Ci a ion: Wand ol, I.; F yd ýšek, K.;
ˇ
Cepica, D. Analysis o he In luence
o The mal Loading on he Beha iou
o he Ea h’s C us . Appl. Sci. 2023,
13, 4367. h ps://doi.o g/10.3390/
app13074367
Academic Edi o : Daniel Dias
Recei ed: 19 July 2022
Re ised: 20 Ma ch 2023
Accep ed: 21 Ma ch 2023
Published: 29 Ma ch 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
applied
sciences
A icle
Analysis o he In luence o The mal Loading on he Beha iou
o he Ea h’s C us
I o Wand ol 1,* , Ka el F yd ýšek 2,3 and Daniel ˇ
Cepica 2,3
1Ins i u e o Physics in Opa a, Silesian Uni e si y in Opa a, Bez uˇco o Námˇes í1150/13,
746 01 Opa a, Czech Republic
2
Depa men o Applied Mechanics, Facul y o Mechanical Enginee ing, VSB—Technical Uni e si y o Os a a,
17. Lis opadu 2172/15, 708 00 Os a a, Czech Republic; [email p o ec ed] (K.F.);
[email p o ec ed] (D. ˇ
C.)
3Ins i u e o Eme gency Medicine, Facul y o Medicine, Uni e si y o Os a a, Syllabo a 19,
703 00 Os a a-Ví ko ice, Czech Republic
*Co espondence: i o.wand [email p o ec ed]
Abs ac :
The a icle ocuses on he de o ma ion and s ain-s ess analysis o he Ea h’s c us unde
ex e nal he mal loading. Mo e speci ically, he in luence o cyclic changes in he su ace empe a u e
ield on he s ess and displacemen inside he c us o e a wo-yea ime span is in es iga ed. The
ini e elemen p og am MSC.Ma c Men a was used o calcula e he s esses and displacemen s.
Fo p ac ical analysis easons, he Ea h’s c us is simpli ied as a plana , piecewise homogeneous,
iso opic model (plane s ain), and ime- a ying empe a u e unc ions o illumina ion ( he mal
adia ion) om he Sun a e conside ed in he local iso opy sec ions o he model. In e ac ion be ween
he Ea h’s c us and man le is de ined by he Winkle elas ic ounda ion. By applying a p obabilis ic
app oach (Mon e Ca lo Me hod), a new s ochas ic model o displacemen s and s esses and new
in o ma ion on c us al displacemen s ela i e o he Ea h’s man le we e ob ained. The esul s p o ed
he hea ing in luence o he Sun on he Ea h’s c us and pla e ec onics.
Keywo ds:
geomechanics; Ea h’s c us ; Fini e Elemen Me hod; Sun hea ing ( adia ion); s ess;
displacemen ; elas ic ounda ion; s ochas ic app oach; Ea h’s c us ; ec onics
1. In oduc ion
The geological s uc u e o plane Ea h is complica ed and ull o ‘unknowns’. The e-
o e, i is sui able o make a disc e iza ion o nume ical modelling; see Figu e 1[1–3].
The Ea h’s c us is a ec ed by many ex e nal in luences, such as idal o ces, cyclic
changes in c us al su ace empe a u es caused by he Sun, ecu en changes in a mo-
sphe ic ai p essu e, he ansmission o ocean and sea wa e mass wa es o he Ea h’s
c us , and geological p ocesses wi hin he co e and man le o he plane [2–9].
This pape ocuses on he e ec o cyclic su ace empe a u e changes on he s ess and
s ain o he Ea h’s c us . The c us (comp ising li hosphe ic pla es) is pe iodically hea ed
( he mally de o med and s essed) by adia ion induced by s ella physical p ocesses inside
he Sun. These a e mainly diu nal pe iods (al e na ion o day and nigh —24 h cycle) o
annual pe iods (al e na ion o seasons in abou 365.4 days). Bo h o hese pe iods ha e
been applied o he inpu o he su ace empe a u e o he Ea h [9–12].
Due o he complexi y o he whole p ocess, he Fini e Elemen Me hod (FEM) is used
and he p oblem is sol ed as a plana one [
2
–
5
]. The Winkle elas ic ounda ion model,
which is commonly used in mechanics, is used o he in e ac ion o he Ea h’s c us
and man le, as shown in Figu e 1[
9
,
13
–
17
]. Howe e , applica ion o he Winkle elas ic
ounda ion in geomechanics is a new way o sol ing he p oblem.
Appl. Sci. 2023,13, 4367. h ps://doi.o g/10.3390/app13074367 h ps://www.mdpi.com/jou nal/applsci
Appl. Sci. 2023,13, 4367 2 o 24
Appl. Sci. 2023, 13, x FOR PEER REVIEW 2 o 25
Figu e 1. S uc u e o he Ea h and he possible disc e iza ion applied (no o scale).
The Ea h’s c us is an aniso opic, he e ogeneous uppe pa o he geoid and is sim-
pli ied in he analysis as a plana , piecewise, homogeneous, iso opic model wi h ime-
a ying empe a u e unc ions o illumina ion, i.e., he mal adia ion, om he Sun. The
la e a e conside ed in he local iso opy sec ions o he model.
The gene ally aniso opic c us is pa i ioned in o piecewise iso opic egions and he
whole p oblem is sol ed as a 2D (plane s ain) p oblem. The heo y o la ge de o ma ions,
i.e., ue (loga i hmic) s ains and ue s esses, was used because i gi es mo e accu a e
esul s, bu leads o he solu ion o a non-linea p oblem. This is a non-s a iona y coupled
( he mal + s uc u al) p oblem and mo e de ails a e gi en in [9,13–15].
A new and mode n p obabilis ic (s ochas ic) app oach is applied, which espec s he
eal a iabili y o he ob ained eco ds o he mal adia ion om he Sun.
The e a e no enough global models o he Ea h’s c us , in connec ion wi h he FEM
and s ochas ic app oaches. This a icle ills he in o ma ion gap. Howe e , 2D models a e
commonly used in geomechanics [9,12].
