ESGI 158 Repo
Sa e ajec o y o a piece mo ed by a obo
No embe 9, 2020
E nes Benedi o, e nes .b[email p o ec ed],
Oli e Bond, [email p o ec ed],
Thomas Babb, [email p o ec ed]x.ac.uk,
Juan R. Pacha, juan. [email protected],
Sandeep Kuma , skuma @bcama h.o g,
Joan Sol`a-Mo ales, jc.sola-mo [email protected].
1 In oduc ion
The company F.EE (www. ee.de/en.h ml) is an in e na ional company supplying au oma ion ech-
nologies o indus y, mos ly in he au omo i e sec o . The company is in e es ed in de e mining he
equa ions o he ajec o ies and he o ien a ions o pieces ha a e mo ed by he ac ion o a obo
wi h se en deg ees o eedom, o which six co espond o o a ions o he a ms and one o longi udinal
ans e s. In many cases, he piece is a cu ed me al shee , and in he o he s, a h ee-dimensional
objec .
A he ini ial momen , he piece is a a ce ain poin /loca ion, o ien a ion, and es posi ion,
hen he obo g abs and lea es i o ano he gi en poin /loca ion, o ien a ion, and es posi ion
co esponding o he o he ime ins an . Du ing i s mo emen , he piece can su e e e sible and
i e e sible de o ma ions caused by bo h mass o ces and su ace o ces. The mass o ces a e due o
ansla ion and o a ions (depending on accele a ions and angula eloci y o he body) while su ace
o ces a e due o ai d ag (depending on he eloci y o he body) ha ac on i . The company wan s
ha he esul ing mo ion does no esul in i e e sible de o ma ions on he piece and ha he ime
be ween he ini ial and inal momen s does no exceed a ce ain h eshold. The ac ual mo emen o
a piece is a consequence o he mo emen s o he obo . This in u n depends on he design o he
ajec o y o be ollowed by he piece, as well as on he obo ’s abili y o ai h ully ollow he desi ed
ajec o y. Any solu ion o he p oblem mus conside no only he condi ions unde which he desi ed
ajec o y canno cause i e e sible de o ma ion bu also, he condi ions unde which he obo can
ollow he ajec o y. The o me depends on he desi ed eloci y, accele a ion, angula eloci y o he
piece, and he la e depends on he so-called je k, o , he i s -de i a i e o he accele a ions.
Ce ainly, he p oblem is complex and is a ely add essed in he scien i ic li e a u e. The e is
abundan li e a u e on he ajec o y design o a piece mo ed by a obo wi h a ying deg ees o
eedom [1, 2] bu he de o ma ions caused by mass o ces o ic ion a e gene ally no aken in o
accoun .
This is ce ainly a ma e o p ac ical in e es o he manu ac u ing indus y and ha i should
be o he scien i ic communi y as well.
1
a Xi :2011.03330 1 [cs.RO] 22 Oc 2020
Figu e 1: Le : Me al piece mo ed by he a m o he obo . Righ : Me al piece as a pa o a ca .
Due o i s complexi y, we ha e unde s ood ha he p oblem canno be sol ed in limi ed ime such
as i e days o he ESGI mee ing, bu in his ime, a concep ual ma hema ical amewo k has been
laid ou , which can be la e adap ed and sol ed. The e o e, du ing he ESGI days, we ha e es ic ed
he objec i es o he p oblem o he ollowing asks:
•Pe o m a li e a u e sea ch in his a ea, including a sea ch o nume ical codes o sol e hese
kinds o p oblems.
•W i e he equa ions o mo ion o he body- luid sys em, in he mos gene al case possible, by
de e mining he s esses and de o ma ions su e ed by a body due o he o ces o mass and
ic ion ac ing on i when being mo ed by ano he .
•Toge he wi h he possible use o comme cial, discuss nume ical algo i hms o sol e hese equa-
ions.
•De ine a e y simple case o he p oblem. This has been he mo emen o a longi udinal piece,
held a one end, o a body ha can mo e in a di ec ion pe pendicula o ha o he o ce o
g a i y, and can o a e he piece in he plane de ined by he di ec ion o mo ion o he body and
ha o he o ce o g a i y.
The esul s o he i s poin will be desc ibed in Sec ion 5 below, oge he wi h conclusions abou
possible u u e wo k, and he nex h ee poin s will be desc ibed in Sec ions 2, 3, and 4.
