# Ozawa’s Intersubjectivity Theorem as justification of RQM’s postulate on
internally consistent descriptions

Andrei Khrennikov
Linnaeus University, International Center for Mathematical Modeling
in Physics and Cognitive Sciences Växjö, SE-351 95, Sweden

###### Abstract

The Ozawa’s Intersubjectivity Theorem (OIT) proved within quantum measurement
theory supports the new postulate of relational quantum mechanics (RQM), the
postulate on internally consistent descriptions. But from OIT viewpoint
postulate’s formulation should be completed by the assumption of probability
reproducibility.

##  1 Introduction

Recently Ozawa’s Intersubjectivity Theorem (OIT) [1] proved within quantum
measurement theory (the indirect measurement scheme for quantum instruments
theory[2, 3]) was used as objection to QBism [4], to its basic
interpretational statement about the individual agent perspective on
measurement’s outcomes [5]-[7]. This privacy component of QBism was also
criticized in my previous articles [8, 9], mainly from the probabilistic
perspective. Now QIT provides the strong argument that measurement’s outcome
is intersubejctive and, although one can still interpret it as a private
experience, such interpretation loses its elegance. QBists replied to my
objection with two papers [10, 11] (the second one presents more elaborated
counterargument). The detailed response two these papers will be prepared by
Ozawa.

The discussions on the intersubjectivity problem in QBism stimulated my
interest to this problem for other information interpretations of QM. A
reviewer of one of my papers pointed that relational quantum mechanics (RQM)
[12] was suffering of the same problem. It seems that it cannot be resolved
within “old RQM”. Its resolution required invention of an additional postulate
on internally consistent descriptions (postulate 6 in [13]). By treating this
postulate within quantum instrument theory, one immediately see that OIT
supports this postulate. So, in contrast to QBism which was challenged by OIT,
RQM’s postulate 6 is mathematically justified via OIT.

However, the OIT view on postulate 6 leads to the conclusion that

  * •

on the one hand, to be supported by the quantum theory, this postulate should
be constrained by the probability reproducibility condition,

  * •

on the other hand, it can be formulated in a stronger form.

We start with a brief recollection of the basics of RQM and the problem of
intersubjectivity in its framework. Here we follow article [13]. Since I am
not an expert in RQM, I shall mainly cite this article.

##  2 Relational quantum mechanics and the problem of intersubjectivity

As was stated by Rovelli [12], RQM is based on the idea that “in quantum
mechanics different observers may give different accounts of the same sequence
of events.” It is crucial that “RQM is built on strong naturalistic
intuitions; therefore, in RQM, the term “observer” is understood in a broad
sense, which allows that any physical system can be an “observer,” so we do
not have to accept that consciousness plays any fundamental role.” [13]. In
the last aspect RQM differs crucially from QBism. As is emphasized in article
[13], RQM has many attractive features and it resolves the basic quantum
paradoxes (but QBism do this as well).

“However, some problems remain; in particular, there is a tension between
RQM’s naturalistic emphasis on the physicality of information and the
inaccessibility of certain sorts of information in current formulations of
RQM. Thus, in this article, we propose a new postulate for RQM which ensures
that all of the information possessed by a certain observer is stored in
physical variables of that observer and thus is accessible by measurement to
other observers. The postulate of cross-perspective links makes it possible
for observers to reach intersubjective agreement about quantum events that
have occurred in the past, thus shoring up the status of RQM as a form of
scientific realism and allowing that empirical confirmation is possible in
RQM.

Adding this postulate requires us to update some features of the ontology of
RQM, because it entails that not everything in RQM is relational.”[13]

For readers convenience, here the list of all RQM’s postulates is presented
[13]:

  1. 1.

Relative facts: Events, or facts, can happen relative to any physical system.

  2. 2.

No hidden variables: Unitary quantum mechanics is complete.

  3. 3.

Relations are intrinsic: The relation between any two systems $A$ and $B$ is
independent of anything that happens outside these systems’ perspectives.

