Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 3, pp. 717-727, Warsaw 2012
50th Anniversary of JTAM

THE CAYLEY VARIATIONAL PRINCIPLE FOR CONTINUOUS-IMPACT

PROBLEMS: A CONTINUUM MECHANICS BASED VERSION IN THE

PRESENCE OF A SINGULAR SURFACE

Hans Irschik

Johannes Kepler University of Linz, Institute of Technical Mechanics, Linz-Auhof, Austria

hans.irschik@jku.at

In 1857, Arthur Cayley presented a variational principle for a class of dynamical problems,
which he designated as continuous-impact problems. Cayley exemplified this class bymeans
of a chain hanging over the edge of a table and being set into motion by its own weight.
In the following, we present a continuum mechanics based version of the Cayley principle.
The moving portion of the chain mentioned by Cayley represents a variable-mass system,
and he assumed that the particles of the chain experience a jump in their velocity when
being taken into connectionwith the moving part.We accordingly study a body containing
a non-material region of transition, within which certain entities suffer considerable changes
of their spatial distribution, and which we replace by equivalent surface growth terms at a
singular surface, in order to derive our version of the principle. The falling chainmentioned
by Cayley is used as an example problem.

Key words: variable-mass systems, singular surface, falling chains

1. Introduction

In a communication transmitted to the Royal Society of London, Arthur Cayley (1857) wrote:
”There are a class of dynamical problemswhich, so far as I am aware, have not been considered
in a generalmanner. The problems referred to (whichmight be designated as continuous-impact
problems) are those inwhich the system is continually taking into connexionwith itself particles
of infinitesimalmass ..., so as not itself to undergo any abrupt change of velocity, but to subject
to abrupt changes of velocity the particles so taken into connexion. For instance, a problem of
the sort arises when a portion of a chain hangs over the edge of the table, the remainder of the
chain being coiled or heaped up close to the edge of the table, the part hanging over constitutes
the moving system, and in each element of time the system takes into connexion with itself,
and sets into motion with a finite velocity an infinitesimal length of the chain.” For this class of
dynamical problems, Cayley presented the following variational principle

Σ
[(d2x

dt2
−X
)

δx+
(d2y

dt2
−Y
)

δy+
(d2z

dt2
−Z
)

δz
]

dm

Σ(∆uδξ+∆vδη+∆wδζ)
1

dt
dµ=0

(1.1)

Cayley noted: ”... the first line requires no explanation, in the second line ξ, η, ζ are the
coordinates at the time t of the particle dµ which then comes into connexion with the system;
∆u, ∆v, ∆w are the finite increments of velocity (or, if the particles is originally at rest, then
the finite velocities) of the particles dµ the instant it has come into connexion with the system;
...The summation extends to the several particles dµwhichcome into connexionwith the system
at time t...”. After he had formulated his principle (1.1), Cayley cast it into a form similar to



