Motivated by the works of Ball (1982) and Pericak-Spector and Spector (1988), we investigate singular solutions of the compressible nonlinear elastodynamics equations.

These singular solutions contain discontinuities in the displacement field and 

can be seen as describing fracture or cavitation.

We explore a definition of singular solution via approximating sequences of smooth functions.

 We use these approximating sequences to investigate the energy of such  solutions, taking into account the energy needed to open a crack or hole.

In particular, we find that the existence of singular solutions and the finiteness of their energy

 is strongly related to the behavior of the stress response function for infinite stretching, i.e.

the material has to display a sufficient amount of softening.

In this note we detail our findings in one space dimension

Giesselmann, Jan

Tzavaras, Athanasios E.

English

ACMAC

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On cavitation in Elastodynamics
Jan Giesselmann and Athanasios E. Tzavaras
Original Citation:
Giesselmann, Jan and Tzavaras, Athanasios E.
(2012)
On cavitation in Elastodynamics.
(Submitted)
This version is available at: http://preprints.acmac.uoc.gr/196/
Available in ACMAC’s PrePrint Repository: March 2013
ACMAC’s PrePrint Repository aim is to enable open access to the scholarly output of ACMAC.
http://preprints.acmac.uoc.gr/Manuscript submitted to Website: http://AIMsciences.org
AIMS' Journals
Volume X, Number 0X, XX 200X pp. X{XX
ON CAVITATION IN ELASTODYNAMICS
Jan Giesselmann
Weierstrass Institute
Mohrenstr. 39, 10117 Berlin, Germany
Athanasios E. Tzavaras
Department of Applied Mathematics, University of Crete and
Institute for Applied and Computational Mathematics, FORTH
Heraclion, Crete, Greece
Abstract. Motivated by the works of Ball (1982) and Pericak-Spector and
Spector (1988), we investigate singular solutions of the compressible nonlinear
elastodynamics equations. These singular solutions contain discontinuities in
the displacement eld and can be seen as describing fracture or cavitation. We
explore a denition of singular solution via approximating sequences of smooth
functions. We use these approximating sequences to investigate the energy of
such solutions, taking into account the energy needed to open a crack or hole.
In particular, we nd that the existence of singular solutions and the niteness
of their energy is strongly related to the behavior of the stress response function
for innite stretching, i.e. the material has to display a sucient amount of
softening. In this note we detail our ndings in one space dimension.
1. Introduction. In the seminal work of Ball [1] compressible, nonlinear elastic-
ity has been used as a model for fracture and cavitation in elastic materials such
as rubber. For a study on the relation between the macroscopic behavior of an
elastic solid and its energy functional, see [5]. Ball constructed singular solutions
to variational problems, which display discontinuities in the displacement eld at
the origin. These solutions which are radially symmetric can be seen as describing
cavitation. Ball computed their energy and compared it to the energy of trivial
solutions with homogeneous strain. The upshot of his study is that for suciently
large prescribed deformations on the boundary the solution displaying cavity is en-
ergetically favorable. Based on his ndings Spector and coworkers [7, 8, 6] studied
the dynamic case getting similar results. One important feature of the singular
solutions constructed in these works is that the normal Cauchy stress vanishes on
the boundary of the cavity such that all the integrals needed to dene weak so-
lutions and energies exist. However, these constructions do not account for any