  4. 4.

Relativity of comparisons: It is meaningless to compare the accounts relative
to any two systems except by invoking a third system relative to which the
comparison is made.

  5. 5.

Measurement: An interaction between two systems results in a correlation
within the interactions between these two systems and a third one; that is,
with respect to a third system $W,$ the interaction between the two systems
$S$ and $F$ is described by a unitary evolution that potentially entangles the
quantum states of $S$ and $F.$

  6. 6.

Internally consistent descriptions: In a scenario where $O_{1}$ measures $S,$
and $O_{2}$ also measures $S$ in the same basis, and $O_{2}$ then interacts
with $O_{1}$ to “check the reading” of a pointer variable (i.e., by measuring
$O_{1}$ in the appropriate “pointer basis”), the two values found are in
agreement.

OIT matches perfectly postulate 6, justifies it mathematically on the basis of
quantum instrument theory. Generally my picture is that RQM can earn a lot
from merging with quantum instrument theory.

As was cited above in “RQM the term “observer” is understood in a broad sense,
which allows that any physical system can be an “observer…”

In quantum instrument theory, “apparatus” plays the key role. I would suggest
to extend its meaning and operate in this theory with the notion observer. The
latter has the RQM meaning, i.e., any physical system can be an observer. I
understand that such a step may induce critique from both sides, RQM and
quantum instrument theory.

##  3 Ozawa Intresubjectivity Theorem

Von Neumann [14] described observables mathematically by Hermitian operators
in Hilbert space ${\cal H}.$ Consider operators with totally discrete spectra
$X\subset\mathbb{R}:A=\sum_{x\in X}xE_{A}(x),$ where $E_{A}(x)$ is projection
on the eigensubspace for the eigenvalue $x.$ The Born rule determines the
probabilities:

$P(A=x|\psi)=\langle\psi|E_{A}(x)|\psi\rangle.$

The indirect measurement scheme involves the following components

  * •

the states spaces ${\cal H}$ and ${\cal K}$ of the systems ${\cal S}$ and the
apparatus ${\cal M}$ for measurement of some observable $A;$

  * •

the evolution operator $U=U(t)$ representing the interaction-dynamics for the
system ${\cal S}+{\cal M};$

  * •

the meter observable $M$ giving outputs of the pointer of the apparatus ${\cal
M}.$

Here the quantum observables $A$ and $M$ can be represented as PVMs,
$E_{A}=(E_{A}(x)),E_{M}=(E_{M}(x)),$ where $E_{A}(x),E_{M}(x)$ are projections
in Hilbert spaces ${\cal H}$ and ${\cal K}$ respectively. It is assumed that
the compound system’s evolution is driven by the Schrödinger equation, so the
evolution operator is unitary.

Formally, an indirect measurement model for an observable $A$, introduced in
[2] as a “measuring process”, is a quadruple

$({\cal K},|\xi\rangle,U,M)$

where $|\xi\rangle\in{\cal K}$ represents the apparatus state.

We explore the Heisenberg picture. To describe meter’s evolution, we represent
it in the state space of the compound system, i.e., as $I\otimes M.$ The meter
observable evolves as

$M(t)=U^{\star}(t)(I\otimes M)U(t).$ (1)

By the Born rule

$P(M(t)=x|\psi\xi)=\langle\psi\xi|E_{M(t)}(x)|\psi\xi\rangle.$ (2)

This is the probability distribution for the outputs of measurements done by
the apparatus and given by the meter.