718 H. Irschik

(but different from) the classical Lagrange equations of analytical dynamics. Arthur Cayley was
one of the leading mathematicians of the 19th century. As was noted by Crelly (1998), ”he
dealt with the mathematics at hand, without much commentary or clues to its development
– much to the consternation of his contemporaries.” Indeed, Cayley’s presentation does not
attempt to derive principle (1.1) from other fundamental relations of mechanics; it obviously
wasmeant as a fundamental principle of its own right. Due to the resulting lack of clue, Cayley’s
principle (1.1) until recently was seldom anticipated in themechanics literature. However, since
in Cayley’s formulations the mass of the dynamical system in hand, the moving part of the
chain, appears to be variable, the contribution of Cayley (1857) represents a fundamental step
in the theory of variable-mass systems. A review on the dynamics of variable-mass systems
was presented by Irschik and Holl (2004). In the following, we only refer to some selected
works that appear to be relevant in the present context. Cayley’s work was mentioned in the
thesis by I.V. Meshchersky on variable-mass systems in 1897. In the introduction to his thesis,
Meshcherky gave an overview on variable-mass problems in astronomy and rocketry. As was
notedbyMeshcherky, the equationofmotion of aplanetwithavariablemass obeying continuous
impacts and separations was obtained by Seeliger (1890). Tait (1895) solved the problem of a
rocket fired vertically, where he attributed the motive power of the rocket to the continual
detachment of a portion of mass of the rocket. Meshscherky noted a relationship between the
latter two solutions andCayley formulation (1.1) for continuous-impact problems. However, the
details of the thesis ofMeshcherkybecameknown toawider audienceonly through the collection
of his papers on themechanics of bodies,Meshcherky (1949). Throughout the twentieth century,
the study of bodies with variable mass formed an active field of research in various fields,
however mostly without reference to Cayley (1857). Several continuous-impact type problems
of the rectilinear motion of chains, ropes, cables and strings were treated in the twentieth-
century literature by means of the classical equation of conservation of energy, as well as by
the Lagrange equations in their classical form, without considering the corrections suggested by
Cayley (1857). The fall of a hanging folded chain and the whip served as model problems in
these studies. Paradoxical effects such as infinite velocities followed as the result of the latter
treatments, which were reviewed by Steiner and Troger (1995). In extension, the problem of a
body deploying along a cable was studied byCrellin et al. (1997). This interest in themotion of
chains, ropes, cables and strings wasmotivated by applications in the field of tethered satellites.
It was pointed out by Steiner and Troger (1995) and by Crellin et al. (1997) that the fall of a
hanging folded chain, the whip and the body deploying along a cable do not represent problems
in which the sum of the kinetic energy and the potential energy of the gravity force can be
assumed to be conserved in general. This was demonstrated by applying the equation of balance
ofmomentum,andbycomparing the respective outcomeswith the results of the classical formsof
the equation of balance of kinetic energy and the classical formsof theLagrange equations,which
assume the system to be conservative. In order to account for the apparent difference in these
results, Steiner andTroger (1995) introduced a Carnot energy loss in the equation of balance of
kinetic energy and in theLagrange equations.However, numerical and experimental studieswith
a tendency towards confirming the assumption of conservative behaviour of falling folded chains
were published afterwards, see Schagerl et al. (1997), and Tomaszewski et al. (2006). Cayley’s
work (1857) was explicitly brought into these discussions on the behaviour of falling chains by
Wong andYasui (2006), who even claimed that the solution presented byCayley for themotion
of a chain hanging over the edge of a table would be incorrect, since it is non-conservative. Their
considerations focused on the related problem of a hanging folded chain. More recently, the
discussion turned again towards the possible correctness of non-conservative solutions, such as
the one given byCayley (1857), seeWong et al. (2007), and the recent considerations byGrewal
et al. (2011). Among other arguments, the latter authors gave attention to fact that O’Reilly
and Varadi (1999), in a concise study of shocks in one-dimensional thermomechanical media,



The Cayley variational principle for continuous-impact problems... 719

also had discussed the example of a falling folded chain from a thermodynamic perspective.
From thework of O’Reilly andVaradi (1999) it is clear that energy conservation in falling chain
problems represents only one limit of a whole solution spectrum. In a recent study on the chain
hanging from a table and set intomotion, de Sousa et al. (2011) presented experimental results,
which indeed do fit better to the non-conservative solution.
It is the scope of the present contribution to bring Cayley’s principle (1.1) into a contempo-

rary light by presenting a corresponding continuummechanics based version. The following plan
of our work is motivated by the definition of the class of continuous-impact problems stated by
Cayley (1857): A system of this type ”is continually taking into connexion with itself particles
of infinitesimal mass..., so as not itself to undergo any abrupt change of velocity, but to subject
to abrupt changes of velocity the particles so taken into connexion.” A region of transition is to
be observed in problems of this type.Within the region of transition, the material particles are
subjected to considerable changes of their velocity. The region of transition represents a non-
material volume, such as the small spatial volume between the coiled and themoving part of the
chain.Correspondingly, a three-dimensional continuummechanics based formulation suitable for
bodies with a non-material region of transition is developed first. In the second step, the region
of transition is replaced by an equivalent singular surface, similar to formulations known from
the field of interfacial transport phenomena. The present considerations are complementary to
our earlier work on rational treatments of the laws of balance and jump, Irschik (2003, 2007).
Particularly in the latter paper, we started from the fundamental local equations of balance
and jump, e.g. for mass and momentum, in order to obtain manipulated forms, such as for the
kinetic energy, or for configurational forms of balance, while in the following we start from the
general balance law in an integral form, which we use to derive continuum mechanics based
formulations of the Cayley variational principle for continuous impact problems, as well as cor-
responding relations for balance of momentum and kinetic energy, respectively. Our exposition
ends with concluding remarks.