We wan o poin ou ha we ha e de o ed ou e o s o he so-called di ec p oblem, a he
ha he mo e di icul , bu pe haps mo e impo an in p ac ice, in e se p oblem. Le us explain he
di e ence: In he di ec p oblem, one assumes ha he ajec o y o he a m and i s o a ions a e
known and gi en, and he p oblem is o calcula e de o ma ions and s esses o he piece. In he in e se
p oblem, he goal is o de e mine ajec o ies and o a ions such ha he induced de o ma ions and
s esses sa is y he es ic ion equi emen s gi en by he na u e o he pieces, o a oid pe manen
de o ma ions. E en i one conside s ha only he in e se p oblem is o p ac ical in e es , i is clea
ha he solu ion o he in e se p oblem equi es he solu ion o many di ec p oblems, o achie e he
op imal candida e.
2 The equa ions
Be o e explaining he equa ions and bounda y condi ions speci ic o pla e heo y, we p o ide a b ie
in oduc ion o he go e ning equa ions in solid mechanics. This explana ion is a condensed e sion
o ha ound in [3], which gi es a much mo e igo ous in oduc ion o he ield.
2
In oduc ion o solid mechanics Suppose ha any poin inside an elas ic objec P ⊂ R3has
an ini ial Lag angian coo dina e ~
X∈ P, which is ixed in he ma e ial. A ime > 0, his
coo dina e changes wi h elas ic de o ma ions o he objec , in o an Eule ian coo dina e ~x(~
X, ),
whe e he coo dina e ~x emains ixed in space. F om hese, we can cons uc he displacemen ield
~u(~x, ) := ~x(~
X, )−~
X, which in pa is ound by o mula ing an equa ion ela ing body and su ace
o ces, and using conse a ion o momen um o ob ain a pa ial di e en ial equa ion known as he
Na ie equa ion in which ~u(~x, ) is he dependen a iable.
Conside an a bi a y olume Vcon ained wi hin he elas ic ma e ial. This small olume expe i-
ences body o ces (e.g. due o g a i y) and su ace o ces. In he case whe e he densi y ρis uni o m
h oughou he ma e ial, one can w i e down exp essions o bo h o hese e ec s, as well as he a e
o change o momen um (whe e momen um is he mass o an objec mul iplied by i s eloci y). In
his case, he i h componen o he displacemen ield, ui, sa is ies
d
d ZZZV
∂ui
∂ ρdP
| {z }
a e o change o momen um
=ZZZV
giρdP
| {z }
o al body o ce
+ZZ∂V
σijnjdS
| {z }
o al su ace o ce
,
whe e giis he i- h componen o he body o ce, and σij is he (symme ic) s ess enso , which ells
us he i h componen o he o ce pe uni a ea ac ing a a poin on he su ace wi h ou wa d no mal
nj. By aking he ime de i a i e inside he i s e m (using he ac ha he densi y is cons an )
and applying he di e gence heo em on he inal e m, we can use he ac ha Vis a bi a y and
he assump ion ha each in eg and is con inuous o ob ain Cauchy’s momen um equa ion
ρ∂2ui
∂ 2=ρgi+∂σij
∂xj
.(1)
To be able o de e mine he displacemen ield elies on us knowing he s ess enso σij. Thank-
ully, he e is a cons i u i e ela ion, known as Hooke’s law, which pos ula es a linea ela ionship
be ween he s ess and ano he quan i y called he s ain, a dimensionless quan i y signi ying he
elas ic ex ension o an objec ela i e o i s o iginal s a e. To iden i y he exac o m o he s ain
enso , we can conside wo pa icles wi h posi ions ~
Xand ~
X+δ~
X, displaced o ~x =~
X+~u(~
X, )
and ~x +δ~x =~
X+δ~
X+~u(~
X+δ~
X, ). Taylo ’s heo em can be used o show ha (upon neglec ing
quad a ic e ms)
|δ~x|2=δ~
X+ (δ~
X· ∇ ~
X)~u(~
X, )
2,
and he e o e ha
|δ~x|2−δ~
X
2= 2εijδXiδXj,
whe e he s ain enso εij is gi en by
εij =1
2∂ui
∂Xj
+∂uj
∂Xi
+∂uk
∂Xi
∂uk
∂Xj∼1
2∂ui
∂Xj
+∂uj
∂Xi.(2)
I he s ess and s ain a e scala s, hen Hooke’s law simply s a es ha σ=Eε, whe e Eis a cons an
known as he Young modulus. Howe e , in his con ex , σij and εij a e bo h ank-2 enso s, so a linea
ela ion be ween hem mus in ol e a ank-4 enso , so ha
σij =Cijklεkl, i, j, k, l = 1,2,3.