Definition. A measurement process $({\cal K},|\xi\rangle,U,M)$ reproduces the
probability distribution for quantum observable $A$ (accurate von Neumann
observable) if

$P(A=x|\psi)=P(M(T)=x|\psi\xi).$ (3)

Following [1], consider two remote observers $O_{1}$ and $O_{2}$ who perform
joint measurements on a system ${\cal S},$ in mathematical terms it means that
the meter quantum observables of the corresponding measurement processes
commute,

$[M_{1}(t),M_{2}(t)]=0.$

Here each apparatus has its own state space, i.e., ${\cal K}={\cal
K}_{1}\otimes{\cal K}_{2}.$ We call such measurements local. In this situation
the joint probability distribution is well defined

$P(M_{1}(t)=x,M_{2}(t)=y|\psi\xi_{1}\xi_{2})=\langle\psi\xi_{1}\xi_{2}|E_{M_{1}(t)}(x)E_{M_{2}(t)}(y)|\psi\xi_{1}\xi_{2}\rangle.$
(4)

Theorem. (OIT [1]) Two observers performing the joint local and probability
reproducible measurements of the same quantum observable $A$ on the system
${\cal S}$ should get the same outcome with probability 1:

$P(M_{1}(T)=x,M_{2}(T)=y|\psi\xi_{1}\xi_{2})=\delta(x-y)P(E=x|\psi)=\delta(x-y)\|E(x)\psi\|^{2}.$
(5)

##  4  Postulate on internally consistent description in the view of Ozawa
Intersubjectivity Theorem

In OIT, as in postulate 6, observer $O_{1}$ measures $S,$ and observer $O_{2}$
also measures $S$ in the same basis. In our terms the later means that they
measure the same observable $A.$ Then $O_{2}$ can interact with $O_{1}$ to
“check the reading” of its pointer variable, the two values found are in
agreement.

The important condition which is missed in postulate 6 is the probability
reproducibility condition, only under this conditions outcomes of measurements
performed by $O_{1}$ and $O_{2}$ coincide. Therefore, it is natural to
complete postulate 6 by this condition. (But this is just my viewpoint and may
be Carlo Rovelli would not support it.)

Moreover, due to OIT observer $O_{2}$ even need not to perform a measurement
on $O_{1}.$ If observers are sure in validity of quantum theory, then they can
be sure that they get the same outcome.

I suggest the following modification of postulate 6:

Postulate 6[OIT]. Under the condition of probability reproducibility, in a
scenario where $O_{1}$ measures $S,$ and $O_{2}$ also measures $S$ in the same
basis, the two values found are in agreement.

## Acknowledgments

This study was partially supported by the EU-project: CA21169 - Information,
Coding, and Biological Function: the Dynamics of Life (DYNALIFE).

## References

  * [1] Ozawa, M. (2019). Intersubjectivity of outcomes of quantum measurements. https://arxiv.org/abs/1911.10893
  * [2] Ozawa, M. (1984). Quantum measuring processes for continuous observables. J. Math. Phys. 25, 79–87
  * [3] Ozawa, M. Soundness and completeness of quantum root-mean-square errors. Quant. Inf. 5, Article number: 1 (2019)
  * [4] A. Khrennikov, Ozawa’s Intersubjectivity Theorem as objection to QBism individual agent perspective. https://arxiv.org/abs/2301.04014
  * [5] Fuchs, C. A., Mermin, N. D. and Schack, R. (2014). An Introduction to QBism with an Application to the Locality of Quantum Mechanics, _Am. J. Phys._ 82, p. 749.
  * [6] Fuchs, C. A. (2017). Notwithstanding Bohr, the reasons for QBism. Mind and Matter, 15(2), 245-300.
  * [7] Fuchs, C. A. (2023). QBism, Where Next?. arXiv preprint arXiv:2303.01446.
  * [8] Khrennikov, A. (2018). External Observer Reflections on QBism, Its Possible Modifications, and Novel Applications, In: _Quantum Foundations_ , STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health; Khrennikov A. and Toni B. Eds.; Springer, Cham, pp. 93–118.
  * [9] Khrennikov, A. (2018). Towards better understanding QBism, _Found. Sc._ , 23 (1), 181–195.
  * [10] Stacey, B. C. Whose Probabilities? About What? A Reply to Khrennikov. arXiv:2302.09475.
  * [11] Schack, R. When will two agents agree on a quantum measurement outcome? Intersubjective agreement in QBism. arXiv:2312.07728 [quant-ph]
  * [12] Rovelli, C. 1996. Relational Quantum Mechanics. Int. J. Theor. Phys. 35, no. 8: 1637–1678.
  * [13] Adlam, E., and Rovelli, C. (2023). Information is physical: Cross-perspective links in relational quantum mechanics. Philosophy of Physics 1(1): 4, 1–19.
  * [14] von Neuman, J. (1955). _Mathematical foundations of quantum mechanics_ (Princeton Univ. Press, Princenton) [Originally published: Mathematische Grundlagen der Quanten-mechanik, Springer, Berlin, 1932].


