Continuous and discrete models in the mechanics of deformable solid bodies
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* Co esponding au ho : glame @eng.unideb.hu
Con inuous and disc e e models in he mechanics o de o mable
solid bodies
Géza Láme
Uni e si y o Deb ecen, Facul y o Enginee ing, Depa men o Enginee ing Managemen and En e p ise, 4028 Deb ecen, Ó eme ő u. 2-
4., Hunga y
Abs ac . The s udy p o ides an o e iew o modelling possibili ies o he mechanical beha iou o media.
The disc e e, con inuous o di e en ial geome ic as well as he disc e e na u e and con inuous desc ip ion
g id con inuum model in pa icula a e highligh ed. We poin ou ha he di e en ial geome ic model is
based on he concep o con inui y and in e p e s a con inuous medium model. We e eal ha he g id
con inuum model is based on he applica ion o nume ical me hod and in e p e s a disc e e medium model.
1 In oduc ion
The sys em o disc e ely loca ed elemen s, classical con-
inuum and gene alised con inuum can be used in li e a-
u e o desc ibe he mechanical beha iou o de o mable
solid bodies. The disc e e sys em can be cha ac e ised as
desc ibe he sys em in bo h physics space and phase space
as disc e e domain unc ions. The classical con inuum can
be cha ac e ised by in e p e ing i as a con inuous
geome ic locus in he physical space, applying
con inuous unc ions in he phase space o desc ibe i s
condi ion. The gene alised con inuums can be desc ibed
by modelling a disc e e geome ic locus e aining he
in e nal s uc u e o he ma e in he physical space (e.g.
wi h „mic o-con inuums” si ing on he g id poin s o
some kind o g id sys em), we cha ac e ise he condi ion
o he in e nal s uc u e ma e wi h con inuous unc ions
o he phase space desc ip ion.
The con inui y o he h ee models di e s in physical
and phase space. We examine he h ee models acco ding
o his dis inc ion in his s udy.
2 Cha ac e isa ion o disc e e modelling
in physical space
2.1 Desc ip ion o he disc e e model
The e a e he bodies, deg ees o kinema ic eedom o
bodies, me hods o in e ac ion exis ing among bodies,
dynamic deg ees o eedom, ma e equa ions, he
condi ions o equilib ium, and he ela ionships desc ib-
ing he equilib ium (mo emen ), he ini ial posi ions o
he bodies and he ini ial alues o dynamic e ec s in he
ini ial posi ions, hose dynamic, occasionally kinema ic
condi ions, which exis when we seek he equilib ium o
he sys em and i s mo emen .
The sys em can be examined limi ed o mass poin s
and igid body si ing in g id poin s (i can be demon-
s a ed ha i s de o mabili y does no play any ole),
solu ion o he sys em can be examined wi hou ac ually
applying equilib ium (mo emen ) equa ions. I is enough
o limi ou sel es o wha deg ee o kinema ic eedom
does an elemen si ing on he g id poin s possesses and
wha equilib ium (mo emen ) equa ions can be applied.
He eina e , he equa ion sys em o he disc e e
mechanic sys em is o mally desc ibed. P o ided he e a e
n numbe s o elemen s. The loca ion o elemen s
( e e ence poin s) is ma ked wi h i, he posi ion o ele-
men s (basis ec o s o angen ial space) is ma ked wi h
ψi. Elemen s can ha e displacemen ui and o a ion φi, (i
= 1,2,3 … n) deg ees o eedom. The case o wo deg ees
o eedom is examined sepa a ely and join ly as well.
Case o displacemen : The in e nal o ce be ween he
i h and j h elemen , Fij and Fji = – Fij, depends on he dis-
ance o he wo elemen s. The dis ance o he wo ele-
men s is ma ked wi h ij = i – j in he ini ial s a e. This
ec o changes o alue Rij wi h he shi o he wo
elemen s, whe e Rij = ij + ui – uj. Acco dingly, he ol-
lowing o ces ac upon he i h elemen :
, , , ,
( 1 , 2,3 , 1, 2,3 ; )
)
.
(
ij ij ij ij i j
ii
c
i nj mm n
F F uu
(1)
He e, he mi ≤ n inequali y e e s o he ac ha he e
is no always a ela ionship be ween all elemen s in ac ,
di e en elemen s can ha e a ela ionship wi h a di e en
numbe o elemen s. The ac ha he magni ude o o ce
be ween he i h and he j h elemen s can depend on a ious
pa ame e s was ma ked wi h cij in he con ex ega ding
Fij o ce. The equilib ium o he sys em o e e y i h
elemen is gi en by o ce equa ions:
1
( , , , ) , ( 1, 2,3 ... ).
i
m
ij ij ij i j i
j
c in
F uu P 0
(2)
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The Pi in he equa ion is he „ex e nal” o ce ac ing
upon he i h elemen .