In he i s s ep, he in luence o g a i a ional and idal effec s om he Sun and
Moon, and he effec o o a ional and cen i ugal o ces on he Ea h’s c us , a e neglec ed
in he model. The eason o his is he he mal hea ing o he c us , which is he ocus and
esea ch o his a icle. We ocused on he mal hea ing o he c us because i is usually
neglec ed o no conside ed, bu i has a signi ican in luence. Ne e heless, a somewha
simple p obabilis ic model has been de eloped which espec s he eal a iabili y o inpu
and ou pu quan i ies [9,15–25].
O he possible app oaches connec ed wi h g oundwa e in es iga ions and unce -
ain y quan i ica ion a e p esen ed in [12,23]. In e e ence [23], he e is a s a e-o - he-a in
app oach o unce ain y quan i ica ion in geomechanics. Howe e , his app oach canno
ye be applied o he complica ed p oblem o c us -man le in e ac ion because o he lack
o global in o ma ion abou he c us .
I has ecen ly been p o ed ha he Sun’s hea ing o he Ea h’s su ace can ha e a
pa ial and impo an in luence on he c ea ion and p opaga ion o pla e ec onics.
Figu e 1. S uc u e o he Ea h and he possible disc e iza ion applied (no o scale).
The Ea h’s c us is an aniso opic, he e ogeneous uppe pa o he geoid and is
simpli ied in he analysis as a plana , piecewise, homogeneous, iso opic model wi h ime-
a ying empe a u e unc ions o illumina ion, i.e., he mal adia ion, om he Sun. The
la e a e conside ed in he local iso opy sec ions o he model.
The gene ally aniso opic c us is pa i ioned in o piecewise iso opic egions and he
whole p oblem is sol ed as a 2D (plane s ain) p oblem. The heo y o la ge de o ma ions,
i.e., ue (loga i hmic) s ains and ue s esses, was used because i gi es mo e accu a e
esul s, bu leads o he solu ion o a non-linea p oblem. This is a non-s a iona y coupled
( he mal + s uc u al) p oblem and mo e de ails a e gi en in [9,13–15].
A new and mode n p obabilis ic (s ochas ic) app oach is applied, which espec s he
eal a iabili y o he ob ained eco ds o he mal adia ion om he Sun.
The e a e no enough global models o he Ea h’s c us , in connec ion wi h he FEM
and s ochas ic app oaches. This a icle ills he in o ma ion gap. Howe e , 2D models a e
commonly used in geomechanics [9,12].
In he i s s ep, he in luence o g a i a ional and idal e ec s om he Sun and Moon,
and he e ec o o a ional and cen i ugal o ces on he Ea h’s c us , a e neglec ed in
he model. The eason o his is he he mal hea ing o he c us , which is he ocus and
esea ch o his a icle. We ocused on he mal hea ing o he c us because i is usually
neglec ed o no conside ed, bu i has a signi ican in luence. Ne e heless, a somewha
simple p obabilis ic model has been de eloped which espec s he eal a iabili y o inpu
and ou pu quan i ies [9,15–25].
O he possible app oaches connec ed wi h g oundwa e in es iga ions and unce -
ain y quan i ica ion a e p esen ed in [
12
,
23
]. In e e ence [
23
], he e is a s a e-o - he-a in
app oach o unce ain y quan i ica ion in geomechanics. Howe e , his app oach canno
ye be applied o he complica ed p oblem o c us -man le in e ac ion because o he lack o
global in o ma ion abou he c us .
I has ecen ly been p o ed ha he Sun’s hea ing o he Ea h’s su ace can ha e a
pa ial and impo an in luence on he c ea ion and p opaga ion o pla e ec onics.
So a , he e ha e been no simple nonlinea geomechanical s ochas ic/p obabilis ic
models inco po a ing c us -man le in e ac ion, wi h a ocus on de o ma ion and s ess
Appl. Sci. 2023,13, 4367 3 o 24
s a es in he Ea h’s c us , induced by sola adia ion. This is he main con ibu ion o
his pape .
Ou wo k builds on simple , p e iously de eloped, models [9,18,19,26–31].
Al e na i ely, he indings o p ocedu es p esen ed in [
32
], which we e ca ied ou o
an a ea in Sweden, can also be used, bu hei applica ion o he en i e su ace o he Ea h
is oo challenging and complica ed.
The nomencla u e o all o he a iables and abb e ia ions used is p esen ed a he
end o his a icle.
2. Ma e ials and Me hods
I he Ea h is o be conside ed as a sphe e in he calcula ions, he solu ion will
be challenging, so i is app op ia e o in oduce a simple plana model in he ini ial
app oxima ion ha can be ex ended in he u u e.
Howe e , i he dimensions o he hea sou ce ( he Sun) and i s dis ance om he Ea h
a e aken in o accoun , i can be concluded ha he hea ays inciden on he Ea h a e
almos pa allel. This app oach is common in mechanics.
An accep able simpli ica ion o he calcula ions can be achie ed by eplacing he
Ea h’s c us by an in ini e hollow cylinde (see Figu e 2), he so-called ‘Ea hcylinde ’, and
de ining he hea ing o cooling o he Ea h’s ou e su ace as a ime- a ying hea unc ion
T= (ϕ, ), whe e ϕis he angle /deg/ and is ime /s/ [9].
Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 o 25
So a , he e ha e been no simple nonlinea geomechanical s ochas ic/p obabilis ic
models inco po a ing c us -man le in e ac ion, wi h a ocus on de o ma ion and s ess
s a es in he Ea h’s c us , induced by sola adia ion. This is he main con ibu ion o his
pape .
Ou wo k builds on simple , p e iously de eloped, models [9,18,19,26–31].
Al e na i ely, he indings o p ocedu es p esen ed in [32], which we e ca ied ou
o an a ea in Sweden, can also be used, bu hei applica ion o he en i e su ace o he
Ea h is oo challenging and complica ed.
