Towards Mechanochemistry of Fracture and Cohesion:

General Introduction and the Simplest Model of Velcro

by Michael A. Grinfeld and Steven B. Segletes

ARL-TR-5309 September 2010

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Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

ARL-TR-5309 September 2010

Towards Mechanochemistry of Fracture and Cohesion:

General Introduction and the Simplest Model of Velcro

Michael A. Grinfeld and Steven B. Segletes

Weapons and Materials Research Directorate, ARL

Approved for public release; distribution is unlimited.
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September 2010 Final

ARL-TR-5309

Approved for public release; distribution is unlimited.

August 2009-March 2010

Towards Mechanochemistry of Fracture and Cohesion: General Introduction and

the Simplest Model of Velcro

Michael A. Grinfeld AH80

Steven B. Segletes

U.S. Army Research Laboratory

ATTN: RDRL-WMP-B

Aberdeen Proving Ground, MD 21005-5066

This is the first in a series of reports.

This project is aimed at a combined theoretical and experimental analysis of adhesives. The theoretical part of it is based on

usage of macroscopic methods and principles: continuum mechanics and thermodynamics. The full extent of the

experimental part will be made more clear in a later report. This part, however, is expected to be as important as the theoretical

part. We plan to begin experimenting with the simplest systems; basically, the macrosystems based on velcro tapes and sheets.

adhesion, coating, film, bond, Griffith model, Frenkel’ model

UU 26

Michael A. Grinfeld

Unclassified Unclassified Unclassified 410-278-7030

ii
Contents

List of Figures iv

1. Novel Possibilities for Application of Mechanochemistry 1

2. Current Theoretical Difficulties in the Thermodynamics of Heterogeneous Systems and

Fracture 3

3. Thermodynamic Model of Velcro 7

3.1 Derivation of Equations 5 and 6........................................................... 8

4. Conditions of Equilibrium of Inextensible Velcro 9

5. Condition of Stability of Inextensible Velcro 11

6. Extensible Velcro Model 11

6.1 Derivation of Equations 20 and 21 ........................................................ 12

7. Conditions of Equilibrium of Extensible Velcro 13

8. Condition of Stability of Extensible Velcro 15

9. Conclusion 15

10. References 16

Distribution List 19

iii
List of Figures

Figure 1. The simplest thermodynamic velcro model.............................................. 7

iv
1. Novel Possibilities for Application of Mechanochemistry

There are several reasons for this project. An acquaintance with any of the annual reviews of the

U.S. Army Research Laboratory (ARL) (see, for instance, reference 1) or almost any issue of the

Advanced Materials and Processes Technology Information Analysis Center (AMPTIAC) (see for

instance, references 2–5 shows that numerous directions of research at the ARL are associated

with consideration of mechanochemical systems). Among those are fracture of solids, analysis

of solid explosives, chemical reactions in solids, environmental stress corrosion and coating with

protective layers, creating composites by adhesives, technologies of the micro-electro-mechanical

systems (MEMS) mounting, growth of semiconductor thin films, development of autonomic

electric batteries, modeling of soft and hard tissues, etc. In fact, a common thermodynamic

background of the macro-analysis of all those systems was not widely recognized mostly because

of the “barbarism of specialization” (6) (the unavoidable socio-scientific phenomenon akin to the

“confusion of tongues at Babel”).

Let us give several examples. First of all, by the very nature of their profession, the Department

of Defense (DoD) and Army researchers and practitioners deal a lot with fracture of solid

structures. Fracture is nothing else but a conversion of the (macroscopic) elastic energy into the

energy of the broken bonds, i.e., the “chemical” energy. Typically, the “chemical” energy of the

bonds is not mentioned explicitly and remains hidden behind the generic term of “surface”

energy. The approach based on macroscopic description of the chemical bonds and their

inclusion into the variational (later thermodynamic) scheme of classical mechanics can be traced

back to the Laplace model of surface tension in liquids. Later, the same approach was used by

Gibbs (7). His use of a thermodynamic variational formalism allowed analysis of heterogeneous

systems and the various physico-chemical transformations therein. Several decades later,

Griffith (8) suggested applying a similar approach to the analysis of fracture. Further

developments along these lines may be found among the publications of Mott (9), Frenkel’ (10),

Barenblatt (11), Raizer (12), and in the comprehensive reviews on fracture in reference 13.

Starting from 2000, similar ideas have received some attention at ARL (14, 15).