He e, Cijkl has 81 en ies in o al. Howe e , by conside ing he s ess ac ing a he su ace o he olume
V(by conside ing a pillbox- ype a gumen ), one can show ha he ank-2 enso s a e symme ic (i.e.
3
Figu e 2: A simple illus a ion o he coo dina e sys em used in Ki cho -Lo e pla e heo y o a ( la )
pla e.
σij =σji and εij =εji). Unde u he modelling assump ions, such as homogenei y and iso opy o
he ma e ial, one can de i e a s ess-s ain ela ion o he o m
σij =λεkkδij + 2µεij,(3)
whe e λis he bulk modulus and µis he shea modulus o he ma e ial. Bo h o hese cons an s a e
collec i ely called he Lam´e cons an s and hey a e ma e ial p ope ies which indica e a ma e ial’s
endency o wi hs and de o ma ion. Upon subs i u ing (3) and (2) in o (1), one ob ains he Na ie
equa ion, w i en in ec o o m as
ρ∂2~u
∂ 2=ρ~g + (λ+µ)∇(∇ · ~u) + µ∇2~u. (4)
By imposing sui able bounda y condi ions, one can sol e equa ion (4) o he displacemen ield ~u(~
X, ).
The usual echniques o applied ma hema ics, such as asymp o ic analysis and nume ical me hods,
can be used o sol e his equa ion, al hough nume ical me hods will be much mo e app op ia e o
a bi a y geome ies (such as hose om CAD iles p o ided by componen manu ac u e s).
Classical pla e heo y A conside able simpli ica ion in ou si ua ion is ha many componen s
mo ed a ound by a obo a e h ee-dimensional bu a e hin; o example, a ca doo . Al hough many
people hink o his ype o componen as s ong, namely due o being made o me al, hin me al shee s
a e p one o elas ic displacemen s which could exceed an elas ic limi , de o ming plas ically and hen
being unsui able o use in any u he manu ac u ing (and hus needing o be ecycled). Thank ully,
a heo y has been de eloped o ackle he case whe e he componen is la ge and hin. This is called
Ki scho -Lo e heo y (o classical pla e heo y), whe e in-plane displacemen s a e dis ega ded, and
ha displacemen s no mal o he plane a e conside ed ela i e o a mid-plane: a su ace equidis an
be ween he op and bo om ace o he plane. This idea is illus a ed in Figu e 4, al hough we
emphasise ha he pla e need no be uni o m e e ywhe e since he coo dina es a e local o he pla e.
We guide he eade h ough he basics o his heo y; and e e hem o [4] o a much mo e de ailed
explana ion. In he speci ic case shown in Figu e 4, an educa ed guess o he displacemen ield is
gi en. This is o he o m
ui(x, y, z, ) = X
j
(z)jφ(j)
i(x, y, ),
4
whe e φ(j)a e unc ions chosen so ha he p inciple o i ual displacemen s is sa is ied and zis he
coo dina e in he di ec ion o hickness. A i ual displacemen o an elas ic sys em is an in ini esimal
displacemen which can one migh suspec o a ise based on he cu en con igu a ion o o ces and
i i was pe u bed sligh ly. Fo example, i an elas ic beam o leng h Lis ixed a a wall a x= 0
and being pulled away by a o ce Fbeing applied a i s ee end, hen he beam can be hough o as
ha ing a “ i ual” displacemen o δu(L) a i s ee end a ising om a “ i ual” o ce δF. The idea
o he displacemen s being “ i ual” is so-called because hey a e ic ional and a e un ela ed o he
displacemen s and ac ual loads on he sys em.
Vi ual wo k and i ual displacemen The idea o “ i ual wo k” a ises in wo possible ways.