The nomencla u e o all o he a iables and abb e ia ions used is p esen ed a he
end o his a icle.
2. Ma e ials and Me hods
I he Ea h is o be conside ed as a sphe e in he calcula ions, he solu ion will be
challenging, so i is app op ia e o in oduce a simple plana model in he ini ial app ox-
ima ion ha can be ex ended in he u u e.
Howe e , i he dimensions o he hea sou ce ( he Sun) and i s dis ance om he
Ea h a e aken in o accoun , i can be concluded ha he hea ays inciden on he Ea h
a e almos pa allel. This app oach is common in mechanics.
An accep able simpli ica ion o he calcula ions can be achie ed by eplacing he
Ea h’s c us by an in ini e hollow cylinde (see Figu e 2), he so-called ‘Ea hcylinde ’,
and de ining he hea ing o cooling o he Ea h’s ou e su ace as a ime- a ying hea
unc ion T= (φ, ), whe e φ is he angle /deg/ and is ime /s/ [9].
Figu e 2. Model o he Ea h as an in ini e hollow cylinde , he so-called ‘Ea hcylinde ’ wi h hea ing
om he Sun (no o scale).
The Ea h’s c us is composed o li hosphe ic pla es and is mainly made up o ocks
and mine als, whe e a ious aul s, ac u es, and o he geological o ma ions occu . In
addi ion o ocks and mine als, he e a e also gases and wa e . I is, he e o e, an inhe -
en ly highly aniso opic, he e ogeneous and inhomogeneous ma e ial [2,32–39].
The c ea ion o such a ma e ial model would be e y complica ed, and so some ap-
p op ia e simpli ica ions a e adop ed in he calcula ions. F om he poin o iew o ma e-
ial p ope ies, ou compu a ional model is a composi e [4,5,9]. The model o he Ea h’s
c us is di ided in o 24 sec ions wi h diffe en ma e ial p ope ies; i is a piecewise iso-
opic homogeneous ma e ial model, which appea s o be aniso opic om he ou side.
On he su ace o each ma e ial sec ion, he e a e empe a u e unc ions T,T…,T =
(φ, ), which co espond o ime- a ying empe a u e alues o e a wo-yea pe iod, ac-
co ding o [6,7], as shown in Figu e 3.
Figu e 2.
Model o he Ea h as an in ini e hollow cylinde , he so-called ‘Ea hcylinde ’ wi h hea ing
om he Sun (no o scale).
The Ea h’s c us is composed o li hosphe ic pla es and is mainly made up o ocks
and mine als, whe e a ious aul s, ac u es, and o he geological o ma ions occu . In
addi ion o ocks and mine als, he e a e also gases and wa e . I is, he e o e, an inhe en ly
highly aniso opic, he e ogeneous and inhomogeneous ma e ial [2,32–39].
The c ea ion o such a ma e ial model would be e y complica ed, and so some app o-
p ia e simpli ica ions a e adop ed in he calcula ions. F om he poin o iew o ma e ial
p ope ies, ou compu a ional model is a composi e [
4
,
5
,
9
]. The model o he Ea h’s c us
is di ided in o 24 sec ions wi h di e en ma e ial p ope ies; i is a piecewise iso opic
homogeneous ma e ial model, which appea s o be aniso opic om he ou side. On he
su ace o each ma e ial sec ion, he e a e empe a u e unc ions
T1
,
T2. . .
,
T24 = (ϕ, )
,
which co espond o ime- a ying empe a u e alues o e a wo-yea pe iod, acco ding
o [6,7], as shown in Figu e 3.
Appl. Sci. 2023,13, 4367 4 o 24
Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 o 25
Figu e 3. ‘Ea hcylinde ’ wi h ma e ials and empe a u e loads (no o scale).
Fo he FEM calcula ion, he p oblem is ea ed as a plana p oblem wi h an assumed
plane s ain. In Figu es 3 and 4, he c us al hickness is highly enla ged ( o cla i y), and
he plana model is shown as being di ided in o 24 sec ions, wi h assigned ma e ial p op-
e ies and loading indica ed om he empe a u e. The angula pe ime e is di ided by
15 deg ees, wi h a di e en ma e ial in each o he 15 deg ees, as shown in Figu es 3 and 4.
Figu e 3. ‘Ea hcylinde ’ wi h ma e ials and empe a u e loads (no o scale).
Fo he FEM calcula ion, he p oblem is ea ed as a plana p oblem wi h an assumed
plane s ain. In Figu es 3and 4, he c us al hickness is highly enla ged ( o cla i y),
and he plana model is shown as being di ided in o 24 sec ions, wi h assigned ma e ial
p ope ies and loading indica ed om he empe a u e. The angula pe ime e is di ided by
15 deg ees, wi h a di e en ma e ial in each o he 15 deg ees, as shown in
Figu es 3and 4
.
The ‘Ea hcylinde ’ is a model o a cylinde eplacing he ea h’s c us wi h cons an
inne and ou e adii, R1=6348 km and R2=6378 km, as shown in Figu es 1–3.
Below he c us , i.e., below he adius
R1
, is he uppe man le, whose assumed cons an
empe a u e is
Tp
. This empe a u e is mos ly ans e ed o he inne adius o he c us by
he mal conduc ion, as shown in Figu es 3and 4.
In he model, he uppe man le is eplaced by a Winkle elas ic ounda ion wi h he
Modulus o he Founda ion K = 10
7
Nm
−3
, as shown in Figu es 1and 3[
9
,
13
–
15
,
19
]. The
elas ic ounda ion is also applied in many enginee ing p oblems [24–27].
The ou e adius o he model
R2
is a ec ed by he empe a u e
T= (ϕ, )
. This
empe a u e depends on he angle
ϕ
and he ime . Such a dependence espec s he ac
ha he empe a u e a ies a di e en loca ions on he Ea h’s c us a di e en imes, i.e.,
he e a e empe a u e di e ences be ween day and nigh o , possibly, seasons [9].