The ideas of general thermodynamics, and of the Gibbs approach, in particular, have been

expanding to include chemical reaction. While not done by Gibbs (the main contribution is due

to Ostwald, van Hoff, and Arrhenius), the main construct of this generalization is associated with

the Gibbs Grand thermodynamic potential. This potential is often (and not always correctly)

1
mentioned as the Gibbs potential or chemical potential. In this very name, the fundamental link

between the mechanics and chemistry is particularly obvious.

An important further step in joining chemistry and mechanics belongs to Bridgman. In the

experiments of 1930s (see the monograph [16]), he noticed the phenomenon of the rapid growth

of the rate of chemical reactions under the action of shear stresses. His studies and observations

have been continued by the Russian group of Enikolopyan (17–19). It is well documented that

sufficiently strong mechanical stresses are able to change the rates of chemical reactions by orders

of magnitude.

The role of mechanochemistry in the thermodynamics of solid explosives is far too obvious to

warrant dwelling on a more detailed discussion. We would, however, like to mention here the

recent studies of triggering hot spots in solid explosives—a project in the stage of rapid further

development (20, 21).

The problem of coatings include two aspects of mechanochemistry: adhesion and stress

corrosion. A corresponding review of the DoD requirements can be found in several issues of the

Advanced Materials Manufacturing & Testing Information Analysis Center (AMMTIAC) (3, 4).

Related studies at the ARL are presented in references 22–25.

The important role of mechanochemistry in the problem of individual and stationary electric

power supply becomes obvious from the fact that the electrostatic force between charges at small

distance is capable of generating huge stresses. The review of various DoD related issues of

power supplies can be found in reference 5.

The scope of applications of mechanochemistry is growing rapidly. The ideas of macroscopic

thermodynamics and of the Gibbs paradigm have penetrated into nanotechnology including

quantum dots and quantum wire growth, epitaxial growth of nano-scale films, etc. Another area

of rapidly growing application of mechanochemistry is the biomechanics of tissue, including soft

and bone tissue. For soft tissue, the mechanochemistry is associated with the mechanism of

adhesion. For bone tissue, the applications of mechanochemistry are associated with the

important and well-documented phenomenon of bone remodeling under load.

So, we are able to see that there are a plethora of promising applications of mechanochemistry.

In Russia, there is a large organization called the Institute of Mechanochemistry of the Russian

Academy of Science in Novosibirsk, which is responsible for several reviews on

mechanochemistry (see, for instance, references 26, 27).

2
There are plenty of novel possibilities for DoD-related studies. The very term

“mechanochemistry” is more and more often heard at ARL. To date, this term is associated with

mechanical milling of solids and the making of micro- and nano- powders. But, in fact, the scope

of applications of mechanochemistry for DoD researchers is much wider. At ARL, the analysis

of adhesives on the nanoscale level is mostly based on molecular dynamics simulations and

similar techniques. As with any other technique, those approaches have several limitations. It is

obvious, however, that one approach to modeling is, mildly speaking, underrepresented at the

ARL—the approach based on macroscopic thermodynamic methods. These methods are deeply

developed in many other branches of science. They are, basically, bare-hands methods leading to

deep penetration into the subject with minimal effort. They provide researchers with intuitively

transparent models, simple formulas, and reliable numerical schemes for subsequent

computer-based modeling (relying, generally, on finite-element and similar techniques). We

would like to fill this currently existing gap at ARL. We hope that, as a result of this effort, a new

direction of research at ARL will be seeded: the thermodynamic analysis of the adhesive

systems.

2. Current Theoretical Difficulties in the Thermodynamics of

Heterogeneous Systems and Fracture

The second target of this project is quite different from the first one. It is connected with the

current state-of-the-art of thermodynamic methods in the theory of phase transformations and

fracture in solids. This state-of-the-art can be characterized by a single word: crisis. It might

look surprising when somebody talks about crisis in the era when thousands of papers appear in

the scientific journals every year. In those sometimes sophisticated and elaborate papers, it is

claimed that theory is in remarkable harmony with the experiment. If so, is there any reason to

talk about crisis? Yes, there is.

The crisis is in the predictive ability of the current thermodynamic theories of heterogeneous

systems with solid phases. The oft-demonstrated agreement of the theory and experiment should

not be exaggerated. It is basically achieved by pumping the models with sufficiently many

parameters. As John von Neumann once paraphrased Cauchy: “With four parameters I can fit

an elephant, and with five I can make him wiggle his trunk.” But try to make an unbiased

prediction based on those allegedly universal theories. In doing so, one will find that those

predictions do not have much in common with reality.