I can be seen om he equa ion sys em (we a e in a
h ee-dimensional space), ha 3n equa ions a e a ailable
o de e mining 3n a iables. S a ing om he known i
posi ions o he indi idual elemen s in he ask, he Ri = i
+ ui (i = 1,2,3 … n) is he new posi ion o elemen s ha
can be de e mined unde he e ec o ex e nal o ces: Pi
(i = 1,2,3 … n).
Case o o a ion: The in e nal o que be ween he i h
and j h elemen s, Mij and Mji = – Mij, depends on he
ela i e posi ion o he wo elemen s. The ela i e posi ion
o he wo elemen s is ma ked wi h ψij = ψi – ψj in he
ini ial s a e. This ec o changes o Ψij alue wi h he
o a ion o he wo elemen s, whe e Ψij = ψij + φi – φj.
Acco dingly, he ollowing o ques ac upon he i h
elemen :
, ,, ,
( 1 , 2
()
,3 , 1 ,2,3 ; ) .
ij ij ij ij i j
ii
d
i nj mm n
MM ψ φφ
(3)
See he anno a ion ega ding he mi ≤ n inequali y
abo e. The ac ha he magni ude o o que be ween he
i h and j h elemen s can depend on a ious pa ame e s was
ma ked wi h dij in he con ex ega ding he Mij o que.
The equilib ium o he sys em o e e y i h elemen is
gi en by o que equa ions:
1( , , , ) , ( 1, 2,3 ... ).
i
m
ij ij ij i j i
jd in
Mψ φφ M 0
(4)
The Mi in he equa ion is he „ex e nal” o que ac ing
upon he i h elemen .
I can be seen om he equa ion sys em ha 3n
equa ions a e a ailable o de e mining 3n a iables.
S a ing om he known ψi posi ion o he indi idual
elemen s in he ask, he Ψi = ψi + φi (i = 1,2,3 … n) is he
new posi ion o elemen s ha can be de e mined unde he
e ec o ex e nal o ques: Mi (i = 1,2,3 … n).
Case o displacemen and o a ion: The in e nal o ce
be ween he i h and j h elemen s, in e nal o que, depends
on he dis ance and ela i e posi ion o he wo elemen s,
see abo e. Acco dingly, he ollowing o ces and o ques
ac upon he i h elemen :
,,,(,, ), ,,
ij ij ij ij i j ij ij i j
cdF F uu ψ φφ
(5)
,,, , , ,, ,
( 1 , 2 , 3 , 1 , 2 , 3 ; .
(
)
)
ij ij ij ij i j ij ij i
ii
j
cd
i nj mm n
M M uu ψ φφ
(6)
See he anno a ions ega ding he mi ≤ n inequali y and cij
and dij abo e. The equilib ium o he sys em o e e y i h
elemen is gi en by o ce and o que equa ions:
1(,,, , , , , ) ,
i
m
ij ij ij i j ij ij i j i
jcd
F uu ψ φφ P 0
(7)
1
(,,, , , , , ) ,
i
m
ij ij ij i j ij ij i j i
j
cd
M uu ψ φφ M 0
(8)
whe e (i = 1,2,3 … n).
The Pi in he equa ion is he „ex e nal” o ce ac ing
upon he i h elemen , while he Mi is he „ex e nal” o que
ac ing upon he i h elemen .
I can be seen om he equa ion sys em ha 2×3n
equa ions a e a ailable o de e mining 2×3n a iables.
S a ing om he known i posi ion and ψi di ec ion o he
indi idual elemen s in he ask, he new posi ion Ri = i +
ui, and i s new di ec ion Ψi = ψi + φi o elemen s (i = 1,2,3
… n) can be de e mined unde he e ec o Pi ex e nal
o ces and Mi ex e nal o ques (i = 1,2,3 … n).