2. A continuum mechanics based version of the principle of Cayley

In order to derive a continuum mechanics based extension of Cayley principle (1.1), we begin
with the general balance law in the form given in Section 157 of Truesdell and Toupin (1960),
see Irschik andHoll (2004) for a review. The general balance law asserts how the rate of change
of the total of some entity ρΨ contained in a material volume V with a closed surface S is to
be balanced by an appropriate combination of the supply s[Ψ] of Ψ within V and the influx
i[Ψ] of Ψ on S

d

dt

∫

V

ρΨ dv=

∫

V

ρs[Ψ] dv−

∮

V

da · i[Ψ] (2.1)

The density of mass is denoted by ρ. The oriented area element da of the material surface S
is a vector pointing outwards of V. The material surface is a surface moving at the velocity of
the material particles located on it, and a material volume is defined correspondingly. The dot
product utilised in the surface integral at the right hand side of (2.1) is defined and explained in
the exposition on tensor fields byEricksen (1960). In the present paper, we restrict the entity Ψ
to scalars and vectors. Hence, the influx i[Ψ] stands for a vector or a tensor of second order,
respectively. The following continuummechanics derivations are motivated by the fact that the
particles of the chain studied in the exposition of Cayley (1857) are subjected to a considerable
change of velocity within a small region, the latter being instantaneously located between the
coiled part, which is at rest, and the hanging part, which is moving. This region of transition
forms a non-material volume. A non-material volume has a closed surface moving at a velocity



720 H. Irschik

different fromthematerial particles located on it.Cayley (1857) andRouth (1898) approximated
the presence of a region of transition in the chain by assuming that a jump in the velocity of
the particles would take place between the coiled and the hanging part. In a refinedmodelling,
it may be necessary to model the region of transition in more detail, at first as a finite part of
the body. In the general balance law (2.1), we therefore need to introduce the rate of change
of the quantity contained in a non-material volume of finite extension. We thus decompose the
material volume V in (2.1) instantaneously into three parts, where t denotes the current time

V = v+(t)∪vΣ(t)∪v
−(t) (2.2)

From these three regions, let vΣ(t) be a finite three-dimensional volume that separates the other
two partial volumes v+(t) and v−(t), see Fig. 1.

Fig. 1. Material volume V in the presence of a three-dimensional region of transition, vΣ(t)

We characterise the location of vΣ(t) by assuming that the spatial distribution of some of
the physical entities under consideration do suffer considerable changes within that region of
transition. The left hand side of (2.1) now may be written as

d

dt

∫

V

ρΨ dv=
d

dt

∫

V+

ρΨ dv+
d

dt

∫

VΣ

ρΨ dv+
d

dt

∫

V−

ρΨ dv (2.3)

where thematerial volume VΣ instantaneously coincides with the non-material region of trans-
ition vΣ(t) in (2.2), and V

+ and V− are analogously defined. In order to relate the rates of
change of the total of ρΨ contained in the material volume VΣ to the rate of change contained
in the corresponding non-material volume vΣ(t), the transport theorem is applied. In the follo-
wing, we use a valuable form first stated in the exposition on the sub-mechanics of the universe
by Reynolds (1903), see again Irschik and Holl (2004) for a review. According to Fig. 1, the
non-material region of transition vΣ(t) is enclosed by the portion SΣ of thematerial surface S,
as well as by the two non-material surfaces Σ+ and Σ−, which separate vΣ(t) from v