energy needed for crack and cavity creation. Therefore, we propose a new solution
concept which accounts for this energy. It takes into account the layer structure of
the solution at the onset of fracture by considering sequences of smooth functions
which approximately solve the equations and approach the discontinuous solutions.
2000 Mathematics Subject Classication. Primary: 74B20; Secondary: 74R99, 35L70, 35Q74.
Key words and phrases. Nonlinear Elastodynamics, Cavitation,
This research was supported by the EU FP7-REGPOT project Archimedes Center for Model-
ing, Analysis and Computation.
12 JAN GIESSELMANN AND ATHANASIOS E. TZAVARAS
We call this type of solution as singular limiting induced from continuum solution
(in short slic solution).
We consider the equations of compressible, nonlinear elastodynamics, searching
for a displacement eld which satises
ytt = div(ry); in Rd  (0;T); t > 0 (1)
for some nite time T > 0; where  : R
dd
+ ! R
dd
+ is the stress response function,
which is related to an energy density function W : R
dd
+ ! R via (F) = @W
@F (F):
We are interested in radially symmetric deformations
y(x;t) = w(jxj;t)
x
jxj
(2)
with prescribed homogeneous tensile deformations in the far eld, i.e.
y(x;t) = x for jxj > rt (3)
for r; > 0: In practice we are interested in the case  >> 1: We focus on solutions
of (1) having the form (2) and satisfying (3), which have a discontinuity at the
origin. Let us note that using the velocity v := yt and the deformation gradient
u := ry we may rewrite (1) as the following system of conservation laws:
ut = rv
vt = div((u)):
(4)
The outline of this contribution is as follows: In Section 2 we describe the basic ideas
and the main technical considerations in the one dimensional case. We determine
their energy dissipation in Section 3. We would like to mention that basically the
same results hold in several space dimensions, but the technical and notational
complexity is considerably higher. We refer to [4] for the details.
2. A special 1-dimensional solution including fracture.
2.1. A special ansatz function. We consider the one dimensional version of (1),
i.e.
ytt = (yx)x ; x 2 R; t > 0 (5)
describing the longitudinal or shearing motion of an one dimensional elastic bar.
We impose a homogeneous deformation far away, i.e.
y(x;t) = x; for jxj > rt; (6)
for some r suciently large. For large values of  the bar under consideration will
break and after breaking the continuum hypothesis which is crucial in deriving (5)
fails. Therefore, (5) will not be an appropriate description of the physical situation.
However, we expect that there is an intermediate range of values of  from a range
of loading where the model is valid to a range where it looses validity. In some
situations it may arguably be possible to give a meaning to (5) in and past this
intermediate regime. In particular, we think that (5) is a reasonable description at
the onset of fracture which is the situation we want to explore.
We assume that the energy density W and the stress function  = W0 satisfy
W 2 C3((0;1);R); 0(u) > 0; 00(u) < 0 8u > 0 (H1)
lim
u&0
(u) =  1; lim
u&0
W(u) = 1: (H2)ON CAVITATION IN ELASTODYNAMICS 3
W
u
τ
u
Figure 1. Sketch of W and  satisfying (H1) and (H2).
Given the rst condition of (H1) the one dimensional version of (4) has the wave
speeds 1;2(u) = 
p
0(u) and is therefore hyperbolic. The second part of (H1) is
related to softening elastic response of the material and is a crucial requirement of
our analysis. The second hypothesis (H2) is placed to prevent a nite volume from
being compressed down to zero. However, as we focus on tensile deformations this
condition will not have any signicant impact.
Smooth solutions of (5) satisfy the conservation law for energy