# On opening crack propagation in viscoelastic solids

B.N.J. Persson  PGI-1, FZ Jülich, Germany, EU  www.MultiscaleConsulting.com

###### Abstract

We show that the Persson-Brener theory of crack propagation in viscoelastic
solids gives a viscoelastic fracture energy factor $G/G_{0}=1+f$ which is
nearly the same as the viscoelastic factor obtained using the cohesive-zone
model. We also discuss finite-size effects and comment on the use of crack
propagation theories for “solids” with a viscoelastic modulus that vanishes at
zero frequency.

Introduction

Crack propagation in viscoelastic solids, or at the interface between a
viscoelastic solid and a counter surface, have many important applications,
e.g., for rubber wear[1], or in adhesion and friction involving rubber-like
materials[4, 6, 5, 2, 3, 7, 8]. Two different approaches have been applied to
crack propagation in viscoelastic solids. One focus on the stress using the
cohesive-zone (or Dugdale–Barenblatt) model, and another is based on an energy
approach. The first approach was used by Knauss[4] and by Schapery[9, 10] and
later by Hui et al[11] and by Greenwood[12]. Since the exact relation between
the stress $\sigma$ and the surface separation $u$ in the cohesive (process)
zone, where the bond-breaking is assumed to occur, is not known in general, it
was assumed that the stress is constant and equal to $\sigma_{0}$ for
$0<u<h_{0}$ and $\sigma=0$ for $u>h_{0}$. The second (energy) approach was
used by de Gennes[13] in a qualitative way, and by Persson and Brener[14] in a
quantitative way. In the latter approach enters a cut-off radius $a_{0}$,
which can be interpreted as the radius of curvature of the crack tip in the
adiabatic limit, and the stress $\sigma_{\rm c}$ at the crack tip (the stress
to break the adhesive or cohesive bonds).

Viscoelastic factor $G/G_{0}$

The stress $\sigma_{0}$ in the cohesive-zone approach is in general not the
same as the stress $\sigma_{c}$ in the theory of Persson and Brener. In Ref.
[12] it was assumed that the stress in the process zone is a constant
$\sigma=\sigma_{0}$ for $0<z<h_{0}$, where $\sigma_{0}$ can be any number as
long as the (adiabatic) work of adhesion is given by $G_{0}=h_{0}\sigma_{0}$.
If one use another wall-wall interaction, e.g., based on the Lennard Jones
potential, one gets another relation between $G_{0}$ and $\sigma_{0}$. Since
the final result depends on $\sigma_{0}$ this approach is in fact somewhat
ill-defined unless the exact relation between the stress and the separation is
known (which in general is not the case) and used in the theory. On the
contrary in the theory of Persson and Brener only well defined (experimental)
quantities occur. Thus the adiabatic crack tip radius $a_{0}$ could in
principle be measured using, e.g., an electron microscope, or calculated from
the adiabatic work of adhesion $G_{0}$ and the stress $\sigma_{\rm c}$ to
break the bonds via $a_{0}=E_{0}G_{0}/(2\pi\sigma_{\rm c}^{2})$. In any case
we expect $a_{0}$ to be of order the length of the polymer chains between the
cross links, i.e., typically of order $1\ {\rm nm}$.