We i s ly ecall ha he wo k done by a o ce is he p oduc o i s p ojec ion (do p oduc ) on o i s
di ec ion o displacemen , and i s magni ude (o , pu simply, “wo k is o ce imes dis ance”). Vi ual
wo k can be conside ed o a ise ei he om (a) an ac ual o ce mo ing h ough a i ual displacemen ,
o (b) a i ual o ce mo ing h ough an ac ual displacemen . I a lowe case del a be o e a a iable
deno es an in ini esimal i ual change in ha a iable, hen he i ual wo k is gi en by
δW =ZZZP
~
F·δ~udP.
Vi ual wo k can be classi ied as ei he ex e nal o in e nal. Fo ces applied by ex e nal sou ces will
do wo k when mo ing h ough i ual displacemen s, and o his e ec one can w i e down he o al
ex e nal i ual wo k. When an elas ic body is subjec ed o body o ces o ~
pe uni olume and
su ace ac ions ~
Tpe uni su ace a ea (whe e Γσis he subse o ∂Pon which s esses a e speci ied),
upon mo ing h ough i ual displacemen s δ~u i will do a o al amoun o i ual wo k δV due o
applied o ces. This is gi en by
δV =−ZZZP
~
·δ~udP+ZZΓσ
~
T·δ~udS.(5)
A de o ming elas ic objec will also unde go in e nal s esses and hen he wo k can be ob ained in
e ms o he o al s ain inside i a he han due o speci ic ec o o ces. In simple e ms, he s ain
o an elas ic body is he a io o he ex ension o an elas ic body o i s o iginal size, and is he e o e
a dimensionless quan i y, and since i in ol es de i a i es o displacemen , i ual s ains δεij can be
exp essed in e ms o i ual displacemen s δu. By conside ing wo k done by no mal s esses and
shea s esses h ough i ual displacemen s, one can show ha he o al in e nal wo k δU is gi en by
δU =
3
X
i=1
3
X
j=1 ZZZP
σijδεijdPwhe e δεij =1
2∂δui
∂xj
+∂δuj
∂xi.(6)
A his poin , we can concisely s a e he p inciple o i ual displacemen s, i.e. ha a con inuous
body in equilib ium will ha e a o al i ual wo k by ac ual o ces h ough i ual displacemen s o
ze o. Pu simply, δV +δU = 0; as each o hese e ms a e in eg als, he p oblem hen becomes abou
minimising hese in eg als ( h ough a weak o mula ion) a he han sol ing o an unknown unc ion
wi h a di e en ial equa ion ( h ough a s ong o mula ion). Mo e speci ically o in e es o us, he
dynamical e sion o he p inciple o i ual displacemen s is
ZT
0
(δU +δV −δK) d = 0,(7)
whe e δK is he i ual kine ic ene gy o he sys em. This me hod o de i ing go e ning equa ions
o an elas ic body is an a ac i e al e na i e o equa ion (4) because (a) i does no need o ely on
he use o cons i u i e laws up on (e.g. Hooke’s law), and (b) he elas ici y p oblem can be posed
5

in he o m o a a ia ional p oblem, he eby making i amenable o bo h analy ical ea men s (e.g.
calculus o a ia ions, Eule -Lag ange equa ions, Hamil on’s p inciple) and con empo a y nume ical
me hods such as ini e elemen me hods (FEM). How one migh app oach his p oblem using FEM is
o be discussed la e .
De i ing he go e ning equa ion We a e in e es ed in Ki cho ’s hypo hesis which p o ides he
elas ic displacemen ield a p io i as
u(x, y, z, ) = u0(x, y, )−z∂w0
∂x ,
(x, y, z, ) = 0(x, y, )−z∂w0
∂y ,
w(x, y, z, ) = w0(x, y, ),
whe e u0, 0and w0a e displacemen s om he mid-plane in he x,yand z-di ec ions espec i ely.
In pa icula , we can se u0≡0 and 0≡0, due o neglec ing in-plane displacemen s. Namely, we
a e in e es ed in he domain [−w, w]× F, whe e F ep esen s he c oss-sec ion o he pla e a z= 0.