1
2
jytj2 + W(yx)

t
  ((yx)yt)x = 0: (7)
This means that any change in mechanical (stored and kinetic) energy in a certain
interval is due to the work performed on the boundary of said interval, as can be
seen from the integral form
d
dt
Z b
a
1
2
jytj2 + W(yx)dx = (yx)ytj
b
a : (8)
In particular, (8) shows that if the deformation is homogeneous outside a certain
interval ( r;r), i.e. y(x;t) = x for jxj > r; the energy inside ( r;r) is conserved.
We focus on solutions in which stress and strain only depend on the self-similar
variable  = x
t which is achieved by the ansatz y(x;t) = tY (x
t). One easily veries
that y satises (5) if Y satises
2Y 00() = (Y 0())0: (9)
Introducing the velocity V () := Y 0()   Y () and the strain U() = Y 0() in the
self-similar variable we can rewrite (9) as
 U0() = V 0()
 V 0() = (U())0:
(10)
We will test a class of self-similar solutions
Y () =
8
> > > <
> > > :
  <  
 Y (0) +    <  < 0
Y (0) +  0 <  < 
  < 
(11)
where , ,  and Y (0) are positive parameters that satisfy  > , and
 = Y (0) + ;
 =
r
()   ()
   
:
(12)4 JAN GIESSELMANN AND ATHANASIOS E. TZAVARAS
Y
ξ
Y (0)
α
λ
α
λ
v
u
(α,Y (0))
(α,−Y (0))
(λ,0)
δ-sh
2-sh
1-sh
Figure 2. Sketches of Y and the Riemann-diagram of the solution
For this deformation the velocity and strain distributions are given by
U() = 2Y (0)=0 + jj< + jj>;
V () = Y (0)0<<   Y (0)0< <:
(13)
Due to the discontinuity of Y (corresponding to a delta distribution in U) it is not
straightforward how such a function can be interpreted as a solution of (5) or (10).
Before we give a denition of solution based on regularizations of the prospective
solutions we like to make some remarks.
Remark 1. 1. The solution ansatz (11) is inspired by the dynamic cavitating
solutions in three dimensions introduced in [7]. The main dierence is that
the solutions in [7] do not contain a delta measure in the strain.
2. For a given  > 0 there is a one parameter family of functions (11) satisfying
(12).
3. The ansatz (11) has singularities at  =  and at  = 0. The former
are shocks. Due to the symmetry the Rankine-Hugoniot conditions at both
singularities amount to
 (   ) =  Y (0)
 ( Y (0)) = ()   ()
and are equivalent to (12). The shock at  =  belongs to the second charac-
teristic family and the Lax shock admissibility criterion reads
2(U ) =
p
0() >  =
r
()   ()
   
>
p
0() = 2(U+):
Thus, it is satised provided  > . The same holds for the singularity at
 =  :
4. At positive times the solution Y has a discontinuity of size 2tY (0) at the
origin, which is associated to a crack whose boundary is moving according to
y(0;t) = tY (0):
In contrast (12)1 ensures the continuity of Y outside the origin.
5. For t & 0 we have y(x;t) ! x and v(x;t) ! 0. Therefore, the initial
data is a conguration with homogeneous deformation which is at rest. The
problem (5),(6) with these initial data obviously admits the trivial solution
y(x;t) = x:
6. In view of (12) a straightforward calculation using  @=0 = =0 in D0
shows
 u0 = v0ON CAVITATION IN ELASTODYNAMICS 5
in D0. Hence, (11) satises (10)1. Thus, we will focus on giving a meaning to
(u) and (10)2 at the origin. This problem is strongly related to the concept
of delta shocks, see [3, Sec. 7.5] and references therein. A main dierence of
our approach compared to those usually employed for delta shocks is that we
exploit the structure of (5) to dene solutions and the denition has a natural
physical interpretation.
7. In one space dimension compressible elasticity coincides with the p-system
in uid dynamics. The solution of the p-system describing vacuum via a
-measure given in [3, Sec. 7.5] can easily be handled in our framework [4].
2.2. Slic-solutions. Our next step is to give a meaning to the equation around
x = 0 by introducing the notion of singular limiting induced from continuum solution
(in short slic-solution). Roughly speaking a discontinuous function is called a slic-
solution provided it can be obtained as the limit of a sequence of smooth functions
that are approximate solutions of the problem.
Denition 2.1. Let y 2 L1
loc(R  ( 1;1)), let  be a mollier:  2 C1
c (R),
  0, supp = B1 (the ball of radius 1),
R
 = 1. Consider n = n
 