In Ref. [14, 15], it was shown that the Persson–Brener theory gives nearly the
same result for the viscoelastic factor $G/G_{0}=1+f(v)$ as the cohesive-zone
model. Thus the numerical results for $G/G_{0}$ obtained by Greenwood and by
Hui et al for the three-element viscoelastic model (see Fig. 1) is nearly the
same as predicted by the Persson–Brener theory if one chooses $\sigma_{0}$ to
get the best possible overlap between the two curves (which means using
$\sigma_{0}\approx 3\sigma_{c}$), see Fig. 2 (see Appendix A for the equations
used in the calculations). Shifting like this is the only meaningful way to
compare the factor $G/G_{0}$ between the two theories, because the velocity
normalization factor in the cohesive-zone approach depends on $\sigma_{0}$ [as
$v_{0}=E_{0}G_{0}/(\tau\sigma_{0}^{2}$)], and is ill-defined (no interaction
force is constant for $0<z<h_{0}$ and zero elsewhere).

Figure 1:  Three-element viscoelastic model used in model calculation of the
crack propagation energy $G(v)$. The low frequency modulus
$E(0)=E_{0}=E_{0}^{\prime}E_{1}/(E_{0}^{\prime}+E_{1})$ and the high frequency
modulus $E(\infty)=E_{1}$ and the viscosity $\eta=1/\tau$ are indicated.
Figure 2:  The crack propagation viscoelastic factor $G/G_{0}=1+f(v)$ as a
function of the crack tip speed (log-log scale) for the three level rheology
model. The red curve is calculated by Greenwood using the cohesive-zone model
with the adiabatic (infinitely slowly) work of adhesion
$G_{0}=h_{0}\sigma_{0}$. The blue curve is from the theory of Persson and
Brener assuming a crack (adiabatic) tip cut-off radius $a_{0}$. The reference
velocity $v_{0}=a_{0}/\tau$ with $a_{0}=E_{0}G_{0}/(2\pi\sigma_{\rm c}^{2})$
in the Persson–Brener cut-off model and $v_{0}=1.41b/\tau$ with
$b=E_{0}G_{0}/\sigma_{0}^{2}$ in the cohesive-zone model.

Finite size effect

At high crack-tip speed (or at low temperatures) the main contribution to the
viscoelastic energy dissipation comes from a region far from the crack tip.
This follows from dimensional arguments: the perturbing deformation
frequencies from the moving crack (velocity $v$) a distance $r$ away from the
crack tip must be of order $v/r$. Thus close to the crack tip the rubber will
effectively be in the glassy state (elastic response) and the dominant
contribution to the energy dissipation will come from regions far from the
crack tip. Hence if the solid has a finite extent, say of linear dimension
$L$, at high enough crack tip speed the solid will, from the point of view of
viscoelastic dissipation, effectively be in the glassy elastic state
everywhere and no viscoelastic energy dissipation will occur during the crack
propagation. Thus one may be tempted to claim that for finite solids
$f\rightarrow 0$ as $v\rightarrow\infty$. This result was used by de Gennes to
argue that for the slab geometry (thickness $d_{0}$), as occur for example in
pressure sensitive adhesives, for large crack tip speed the crack propagation
energy $G(v)$ will decrease with increasing $v$ which may result in mechanical
instabilities[13]. However, I will now shown in that this argument is in fact
not correct and there exist no finite size effect (see also Ref. [16]).

Figure 3:  Crack propagation in a viscoelastic slab clamped between two rigid
flat surfaces. (a) The slab is infinite long ($L=\infty$) in the crack
propagation direction ($x$-direction). (b) The slab is of finite length $L$.
In (a) the segment (of width $\Delta x$) at A is stretched with the strain
$\epsilon$. At B the strain is reduced (high frequency relaxation) by
$\Delta\epsilon=\sigma_{0}/E_{1}$ due to the abrupt decrease in the tensile
stress from $\sigma_{0}$ to $0$ when the crack tip pass the segment. At C, far
away from the crack tip, the strain vanish due to (slow) viscoelastic
relaxation.

