directions transverse to the loading axis be  2  3  1 / 2 . Inserting into the above stressstretch relation, we obtain the relation under the uniaxial stress:
   2  1 .
Recall that stretch relates to the engineering strain as
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  l / L  1 e .
When the strain is small, namely, e  1, the above stress-stretch relation reduces to
  3e .
Thus, we interpret 3 as Young’s modulus and  the shear modulus.
The form of strain-energy function of Neo-Hookean materials has also emerged from a model in
statistical mechanics. The model gives   NkT , where N is the number of polymer chains per
unit volume, and kT the temperature in units of energy. See Treloar.
The Mooney materials are those that can be fitted by




W  c1 12  22  32  3  c 2 12  22  32  3 .
M. Mooney (1940) A theory of large elastic deformation, Journal of Applied Physics 11, 582592.
The Ogden materials are those that can be fitted by a series of more terms
W 
n
n 

1  2  3  3 ,
n
n
n
n
where  n may have any values, positive or negative, and are not necessarily integers, and  n are
constants. The stress-stretch relations are
 1   3    n 1  3  ,
n
n
n
 2   3    n 2  3  .
n
n
n
R.W. Odgen (1972) Large deformation isotropic elasticity – on the correlation of theory and
experiment for incompressible rubberlike solids. Proceedings of the Royal Society of London
A326, 567-583.
For the literature on other functional forms, theoretical motivations, and experimental data, see
Treloar for a classic treatment, and Section 6.5 of Holzapfel for an update.
Inflation of a balloon (or bursting of an aneurysm?) A spherical balloon, radius R and
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thickness H in the undeformed state, is subject to a pressure p, and expands to radius r while
reduces its thickness to h. We would like to determine the relationship between the radius and
the pressure.
As always in solid mechanics, we invoke the 3 ingredients.
Deformation geometry. The stretch in the thickness direction is 3  h / H . The stretches in the
circumferential directions are 1  2  r / R , which we will denote as  . The material of the
skin is taken to be incompressible, so that hr 2  HR 2 , or 3  2 .
Force balance. The stress normal to the skin is between p and zero, and is small compared to the
stress in the circumferential direction,  . Draw the free-body diagram of a half balloon, and
balance the forces, 2rh  r 2 p . Thus
p
2h
.
r
Because r  h , we confirm that   p .
Materials law. The material law will take the form
     .
Put the three ingredients together, and we obtain that
p
2 H 3
    .
R
This is the desired relation between the pressure and the current radius, if we take   r / R as
the normalized current radius. Assume the skin is made of a neo-Hookean material. under the
stretches 1  2   , 3  2 , so that    2  4 . Sketch the function p  . Note


p0  0 and p = 0. As the balloon expands, the material stiffens, but the skin thins. At
certain stretch, the geometric thinning prevails over material stiffening, and the pressure reaches
a peak. This peak is reached at c  7 1 / 4 .
What will happen after this peak? See an analysis by Needleman (1977, Inflation of spherical
rubber balloons, International Journal of Solids and Structures 13, 409-421). Also see
experiments by D.K. Bogen and T.A. McMahon (1979, Do cardiac aneurysms blow out?
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Solid Mechanics
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Biophysics Journal, 27, 301-316)
Exercise. Study the problem for a cylindrical balloon.
Inhomogeneous field. When the state of deformation in a body is inhomogeneous, we must
establish differential equations. Balancing forces on a material particle in the current state, we
find the equilibrium equation in the familiar form:
 ij
x j
 0.
The deformation geometry is complicated in general. However, for a problem with such a
symmetry that the principal directions of stretch are obvious, when an element of length in the
reference state, dL, becomes dl in the current state, the stretch is