nx

and
dene
yn(t;x) = n ? y =
Z 1
 1
n(x   z)y(t;z)dz:
Then y is called a slic-solution of (5) provided
Z
R
Z
R
yn tt + (yn
x) x dtdx ! 0
for any   2 C2
c(R  ( 1;1)):
Remark 2. 1. The denition of slic-solution is in fact independent of the par-
ticular choice of mollier :
2. In general a convolution in both space and time may be necessary. In the
present context the tested solution is self-similar and it suces to regularize
in space to smooth it.
3. The denition is in fact local and it could be stated for a solution y 2 L1
loc(O)
with O an open set in space-time. Moreover, it can be adapted in the obvious
way to account for initial conditions (see [4] for further details).
4. A straightforward calculation shows that any standard W1;1 weak solution of
(5), which satises the correct initial and boundary data, is also a slic-solution.
Proposition 1. Let y dened in (11) satisfy (12) and be extended to t < 0 by
y(x;t) = x, then it is a slic-solution of (5) if and only if
L := lim
u!1
(u)
u
= 0: (H3)
Proof. We only give the main idea of the proof, for details see [4]. Note that y is
a weak solution of (5) on any open subset of R2nf(x;t) : t > 0;x = 0g. Moreover,
the sequences (yn)tt and (yn
x)x restricted to f(x;t) : jxj > 1
n _t < 0g are uniformly
bounded and converge pointwise. Thus, it is sucient to check whether
lim
n!1
Z 1
0
Z 1
n
  1
n
(yn
x(x;t)) x(x;t)dxdt = 06 JAN GIESSELMANN AND ATHANASIOS E. TZAVARAS
for all   2 C1
c(R  R;R): We compute
lim
n!1
Z 1
n
  1
n
(yn
x(x;t)) x(x;t)dx = 2LtY (0) x(0;t);
which proves the lemma.
This shows that there only exist discontinuous slic-solutions of the form (11) to
(5), (6) in case the material exhibits softening elastic response. In the sequel we
always assume (H3) to be satised.
3. The energy needed to open a crack. It seems reasonable to consider the
energies of the approximate solutions yn as an approximation of the energy of the
slic-solution y. As the approximate solutions are smooth they satisfy the energy
identity 
1
2
(vn)2 + W(un)

t
  (vn(un))x = ((vn)t   (un)x)vn; (14)
where vn := yn
t and un = yn
x: Let us dene the residual of yn in the wave equation
(5) as fn, i.e.
fn := (vn)t   (un)x: (15)
This quantity may be interpreted as an exterior force which (if it were applied)
would act on a neighborhood of the origin and make yn the exact solution. Let
B = ( r;r) some interval containing a neighborhood of the whole wave fan at time
t. Then (for suciently large n) vnj@B = 0: The energy of the solution inside B
EB[yn](t) :=
Z
B
1
2
(vn(x;t))2 + W(un(x;t))dx (16)
evolves according to
d
dt
EB[yn] =
Z
B
((vn)t   (un)x)vn dx: (17)
Proposition 2. Let vnj@B = 0 and n > 2
t then the energy change rate is given by
d
dt
EB[yn](t) = Y (0)2 + 2(W()   W())
+ 2
Z 1
n
0
( + 2n(x)tY (0))2n(x)Y (0)dx: (18)
Proof. We have
d
dt
EB[yn] =
Z r
 r
vn(vn)t + (un)(un)t dx = 2
Z r
0
vn(vn)t + (un)(un)t dx:
For nt > 2 and x > 0; we calculate
vn(x;t) =  Y (0)
Z 0
 t
n(x   z)dz + Y (0)
Z t
0
n(x   z)dz
un(x;t) =
8
> > <
> > :
 + 2tY (0)n(x) : x < 1
n
 : 1
n < x < t   1
n

R t
0 n(x   z)dz + 
R 1
t n(x   z)dz : t   1
n < x < t + 1
n
 : x > t + 1
n
(19)ON CAVITATION IN ELASTODYNAMICS 7
such that
(vn)t(x;t) =  Y (0)n(x + t) + Y (0)n(x   t)
(un)t(x;t) = 2Y (0)n(x) + (   )n(x   t)
(20)
and thus, as n > 2
t;
d
dt
EB[yn] = 2
Z t+ 1
n
t  1
n
Y (0)2
Z t
0
n(x   z)dz

n(x   t)dx
+ 2
Z 1
n
0
( + 2tY (0)n(x))2Y (0)n(x)dx
+ 2
Z t+ 1
n
t  1
n


 + (   )
Z 1
t
n(x   z)dz

(   )n(x   t)dx
=: In
1 + In
2 + In
3
(21)
We nd
In
1 =  
Z t+ 1
n
t  1
n
Y (0)2
d
dx
Z t
0
n(x   z)dz
2
dx =
1
2
Y (0)2 (22)
and
In
3 =  
Z t+ 1
n
t  1
n
d
dx
W