2.5 Possibili ies and limi a ions o he disc e e
model
The disc e e me hod p o ides an oppo uni y o examine
he mechanical condi ion o a sys em consis ing o a ini e
numbe o mass poin s o igid bodies wi h assump ion
ha he mass poin has h ee displacemen , and he igid
body has h ee displacemen and h ee o a ion deg ees o
eedom, he known posi ion ec o s and he di ec ion o
he indi idual igid bodies enables he de e mina ion o
o ce be ween wo- wo mass poin s and he de e mina ion
o o ce and o que be ween wo- wo igid bodies. These
in e nal o ces depend on he displacemen o mass poin s,
and he in e nal o ces and o ques depend on he
displacemen and o a ion o igid bodies. The equa ions
o equilib ium (mo emen ) o mass poin and igid body
p o ide he same numbe o equa ions as he numbe o
kinema ic a iables. The solu ion o equilib ium
equa ions p o ides he loca ions o mass poin s and he
loca ion and di ec ion o igid bodies depending on
ex e nal o ces, o ex e nal o ces and o ques.
The sys em p o ides a solu ion ega ding he disc e e
poin s o space. Tha means ha he heo y o unknown
quan i ies (as unc ions) whe he hey a e displacemen s,
o a ions, o ces o o ques can only de e mine alues in
he disc e e poin s o he physical space.
3 Con inuous in physical space model
3.1 Desc ip ion o con inuous in physical space
model
A body is gi en as a con inuous model, along wi h i s
loca ion in space, posi ion, shape, cha ac e is ics o
ma e ial beha iou , suppo condi ions o he body, e -
ec s on he body, kinema ic and dynamic alues in he
ini ial posi ion o he body.
We iden i y a de o mable body in he con inuous
model wi h i s egion occupied in he Euclidean space. Le
he e be gi en he {qi} coo dina e sys em. The posi ion
ec o is ma ked wi h be o e he de o ma ion, and wi h
an R a e he de o ma ion.
Fi s ly, o e iew he quan i ies desc ibing he
geome y; see he ela ionships hemsel es e.g. [1,2].
– Basis ec o s in he posi ion ec o poin .
– Componen s o he me ic enso .
– Pa ial di e en ials o basis ec o s.
– A ine coe icien s (Ch is o el symbols).
– Ch is o el–Riemann cu a u e enso o exp essing
he cu a u e o he Riemann space.
2
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The ollowing geome ic e ms can be exp essed wi h
he abo emen ioned quan i ies:
– scala p oduc s o ec o s a and b,
– angles be ween ec o s a and b (cosine),
– egula a c leng h be ween wo poin s in space,
– absolu e di e en ials,
– pa allel shi along he cu e.
The basis o a con inuous model is he con inui y
i sel . This enables he applica ion o Taylo Se ies, which
is applied in p ac ice in he ollowing o m:
( ,,) (,,) (,,)/ .Fx xyz Fxyz Fxyz x x
(9)
The consequence o his is ha di e en ial
ela ionships a e es ablished among he unknown
quan i ies.
Acco ding o he con inuous model, he posi ion
ec o o all poin s o he de o mable igid body changes
unde ex e nal e ec . In acco dance wi h his, he
di e en ial geome ic desc ip ion o space can be used o
he kinema ic cha ac e isa ion o he body. The
in e p e ed e ms a e he ollowing [1,2].
The di e ence o posi ion ec o s is he displacemen
ec o :
, u R
his also desc ibes he changes o he
coo dina e sys em a he same ime. The componen s o
he me ic enso in he de o med s a e a e as he scala
p oduc o basis ec o s. The di e ence o he me ic
enso s can be ela ing o he de o ma ion. The di e ence
o he wo enso s is he measu emen enso o he
de o ma ion:
. γGg
The ela i e s e ch o a di ec ion
ec o is desc ibed by he main diagonal elemen s, he
change in he angle be ween wo di ec ion ec o s is
desc ibed by he o -diagonal elemen s o he
measu emen enso [2], when he ela i e s ains
(s e ches and angel changes) a e small [3-5]. This leads
o he enso o small s ains:
( ) / 2. ε Gg
The change
o a ine connexion coe icien ,
,
k kk
ij ij ij
Γ
he
ela ionships equi ed o he equilib ium equa ions o he
de o med s a e. The change o he cu a u e enso
p o ides he compa ibili y equa ions (i we make i equal
o ze o). The exp ession o di e en geome ic objec s
wi h he componen s o he g adien enso o
displacemen ec o is equi ed o es ablishing he
heo y. He e, we only p o ide a ela ionship o one
geome ic objec ega ding he measu emen enso o he
s ain:
()()().
γ u u uu
The o he
ela ionship can also be exp essed wi h he g adien enso
o he displacemen , see e.g. [1-5].