3
The problem is not only with the number of tuning parameters per se. More importantly, those

parameters require permanent and essential adjustments even when dealing with very close

experimental settings. Of course, this situation is intolerable. We heard many complaints from

fracture theorists that, in its current stage, fracture models are not able to predict numerous

phenomena. For example, they are not able to predict the curvilinear trajectories of the

propagating cracks, among many other phenomena.

Many critical words have especially come from theories based on the Griffith energetic approach,

initiated in the classical paper (8) and continued in numerous other studies (9–13). In the Griffith

model, the release of the bonds’ energy is fulfilled across a single surface, called a crack. The

Griffith “growing-crack” model was developed in various directions and has become the basis of

numerous applications, despite its essential shortcomings.

We can add one more example to those complaints. One of us (Michael Grinfeld), over a lengthy

period of time, was involved in the development of the Gibbs paradigm of analysis of

heterogeneous systems with phase transformations.1 The methods developed in reference 28

allow one to explore not only the condition of equilibrium of heterogeneous systems but also their

stability and, in particular, the morphological instabilities of phase interfaces. In addition to

many useful results, the theory, developed in reference 28, leads also to multiple results that have

nothing in common with physical reality. These methods are already used in several research

areas at ARL. It is clear that the cases in which the theory outlined in reference 28 appears to be

misleading should be explored and made transparent for ARL users.

The unsatisfactory state-of-the-art in the theories of phase transformations and fracture is the

second reason of the current (and future) study. To make some progress in understanding the

existing difficulties, we propose exploring the strong features and weaknesses of the energetic

(thermodynamic) methods by making theoretical and experimental studies of macroscopic

adhesive systems. Basically, we plan to explore the mechanisms of attachment-detachment of the

macroscopic adhesive tapes similar to the widely available Velcro2-tapes.

We strongly believe that by dealing with the macroscopic adhesives we will have a better chance

of understanding of adequateness of the current theories of fracture, phase transformations,

crystal growth, etc. What are our reasons for such aspirations?

1Griffith fracture theory is nothing more than the very special heterogeneous system “solid - vacuum”.

2Velcro is a hook-and-loop fastener system, which is a registered trademark of Velcro Industries B.V. In this report,

we use the lowercase term “velcro” to refer to generic hook-and-loop fasteners in the style of Velcro.

4
Our hopes are mostly based on the larger likelihood of experimental verification of the theoretical

predictions. Many experiments with adhesives can be made with bare hands and naked eyes.

Similar experiments for heterogeneous systems, undergoing phase transformations and fracture,

for example, would require very sophisticated experimental skills and expensive equipment. We

realize that many researchers will ask us: “Is there anything in common between adhesion,

fracture, and phase transformations?” The answer is affirmative. From the standpoint of

thermodynamics, all these systems include, as an essential feature, the energy exchange between

two storages of the accumulated energy—the energy accumulated due to reversible elastic

deformations and the “chemical” energy stored in the chemical bonds. It is for these reasons that

the systems require almost identical thermodynamic methods.

Thermodynamic methods are essentially macroscopic by origin and nature. They appear in the

analysis of macroscopic engineering systems. They have been reliably validated in numerous

macroscopic experiments and observations. Naturally, researchers later applied those methods to

the systems way beyond the area in which they had been originally validated. In particular, the

approaches are widely used in “all” physics at the nano-, micro-, and macro- levels. It is typical

these days for macro-language and macro-cartoons to be widely used in modeling even

nano-systems. There is nothing wrong with that as long as we realize that, while some aspects of

the nano-system are like those of their macroscopic counterparts, others are completely different

in nature. This unavoidable duality is emphasized in the very name “Quantum Mechanics” of

this prototypical discipline.

In our opinion, mechanochemistry shares similar methodological features with quantum

mechanics in what concerns the above-mentioned duality.

The mechanical component of mechanochemistry encourages the usage of macroscopic methods,

and of thermodynamics, in particular. The chemical component encourages us to think on the

nano- and atomic levels and use discrete modeling. We can capitalize, however, on the

experience of our predecessors, who long ago made progress extracting those macroscopic

features of the chemical subsystems, which can be analyzed by means of macroscopic methods.

As a result, there can presumably be found areas that permit analysis of mechanochemical

systems by means of relatively simple thermodynamic methods. From a purely thermodynamic

point of view, the central problem of mechanochemistry is the exchange of energy between the

(long-range) elastic energy and the (short-range) energy accumulated in individual bonds.