Subs i u ing he abo e o m o he displacemen ield in o he linea ised s ains (2), neglec ing in-plane
displacemen s, i is s aigh o wa d o show ha he s ain enso is gi en by
11 =1
2∂w0
∂x 2
−z∂2w0
∂x2
22 =1
2∂w0
∂y 2
−z∂2w0
∂y2
12 =1
2∂w0
∂x
∂w0
∂y −2z∂2w0
∂x∂y 
13 =1
2−∂w0
∂x +∂w0
∂x = 0
23 =1
2−∂w0
∂y +∂w0
∂y = 0
33 = 0,
(8)
whe e we ha e gi en six en ies ins ead o nine, owing o he ac ha he s ain enso is symme ic
(so ha ij =ji). This pu s us in a posi ion o calcula e he in e nal i ual wo k δU, gi en by
δU =Zw
−wZF
(σ11δ11 + 2σ12δ12 +σ22δ22) dFdz
=−Zw
−wZFσ11z∂2δw0
∂x2+ 2σ12z∂2δw0
∂x∂y +σ22z∂2δw0
∂y2dFdz+ h.o. .
=−ZFM11
∂2δw0
∂x2+ 2M12
∂2δw0
∂x∂y +M22
∂2δw0
∂y2dF+ h.o. .
whe e “h.o. .” is an abb e ia ion o “highe -o de e ms”; e ms which a e quad a ic in he pa ial
de i a i es o δw0and can he e o e be neglec ed (al hough his is no longe app op ia e i on Ka man
s ains a e conside ed). We also ha e
Mij =Zw
−w
σijzdz, (9)
a e he s ess momen esul an s. We emphasise ha in gene al, he e will also be s ess esul an s
Nij =Zw
−w
σijdz,
6
bu neglec ing he e ms which a e quad a ic in he de i a i es o w0leads o hem being absen om
he exp ession o he in e nal i ual wo k.
Meanwhile, he ex e nal i ual wo k is gi en by
δV =−Zw
−wZF
~
·δ~udFdz−Zw
−wZ∂F
~
T·δ~udSdz(10)
=−Zw
−wZF
(q−kw0)δw0dxdy. (11)
The second e m in equa ion (10) disappea s because he ac ion o ce, Ti=σijnjwhe e ~n is
he no mal o ∂F(a closed cu e in he (x, y)-plane), is pe pendicula o he z-di ec ion in which he
displacemen is solely assumed o ake place. In equa ion (11), q(x, y) is he ne load on F, and he
−kw0con ibu ion a ises om Hooke’s law, whe e kis he s i ness cons an o he pla e ma e ial.
The in e nal kine ic ene gy δK is gi en by
δK =ZPZw
−w
ρ( ˙uδ ˙u+ ˙wδ ˙w+ ˙wδ ˙w) dzdxdy
=−I0ZP
( ˙u0δ˙u0+ ˙w0δ˙w0+ ˙w0δ˙w0) dxdy−I2ZP∂˙w0
∂x
∂δ ˙w0
∂x +∂˙w0
∂y
∂δ ˙w0
∂y dxdy(12)
whe e he momen s o ine ia a e gi en by
Ik=Zw
−w
zkρdzwhe e k= 0,1,2.
We do no p esen he ull de i a ion o he go e ning equa ions he e due o he amoun o algeb a
in ol ed, bu b ie ly desc ibe how i is done (see pages 103-105 o [4] o a ull de i a ion). By
subs i u ing esul s (6), (11) and (12) in o (7), making use o he i ual s ains, employing echniques
om he calculus o a ia ions and se ing he coe icien o δw0 o ze o, one can show ha
∂2M11
∂x2+ 2∂2M12
∂x∂y +∂2M22
∂y2−kw0+q=I0
∂2w0
∂ 2−I2
∂2
∂ 2∂2w0
∂x2+∂2w0
∂y2.
Finally, i is desi able o exp ess his go e ning equa ion in e ms o displacemen s a he han mo-
men s. Fo a homogeneous, iso opic pla e, a cons i u i e law ells us ha


σ11
σ22
σ12

=E
1−ν2

1ν0
ν1 0
0 0 1 −ν



11
22
12

,(13)
whe e Eis he Young modulus o he ma e ial, and νis he Poisson a io. Combining he cons i u i e
law (13) wi h (9) and he s ain-displacemen ela ions (8), one ob ains
D∇2∇2w0=−q(x, y, )−2ρh∂2w0
∂ 2,(14)
whe e he bending s i ness,D, is gi en by
D=2h3E
3(1 −ν2),(15)
which will be loosely e e ed o as he “Ki cho -Lo e equa ion”.