Appl. Sci. 2023,13, 4367 5 o 24
Appl. Sci. 2023, 13, x FOR PEER REVIEW 5 o 25
Figu e 4. ‘Ea hcylinde ’ wi h empe a u e loads (no o scale).
The ‘Ea hcylinde ’ is a model o a cylinde eplacing he ea h’s c us wi h cons an
inne and ou e adii, R= 6348 km and R= 6378 km, as shown in Figu es 1–3.
Below he c us , i.e., below he adius R, is he uppe man le, whose assumed con-
s an empe a u e is T. This empe a u e is mos ly ans e ed o he inne adius o he
c us by he mal conduc ion, as shown in Figu es 3 and 4.
In he model, he uppe man le is eplaced by a Winkle elas ic ounda ion wi h he
Modulus o he Founda ion K = 107 Nm−3, as shown in Figu es 1 and 3 [9,13–15,19]. The
elas ic ounda ion is also applied in many enginee ing p oblems [24–27].
The ou e adius o he model R is affec ed by he empe a u e T= (φ, ) . This
empe a u e depends on he angle φ and he ime . Such a dependence espec s he ac
ha he empe a u e a ies a diffe en loca ions on he Ea h’s c us a diffe en imes,
i.e., he e a e empe a u e diffe ences be ween day and nigh o , possibly, seasons [9].
The ma e ial p ope ies o indi idual sec ions o he Ea h’s c us , as shown in Figu e
3, a y wi hin gi en anges and a e de e mined by andom combina ions:
• Young’s modulus E ∈〈3×10; 4.5 × 10〉 Pa;
• Poisson numbe µ ∈ 〈0.30; 0.35〉;
• Densi y (mean alue) ρ = 2760 kgm−3;
• Conduc i i y ( he mal conduc i i y coefficien ) λ = 3 Wm−1K−1;
• Speci ic hea c = 1100 Jkg−1K−1;
• The mal expansion coefficien α ∈ 〈3.5×10; 4.5 × 10〉 K
−1.
Figu e 4. ‘Ea hcylinde ’ wi h empe a u e loads (no o scale).
The ma e ial p ope ies o indi idual sec ions o he Ea h’s c us , as shown in Figu e 3,
a y wi hin gi en anges and a e de e mined by andom combina ions:
•Young’s modulus E ∈3×1010; 4.5 ×1010Pa;
•Poisson numbe µ∈h0.30;0.35i;
•Densi y (mean alue) ρ= 2760 kgm−3;
•Conduc i i y ( he mal conduc i i y coe icien ) λ= 3 Wm−1K−1;
•Speci ic hea c = 1100 Jkg−1K−1;
•The mal expansion coe icien α∈3.5 ×10−5; 4.5 ×10−5K−1.
Fo he indi idual sec ions 1–24 in Figu e 3, he ma e ial p ope ies a e de e mined
acco ding o he anges men ioned abo e. The ma e ial p ope ies a e gi en in Table 1. The
eal ma e ial p ope ies o he ma e ials con ained in he ea h’s c us a e gi en in [
9
,
37
–
41
].
2.1. Tempe a u e Loading
A cons an ( ime independen ) mean empe a u e
Tp
= 648 K is conside ed o be on
he inne su ace o he cylinde , i.e., he Ea hcylinde , a a dep h o 30 km in he Ea h’s
c us , see [9,41].

Appl. Sci. 2023,13, 4367 6 o 24
Table 1. Ma e ial p ope ies o indi idual sec ions o he plana c us al model o he Ea h’s c us .
Sec ion Angle φ/deg/ Young’s
Modulus E/Pa/
Poisson Numbe
/1/
Densi y
/kg·m−3/
Conduc i i y
/W·m−1·K−1/
Speci ic Hea
/J·K−1·kg−1/
The mal Expansion
/K−1/
1 345–360 3.00 ×1010 0.30 2760 3 1100 3.50 ×10−5
2 0–15 3.50 ×1010 0.35 2760 3 1100 4.00 ×10−5
3 15–30 4.00 ×1010 0.30 2760 3 1100 4.50 ×10−5
4 30–45 4.50 ×1010 0.35 2760 3 1100 3.50 ×10−5
5 45–60 3.00 ×1010 0.30 2760 3 1100 4.00 ×10−5
6 60–75 3.50 ×1010 0.35 2760 3 1100 4.50 ×10−5
7 75–90 4.00 ×1010 0.30 2760 3 1100 3.50 ×10−5
8 90–105 4.50 ×1010 0.35 2760 3 1100 4.00 ×10−5
9 105–120 3.00 ×1010 0.30 2760 3 1100 4.50 ×10−5
10 120–135 3.50 ×1010 0.35 2760 3 1100 3.50 ×10−5
11 135–150 4.00 ×1010 0.30 2760 3 1100 4.00 ×10−5
12 150–165 4.50 ×1010 0.35 2760 3 1100 4.50 ×10−5
13 165–180 3.00 ×1010 0.30 2760 3 1100 3.50 ×10−5
14 180–195 3.50 ×1010 0.35 2760 3 1100 4.00 ×10−5
15 195–210 4.00 ×1010 0.30 2760 3 1100 4.50 ×10−5
16 210–225 4.50 ×1010 0.35 2760 3 1100 3.50 ×10−5
17 225–240 3.00 ×1010 0.30 2760 3 1100 4.00 ×10−5
18 240–255 3.50 ×1010 0.35 2760 3 1100 4.50 ×10−5
19 255–270 4.00 ×1010 0.30 2760 3 1100 3.50 ×10−5
20 270–285 4.50 ×1010 0.35 2760 3 1100 4.00 ×10−5
21 285–300 3.00 ×1010 0.30 2760 3 1100 4.50 ×10−5
22 300–315 3.50 ×1010 0.35 2760 3 1100 3.50 ×10−5
23 315–330 4.00 ×1010 0.30 2760 3 1100 4.00 ×10−5
24 330–345 4.50 ×1010 0.35 2760 3 1100 4.50 ×10−5
2.1.1. Ini ial Condi ions
The ini ial p ehea ing condi ion o he model de e mines he empe a u e a he
beginning o he analysis in MSC.Ma c Men a 2006 so wa e. On he inne su ace, he em-
pe a u e
Tp
is p esc ibed, and on he ou e su ace o he c us , he empe a u e
Tc
= 287 K
is p esc ibed. F om he men ioned ini ial empe a u es
Tp
and
Tc
, he ini ial empe a u e
dis ibu ion T
ini ial ∈Tc;Tp
is acqui ed. The ini ial empe a u e is independen o he
angle ϕbu dependen on he adius R, as shown in Figu e 5.