Fortunately, such an approach to the analysis of the mechanochemistry of fracture has been

developed by the Russian theorists (10–12) and has received worldwide attention. It was initiated

5
by Frenkel’ (10) who revealed conceptual problems in the Griffith paradigm and suggested the

approach to fixing them by adding a more elaborate model of the subsystem of chemical bonds.

He noticed, for instance, that the Griffith theory does not give the answer to the key question of

the origin and disappearance of micro-cracks. In Griffith theory, the influence of the chemical

bonds is modeled by means of adding the surface energy term (as was earlier suggested by

Laplace in his variational theory of capillarity). Frenkel’ suggested not to limit the analysis with

a single constant to describe the surface energy density but, rather, to include explicitly a model of

the chemical bonds, described as a system of springs.

The Frenkel’ spring-model of the chemical bond should not be taken too literally. Actually, it

substitutes for various mechanisms like electrostatic, van der Waals, and other types of physical

forces. But those more elaborate models of forces cannot be directly included in the

thermodynamic scheme, whereas the spring model can.

In this project, we will develop further the approach of Frenkel’ (10) and his followers (11, 12).

In his original paper, Frenkel’ modeled the solid with the help of a bending-plate model. His

followers used the model of an isotropic, linear-elastic solid instead of a bending plate. From a

technical point of view, this implies significant added complexity because of the transition from

ordinary differential equations to partial differential equations. More complicated models

obviously diminish the depth of research attainable from analytical methods. For instance, the

problem of stability of the equilibrium configurations has not been resolved in the

studies (8, 10–12). Being quite difficult in its present form, these models (8, 10–12) are not

precise enough to address the issues of stability. It is because their “elasticity” component is

based on the linear theory—and linear elasticity is known to give misleading results in what

concerns the analysis of stability.

In view of the facts mentioned previously, we choose the following approach. First, we will limit

ourselves to the models adequately described by ordinary differential equations. Second, we will

take into account only the membrane energy of the adhesive tape. That will allow us to avoid any

assumptions of linearity at the stage of formulating the problems. Certain simplifying

assumptions will appear later, at the stage of mathematical solution of the equations. As a result,

we will get two advantages. First, we will have full confidence of the chosen model, since no

uncontrollable assumptions have been made. In other words, we are guaranteed that our model

will be consistent from the standpoint of the basic principles of mechanics, physics, and

thermodynamics. Secondly, we will consistently address the issue of stability of the equilibrium

configurations. This is very important because instability is a qualitative feature, which, if it

exists, can be easily detected.

6
In this first report, we dwell on the simplest problem, the analysis of which does not require even

differential equations—elementary algebra is sufficient for our purposes.

3. Thermodynamic Model of Velcro

State-of-the-art in modeling adhesives can be found in monographs (29–31). More specific

studies of Velcro are presented in references 32 and 33. In the 1990s, velcro-ceramic composites

were developed by the U.S. Army and U.S. Air Force for the purpose of providing armor

protection while maintaining flexibility of manufacture and assembly. The Velcro component of

the composite tiles play two roles: (1) it provides the fastening capability and (2) it gives a degree

of shock resistance essential for the survival of the brittle ceramic. An analysis of these tiles, in

the current context, would require a two-dimensional (2-D) approach using partial differential

equations and multi-dimensional calculus of variations. Such an analysis goes beyond the scope

of this report, which is based on one-dimensional (1-D) analysis and elementary calculus.

Consider, instead, the system shown in figure 1. Velcro is attached to a rigid flat substrate. The

velcro fabric is assumed to be an inextensible thread of length L. The thread is loaded by the

weight mg. Let xf and xR be the coordinates of the left and right ends of the attached part of the

velcro. We assume that the velcro has negligible resistance to bending. However, despite the

fact that the velcro is modeled to contain no (long-range) elastic energy, it nonetheless carries a

(short-range) energy of distributed chemical bonds between the velcro and its attached substrate.

x

z

w h z

f

x R

x

 

Figure 1. The simplest thermodynamic velcro model.