By desc ibing de o ma ion wi h displacemen wo
u he cha ac e isa ions o changes in space can be
p o ided. The igid body o a ion o he neighbou hood o
a poin s ( he angen ial basis ec o s o he poin ) is gi en
by o hogonal componen o he pola decomposi ion o
he g adien enso o he de o med posi ion ec o . The
o he is he o a ion o a di ec ion. This is p o ided by he
whole g adien enso o he de o med posi ion ec o .
The displacemen o he poin s o he con inuum
clea ly de e mines he o a ion o neighbou hood o all he
poin s. This canno be an independen quan i y. Con-
sequen ly, nei he he o a ion o a di ec ion, no he o-
a ion o he neighbou hood o a poin can be in e p e ed
as an independen kinema ic a iable.
The kinema ic cha ac e isa ion o he con inuum can
be summa ised as ollows. The p ima y kinema ic a i-
able o he con inuum is he displacemen ield, while he
seconda y kinema ic a iable is he symme ical s ain
enso . The displacemen o he con inuum pe poin s
clea ly p o ides he ela i e s e ch and o a ion o e e y
uni ec o , he angle change pe poin be ween wo- wo
uni ec o s and he igid-body like o a ion o he neigh-
bou hood o e e y poin ( ha is o say he angen ial basis
ec o s belonging o he poin ).
No e: We ha e 3 + 6 kinema ic a iables, wi h six
s ain-displacemen ela ionships ( he kinema ic equa-
ions). The kinema ic desc ip ion o he con inuum inde -
ini e.
We wish o in e p e in e nal o ces in he con inuous
model in physical space as con inuous. As a poin has no
dimension, no weigh and only one concen a ed o ce can
be assigned o i in he physical space, he New onian
model has o be modi ied so ha we assign dynamic
quan i y o a small bu ini e olume a he han o a poin .
A oiding de ails, le he e be σn(x,y,z) ec o quan i y on
an su ace wi h n ou wa d no mal, which i we sum
(in eg a e) on ini e size su ace igu e, hen ul ima ely we
ob ain concen a ed o ce ( o ce ec o ):
,
,,)(().,,
A
A
dAxyz xyz
nn
σF
(10)
The σn(x,y,z) quan i y in oduced his way is called s ess
ec o [2]. The concen a ed o que on a small bu ini e
size su ace igu e can be in e p e ed simila ly [2]:
,
,,) ,( ,( ).
A
A
x A yzdyzy x
nn
σM
(11)
The equilib ium o elemen al e ahed on can be
examined wi h he help o o ces and o ques in e p e ed
on a plane igu e. I can be e i ied wi h o ce equa ions
ha s esses o m a second o de enso quan i y, and i
can be e i ied wi h o que equa ions ha he s ess enso
is symme ical [2].
The equilib ium o he elemen al cuboid can be ex-
amined wi h he help o o ces and o ques in e p e ed on
a su ace igu e. The gene al o m o o ce equa ions is
0, ( , , ),
yi
xi zi
i xyz
xyz
(12)
which can be ex ended wi h body o ce and he ine ia
pa ame e [2], he o que equa ions a e me due o he
symme y o he s ess enso .
No e: We ha e 6 dynamic a iables, wi h h ee ela-
ionships ( he equilib ium equa ions) among hem. The
dynamic desc ip ion o he con inuum is inde ini e.
The e a e ela ionships and ma e ial equa ions
be ween quan i ies cha ac e ising he kinema ic and dy-
namic side and p o ide exac ly as many equa ions as he
numbe o a iables in he sys em.
3
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3.2 Possibili ies and limi a ions o he con inu-
ous in physical space model
The con inuous model is based on many, p ima ily
opological and me ic assump ions (see [6,7]).
The model can desc ibe he change in shape o he
de o mable solid body (locally he ela i e s e ches and
also locally he angle change be ween wo cu es), and
he in e nal o ces (s esses) dis ibu ed in he body.
The model canno exp ess he o a ion o a poin , he
o ce ac ing upon a poin , and he o que ac ing upon a
poin as a poin has no dimensions and su aces ac upon
each o he a he han poin s. A he same ime he model
is sui able o exp essing he o a ion o a di ec ion ec o
and he neighbou hood o a poin as well as assigning
o ce and o que o a small bu ini e size su ace igu e
wi h any ou wa d no mal ec o in he poin .