dl
dL
We have now established the 3 ingredients in solid mechanics for finite deformation.
Cavitation. Consider a cavity in an infinite medium, subject to a remote hydrostatic tension S.
For a brittle material, the cavity concentrates stress, so that a crack may emanate from the cavity.
In such a case, deformation is typically small when the stress reaches a critical value, so that we
can analyze the problem assuming that the material is linearly elastic and deformation is
infinitesimal. Formulated this way, the problem is known as the Lame problem, as we have seen
before. The key result of the analysis is that the hoop stress at the surface of cavity is 3/2 times
the remote stress.
For a material capable of large deformation, such as a ductile metal or a rubber, however, the
cavity may cause another mode of failure. Under the hydrostatic stress remote from the cavity,
the cavity may expand indefinitely when the applied stress reaches some finite value, a
phenomenon known as cavitation. To study this phenomenon, we need to determine the radius
of the cavity as a function of the remote tension. The Lame solution clearly gives an erroneous
prediction in this case because it says that the radius of cavity increases linearly with the applied
stress.
Deformation geometry. In the unstressed state, let the radius of the cavity be A, and the distance
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Solid Mechanics
Fall 2011
between a material particle and the center of the cavity be R. The deformation make the radius of
the cavity become a, and move the material particle to a new position r. Deformation of the
medium is characterized by the function r R  . The hoop stretch is
  r / R .
The radial stretch is
r  dr R / dR .
We assume that the material is incompressible, so that
r 3  a 3  R 3  A3 .
Consequently, once the current radius of the cavity a is determined, the entire field of
deformation is determined. Because the independent variable in the equilibrium equation is
current position r, we express the R in terms of r:

Rr   r 3  A3  a 3

1/ 3
Incompressibility dictates that
r  2  R / r 2 ,
Insert the expression of Rr  , we express the stretch as a function of r:
 r 3  A3  a 3 

 r  
3
r


2/3
.
Force balance. The stress in the medium is nonuniform: remote from the cavity, the state of
stress is hydrostatic; near the cavity, the state of stress is equal-biaxial. A material particle at
location r in the current state is subject to a triaxial state of stress,  r r ,   r     r  . We
balance force for a material particle in the deformed state, so that
 
d r
2 r
 0.
dr
r
This equation takes the same form as that used in the Lame problem.
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Material law. Suppose that we have conducted a test of the material under unaxial stress states.
The uniaxial stress-stretch curve is
  g   .
In the medium around the cavity, a material particle is under a triaxial stress state: one radial
component and two hoop components: (  r ,   ,   ). Because we have assumed that the
material is incompressible, superposing a hydrostatic stress on the material particle will not
change the state of deformation of the particle. For example, we can superimpose
   ,  ,   on the material particle,
 r    ,0,0 . This is a uniaxial stress state.
so that the stress state of the particle becomes
Thus,
 r     g r  .
Putting the three ingredients together, and integrating from the surface of the cavity to the remote
point, we obtain that

S  2 
g  r d

1
,
where   r / a ,
  3   A / a 3  1 

r  
3




2/3
,
and for Neo-Hookean material,


g     2  1 .
This set of equations define S as a function of a/A. The integral can be evaluated analytically,
giving
S

1

4
5
1 a 
a
 2     .
2
2 A
 A
Sketch this result and compare it with the linear elastic solution. The cavity can expand
indefinitely when the applied stress is still finite. The stress needed to cause the cavity to expand
indefinitely is called the cavitation limit. The cavitation limit for a Neo-Hookean material is
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S c  5 / 2 .
Cavitation limit should exist for other material laws. To see this, set A / a  0 , so that
  3 1
r   3 
  
2/3
We need to see if the above integral is finite. As    , the material is under hydrostatic state,
and r  1 , so that g r   Er  1 , were E is Young’s modulus. Thus
2/3


1 
2E

g r   E 1  3   1   3 .
  

3


The integral must converge.
5
Large Deformation
Small Deformation
4.5
4
3.5
S/ 
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
a/A
6
7
8
9
10
This problem was solved for metals by R. Hill (1949, Journal of Applied Mechanics16, 259), and
for rubbers by A.N. Gent and D.A. Tompkins (1969, Nucleation and growth of gas bubbles in
elastomers, Journal of Applied Physics 6, 2520-2525). For more recent studies of cavitation in
metals, see Y. Huang, J.W. Hutchinson and V. Tvergaard (1991, Journal of Mechanics and
Physics of Solids 39, 223); A.G. Varias, Z. Suo and C.F. Shih (1992, Mode mixity effect on the
damage of a constrained ductile layer, Journal of Mechanics and Physics of Solids 40, 485-509).
Exercise. Plot the hoop stress at the surface of the cavity as function of a/A. Compare the result
with the Lame solution.
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Exercise. Plot S as a function of a/A for a power-law material. Numerical integration might be
needed.
Exercise. Study the problem for a cavity in a sphere of material of finite radius.
Exercise. Study the phenomenon of cavitation under the plane strain conditions.
Exercise. In the above formulation, we have used the equilibrium equation formulated in terms
of Cauchy stresses as functions of r. Derive the equilibrium equation in terms of the nominal
stresses as functions of R. Solve the problem using this alternative formulation.
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