7
Let Lat be the length of an attached part of the velcro with the distributed chemical energy per

unit length equal to σat, and Lde = L − Lat = L − xR − xf is the length of the detached part of

the velcro with the distributed chemical energy per unit length equal to σde. Then, the total

accumulated chemical energy Echem is given by the formula

Echem = σatLat + σdeLde . (1)

Neglecting the radius of the pulley, we obviously have the following geometric formula for the

detached part of the thread:

Lde = h − zw +

q

h

2 + x

2

f

. (2)

In addition to the accumulated chemical energy equation 1, the system also possesses gravitational

energy Egrav. Thus, the total energy Etotal of the system can be presented in the form

Etotal = Egrav + Echem = mgzw + σatLat + σdeLde (3)

or, alternately, as

Etotal = Egrav + Echem = mgzw + σdeLde + σat(xR − xf ) . (4)

In the presented model, the quantities σde, σat, g, xR, L, and h are fixed. In essence, the total

energy Etotal can be presented as a function of a single variable xf :

Etotal = Etotal(xf ) = mg q

h

2 + x

2

f − xf



+ σeffxf + C , (5)

where

σeff = σde − σat (6)

and C is a constant.

3.1 Derivation of Equations 5 and 6

Let us derive the key formulas of the analysis, namely equations 5 and 6. With the help of the

relationship 2 we can rewrite the total energy in the form

Etotal = mgzw + σde h − zw +

q

h

2 + x

2

f



+ σat(xR − xf ) . (7)

8
The coordinate zw can be excluded from the expression of the total energy, equation 7, with the

help of the geometric formula for the total length:

h − zw +

q

h

2 + x

2

f + xR − xf = L , (8)

implying

zw = h +

q

h

2 + x

2

f + xR − xf − L . (9)

Eliminating zw between equations 7 and 9, we arrive at the expression 5; the constants σeff and C

are given by the formulas

σeff ≡ σde − σat

C = mg(h + xR − L) + σdeL − (σde − σat)xR

. (10)

4. Conditions of Equilibrium of Inextensible Velcro

The condition of equilibrium of the velcro with respect to the coordinate xf of detachment can be

found by equating to zero the first derivative of the function Etotal(xf ):

dEtotal(xf )

dxf

= mg





xf

q

h

2 + x

2

f

− 1



 + σeff = 0 . (11)

In terms of the angle θ, shown in the figure 1, the equation 11 can be rewritten as follows:

1 − cos θ =

σeff

mg

. (12)

We can also write equation 12 as follows:











sin

θ

2











=

s

σeff

2mg

. (13)

In the system described, the loading of the velcro was made by gravity. In this very system,

however, the force mg could obviously have been replaced by an equivalent tension T applied to

9
the velcro. Thus, it is reasonable to rewrite equation 13 as











sin

θ

2











=

rσeff

2T

. (14)

So far, we have established formula 14 for the case of gravitational loading only. It is rather

appealing to assume, though, that formula 14 has much wider applications for different loading.

This is not as obvious as one might think. In fact, the concept of force, and of tension, in

particular, remains quite controversial. This is especially true when electro- and magnetostatic

forces are taken into account. This is also true when the influence of the chemical bonding

should be taken into account. For instance, the distinctions between the surface tension and the

surface energy remain an issue of intense debate3.

The equation 14 is the central formula of the purely thermodynamic model of the velcro

detachment and of this report. This formula shows that the angle θ diminishes when the tension

T grows. In particular, the angle approaches zero when the tension approaches infinity—this

qualitative conclusion seems intuitively transparent. The quantitative validity of this formula

requires, as always, experimental verification.

For the angle θ = π/2, the formula 14 implies

σeff = mg . (15)

Its physical meaning is absolutely transparent. At this very angle, the incremental detachment

dxf , causing the release the chemical energy dEchem = (σde − σat)dxf , completely converts into

the decrease of the load height −dzf = dxf , causing consumption of the gravitational potential

dEgrav = −mg dzf = mg dxf . By equating these energy increments, we arrive at formula 15.

From the standpoint of the physical intuition, equation 15 is not transparent at all. It can be used,

however, for the macroscopic interpretation and measurement of the microscopic (or, if one

wishes, nanoscale) concept σeff.

3From the conceptual standpoint, it is remarkable and instructive that in our thermodynamic analysis the concept of

tension within the velcro was completely eliminated from the derivation of the key formulas 12 and 13. The readers

can legitimately argue that instead of basing it on the concept of tension we based our analysis on the much more vague

concept of energy. This is a valid criticism that requires immediate justification. Our justification is the following:

the vagueness of the concept of force often implies logical, mathematical, and computational inconsistencies of the

physical models. In this respect, the development of models based on the energy approach carries a huge advantage.

Even outstanding researchers quite often confuse this logical advantage with the universality of the law of energy

conservation. We will not dwell here on a more detailed discussion of this somewhat metaphysical issue.