Fig. 3(a) shows a fast moving opening crack in a thin viscoelastic slab
(thickness $d_{0}$) under tension. The slab is assumed to be infinite long in
the crack propagation direction. The slab is elongated by $d_{0}\epsilon_{0}$,
and we wait until a fully relaxed state is formed before inserting the crack.
Thus the elastic energy stored in the strip A of width $\Delta x$ and volume
$\Delta V=wd_{0}\Delta x$ (where $w$ is the width of the solid in the
$y$-direction) is

$U_{0}={1\over 2}\sigma_{0}\epsilon_{0}\Delta V={\sigma_{0}^{2}\over
2E_{0}}\Delta V.$

This energy is partly used to break the interfacial bonds and partly
dissipated due to the material viscoelasticity. The crack propagation energy
$G=U_{0}/(w\Delta x)=\sigma_{0}^{2}d_{0}/(2E_{0})$.

Consider now the slab $\Delta x$ as it moves from one side of the crack to the
other side. During this transition it will experience an (elongation) stress
$\sigma(t)$ which for a very fast moving crack can be considered as a step
function where $\sigma=\sigma_{0}$ for $t<0$ and $\sigma=0$ for $t>0$, where
$t=0$ correspond to the case where the segment $\Delta x$ is at the crack tip.
The viscoelastic material will respond to this step-like change in the stress
with its high frequency modulus $E_{1}$ so the strain in the segment $\Delta
x$ will abruptly drop by $\Delta\epsilon=\sigma_{0}/E_{1}$ as the crack pass
the segment. The drop in the elastic energy

$\Delta U={1\over 2}\sigma_{0}\Delta\epsilon\Delta V={\sigma_{0}^{2}\over
2E_{1}}\Delta V,$

is used to break the interfacial bonds (energy $\Delta\gamma=G_{0}$ per unit
surface area), i.e. $G_{0}=\Delta U/(w\Delta x)$ or

$G_{0}={\sigma_{0}^{2}\over 2E_{1}}d_{0}={\sigma_{0}^{2}d_{0}\over
2E_{0}}{E_{0}\over E_{1}}=G{E_{0}\over E_{1}},$

so that $G=G_{0}E_{1}/E_{0}$. The remaining elastic energy stored in the
segment $\Delta x$,

$U=U_{0}-\Delta U={1\over 2}\sigma_{0}^{2}\left({1\over E_{0}}-{1\over
E_{1}}\right)\Delta V,$

is dissipated in the slow viscoelastic relaxation occurring far away from the
crack tip so that finally the material reach its fully relaxed state (zero
strain and stress).

For a solid with a finite extent in all directions the fast relaxation process
at the crack tip is the same as above so the result $G=G_{0}E_{1}/E_{0}$ still
holds. However, for a fast moving crack the time it takes for the crack to
fracture the whole interface, $\Delta t=L/v$, is so short that, in accordance
with the discussion presented earlier, negligible viscous energy relaxation
has occurred during the crack propagation act. Thus when the crack separate
the two solids the viscoelastic solid is still in a strained state. Only after
a possible long time period will it return to a strain (and stress) free
state. Thus in this case the viscoelastic energy dissipation occur in a
process separated from the actual crack propagation [see Fig. 3(b)], and this
fact was overlooked in the earlier energy based discussions of finite size
effects. We conclude there exist no finite size effects and the theory
developed in Ref. [17, 18, 19] is not relevant for crack propagation in
viscoelastic solids.

Figure 4:  The real part of the viscoelastic modulus as a function of
frequency (log-log scale) for a physically crosslinked polymer, which behaves
as a fluid for small frequencies, corresponding to long times (schematic). The
modulus $E_{0}$ in the rubbery region $\omega_{\rm a}<\omega<\omega_{\rm b}$
is approximately constant and much smaller than the modulus $E_{1}$ in the
glassy region (high frequencies).