+(t)
and v−(t), respectively. The oriented area element of vΣ(t) on Σ

+ is a vector da+
Σ
pointing

outwards to vΣ(t). Thenon-material surface Σ
+ moves at the velocity u+

Σ
. Analogously, da−

Σ
is

the oriented area element of vΣ(t) at Σ
−, the latter surface moving at the velocity u−

Σ
. The

Reynolds transport theorem then yields

d

dt

∫

VΣ

ρΨ dv=
duΣ
dt

∫

vΣ(t)

ρΨ dv+

∫

Σ+

da+
Σ
· (ṗ−u+

Σ
)ρΨ+

∫

Σ−

da−
Σ
· (ṗ−u−

Σ
)ρΨ (2.4)

The notation duΣ/dt indicates that the differentiation refers to the motion of the region of
transition vΣ(t), see Sect. 81 of Truesdell and Toupin (1960). The relation presented in (2.4)



The Cayley variational principle for continuous-impact problems... 721

may be derived by subtracting equation (81.4) from equation (81.3) of Truesdell and Toupin
(1960). The relation stated in (2.4) dates back to the 1903 exposition of Reynolds (1903), but it
was seldom utilised in this form in the subsequent literature. For a contemporary presentation
containing formulation (2.4), see Chapter 1.12 of the book on fluid dynamics by Warsi (1999).
For balance ofmass, linear and angularmomentum, seeChapter 7 of the book on themechanics
of solids and fluids by Ziegler (1998). We now turn to the rate of change of the quantities
contained in the remaining twomaterial volumes in (2.3), V+ and V−. Note that some less well
specified regions of the bodymay be encountered in continuous-impact problems, or that there
may be some parts of the body that are ofminor interest for the problems in hand. Such a part,
for instance, is represented by the coiled part of the hanging chain, the latter part being not
precisely specified in the above cited problem statement byCayley (1857). Itmay bedesirable to
exclude unspecified configurations like the coiled part of the chain from the formulation. This is
performed as follows. Let the volume to be excluded be represented by the partial volume v−(t)
in the above formulas. For the material volume V− instantaneously coinciding with v−(t), we
write down the general equation of balance as

d

dt

∫

V−

ρΨ dv=

∫

V−

ρs[Ψ] dv−

∫

S−

da · i[Ψ]+

∫

Σ−

da−
Σ
· i[Ψ]− (2.5)

see Fig. 1. The volume V− is enclosed by the portion S− of the material surface S and by the
part Σ− of the surface of the region of transition vΣ(t). Note that (−da

−

Σ
) is the oriented area

element of v−(t) at Σ−. The relation given in (2.5) now is subtracted from general balance law
(2.1), thus omitting the volume V− from our formulation. In order to derive a counterpart to
Cayley variational statement (1.1), wemay furthermore state that, for thematerial volume V+

in (2.3), there is

d

dt

∫

V+

ρΨ dv=

∫

V+

Ψ̇ρ dv (2.6)

sincewedonot take into account distributed sources ofmass in our present formulation. Putting
(2.3)-(2.6) into general balance law (2.1), we obtain

∫

V+

Ψ̇ρ dv=
duΣ
dt

∫

vΣ(t)

ρΨ dv+

∫

Σ+

da+
Σ
· (ṗ−u+

Σ
)ρΨ+

∫

Σ−

da−
Σ
· (ṗ−u−

Σ
)ρΨ

=

∫

V+

ρs[Ψ] dv+

∫

VΣ

ρs[Ψ] dv−

∫

S+

da · i[Ψ]−

∫

SΣ

da · i[Ψ]+

∫

Σ−

da−
Σ
· i[Ψ]−

(2.7)

where S+ denotes that part of the surface of V+ which is located on thematerial surface S, see
Fig. 1. In theCayley problem of the hangingmoving chain, this region V+ is represented by the
moving part of the chain. For subsequentuse, it appears to be convenient to represent the rate of
change contained in the non-material region of transition vΣ(t) and its supply in (2.7) by some
quantity attributed to a certain non-material surface. Subsequently, we call this singular surface
a surface of transition.Without loss of generality, let this surface of transition be chosen as the
surface Σ+ of vΣ(t). Accordingly, we introduce an equivalent influx j[Ψ] across the surface Σ