10
The magnitude of σeff = σde − σat can, in fact, change in time due to various micro- and

nanoscale phenomena like chemical contamination, absorption, desorption, environmental

changes, etc. The result will be indicated by the macroscopic variation in the detachment length.

5. Condition of Stability of Inextensible Velcro

The issue of stability of the equilibrium configuration can be addressed by calculating the second

derivative of the function Etotal(xf ). The direct calculation gives us

d

2Etotal(xf )

dx2

f

=

mg

(h

2 + x

2

f

)

3/2

. (16)

According to equation 16, the second energy variation is always greater than zero. Thus, in the

framework of thermodynamic theory, the equilibrium configuration is always stable.

6. Extensible Velcro Model

We can make one more important extension to the model without any necessity for considering

ordinary or partial differential equations.

We now assume that the total internal energy Ei of the system consists of the distributed

“elongation” energy Eel, accumulated in the detached part of the stretched thread, and the

distributed “chemical” energy Echem of the velcro bonds:

Ei = Eel + Echem . (17)

We assume that the attached part of the velcro cannot undergo any stretching. Let Eel be defined

by the following relationship:

Eel = L

o

de e(λ) , (18)

where L

o

de is the “reference” length of the detached part of the thread and λ is the elastic

elongation of the thread.

11
The total chemical energy of the system is equal to

Echem = σde(λ) L

o

de + σat(xR − xf ) (19)

where σde(λ) is a positive material function and σat is a positive material constant.

In this case, the total energy E

ext

total(xf , λ) is given by an expression analogous to equation 5:

E

ext

total(xf , λ) = mg hq

h

2 + x

2

f + (xR − xf − L

o

)λ

i

−

[σde(λ) + e(λ)] (xR − xf − L

o

) − σatxf + C

∗

, (20)

where L

o

is the velcro’s full unstretched “reference” length and

C

∗ ≡ σatxR + mgh . (21)

In the case when the surface energy σde of the detached film is independent of λ, the equation pair

20, 21 can be replaced with the following expression:

E

ext

total(xf , λ) = mg hq

h

2 + x

2

f + (xR − xf − L

o

)λ

i

−

(xR − xf − L

o

)e(λ) + (σde − σat)xf + C

∗∗

, (22)

where

C

∗∗ ≡ −Loσde + (σat − σde)xR + mgh . (23)

6.1 Derivation of Equations 20 and 21

With the help of the relationships 17–19, we can rewrite the total energy in the form

E

ext

total = mgzw + σde(λ)

1

λ



h − zw +

q

h

2 + x

2

f



+ σat(xR − xf ) (24)

The coordinate zw can be eliminated from the expression for the total energy, equation 7, with the

help of the geometric formula of the total length:

1

λ



h − zw +

q

h

2 + x

2

f



+ xR − xf = L

o

(25)

implying

zw = h +

q

h

2 + x

2

f + (xR − xf − L

o

)λ . (26)

12
Eliminating zw between equations 7 and 9, we arrive at the following chain of the relationships,

E

ext

total = mgzw + σde(λ)

1

λ



h − zw +

q

h

2 + x

2

f



+ σat(xR − xf )

=



mg − σde(λ)

1

λ



zw + σde(λ)

1

λ



h +

q

h

2 + x

2

f



+ σat(xR − xf )

=



mg − σde(λ)

1

λ

 h

h +

q

h

2 + x

2

f + (xR − xf − L

o

)λ

i

+

σde(λ)

1

λ



h +

q

h

2 + x

2

f



+ σat(xR − xf )

= mg hh +

q

h

2 + x

2

f + (xR − xf − L

o

)λ

i

−

σde(λ)

1

λ

h

h +

q

h

2 + x

2

f + (xR − xf − L

o

)λ

i

+

σde(λ)

1

λ



h +

q

h

2 + x

2

f



+ σat(xR − xf )

= mg hq

h

2 + x

2

f + (xR − xf − L

o

)λ

i

− σde(λ)(xR − xf − L

o

) −

σatxf + σatxR + mgh

, (27)

which will ultimately result in equations 20 and 21.

7. Conditions of Equilibrium of Extensible Velcro

Equating to zero the first derivative of the energy E

ext

total(xf , λ), we arrive at the following

equations of equilibrium:

∂Eext

total(xf , λ)

∂λ = mg(xR − xf − L

o

) −

 

dσde

dλ +

de

dλ!