Application to “solids” with $E(\omega=0)=0$

Schapery applied his viscoelastic crack propagation theory to “solids”, with
has a viscoelastic modulus that vanish for zero frequency (i.e., a relaxation
modulus which vanish for long times)[9, 10]. Such a “solid” is really a liquid
with a non-Newtonian and possible complex rheology, and in this case no
rigorous crack propagation theory can be developed. Thus, for example, the
Johnson, Kendall, and Roberts (JKR) adhesion theory[20] assumes that the solid
deformations far from the crack tip (here the line separating the contact area
from the non-contact area), is characterized by the low frequency (fully
relaxed state) elastic modulus $E_{0}=E(\omega=0)$, which would vanish in the
present case. We note, however, that physically crosslinked polymer materials
may have a rubbery plateau (see Fig. 4 and Ref. [21, 22]) for $\omega_{\rm
a}<\omega<\omega_{\rm b}$, and behave as an elastic solid for all practical
time scales $1/\omega_{\rm b}<t<1/\omega_{\rm a}$, in which case the Persson-
Brener (or cohesive-zone) theory could still be applied but with $E_{0}$ being
the modulus in the rubbery plateau region.

If the adiabatic crack tip radius $a_{0}$ is treated as a constant as the low-
frequency modulus $E_{0}$ is varied then the viscoelastic factor $G/G_{0}$ is
independent of $E_{0}$ for low enough velocities and hence well-defined even
in the limit $E_{0}=0$ (see Appendix B). Nevertheless, the quantities $a_{0}$
and $G_{0}$ are not well defined in the limit $E_{0}=0$ since they both refer
to the adiabatic limit where the “solid” responds as a fluid, where no crack-
like defects can occur.

Discussion

In a recent series of papers Ciavarella et al [23, 24] have criticized the
Persson-Brener theory and claimed that the Persson-Brener theory[14] give a
viscoelastic factor $G/G_{0}=1+f$ which differ strongly from what Ciavarella
et al denote as the exact viscoelastic factor obtained using the cohesive-zone
model[12, 11]. We have shown above that both theories gives nearly the same
result for $G/G_{0}$ if the ill-defined quantity $\sigma_{0}$ in the cohesive-
zone theory is chosen appropriately!

In another paper Popov[25] claim that viscoelasticity increases the JKR pull-
off force with a factor $E_{1}/E_{0}$ in the quasi-static limit. This result
is incorrect: in the quasi-static case the work of adhesion is not influenced
by the viscoelasticity and only in the limit $v\rightarrow\infty$ is the work
of adhesion increased by the factor of $E_{1}/E_{0}$. The result of Popov
would hold if there would be no shortest length scale in the problem but in
reality there is a short distance cut-off denoted by $a_{0}$ above.

Acknowledgments: I thank J.A. Greenwood for supplying the numerical data used
for the $G(v)/G_{0}$ curve for the cohesive-zone model (the red curve in Fig.
1). I thank M. Müser for comments on the manuscript.

Appendix A

Here we give the equations for viscoelastic crack propagation used in the
calculations in Fig. 2.

When a strip of a viscoelastic material is exposed to an oscillating strain
$\epsilon(\omega){\rm exp}(-i\omega t)$ the amplitude of the oscillating
stress

$\sigma(\omega)=E(\omega)\epsilon(\omega)$

This equation define the viscoelastic modulus $E(\omega)$, which is a complex
quantity, where the imaginary part is related to energy dissipation (transfer
of mechanical energy into the disordered heat motion). In the calculation in
Fig. 2 we used the three-element rheological model illustrated in Fig. 1. For
this model the viscoelastic modulus

$E={E_{0}E_{1}(1-i\omega\tau)\over E_{1}-i\omega\tau E_{0}}$ $None$

The ratio between the high frequency and low frequency modulus, $E_{1}/E_{0}$,
is typically very large, e.g., $\sim 1000$.