2.1.2. Bounda y Condi ions
The empe a u e bounda y condi ion
T= (ϕ, )
(i.e.,
T1
,
T2. . .
,
T24 = (ϕ, )
, as
shown in Figu es 2–4), depends on ime and he angula dimension o he Ea h and ac s
on he ou e su ace o he c us . This bounda y condi ion simula es he cyclic empe a u e
changes based on he daily cycle o he Ea h’s o a ion and he annual cycle o he plane ’s
o bi a ound he Sun. An example o
T2
dependence loading o 2 yea s and 5 days is
shown in Figu e 6.
Appl. Sci. 2023,13, 4367 7 o 24
Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 o 25
Figu e 5. Ea hcylinde and i s ini ial empe a u e dis ibu ion Tini ial.
2.1.2. Bounda y Condi ions
The empe a u e bounda y condi ion T= (φ, ) (i.e., T,T…,T = (φ, ) , as
shown in Figu es 2–4), depends on ime and he angula dimension o he Ea h and ac s
on he ou e su ace o he c us . This bounda y condi ion simula es he cyclic empe a u e
changes based on he daily cycle o he Ea h’s o a ion and he annual cycle o he plane ’s
o bi a ound he Sun. An example o T dependence loading o 2 yea s and 5 days is
shown in Figu e 6.
Figu e 5. Ea hcylinde and i s ini ial empe a u e dis ibu ion Tini ial.
Fo he calcula ion, su ace empe a u es om he ALA moni o ing sys em [
10
,
11
]
(in addi ion o su ace empe a u e and ai empe a u e) we e aken om s a ions loca ed
be ween la i udes o app oxima ely 40 and 50 deg ees no h o he pe iod Oc obe 2007 o
Augus 2009, a 15 min inc emen s.
The basic cha ac e is ic alues o he s a is ical da a o empe a u e measu emen s
T1,T2. . . ,T24 = (ϕ, )a e:
•Minimum empe a u e = 263.65 K,
•Mean empe a u e = 282.20 K,
•Median empe a u e = 282.15 K,
•S anda d de ia ion o empe a u e = 7.68 K,
•Maximum empe a u e = 303.35 K.
The da a we e hen p ocessed o impo ing in o he MSC.Ma c Men a 2006 so -
wa e [
42
]. The esul ing ime-dependen empe a u e se ies con ained 35,424 alues, wi h
110 eal empe a u e anomalies whe e he empe a u e con inui y check (la ge a iabili y
be ween wo measu emen s) was no me . These we e mainly la ge luc ua ions in ai
empe a u e, which could ha e been caused by he ou b eak o a se e e s o m, e c. The
su ace empe a u e ollowed he ai empe a u e pa e n. In his case, he abo e-men ioned
anomalies occu ed only in he sp ing and summe pe iods, om 3 Ap il 2008 o 30 Augus
2008. O cou se, hese anomalies, which a e eal in na u e, we e also used in he solu ion.
Appl. Sci. 2023,13, 4367 8 o 24
Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 o 25
(a)
(b)
Figu e 6. Tempe a u e T dis ibu ion (a) Bounda y condi ion— o whole 2 yea s a node 1; (b)
Bounda y condi ion a node 1 and calcula ed empe a u e a node 54 o 99 h o 105 h day o solu-
ion.
Fo he calcula ion, su ace empe a u es om he ALA moni o ing sys em [10,11] (in
addi ion o su ace empe a u e and ai empe a u e) we e aken om s a ions loca ed
be ween la i udes o app oxima ely 40 and 50 deg ees no h o he pe iod Oc obe 2007
o Augus 2009, a 15 min inc emen s.
The basic cha ac e is ic alues o he s a is ical da a o empe a u e measu emen s
T,T…,T = (φ, ) a e:
• Minimum empe a u e = 263.65 K,
• Mean empe a u e = 282.20 K,
• Median empe a u e = 282.15 K,
Figu e 6.
Tempe a u e
T2
dis ibu ion (
a
) Bounda y condi ion— o whole 2 yea s a node 1; (
b
) Bound-
a y condi ion a node 1 and calcula ed empe a u e a node 54 o 99 h o 105 h day o solu ion.
I can, he e o e, be concluded ha he empe a u e a iabili y is signi ican ly highe in
he ising empe a u e phase han in he alling empe a u e phase. A simila phenomenon
could be obse ed in he diu nal ime cycle. Fo his eason, he idea o modelling he
empe a u e unc ion using goniome ic unc ions was abandoned as i would in oduce
e o s ( he applica ion o Fou ie se ies will no b ing accu acy and is no necessa y).
The p ocessed da a we e used o he sec ion
ϕ∈h345◦;360i
. Fo all subsequen
sec ions, he en i e ime se ies was successi ely shi ed by one hou . Fo Sec ion 2o
Figu e 3(
ϕ∈h0◦;15i
), he da e 1 Feb ua y 2008 0:00:00 co esponded o he o iginal da e
Appl. Sci. 2023,13, 4367 9 o 24
o 1 Feb ua y 2008 01:00:00 and, o each subsequen sec ion, he ime se ies is shi ed by
ano he hou , as al eady men ioned.
The di e en empe a u e bounda y condi ions in each sec ion ep esen a i ual
o a ion o he model, i.e., he change in empe a u e simula es he o a ion o he Ea h
a ound he Sun.