7
Ini ial and bounda y condi ions Along wi h he equa ion o mo ion (14), se e al bounda y
condi ions based on likely physics in a ac o y se ing need o be imposed. Le C ⊂ ∂Fbe he pa o
he la ge ace bounda y which is clamped by he obo , hen he pla e
1. is ini ially unde o med, i.e.,
w0(x, y, = 0) = 0,
2. is ini ially s a iona y, i.e., ∂w0
∂ (x, y, = 0) = 0,
3. does no de o m whe e i is clamped, i.e.,
w0(x, y, ) = 0, x, y ∈ C;
4. and no bending momen s o loads a he ee bounda y:
∇2w0(x, y, )=0,∂
∂n(∇2w0(x, y, )) = 0,(x, y)∈∂F C
Changing he ame As i s ands, i su ices o sol e equa ion (14) o he ans e se elas ic
displacemen s o he pla e, p o ided ha he pla e is no being mo ed ex e nally. Conside ing how
he pla e is being mo ed a ound by a obo , his is no sa is ac o y; i he pla e is being mo ed in
a non-ine ial ame, i will expe ience “ ic i ious” o ces, namely he Eule , Co iolis and cen i ugal
o ces. These o ces a e so-called because hey a ise om a change o ame a he han a physical
mechanism. They a e ma hema ical in na u e and should he e o e be inco po a ed wi hin he ex e nal
load q(x, y, ) as i appea s in equa ion (14). The ma hema ical echniques o changing ames a e
s anda d and can be ound in classical mechanics ex s such as [5].
We use ˆ
S o deno e an ine ial ame which emains ixed o e ime, wi h o igin ˆ
Owhich is ixed
in space, and o hono mal basis {ˆ
~e1,ˆ
~e2,ˆ
~e3}. We use S o deno e he non-ine ial ame, which is ixed
local o he pla e and whose o igin Omo es wi h angula eloci y ~ω ela i e o ˆ
O. The ela ionship
be ween he wo coo dina e sys ems is illus a ed in Figu e 3.
F.EE ha e comple e con ol o e he o a ional mo emen o he obo ic a m, as well as ansla-
ional mo ion along a s aigh line. In p ac ice, he o a ional mo ion o he obo mo ion may be
p esc ibed by a o a ion ma ix Rij( ) which ac s as a ime-dependen linea ans o ma ion om ˆ
S
o Sand may be exp essed in e ms o Eule angles. I ˆ
D and D deno e ime de i a i es in he ine ial
and non-ine ial ames espec i ely, and ~ is he posi ion ec o o a poin in S ixed on he pla e
ela i e o O, hen he Co iolis o mula is gi en by
ˆ
D~ = D~ +~ω ×~ ,
whe e ~ω is he angula eloci y ec o whose en ies sa is y
Rij ˙
Rji =
3
X
k=1
ijkωk.(16)
Applying he Co iolis o mula wice can be used o ob ain an exp ession o he accele a ion ˆ
~a in he
ine ial ame in e ms o he accele a ion ~a in he non-ine ial ames:
ˆ
~a =~a + (D~ω)×~ + 2~ω ×D~ +~ω ×(~ω ×~ ) + ~
A, (17)
whe e ~
A=ˆ
D2~x is he accele a ion o O ela i e o ˆ
S.
8
Figu e 3: An illus a ion o pla e mo ion in wo ames o e e ence.
This is use ul because i can be used in p inciple o e-exp ess he Ki cho -Lo e equa ion (14) in
a non-ine ial ame local o he pla e, as he accele a ion a ises na u ally he e. The accele a ion
o mula (17) can hen be used o o e-exp ess New on’s second law in S a he han ˆ
S:
mˆ
~a =~
Fin ˆ
S=⇒m~a =~
F−m(D~ω)×~
| {z }
“Eule o ce”
−2m~ω ×D~
| {z }
“Co iolis o ce”
−m~ω ×(~ω ×~ )
| {z }
“Cen i ugal o ce”
+m~
Ain S.(18)
Since we a e only conce ned wi h elas ic oscilla ions o he pla e in he ~e3-di ec ion, i su ices o
ake he do p oduc o equa ion (18) wi h ~e3in o de o ob ain he equi alen o he Ki scho -Lo e
equa ion (14). We do no w i e down he equa ion in ull, since i depends on he speci ic o a ions
being applied. Howe e , we simply s a e ha he e m in ol ing he second de i a i e o w0wi h
espec o ime co esponds o a·e3and ha he ex e nal loads q(x, y, ) co espond o ~
F.