+.
We do this such that the integrals over Σ− formally do not appear in our expressions anymore.
This yields

∫

V+

Ψ̇ρ dv+

∫

Σ+

da+
Σ
· [(ṗ+−u+

Σ
)ρ+Ψ+− (ṗ−u+

Σ
)ρ−Ψ−]

=

∫

V+

ρs[Ψ] dv−

∫

S+

da · i[Ψ]−

∫

SΣ

da · i[Ψ]−

∫

Σ+

da+
Σ
· (j[Ψ]+ i[Ψ]−)

(2.8)



722 H. Irschik

The equivalent influx j[Ψ] thus also attributes certain entities defined at Σ− to the equivalent
singular surface Σ+. In order to perform this strategy in practice, more information about the
problem in hand is necessary. Particularly, formulation (2.8) appears to be straightforward in
the case of a region of transition having parallel surfaces Σ+ and Σ−, with the oriented surface
elements pointing in opposite directions. The region of transition then has the geometrical form
of a shell. For the corresponding treatment of a three-dimensional thin interfacial region,we refer
to the comprehensive exposition on interfacial transport phenomena by Slattery (1990). In fluid
mechanics, onegenerally talks abouta shock layerwhendiscussinga thin shell-type regionwithin
which considerable changes of someof the field variables take place.Ahistorical exposition anda
review on contemporary developments on shock layers was presented byKluwick (2000). For the
classical treatment of singular surfaces, see Sect. 192 ofTruesdell andToupin (1960). It is evident
from the example of shock layers in gas dynamics that the constitutive behaviour of the specific
material under consideration must be taken into account in the framework of thermodynamics
in order to estimate, whether a layer of transition may be treated by the classical formulation
for singular surfaces. Since such a detailed study is beyond our present considerations, we have
introduced the equivalent influx j[Ψ] in our subsequent formulations. The equivalent surface
influx j[Ψ] is denoted as a surface growth term in the following. By means of surface growth
terms, the present formulation can include both, a region of transition with a finite extent, and
a region of transition with a vanishing extension in the thickness direction, and it enables one to
consider non-vanishing surface growth terms also in the latter case. General balance law (2.8)
now is written as

∫

V+

Ψ̇ρ dv+

∫

Σ+

da+
Σ
· [[(ṗ−u+

Σ
)ρΨ]]

=

∫

V+

ρs[Ψ] dv−

∫

S+

da · i[Ψ]−

∫

SΣ

da · i[Ψ]−

∫

Σ+

da+
Σ
· (j[Ψ]− i[Ψ]−)

(2.9)

where we have introduced the jump term

[[(ṗ−u+
Σ
)ρΨ]] = (ṗ+−u+

Σ
)ρ+Ψ+− (ṗ−u+

Σ
)ρ−Ψ− (2.10)

which iswell known fromthe classicalKotchine theorem for a singular surfacemovingat thenon-
material velocity u+

Σ
, see Section 193 of Truesdell andToupin (1960).Whenwe let thematerial

volume V+ shrink to zero, such that S+ →Σ+, we obtain the generalised jump condition

∫

Σ+

da+
Σ
·

(

[[(ṗ−u+
Σ
)ρΨ]]+ j[Ψ]+ [[i[Ψ]]]

)

=−

∫

SΣ

da · i[Ψ] (2.11)

where we have assumed s[Ψ] and Ψ̇ρ to be boundedwithin V+.We have used the abbreviation

[[i[Ψ]]] = i[Ψ]+− i[Ψ]− (2.12)

Putting j[Ψ] = 0, and setting i[Ψ] = 0 on SΣ in (2.11), there follows the classical form of the
Kotchine jump conditions at a singular surface, see Section 193 of Truesdell andToupin (1960).