The empe a u e ime se ies, howe e , had o be educed o impo ing in o he
MSC.Ma c Men a so wa e due o he la ge amoun o da a, so ha he o iginal ime s ep
o 15 min had o be inc eased o a ime s ep o 3519 s (i.e., 58 min 39 s). This leads o he
as e solu ion by FEM.
Since he ini ial su ace empe a u e o he model is cons an a ound he pe ime e
and he bounda y empe a u e condi ions a e angle dependen , i is necessa y o le he
model se le a he beginning o he analysis. Acco ding o he esul s ob ained, he model
is conside ed o be s eady a e h ee days o loading, i.e., he alues om 4 Feb ua y 2008
00:48:37 a e aken o p ocessing he esul s. Mo e in o ma ion ega ding he bounda y
condi ions is gi en in [9].
2.2. Fini e Elemen Mesh
Figu e 7shows he ini e elemen mesh used o he calcula ion. A he ansi ion
poin s be ween he wo sec ions, i.e., di e en ma e ial p ope ies and su ace empe a u es,
bias subdi ision is used wi h e inemen o he mesh owa ds he ou e su ace, since la ge
empe a u e a ia ions a e expec ed nea he ou e su ace han a g ea e dep hs. The FE
mesh con ains 8973 quad a ic elemen s and 10,116 nodes.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 o 25
Figu e 7. De ail o biased mesh ( e inemen owa ds he su ace o he Ea h).
3. Resul s
F om he FE calcula ion pe o med on an Ea hcylinde , he mechanical s ess and
displacemen dis ibu ions we e e alua ed. The equi alen on Mises mechanical s ess
(HMH) σ /Pa/ is calcula ed acco ding o Equa ion (1)
σ =σ
σ
σ
−(σσσσσσ), (1)
whe e σ,, /Pa/ a e he p incipal s esses, see [43].
The o al displacemen 𝑢/m/ can be exp essed by
𝑢=𝑢
𝑢
, (2)
whe e 𝑢 /m/ is he displacemen in he X-axis di ec ion, and 𝑢/m/ is he displacemen
in he Y-axis di ec ion, see Figu e 2 o Figu e 3.
Equa ions (1) and (2) we e applied in he ini e elemen analysis (FEA).
3.1. Resul s om FEA
As o he ini e elemen model, he accu acy was checked by compa ison. Fi s , a
coa se ini e elemen mesh wi h a longe ime s ep was sol ed. Then a ine mesh wi h a
sho e ime s ep was sol ed, e.g., his ini e elemen id desc ibed in Figu e 7. The e was
no signi ican diffe ence be ween hese solu ions and he p esen ed solu ion can hen be
decla ed sufficien ly accu a e. Because o p oblems whe e he analy ical solu ion is no
Figu e 7. De ail o biased mesh ( e inemen owa ds he su ace o he Ea h).
Appl. Sci. 2023,13, 4367 16 o 24
Table 2. S a is ical pa ame e s o angen ial displacemen s.
Node Angle φ/deg/ Min /m/ Mean /m/ Median /m/ Max /m/
1 0.0 −22.405145 −4.016917 −4.155867 19.706385
7646 7.5 −13.167980 1.980264 2.203522 12.992962
7094 15.0 −29.331777 7.971764 8.205548 40.070023
51 22.5 −6.667464 1.772484 1.822645 8.890464
3 30.0 −8.410809 0.015862 0.108814 8.880879
53 37.5 −2.733061 −0.369284 −0.444654 2.724747
4 45.0 −12.265832 −2.029589 −2.114735 11.494079
55 52.5 −1.619803 0.019862 0.053834 1.584367
5 60.0 −11.991985 2.225961 2.311249 13.202844
57 67.5 −1.703642 0.148204 0.202752 1.321986
6 75.0 −8.316151 −1.156488 −1.263755 9.357056
59 82.5 −2.202902 0.168749 0.243539 1.710141
7 90.0 −10.964751 1.914396 2.002252 11.641080
61 97.5 −1.717808 −0.088355 −0.117302 1.903794
11491 105.0 −14.292981 −2.386571 −2.480774 13.402903
63 112.5 −2.136792 −0.321966 −0.361097 2.297582
11585 120.0 −6.676715 −0.051041 −0.041540 6.316006
65 127.5 −2.620005 0.196727 0.304310 1.918440
11782 135.0 −10.374590 1.212906 1.450878 8.992166
67 142.5 −3.431526 0.537606 0.601411 3.128304
11925 150.0 −12.217613 2.033221 2.138069 12.652762
69 157.5 −3.368985 −0.578049 −0.604307 3.254372
12044 165.0 −23.284528 −4.365703 −4.469298 18.689545
71 172.5 −2.412931 −0.367422 −0.407727 2.594586
13 180.0 −10.843711 2.072278 2.139710 12.114409
73 187.5 −2.344523 0.396470 0.427631 2.438536
12234 195.0 −9.007515 0.961226 1.080069 7.542674
9225 202.5 −2.045828 0.180401 0.245550 1.597627
12424 210.0 −6.245059 −0.379791 −0.502522 7.471727
9372 217.5 −3.101930 −0.514486 −0.545275 3.085332
12500 225.0 −12.566509 −2.093027 −2.173286 11.838935
9471 232.5 −2.061168 0.018279 0.060519 1.922392
12599 240.0 −13.427477 2.207517 2.396578 13.803798
9594 247.5 −2.078801 0.147412 0.237499 1.551424
12674 255.0 −8.193241 −1.127393 −1.232754 8.960823
9736 262.5 −3.634378 0.168002 0.216868 2.734815
19 270.0 −18.844958 1.926130 1.995067 16.550423
9910 277.5 −2.515986 −0.117068 −0.148072 2.376055
10004 285.0 −14.114205 −2.368588 −2.561178 14.466366
12796 292.5 −2.534931 −0.399108 −0.459958 2.579560
12919 300.0 −6.537996 −0.083045 −0.093447 6.221886

Appl. Sci. 2023,13, 4367 17 o 24
Table 2. Con .