(xR − xf − L

o

) = 0 (28)

and

∂Eext

total(xf , λ)

∂xf

= mg





xf

q

h

2 + x

2

f

− λ



 + σde − σat + e = 0 . (29)

The force equilibrium condition 28 implies the relationship

de

dλ = mg −

dσde

dλ . (30)

13
Traditionally, the derivative de/dλ would be identified with the tension T of the extensible thread.

Formula 30 indeed confirms this vision, provided that the surface tension σde is zero or that it is

independent of the elongation λ. However, that is not the case. There is a nontrivial dependence

on σde(λ). It is wrong to think that such a dependence is somehow exotic. On the contrary, it is

a typical situation in physical chemistry, and in mechanochemistry, in particular.

Let us turn now to the condition of chemical equilibrium 29. With the help of the condition of

mechanical equilibrium 30, we can rewrite equation 29 as follows:

mg





xf

q

h

2 + x

2

f

− λ



 + σde − σat + e = 0 (31)

or alternately as

d

dλ

e + σde − σat

λ

= 0 . (32)

Consider the model of extensible velcro described by the following conditions

σde = const.

e =

K

2

(λ − 1)2

(33)

In this case, the equilibrium equations 30 and 31 provide the relationships

K(λ − 1) = mg (34)

and

mg





xf

q

h

2 + x

2

f

− λ



 +

K

2

(λ − 1)2 + σde − σat = 0 . (35)

With the help of equation 34, equation 35 can be rewritten as follows:

λ − cos θ =

mg

2K

+

σde − σat

mg

, (36)

which generalizes equation 12 and allows the calculation of the corrections for the finite

magnitude of the elasticity module K.

14
8. Condition of Stability of Extensible Velcro

Calculating the second derivative of energy in vicinity of equilibrium configuration, we get

∂

2Eext

total(xf , λ)

∂λ2

=

d

2

(σde + e)

dλ2

(L

o − xR + xf )

∂

2Eext

total(xf , λ)

∂λ∂xf

= 0

∂

2Eext

total(xf , λ)

∂x2

f

= mg

1

(h

2 + x

2

f

)

3/2

. (37)

In the case of constant surface energy density, equation 37 implies stability of the equilibrium

configurations.

9. Conclusion

In this first report on the mechanochemistry of fracture and adhesion, we discussed possible

applications of this discipline for the needs of Army researchers.

We formulated the simplest 1D model of a velcro-like adhesive and applied it to an analysis of

equilibrium and stability, using an energy (variational) approach. The velcro was assumed to be

partially attached to a rigid flat substrate. No assumption of small displacement of the velcro was

made. For both the cases of inextensible and extensible velcro, the analysis appeared to be rather

elementary and required only the standard techniques of multivariate calculus. No special

knowledge of differential equations was required. We established the conditions of equilibrium

for the system and proved that those configurations are always stable.

The model used here is a direct analogy of the Griffith crack model of brittle fracture. In the

Griffith approach, the energy of the bond is modeled by introducing a constant called surface

tension. The Griffith model was justly criticized by Iakov Frenkel’ (10), who suggested a remedy

based on introducing a finite zone of partial decohesion. This idea was further developed by

many researchers, starting with the pioneering publications of Barenblatt et al.(11). In the zone

of partial decohesion, the molecular interaction is modeled with extensible springs. One of our

further goals will be to combine the model in this report with Frenkel’s spring model.

15
10. References

1. U.S. Army Research Laboratory Annual Review, 2008.

2. Army Materials Research: Transforming Land Combat Through New Technologies.

AMPTIAC Quarterly 2004, 8 (4).

3. Corrosion Conscious Design. AMPTIAC Quarterly 2005, 9 (3).

4. New Technologies & Training Methods Revolutionize the Coating Process. AMMTIAC

Quarterly 1 (4).

5. Powering the Future Force: New Power & Energy Technologies for the Warfighter.

AMMTIAC Quarterly 4 (1).

6. Ortega-y Gasset, J. The “Barbarism” of Specialization. In Revolt of the

Masses; http://www.ellopos.net/notebook/masses/index.htm: (accessed 2010).

7. Gibbs, J. On the Equilibrium of Heterogeneous Substances. Trans. Connecticut Acad.

1875–1878, 3, 108–248, 343–524.

8. Griffith, A. The Phenomenon of Rupture and Flow in Solids. Philosophical Transactions of

the Royal Society of London (A.) 1921, 221, 163–198.

9. Mott, N. Fracture of Metals: Theoretical Considerations. Engineering 1948, 165 (4275,

4276).

10. Frenkel’, Y. I. Theory of reversible and irreversible cracks in solid bodies. Zhural

Tekhnicheskoy Fiziki 1952, 22 (11).