We now study the special case of balance of mass in the presence of a region of transition,
and we apply the result to the hangingmoving chain afterwards.We thus put Ψ =1 in (2.11),
and, since we do not consider distributed sources of mass to be present, we set i[Ψ] = 0 . By a
standard argument, this yields the local jump condition

da+
Σ
·

(

[[(ṗ−u+
Σ
)ρ]]+ j[Ψ =1]

)

=0 (2.13)



The Cayley variational principle for continuous-impact problems... 723

where j[Ψ = 1] denotes the surface mass growth at Σ+. Particularly, in the problem of the
hanging moving chain introduced by Cayley (1857), we set ρ− = ρ+ = ρ, since the chain
is assumed to be inextensible. Furthermore, we write u+

Σ
= 0, since the location of the table,

which forms the upper end of themoving part of the chain, is fixed.Also, we have ṗ− =0 in the
coiled part of the chain.Hence, for thehangingmoving chain,weneed anon-vanishing equivalent
surfacemass growth in (2.13) in order to assure the balance ofmass, j[Ψ =1]=−ṗ+ρ. Through
this equivalent surface mass growth term, we have again found a context between the Cayley
class of continuous impact problems and the class of variable-mass problems.When we neglect
the equivalent mass influx, j[Ψ] = 0 in (2.13), we obtain the classical Stokes-Christoffel jump
condition for balance of mass at a singular surfacemoving at the non-material velocity u+

Σ
, see

Section 189 of Truesdell and Toupin (1960). Putting (2.13) into (2.9), the following formulation
of the general balance law is obtained

∫

V+

Ψ̇ρ dv+

∫

Σ+

da+
Σ
· (ṗ+−u+

Σ
)ρ+[[Ψ]]

(2.14)

=

∫

V+

ρs[Ψ] dv−

∫

S+

da · i[Ψ]−

∫

SΣ

da · i[Ψ]−

∫

Σ+

da+
Σ
· (j[Ψ]− j[Ψ =1]Ψ−− i[Ψ]−)

From (2.14), we now deduce various specialised equations of balance in the presence of a layer
of transition. These equations are then applied exemplary to the hanging moving chain. We
start with the equation of balance of linear momentum by setting Ψ = ṗ in (2.14). The supply
within V+ is identified with the assigned body force per unit mass, s[ṗ] =b, and the influx is
taken as the negative of the Cauchy stress tensor, i[ṗ] =−t. We thus obtain

∫

V+

p̈ρ dv+

∫

Σ+

da+
Σ
· (ṗ+−u+

Σ
)ρ+[[ṗ]]

(2.15)

=

∫

V+

ρb dv+

∫

S+

da · t+

∫

SΣ

da ·t−

∫

Σ+

da+
Σ
· t
−
−

∫

Σ+

da+
Σ
· (j[ṗ]− j[Ψ =1]ṗ−)

This relation now is applied to the Cayley problem of the hanging moving chain. The moving
part V+ of the chain is assumed to have the instantaneous length s(t), such that the particles
in V+ move at the common velocity ṗ= ṡn+

Σ
. The assigned body force is the weight per unit

mass, b = gn+
Σ
. Furthermore, we set da · t = 0 at S+, since the free end of the chain is not

loaded by external forces, and we use da · t = 0 at SΣ, since we assume no friction between
the table and the chain to be present. Also, there is da+

Σ
· t− = 0, certainly since the chain is

assumed to be coiled up loosely. As explained above, we also have ρ− = ρ+ = ρ, u+
Σ
= 0 and

ṗ− = 0. When we neglect the equivalent surface momentum growth in (2.15), da+
Σ
· j[ṗ] = 0,

we obtain the result derived by Cayley (1857)

(ρAs)s̈+(ρAṡ)ṡ=(ρAs)g (2.16)

where A denotes the cross-section of the chain. However, a whole spectrum of solutions can be
obtained when taking into account non-vanishing equivalent surface momentum growth terms.
Note also that a further mechanical meaning of the surface momentum growth term can be
assigned by comparison of the above methodology with treating the moving part of the chain
only, excluding the region of transition. This brings into the play the stress in the moving part
of the chain in its upper cross section, see Irschik and Holl (2002) for a corresponding solution.
We now proceed to the equation of balance of kinetic energy.We thus set

Ψ =
1

2
ṗ · ṗ (2.17)