The model only desc ibes he change kinema ically
co ec ly in case o ixed a omic-molecula o de o
s ained (unchanged opology) pa icles. The e o e e.g.: in
case o a omic s uc u e he s ain o he c ys alline g id is
desc ibed accu a ely by he con inuous model. I canno
desc ibe he ea angemen o he a omic o de accu a ely
in case o plas ic low as he ea angemen is
accompanied by he damage o he opologic o de .
Simila ly, he model does no e eal he ea angemen o
a oms in luids, he collisions o a oms-molecules in gases
and hei changeo e s ei he .
The model can only desc ibe he collec i e mo emen
o molecules. The e o e he model is sui able o
desc ibing acous ic ib a ions no o he op ical b anch.
I is capable o desc ibing lamina low in luids and gases
wi h adequa e accu acy bu no he c oss di usion.
The model is no sui able o de e mining o ce
be ween a oms-molecules and pa icles as i can only
in e p e de ined o ce and o que in he in eg a ed sense
assigned o he su ace igu es o a small bu ini e size.
The model does no e eal kinema ic and dynamic
pa ame e s ela ed o dimension o elemen s (a oms, mo-
lecule, pa icle) conside ed o be ini e. The e o e, he
alues o o ces be ween a oms and molecules cons i u-
ing he ma e canno be de e mined in he model no he
alues o o a ion o pa icles and he o ques occu ing
du ing o a ion.
4 Disc e e in he physical space, con-
inuous in he phase space model
4.1 Desc ip ion o he disc e e in he physical
space, con inuous in he phase space model
The ma e is s ill conside ed o be he sys em o ini e
dimension elemen s in he physical space. (The da a lis ed
in he disc e e model a e known o he ask; see he i s
pa ag aph o poin 2.1.) We p esume abou his model ha
i is pe iodical and ixed. Because i is ixed, he beha iou
o he medium can be cha ac e ised wi h he ac ha he
elemen s cons i u ing he medium do no change places,
hey only mo e a ound hei es posi ion a mos . We
p esume in he s udy ha in he g id poin s a mass poin ,
a igid body o a de o mable solid body a e loca ed.
We assigned one s a e unc ion o all elemen s col-
lec i ely and no o each and e e y elemen in he phase
space. We conside displacemen as he example. Ins ead
o he
( ( , , ))
i jk
Px y zu
displacemen ec o s assigned o
he a ious
(, , )
i jk
Px y z
poin s, we conside one, he dis-
placemen ec o ield
(, ,)xyzu
. Consequen ly, ins ead
o he
( ( , , ))
i jk
Px y zu
„exac ” solu ion, we conside he
„app oxima e” solu ion
(, ,)xyzu
. The „ ansi ion” is
b ie ly ma ked:
( ( , , )) ( , .,)
i jk
Px y z xyzuu
(13)
In his case we ha e no ye made he heo e ical
desc ip ion con inuous in phase space. The mechanical
ela ionships in e p e ed in disc e e poin s a e s ill
in e p e ed in disc e e poin s, bu i is no w i en in he
( ( , , ))
i jk
Px y zu
o m, bu in he
(, , )
(, ,)
i jk
Px y z
xyzu
o m.
(The e a e he same numbe s o a iables in bo h o ms.)
The nex s ep o con inuous model cons uc ion in he
phase space is o ind such con inuous equa ions, which
desc ibe he mechanical s a e o he disc e e sys em o he
con inuous kinema ic u(x,y,z) and dynamic analogue wi h
his a iables, as well as he ma e ial ela ions. I has o be
no iced ha we app oach he „ ansi ion” analogue unde
(13) wi h one o he s eps applied in he nume ical
me hod, he selec ion o app oaching unc ions: he
unknown unc ions a e app oached by he linea
combina ion o known basis ( unc ions), and we se up
algeb aic equa ions o de e mina ion o unknown eal
numbe s (namely coe icien s o he known basis
unc ions). The nex s ep o model cons uc ion in he
phase space has o be analogue wi h he o he s ep o
nume ical me hod: he dis ance o he exac and app ox-
ima e solu ion has o be de e mined and his dis ance has
o be minimised o made o hogonal o any (comple e)
ec o o he space o he applied basis unc ion. The
nume ical me hod applies o he app oxima e solu ion o
he known equa ion. The ask du ing he es ablishmen o
he con inuous model in he phase space is o se up
(con inuous) s a e equa ion ela ed o he s a e unc ions
in e p e ed as con inuous one in he phase space. As we
do no ha e a con inuous ope a o in he phase space, we
canno in e p e he dis ance o he exac and app oxima e
solu ion o he nume ical me hod in an analogue way.