I is no qui e enough o simply apply a change o ame o he Ki scho -Lo e equa ion o ake
all physical conside a ions in o accoun . The e is also he weigh o he pla e, he ine ia, he in e nal
elas ic o ces inside he pla e, and also he ai d ag. Howe e , in a ac o y se ing, he e ec s o hese
e ms would be ques ionable, and may make he equa ions signi ican ly mo e coupled wi hou adding
much insigh . This is pa icula ly he case o he ai d ag because a comple e desc ip ion would ely
on coupling he elas ic displacemen equa ions discussed hi he o wi h he Na ie -S okes equa ions o
luid dynamics. A simpli ied s a ing poin could be o in oduce a simple d ag law, whe e he d ag
o ce on an objec in a luid is p opo ional o he squa e o he speed a which objec is being passed
by he luid. The exac na u e o he physical e ec s is well beyond he scope o his epo , bu would
be in e es ing none heless.
Ano he physical phenomenon o conside is ha many elas ic ma e ials de o m plas ically when
hei displacemen s pass a limi ha is la ge enough in size. This is o en called yield, and a hypo hesis
go e ning an ins ance in which a ma e ial de o ms is known as a yield c i e ion. Yield hypo heses a e
equen ly gi en in e ms o unc ions o he en ies o he s ess enso σ; o example, he on Mises’
9
g an MTM PGC2018-100928-B-I00. S.K. is suppo ed by he g an Se e o Ochoa SEV-2017-0718.
J.S.-M. acknowledges pa ial suppo by MINECO (Spain) g an MTM2017-84214-C2-1-P.
Re e ences
[1] D. E. Whi ney, “The ma hema ics o coo dina ed con ol o p os he ic a ms and manipula o s,”
1972.
[2] V. B aiban and M. Ge adin, “Op imum pa h planning o obo a ms,” Robo ica, ol. 5, no. 4,
pp. 323–331, 1987.
[3] P. Howell, G. Kozy e , and J. Ockendon, Applied solid mechanics. Camb idge Uni e si y P ess,
2009, ol. 43.
[4] J. Reddy, Theo y and Analysis o Elas ic Pla es and Shells, 2nd ed. CRC P ess, 2007.
[5] H. Golds ein, C. P. Poole, and J. L. Sa ko, “Classical mechanics hi d edi ion, chap e 7,” 2002.
[6] R. . Mises, “Mechanik de es en k¨o pe im plas isch-de o mablen zus and,” Nach ich en on
de Gesellscha de Wissenscha en zu G¨o ingen, Ma hema isch-Physikalische Klasse, ol. 4, pp.
582–592, 1913.
[7] O. C. Zienkiewicz and R. L. Taylo , The Fini e Elemen Me hod, 5 h ed. Ox o d: Bu e wo h
Heinemann, 2000, ol. 2.
[8] A. M. Lush, “Coupling o beam and shell ini e elemen s o he apid analysis o ubula s uc-
u es,” MsC Thesis, School o Enginee ing and Compu ing Sciences. Du ham Uni e si y, 2014.
[9] K. Ba he, Fini e Elemen P ocedu es. P en ice Hall, 2006.
[10] A. Moysidis and V. Koumousis, “Hys e e ic shell ini e elemen ,” Jou nal o Enginee ing Mecha-
nics, ol. 145, no. 5, 2019.
[11] Y. W. Kwon and H. Bang, The Fini e Elemen Me hod using MATLAB, 2nd ed., se . CRC
Mechanical Enginee ing Se ies. Chapman & Hall/CRC, Boca Ra on, FL, 2000.
[12] Wikipedia con ibu o s, “Ansys — Wikipedia, he ee encyclopedia,” h ps://en.wikipedia.o g/
w/index.php? i le=Ansys&oldid=963235585, 2020, [Online; accessed 23-June-2020].
[13] ——, “ADINA — Wikipedia, he ee encyclopedia,” h ps://en.wikipedia.o g/w/index.php?
i le=ADINA&oldid=875541407, 2018, [Online; accessed 23-June-2020].
[14] ——, “FEniCS P ojec — Wikipedia, he ee encyclopedia,” h ps://en.wikipedia.o g/w/index.
php? i le=FEniCS P ojec &oldid=957113708, 2020, [Online; accessed 23-June-2020].
16