724 H. Irschik

in (2.14). The corresponding supplywithin V+ consists of the power of the assigned body forces
per unit mass and of the power density of internal forces

s
[1

2
ṗ · ṗ
]

=b · ṗ−
1

ρ
t · gradṗ (2.18)

while the influx term gives the surface power density of the surface tractions

da · i
[1

2
ṗ · ṗ
]

=−(da · t) · ṗ (2.19)

We thus obtain from the general equation of balance, (2.14), that

∫

V+

p̈ · ṗρ dv+

∫

Σ+

da+
Σ
· (ṗ+−u+

Σ
)ρ+
[[1

2
ṗ · ṗ
]]

=

∫

V+

ρb · ṗ dv+

∫

S+

(da · t) · ṗ+

∫

SΣ

(da ·t) · ṗ−

∫

Σ+

(da+
Σ
·t
−) · ṗ−

−

∫

V+

t · gradṗ dv−

∫

Σ+

(da+
Σ
·

(

j
[1

2
ṗ · ṗ
]

−
1

2
j[Ψ =1]ṗ− · ṗ−

)

(2.20)

where j
[

1
2
ṗ · ṗ

]

represents the equivalent influx of kinetic energy across the surface of transi-

tion Σ+. This represents the equation of balance of kinetic energy in the presence of a region
of transition.We now apply (2.20) to the Cayley problem of a hangingmoving chain, using the
modelling given above for the derivation of the equation of balance of momentum, (2.16). We
thus obtain the result

(ρAs)s̈ṡ+(ρAṡ)
1

2
ṡ2 =(ρAs)gṡ−An+

Σ
· j
[1

2
ṗ · ṗ
]

(2.21)

wherewehave set thepowerof the internal forces to zerowithin themovingpartof the chain, V+,
since the chain is assumed to be inextensible and to remain straight in its hanging part. From
a comparison of (2.21) and (2.16), we see that an equivalence of the latter relations is assured,
when we introduce an equivalent surface influx of kinetic energy of amount

n
+
Σ
· j
[1

2
ṗ · ṗ
]

=
1

2
ρṡ3 (2.22)

in (2.21). The non-vanishing surface growth term j
[

1
2
ṗ · ṗ
]

of (2.22) is in analogy to the Carnot

loss terms thatwere introducedbySteiner andTroger (1995) andbyCrellin et al. (1997) in order
to explain the differences between the equation of balance of linearmomentumand the equation
of balance of kinetic energy in the problems of the falling folded chain and of the bodydeploying

alongacable. Inourpresentcontribution, j
[

1
2
ṗ·ṗ
]

is anoutcomeof consideringthe rateof change

of the kinetic energy and the supplies contained in the non-material region of transition vΣ(t),
and of attributing certain entities defined at Σ− to the surface of transition Σ+. We note that
our strategy does not allow judging whether a non-vanishing equivalent momentum influx j[ṗ]

should be introduced in the problem of the hangingmoving chain, in addition to j
[

1
2
ṗ ·ṗ
]

. This

question must be answered by experiments, or by amore detailed modelling.What our present
three-dimensional formulation does provide, is the form of the equations of balance that have
to be used in the problems in hand. How these equations must be brought into coincidence by
means of equivalent surface growth terms has been exemplary demonstrated for the hanging
moving chain, see (2.16) and (2.21). A justification for the necessity of surface growth terms



The Cayley variational principle for continuous-impact problems... 725

in order to ensure the consistency of jump relations at a singular surface was given by Irschik
(2003), see also Irschik (2007).