Consequen ly, i is no su icien o conside he s a e
unc ions as con inuous when c ea ing a con inuous in he
phase space model, bu he ela ed s a e equa ions also
ha e o be „selec ed” con inuous. The selec ion o he
equa ion canno be a bi a y as i is o be applied o
desc ibe a gi en mechanical sys em. The only possibili y
is o apply analogy. A oiding de ails, in o de o c ea e a
con inuous in he phase space model, he Lag ange-
unc ion o he disc e e sys em has o be made con inuous
as he example in (13):
( ( , , ), ( , , ), , ( , , ))
( ( , , ), ( , , ), , ( , , )).
i jk i jk i jk
L xyz xyz Pxyz
L xyz xyz Pxyz
uF
uσ
(14)
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MATEC Web o Con e ences 184, 01004 (2018) h ps://doi.o g/10.1051/ma eccon /201818401004
Annual Session o Scien i ic Pape s IMT ORADEA 2018
No he di e ence o he unc ion alues aken up in
he disc e e poin s bu i s con inuous coun e pa has o be
included in he con inuous Lag ange- unc ion; applying
he Taylo Se ies, o displacemen :
1
( ( , , )) – ( ( , , )) ( , , ) / .
i jk i jk
Px y z Px y z xyz x
uu u
(15)
In e p e ing displacemen (u), o a ion (φ) and he
s ain modus (ε) as p ima y kinema ic a iable, he
seconda y kinema ic a iables a e in e p e ed wi h he
ela ionships below:
,,) ,,( )( ,xyz xyz
u
εu
(16)
,,) ,,),( (xyz xyz
φ
εφ
(17)
,,) ,,( )( .xyz xyz
ε
εε
(18)
Fo mally, o he analogy (13) o he disc e e-con-
inuous „ ansi ion”, a
( ( , , )) ( , , )
i jk
Px y z xyzFσ
ype
disc e e-con inuous „ ansi ion” should be also applied in
he case o in e nal o ces. In p ac ice, we conside he
pa ial di e en ial quo ien s acco ding o kinema ic a i-
ables o he Lag ange- unc ion as (gene alised) dynamic
a iables. Fo hese, in he case o he u, φ and ε a iables
,,) ( ,,)( ((, , , ), , ) / ,xyz L xyz xyz
u uu
σ u ε ε
(19)
,,) ( ,,)( ((, , , ), , ) / ,xyz L xyz xyz
φ φφ
σ φ ε ε
(20)
,,) ( ,,), ,,), ,)/( ((xyz L xyz xyz
ε εε
σ ε ε ε
(21)
ela ionships can be se up. The gene alised Hooke’s law
o he gene alised con inuum can be se up in he o m:
.
u uu uφ uε u
φ φu φφ φε φ
ε εu εφ εε ε
σ CCC ε
σ CCC ε
σ CCC ε
(22)
Bo h in (19-21), and (22) we se ou om he ac ha
he ma e ial equa ions, namely ela ions be ween he
gene alised s ains and he gene alised in e nal o ces, a e
known.
The s a e equa ions ( he desc ip ion limi ed o
equilib ium), in acco dance wi h he me hod o he model
c ea ion, a e p o ided wi h by de e mining he maxima o
he L unc ional desc ibing he s a e. Fo mally, he
ollowing h ee equa ions a e se up o a iables u, φ and
ε:
( ( , , ), ( , , ), , ( , , )) /
( ( , , ), ( , , ), , ( , , )) / ,
L xyz xyz Pxyz
L xyz xyz Pxyz
uu
u
uεε
uεu
(23)
( ( , , ), ( , , ), , ( , , )) /
( ( , , ), ( , , ), , ( , , )) / ,
L xyz xyz Pxyz
L xyz xyz Pxyz
φφ
φ
φε ε
φε φ
(24)
( ( , , ), ( , , ), , ( , , )) /
( ( , , ), ( , , ), , ( , , )) / .
L xyz xyz Pxyz
L xyz xyz Pxyz
εε
ε
εε ε
εε ε
(25)
Finally, he unc ional – he Lag ange- unc ion o he
sys em –, which desc ibes he s a e o he sys em also has
o be gi en. The con inuous Lag ange- unc ion in he
phase space can only be se up based on analogy.