 + (   )
Z 1
t
n(x   z)dz

dx = (W()   W()):
(23)
Inserting (22) and (23) into (21) we obtain the assertion of the Lemma.
The solution contains three waves, all of which contribute to the energy rate.
There are the two shocks located at  =  and the discontinuity at 0. Both
shocks satisfy the Lax criterion and therefore there is energy dissipation along them.
Moreover, the energy dissipation at the shocks ;  can be calculated by classical
Riemann problem theory, see [3, Sec. 8.5]. This gives
 =  

 
1
2
Y (0)2 + W()   W()

+ Y (0)()
= Y (0)

 
W()   W()
   
+
1
2
(() + ())

< 0; (24)
because 00 < 0; and
  = 

1
2
Y (0)2 + W()   W()

+ Y (0)() < 0: (25)
As there is no energy dissipation/creation away from the three waves the energy
balance can be expressed as
d
dt
EB[yn] =   + pn
c +  (26)
where
pn
c = 2
 Z 1
n
0
( + 2n(x)tY (0))2n(x)Y (0)dx   Y (0)()
!
(27)8 JAN GIESSELMANN AND ATHANASIOS E. TZAVARAS
can be interpreted as the work performed by the force fn at the crack. Its limiting
contribution is given by
lim
n!1
pn
c = lim
n!1
2
 Z 1
n
0
( + 2n(x)tY (0))2n(x)Y (0)dx   Y (0)()
!
= lim
n!1
2
Z 1
0
( + 2n(z)tY (0))2(z)Y (0)dz   Y (0)()

=

1 if limu!1 (u) = 1
2(1   ())Y (0) if limu!1 (u) =: 1 < 1:
(28)
Thus, the energy rate needed to open a crack is innite in case limu!1 (u) = 1
while it is nite in case limu!1 (u) < 1: In that case the energy needed to create
a crack can be computed as follows:
Proposition 3. Let 1 := limu!1 (u) < 1 then the total energy of the wave fan
satises
d
dt
EB[yn] =   + pn
c + 
n!1  !   + 2(1   ())Y (0) +  =: T > 0; (29)
where  denotes the energy dissipation at the shocks and 2(1 ())Y (0) is the
cost for opening the crack. The total energy rate T is positive, i.e. more energy is
needed to open the crack then is dissipated at the shocks.
Proof. Given Proposition 2 and (28) it only remains to show T > 0: Inserting (24)
and (25) into the denition of T we get
T = Y (0)2 + 2Y (0)

1  
W()   W()
   

> 2Y (0)(1   (u)) > 0
for some u 2 (;):
REFERENCES
[1] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos.
Trans. Roy. Soc. London Ser. A, 306, (1982) 557{611.
[2] P.G. Ciarlet, "Mathematical Elasticity, Vol. I, Three-dimensional elasticity,\ North Holland,
Amsterdam, 1988.
[3] C. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,\ 3rd edition.
Grundlehren der Mathematischen Wissenschaften, 325. Springer Verlag, Berlin, 2010.
[4] J. Giesselmann and A.E. Tzavaras, Singular limiting induced from continuum solutions and
the problem of dynamic cavitation (in preparation).
[5] R.W. Ogden, Large deformation isotropic elasticity: On the correlation of theory and exper-
iment for compressible rubberlike solids, Proc. Roy. Soc. London Ser. A, 328 (1972), 567{583.
[6] K.A. Pericak-Spector, J. Sivaloganathan and S.J. Spector, An Explicit Radial Cavitation
Solution in Nonlinear Elasticity, Math. Mech. Solids. 7 (2002), 87{93.
[7] K.A. Pericak-Spector and S.J. Spector, Non-uniqueness for a hyperbolic system: cavitation
in nonlinear elastodynamics, Arch. Rational Mech. Anal. 101 (1988), 293{317.
[8] K.A. Pericak-Spector and S.J. Spector, Dynamic cavitation with shocks in nonlinear elasticity,
Proc. Royal Soc. Edinburgh Sect A, 127 (1997), 837{857.
Received xxxx 20xx; revised xxxx 20xx.
E-mail address: jan.giesselmann@wias-berlin.de
E-mail address: tzavaras@tem.uoc.gr

On cavitation in Elastodynamics

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On cavitation in Elastodynamics

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