We now turn to Cayley principle (1.1). In order to obtain coincidence, in (2.14) we set

Ψ = ṗ · δp (2.23)

whereweassumethevirtual change ofposition δp tobea smoothvector field.Thecorresponding
supply within V+ is given by the sum of the virtual virials of the assigned body forces per unit
mass and of the internal forces

s[ṗ · δp] =b ·δp−
1

ρ
t · gradδp (2.24)

while the influx term is represented by the virtual virial of the surface tractions

da · i[ṗ ·δp] =−(da ·t) ·δp (2.25)

see Irschik (2000) for thenotionof virtual virials.Weeventually obtain our counterpart toCayley
principle (1.1), written for a material volume containing a non-material region of transition or
a singular surface. The result is

∫

V+

(p̈−b) ·δpρ dv−

∫

S+

(da · t) ·δp−

∫

SΣ

(da ·t) ·δp+

∫

Σ+

(da+
Σ
· t) · δp−

+

∫

V+

t · gradδp dv+

∫

Σ+

da+
Σ
· (ṗ+−u+

Σ
)ρ+[[ṗ]] · δp

+

∫

Σ+

da+
Σ
· (j[ṗ ·δp]− j[Ψ =1]ṗ− · δp)= 0

(2.26)

Applying (2.26) to the hangingmoving chain and setting the equivalent influx term j[ṗ ·δp] to
zero, we again arrive at the Cayley result stated in (2.16). It is noted again, however, that this
solution is only a limiting case of solutions following fromnon-vanishing equivalent influx terms.

3. Conclusions

With respect to Cayley’s original formulation (1.1), we can say the following. Without further
need of explanation, the first two lines of (2.26) are seen to be analogous to the first line of (1.1).
In the third line of (2.26), the term (ṗ+−u+

Σ
)ρ+ represents the influxofmass through the surface

element (−da+
Σ
) of the surface of transition Σ+ into the non-materialmoving region v+(t), and

[[ṗ]] is the jump of velocity across the region of transition. Thus, a direct analogy between the
third line of (2.26) and the second line of (1.1) is seen to exist. The last line of (2.26) accounts
for the behaviour of the particles within the region of transition, cast into the form of equivalent
surface growth terms defined at Σ+. As already pointed out, the latter termsmust be evaluated
in amore detailed and problem-oriented study. The present formulation does not provide more
than the general form of the relations to be encountered. In this sense, however, our present
contribution can serve as a rational extension of the Cayley principle. Neglecting some or all of
the terms in the last line in (2.26) may provide a good first estimate of the solution in some
cases, such as in the above modelling of the Cayley chain problem. It is hoped, that extended
formulation (2.26) might contribute to the understanding of continuous impact problems in
which it is not appropriate to neglect the quantities that are equivalent to the above surface
growth terms.



726 H. Irschik

Acknowledgements

Support of the present contribution in the frameworkof theAustrianFWF-ProjectP14866TEC,”Vi-

brations of Bridges with VariableMass”, in the framework of which preliminary studies were performed,

and of the Austrian Center of Competence inMechatronics (ACCM), to the strategic research of which

the present paper is devoted, are gratefully appreciated.

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Zasada wariacyjna Cayley’a w zagadnieniu ciągło-uderzeniowym: wersja mechaniki

kontinuum z obecnością powierzchni osobliwej

Streszczenie

W 1857 rokuArthur Cayley zaprezentował zasadę wariacyjną dla pewnej klasy zagadnień dynamicz-
nych,którąnazwałciągło-uderzeniową.Cayleyzegzemplifikowałtę klasęproblememłańcuchazwisającego
z krawędzi stołu i wprowadzanegow ruch własnym ciężarem.W niniejszej pracy zasadę Cayley’a przed-
stawiono w wersji opartej na mechanice kontinuum. Dawny uczony opisał ruchomą część spadającego
łańcucha układem o zmiennej masie i założył, że jego fragmenty spoczywające jeszcze na stole doświad-
czają skokowego przyrostu prędkości wmomencie przyłączania się do części wprawionej już w ruch. Na-
wiązując do podejścia Cayley’a, w obecnym artykule przeprowadzono analizę ruchu ciała zawierającego
niematerialny obszar przejściowy, w którym pewne wielkości podlegają znacznym zmianom konfiguracji
przestrzennej i które wyrażono zastępczymi przyrostami na powierzchni osobliwej. Takie sformułowa-
nie pozwoliło wyprowadzić własną wersję zasady Cayley’a, a spadający łańcuch posłużył jako przykład
przedstawionej analizy.

Manuscript received April 12, 2012; accepted for print April 30, 2012