Node Angle φ/deg/ Min /m/ Mean /m/ Median /m/ Max /m/
13017 307.5 −2.130202 0.200543 0.266355 1.685076
13116 315.0 −9.207633 1.157171 1.258941 8.282640
13259 322.5 −3.271788 0.624200 0.663308 3.667678
13384 330.0 −12.221221 2.082060 2.173141 12.821112
13569 337.5 −2.657303 −0.430821 −0.459640 2.697734
10258 345.0 −23.800367 −4.473265 −4.585868 18.998453
10135 352.5 −8.195234 −1.552937 −1.599643 6.756131
TOTAL −29.331777 0.063933 0.055363 40.070023
Table 3. S a is ical pa ame e s o adial displacemen s.
Node Angle φ/deg/ Min /m/ Mean /m/ Median /m/ Max /m/
1 0.0 −0.247709 0.045662 0.048070 0.264486
7646 7.5 −1.069984 −0.220166 −0.225239 0.744611
7094 15.0 −0.225554 0.050253 0.052106 0.273853
51 22.5 −2.949429 −0.597693 −0.609791 2.074209
3 30.0 −2.840615 −0.575473 −0.589606 1.996382
53 37.5 −2.451921 −0.498885 −0.509240 1.719541
4 45.0 −2.948489 −0.593957 −0.604933 2.117883
55 52.5 −3.122978 −0.632979 −0.646084 2.204687
5 60.0 −3.688951 −0.746602 −0.762182 2.596254
57 67.5 −3.284413 −0.665639 −0.679903 2.317962
6 75.0 −3.600448 −0.728494 −0.744516 2.538881
59 82.5 −2.910104 −0.590158 −0.600756 2.054802
7 90.0 −3.126893 −0.632771 −0.646967 2.203372
61 97.5 −3.260287 −0.660954 −0.672389 2.301363
11491 105.0 −3.269007 −0.658896 −0.672805 2.344291
63 112.5 −3.078366 −0.623752 −0.635934 2.171800
11585 120.0 −3.613587 −0.732255 −0.748583 2.538011
65 127.5 −3.173468 −0.643556 −0.659318 2.232398
11782 135.0 −3.171930 −0.642370 −0.657984 2.232703
67 142.5 −3.036209 −0.616722 −0.629598 2.138323
11925 150.0 −3.896294 −0.789001 −0.806090 2.740027
69 157.5 −3.991671 −0.808510 −0.825623 2.816905
12044 165.0 −3.199840 −0.645226 −0.658717 2.294331
71 172.5 −2.829153 −0.573808 −0.584828 1.996196
13 180.0 −2.850070 −0.577022 −0.589402 2.007131
73 187.5 −3.290911 −0.667605 −0.679138 2.323781
12234 195.0 −3.451112 −0.699088 −0.712300 2.430043
Appl. Sci. 2023,13, 4367 18 o 24
Table 3. Con .
Node Angle φ/deg/ Min /m/ Mean /m/ Median /m/ Max /m/
9225 202.5 −3.446525 −0.698456 −0.712473 2.433441
12424 210.0 −3.481797 −0.705072 −0.719768 2.452611
9372 217.5 −3.153902 −0.639660 −0.651908 2.224569
12500 225.0 −2.889573 −0.583077 −0.595777 2.063464
9471 232.5 −3.225099 −0.653850 −0.670153 2.270460
12599 240.0 −3.643746 −0.739617 −0.755094 2.553612
9594 247.5 −3.900572 −0.792134 −0.808253 2.746832
12674 255.0 −3.421864 −0.694008 −0.705438 2.417951
9736 262.5 −2.969488 −0.607632 −0.618616 2.114656
19 270.0 −3.264306 −0.662024 −0.676559 2.271688
9910 277.5 −3.891178 −0.791065 −0.807631 2.689200
10004 285.0 −3.323049 −0.665437 −0.681295 2.371287
12796 292.5 −3.731120 −0.756405 −0.773357 2.623065
12919 300.0 −3.443254 −0.697380 −0.711933 2.418230
13017 307.5 −3.195928 −0.648019 −0.661250 2.255990
13116 315.0 −3.107899 −0.628968 2.190442 2.190442
13259 322.5 −3.210917 −0.650901 −0.664147 2.267816
13384 330.0 −3.473598 −0.702698 −0.716477 2.446307
13569 337.5 −3.841410 −0.777634 −0.794236 2.709838
10258 345.0 −1.664193 −0.333632 −0.341247 1.236957
10135 352.5 −1.383713 −0.281419 −0.288036 0.969561
TOTAL −3.991671 −0.613224 −0.565425 2.816905
The indings in Table 3show ha he displacemen a he Ea h’s su ace can be as
la ge as
h−3.992;2.817i
m o adial displacemen o e wo yea s. This is in e es ing and,
again, is no inconsis en wi h c us al mo ion.
The abo e-men ioned displacemen s a e usually quasi-s a ic bu , in ex eme si ua ions
associa ed wi h ea hquakes, hey can also occu in a dynamic manne .
The his og am o he esul ing adial and angen ial displacemen s (s ochas ic e alua-
ion o angen ial and adial displacemen ) is shown in Figu e 17.
A la ge numbe o di e en ela ionships can be used o es ima e he e o o he Mo e
Ca lo me hod. Fo example, consis en wi h he li e a u e [
45
,
46
], he e o o he Mon e
Ca lo me hod can be app oxima ely es ima ed by he ela ion:
e o Mon eCa lo =S De ia ion
pS eps , (3)
whe e ‘S De ia ion’ is he s anda d de ia ion and ‘S eps’ is numbe o Mon e Ca lo simula ions.
Appl. Sci. 2023,13, 4367 19 o 24
1
Figu e 16.
His og ams o he angen ial componen o he displacemen a nodes on he Ea h’s ou e
su ace induced by he mal adia ion om he Sun.