11. Barenblatt, G. I.; Christianovich, S. A. “On the Fracture of Roofs in Mine Working”,

Izvestiya (Bulletin), USSR Academy of Sciences, Division of Technical Sciences, 1955 No.

11, 73–86 (in Russian).

12. Raizer, Y. P. Physical Principles of the Theory of Brittle Fracture Cracks. Soviet Physics

Uspekhi 1970, 13 (1).

13. Lebowitz, H. Fracture: An Advanced Treatise; Academic Press: New York, 1968–1972

(in 7 volumes).

16
14. Grinfeld, M. A.; Wright, T. W. Thermodynamic of Brittle Fracture; ARL-TR-3659; U.S.

Army Research Laboratory: Aberdeen Proving Ground, MD, 2005.

15. Grinfeld, M. A.; Schoenfeld, S. E.; Wright, T. W. Failure Fronts in Brittle Materials and

Their Morphological Instabilities; ARL-TR-3660; U.S. Army Research Laboratory:

Aberdeen Proving Ground, MD, 2005.

16. Bridgman, P. W. The Physics of High Pressure; Dover: New York, 1970.

17. Enikolopyan, N. S.; Vol’eva, V. B.; Khzardzhyan, A. A.; Ershov, V. V. Explosive Chemical

Reactions in Solids. Doklady Akad. NAUK 1987, 292 (177), 1168-1168.

18. Enikolopyan, N. S. Detonation, Solid-Phase Chemical Reaction. Doklady Akad. NAUK

1988, 302 (862).

19. Jaroslavsky, M. A. Rheological Explosion; Institute of Geophysics: Moscow, 1982 (in

Russian).

20. Grinfeld, M.; Bjerke, T. The Impact-Induced Triggering of Hot Spots in Energetic/Explosive

Materials Part 2: Adiabatic Temperature Distribution Near a Spherical Hole;

ARL-TR-4965; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, 2009.

21. Grinfeld, M. The Impact-Induced Triggering of Hot Spots in Energetic/ Explosive Materials

Part 3: An Elliptic Air-Filled Inclusion Within an Unbounded Isotropic Plane;

ARL-TR-4963; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, 2009.

22. Sands, J. M.; McKnight, S. H.; Fink, B. K. Formulation of Toughened Paste Adhesives for

Reduced-Pollutant Electron Beam Repair and Assembly of Composite Structures;

ARL-TR-2270; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, 2000.

23. Sands, J. M.; McKnight, S. H.; Fink, B. K. Interpenetrating Polymer Network (IPN)

Adhesives for Electron Beam Cure; ARL-TR-2321; U.S. Army Research Laboratory:

Aberdeen Proving Ground, MD, 2000.

24. Grendahl, S. M. Environmentally Assisted Cracking Concerns for Cadmium Replacement;

ARL-TR-3099; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, 2003.

25. Pappas, D.; Copeland, C.; Jensen, R. Wettability Tests of Polymer Films and Fabrics and

Determination of Their Surface Energy by Contact-Angle Methods; ARL-TR-4052; U.S.

Army Research Laboratory: Aberdeen Proving Ground, MD, 2007.

17
26. Levitas, V. Continuum Mechanical Fundamentals of Mechanochemistry. In High Pressure

Surface Science and Engineering; Gogotsi, Y., Domnich, V., Eds.; IOP

Publishing: Bristol, 2004.

27. Beyer, M.; Clausen-Schaumann, H. Mechanochemistry: The Mechanical Activation of

Covalent Bonds. Chem. Rev. 2005, 105 (8), 2921–2948.

28. Grinfeld, M. A. Thermodynamic Methods in the Theory of Heterogeneous

Systems; Longman: Sussex, 1991.

29. Brockmann, W. Adhesive Bonding: Materials, Applications and Technology; Wiley:

Hoboken, 2009.

30. Ebnesajjad, S. Adhesives Technology Handbook; William Andrew Pub.: Norwich, 2008.

31. Licari, J. J. Adhesives Technology for Electronic Applications: Materials, Processes,

Reliability; William Andrew Pub.: Norwich, 2005.

32. Pugno, N. Velcro nonlinear mechanics. Applied Physics Letters 2007, 90, 121918.

33. Neuenschwander, W.; Fua, P.; Szekely, G.; Kubler, O. Deformable Velcro Surfaces;

Proceedings of the Fifth International Conference on Computer Vision: Cambridge, MA,

20–23 June 1995.

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