In the Persson-Brener theory the crack propagation energy
$G(v)=G_{0}a(v)/a_{0}$, where $a(v)$ is the velocity dependent effective crack
tip radius, and where $a_{0}=E_{0}G_{0}/(2\pi\sigma_{\rm c}^{2})$. The
fracture energy $G(v)$ satisfy[14]

${G_{0}\over G}=1-E_{0}{2\over\pi}\int_{0}^{\omega_{c}}d\omega\
F(\omega){1\over\omega}{\rm Im}{1\over E(\omega)}$ $None$

where

$F(\omega)=\left[1-\left({\omega\over\omega_{c}}\right)^{2}\right]^{1/2}$

where $\omega_{c}=2\pi v/a$. For numerical calculations it is more convenient
to reformulate (A2) into[6, 17]

${G_{0}\over
G}=1-{E_{1}{2\over\pi}\int_{0}^{\omega_{c}}d\omega{1\over\omega}F(\omega){\rm
Im}{1\over E(\omega)}\over
1+E_{1}{2\over\pi}\int_{0}^{\infty}d\omega{1\over\omega}{\rm Im}{1\over
E(\omega)}}.$ $None$

Since $\omega_{c}$ depends on $a$ (and hence on $G$ since $G=G_{0}a/a_{0}$)
this is an implicit equation for $G(v)$. Thus the theory gives both crack
propagation energy $G(v)$ and the (velocity-dependent) radius of the crack
tip,

$a(v)=a_{0}{G(v)\over G_{0}}={E_{0}G\over 2\pi\sigma_{\rm c}^{2}}.$ $None$

Since $E(\omega)$ typically varies with $\omega$ over very many decades in
frequency, for the numerical evaluation of the integrals in (A3) it is
convenient to write (see Ref. [6]) $\omega=\omega_{0}e^{\xi}$, so that if
$\omega$ varies over $\sim 30$ decades, $\xi$ varies only by a factor $\sim
100$.

In the cohesive-zone model assuming that the stress is constant and equal to
$\sigma_{0}$ for $0<u<h_{0}$ and $\sigma=0$ for $u>h_{0}$, and assuming the
viscoelastic modulus (A1), one get[12]:

${G_{0}\over G}={E_{0}\over E_{1}}+{1\over 2}\left(1-{E_{0}\over
E_{1}}\right)\alpha\int_{0}^{1}d\xi\ H(\xi)e^{-\alpha(1-\xi)},$ $None$

where $\alpha=l/(v\tau)$ and

$H(\xi)=2\xi^{1/2}-(1-\xi){\rm ln}\left({1+\xi^{1/2}\over
1-\xi^{1/2}}\right),$

and the width $l$ of the crack tip process zone

$l={\pi\over 4}{E_{0}G\over\sigma_{0}^{2}}$ $None$

Here we have assumed plain stress; for plain strain $E$ must be replaced with
$E/(1-\nu^{2})$, where $\nu$ is the Poisson ratio. Since $l$ depends on $G$,
(A5) is an implicit equation for $G(v)$.

Figure 5:  The viscoeladstic factor $G/G_{0}$ as a function of the crack tip
speed (log-log scale) for the case $E_{1}=100$, $\tau=1$ and $E_{0}=1$ (red
curve) and $E=0.1$ (blue curve). In the calculation we have used the same
adiabatic crack tip radius $a_{0}=0.03$.

Appendix B

Using the viscoelastic modulus (A1), which we can also write as

${1\over E}={1\over E_{1}}+\left({1\over E_{0}}-{1\over E_{1}}\right){1\over
1-i\omega\tau},$

in (A2) or (A3) in Fig. 5 we show the viscoelastic factor $G/G_{0}$ as a
function of the crack tip speed on a log-log scale. We have used $E_{1}=100$,
$\tau=1$ and $E_{0}=1$ (red curve) and $E=0.1$ (blue curve), and the same
adiabatic crack tip radius $a_{0}=0.03$. Note that for low crack tip speed
$G/G_{0}$ is independent of $E_{0}$ and hence well-defined also in the “fluid”
limiting case $E_{0}=0$.

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