Regula ly, we se ou om he ac ha kine ic ene gy is
p opo ional o he squa e o eloci y, while elas ic ene gy
is p opo ional o he squa e o he s ain. Taking his in o
conside a ion, based on analogy, he Lag ange- unc ion
can be cons uc ed. The s a e equa ions o he sys em o
a iables u, φ and ε can be se up in he ollowing ope a o
o m:
( )( )( ) .I
uu u uu uφ uε u u
u Cu Cφ Cε Q
(26)
( )( )( ) .I
φφ φ φu φφ φε φ φ
φ Cu Cφ Cε Q
(27)
( )( )( ) .I
εε ε εu εφ εε ε ε
ε Cu Cφ Cε Q
(28)
The Iij quan i ies a e ine ias o he „sp ead” ma e in
he gene alised con inuum e sus he displacemen ,
o a ion and s ain. The analogy is pe ec wi h he mo ion
equa ions o he classic con inuum because he o mulas
we e compiled acco ding o i s o malism. Such
condi ions om ma hema ical aspec s exis , which ensu e
he co ec ness o he a ge ed bounda y alues ( [8]).
4.2 Cha ac e isa ion o disc e e in he physical
space, con inuous in he phase space model
The con inuous in he phase space model is cons uc ed
based on analogy. The con inuous, seconda y kinema ic
a iables, he con inuous Lag ange- unc ion o he sys em
and he con inuous in e nal o ces a e in e p e ed based on
analogy. Following his, we se up he equa ions
desc ibing he s a e o he sys em based on he known
ma hema ical algo i hms. A peculia i y o he model is
ha no such expe imen s exis , which would enable di ec
measu emen o ma e ial cons an s. While in he case o
classic con inuum, om he di e en ial geome ic
desc ip ion i can be concluded such expe imen al
a angemen (uniaxial pull and shea ), which connec s he
six s ains o an elemen al cube o s ess assigned o h ee
su aces o he elemen al cube can be in e p e ed, in he
case o he gene alised con inuum no such geome ical
shape exis s, whe e he dynamic a iables could be clea ly
assigned o i s displacemen s. In case o gene alised
con inuums no simply single-pa ame e expe imen s a e
equi ed o he e i ica ion o he model. Fi s ly, he
in oduced kinema ic and dynamic a iables ha e o be
in e p e ed, secondly, such expe imen al a angemen s
ha e o be p epa ed, whe e hese phenomena can be
de ec ed, measu ed and he expe imen s ha e o be ca ied
ou , hi dly, he heo e ical asks ega ding he a ge ed
expe imen al a angemen ha e o be se and ha e o be
sol ed. Finally, he expe imen al esul s ha e o be
compa ed wi h he heo e ically de e mined esul s. As
many expe imen al s a es ha e o be in e p e ed as he
numbe o ma e ial cons an s exis in he heo y. I hese
ma e ial cons an s a e de e mined in acco dance wi h he
expe imen s, ha is o say he pa ame e s in he heo y a e
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MATEC Web o Con e ences 184, 01004 (2018) h ps://doi.o g/10.1051/ma eccon /201818401004
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i ed o he measu ed esul s hen he heo y can be used
o he ask o be sol ed.
The model desc ibes he s a e o poin s loca ed on g id
poin s; he alues o con inuous unc ions aken up on g id
poin s ha e ma hema ical and mechanical senses.
The model can be c ea ed p e e ably when he
mechanical ela ionship among elemen s loca ed on g id
poin s is known.
5 Summa y
We o e iewed h ee possibili ies o modelling he
mechanical beha iou o media. The i s possibili y, is
ha he mo ion o each and e e y mass pa icle cons i-
u ing he medium desc ibed by an indi idual equa ion.
The unc ions desc ibing he s a e o he medium ob ain a
alue in he loca ion o mass pa icles cha ac e ised by i .
The sys em is disc e e om his aspec . This possibili y
can be es ic ed o he concu en examina ion o a ew
housand o en housand mass pa icles; i is no sui able
o he examina ion o millions o pa icles o a a omic
le el o he concu en examina ion o he scale o 1023
gas molecules p esen in one cubic decime e. As a second
possibili y, he applica ion o some con inuous unc ions
has o be men ioned ins ead o he disc e e domain
unc ions. This in he i s app oach means ha he s a e
o he sys em desc ibed by a ew unc ions in e p e ed in
he con inuous domain o some con inuum consis ing o
many poin s including independen poin s ins ead o
unc ions in e p e ed in a ini e numbe o independen
poin s. Two such spaces can be iden i ied. One is he
physical space, whe e he elemen s o he medium mo e,
while he o he is he phase space, whe e he s a e o he
elemen s o he medium a e ma hema ically cha ac e ised.
The e o e, he second possibili y is o conside a egion o
he Euclidean space (which has con inuum ca dinali y)
ins ead o he examined ini e numbe o pa icles, and we
ex end he mechanical ela ionships ela ed o he
pa icles o his domain. In his model cons uc ion, we
„sp ead” all mass p esen in poin s by de aul in one
con inuous egion o he Euclidean space modelling he
physical space, we ha e no mass pa icles bu a medium
wi h con inuous dis ibu ion exis s, along wi h his he
unc ions cha ac e ising he sys em a e con inuous
unc ions in e p e ed on a con inuum. This model c ea ion
is based on di e en ial geome y. This me hod leads o
he e m o classic con inuum and as such only one exis s.
The hi d possibili y is o keep he ini e numbe o many
poin s wi h hei independen , unique size, cha ac e is ics
as models, a he same ime we embed hem in o he ini e
egion o he Euclidean space wi h he help o hei
e e ence poin s, hen we in e p e he con inuous
unc ions desc ibing he sys em on his ange and ex end
ela ionships ela ed o he disc e e mechanical sys em o
hese con inuous domain unc ions. In his model con-
s uc ion, we „sp ead” he unc ions desc ibing he s a es
in he phase space by de aul : con inuous domain (con
inuous) unc ions a e included ins ead o disc e e
domain unc ions. We c ea e he con inuous domain
Lag ange- unc ion o he sys em and de e mine he s a e
indica o s and s a e equa ions o con inuous unc ions.
This model c ea ion applies s eps o he nume ical me hod
(selec ion o basic unc ions, c ea ion o e o p inciple).
This me hod is sui able o desc ibing ixed, pe iodical
s uc u e igid bodies and leads o he gene alised
con inuums. The gene alised con inuum, con a y o i s
name, is no con inuous bu a disc e e sys em wi h ypical
disc e e s a es (e.g.: op ical ib a ion b anch o a oms),
which do no exis in he classic con inuum. Many
gene alised con inuums exis , hese o m a hie a chy.
Re e ences
[1] Lo e, A.E.H. A T ea ise on he Ma hema ical
Theo y o Elas ici y. Fou h ed. Camb idge, A he
Uni e si y P ess (1927)
[2] Lu ’e, A.I.: Theo y o Elas ici y. In Russian. Nauka,
М. (1970)
[3] Láme G.: No es on he Theo y o La ge Displace-
men wi h Small S ain = Pe iodica Poli echnica 29
(1-2), pp. 53-65 (1985)
[4] Láme G.: Ma hema ical Founda ions o he
Theo ies o Pe ec Elas ic Shells and Rods Un-
de doing La ge Displacemen wi h Small S ains.
PhD Disse a ion. (A kis alak ál ozások melle
nagy elmozdulásoka égző ökéle esen ugalmas
héjak és udak elméle einek ma ema ikai alapjai)
Budapes (1990)
[5] Láme G.: On he Kinema ics o he Con inuum
Unde doing La ge Displacemen wi h Small
S ains. (Kis alak ál ozások melle nagy elmoz-
dulás égző kon inuum kinema ikájá ól) = Épí és-
, Épí észe udomány XXIII (1-2), pp. 35-59 (1992-
93)
[6] Láme G.: Oppo uni y and Limi o Applica ions
o he Topological Tools in he Mechanical Models
o he Media. (Topológiai eszközök alkalma-
zásának lehe őségei és ko lá ai a közegek mecha-
nikai modellezésében). P oceedings o XII.
MAMEK (Miskolc, 2015. aug. 25-27.) Ed.: Baksa
A. – Be ó i E. – Szi bik S. pape 211. p. 13 (2015)
[7] Láme G.: Oppo uni y and Limi o Applica ions
o he Me ical Tools in he Mechanical Models o
he Media. (Me ikus eszközök alkalmazásának
lehe őségei és ko lá ai a közegek mechanikai
modellezésében. P oceedings XII. MAMEK (Mis-
kolc, 2015. aug. hó 25-27.) Ed.: Baksa A. – Be ó i
E. – Szi bik S. pape 326. p. 13 (2015)
[8] Kunin, I.A.: Theo y o Elas ic Media wi h
Mic os uc u e. In Russian. Nauka, М. (1975)
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MATEC Web o Con e ences 184, 01004 (2018) h ps://doi.o g/10.1051/ma eccon /201818401004
Annual Session o Scien i ic Pape s IMT ORADEA 2018