F om he da a in Figu e 17a, Equa ion (3) gi es
3.90418181
√2000000 =
0.00276 and, om he
da a in Figu e 17b, Equa ion (3) gi es
1.05741154
√2000000 =
0.00075. The e o is oo small, and he
s ochas ic app oach is eliable.
Appl. Sci. 2023,13, 4367 20 o 24
Appl. Sci. 2023, 13, x FOR PEER REVIEW 20 o 25
Radial displacemen s, which a e no p esen ed in his a icle, can also be e alua ed
in a simila way [9].
The indings o Table 2 show ha he he mally induced displacemen s on he Ea h’s
su ace can be as la ge as he in e al 〈−29.332; 40.070〉 m o angen ial displacemen
o e wo yea s, an in e es ing inding ha is no gene ally inconsis en wi h c us al mo-
ion. Some places on Ea h mo e less and some mo e mo e, e.g., in a eas o ec onic pla e
con ac o aul ing.
The indings in Table 3 show ha he displacemen a he Ea h’s su ace can be as
la ge as 〈−3.992;2.817〉 m o adial displacemen o e wo yea s. This is in e es ing and,
again, is no inconsis en wi h c us al mo ion.
The abo e-men ioned displacemen s a e usually quasi-s a ic bu , in ex eme si ua-
ions associa ed wi h ea hquakes, hey can also occu in a dynamic manne .
The his og am o he esul ing adial and angen ial displacemen s (s ochas ic e alu-
a ion o angen ial and adial displacemen ) is shown in Figu e 17.
(a)
(b)
Figu e 17. His og ams and dis ibu ion unc ions o (a) angen ial componen o o al c us al su ace
displacemen , (b) adial componen o o al c us al su ace displacemen (sw MSC.Ma c Men a and
An hill).
A la ge numbe o diffe en ela ionships can be used o es ima e he e o o he
Mo e Ca lo me hod. Fo example, consis en wi h he li e a u e [45,46], he e o o he
Mon e Ca lo me hod can be app oxima ely es ima ed by he ela ion:
e o   =S De ia ion

S eps , (3)
whe e ‘S De ia ion’ is he s anda d de ia ion and ‘S eps’ is numbe o Mon e Ca lo simu-
la ions.
F om he da a in Figu e 17a, Equa ion (3) gi es .
√ = 0.00276 and, om he
da a in Figu e 17b, Equa ion (3) gi es .
√ = 0.00075. The e o is oo small, and he
s ochas ic app oach is eliable.
Figu e 17.
His og ams and dis ibu ion unc ions o (
a
) angen ial componen o o al c us al su ace
displacemen , (
b
) adial componen o o al c us al su ace displacemen (sw MSC.Ma c Men a
and An hill).
4. Discussion
An ini ial new model o he Ea h’s hea loading has been de eloped. I s ad an age is
he simplici y o he plane s ain p oblem and he o iginal combina ion o he ini e elemen
me hod and he p obabilis ic Mon e Ca lo Me hod. FEA is pe o med o he non-s a iona y
and non-linea coupled geomechanical p oblem (i.e., empe a u e ask and s uc u al ask).
A simple measu e o he ma e ial Inhomogenei y o he Ea h’s c us (pa i ioned in o
24 ma e ials) was espec ed in a simple way.
The in e ac ion be ween he c us and he uppe man le is eplaced by he bila e al
Winkle elas ic ounda ion bounda y condi ion.
This esul s in a simpli ied model, which espec s many andom (pseudo- andom)
inpu s, i.e., 2
×
10
6
Mon e Ca lo simula ions. Nonlinea geomechanics asks, in he case o
la ge de o ma ion caused by Sun hea ing, we e sol ed.
F om he s ochas ic modelling, he angen ial and adial displacemen s o he Ea h’s
su ace and c us al s esses a e ob ained, and hei alues can ini ia e he possible ac u es
o he Ea h’s c us connec ed wi h i s ec onics.
The model is conside ed o be con inuous and, he e o e, does no allow o he
o ma ion o c acks in he Ea h’s c us . Howe e , he ob ained s esses and displacemen s
clea ly con i m su icien capaci y o induce ec onic changes in he Ea h’s su ace. The
pe iodic and inhomogeneous he mal adia ion om he Sun ac ing on he Ea h’s su ace is
su icien o induce hese ec onic changes. This is he impo an inding o unde s anding
he complexi y o he Ea h’s c us beha io .
I can be said ha ec onic changes in he Ea h’s c us a e also caused by he mal
adia ion om he Sun (ex e nal hea ing o he Ea h), which is s ill a new idea. Ou esul s
a e no inconsis en wi h ea lie o eign wo ks [26,28,29] o ou wo ks [9,15,18,19,30,31].
As is well known, ec onic changes in he Ea h’s c us a e also due o idal e ec s om
he g a i a ional pull o he Sun and Moon, cen i ugal o ces om he Ea h’s o a ion,
Appl. Sci. 2023,13, 4367 21 o 24
and in e nal magma ic p ocesses wi hin he Ea h. The in luence o idal, cen i ugal, and
in e nal magma ic e ec s is no add essed in his pape .
In he u u e, he dynamic and cyclic loading o he Ea h’s c us om idal e ec s and
magma ic p ocesses can also be conside ed, o a mo e complex spa ial (3D) model can be
de eloped ha could also accoun o he o he geological and geomechanical phenomena
men ioned abo e. In he u u e, by applying mechanical con ac s be ween he Ea h’s
pla es, i will be possible o deal wi h ec onic phenomena on ou own o o eign plane s in
a simila bu mo e complex and complica ed way.
Theimpo an in luenceo he mal adia ionon heEa h’s ec onicsa enew indings,p o ed
by ou wo k and by he di e en models, bu consis en wi h he li e a u e [9,15,18,19,26,28–31].
5. Conclusions
Using a simple model o he hea ing o he Ea h om a hea sou ce, i.e., he Sun,
o e a wo-yea pe iod, he displacemen ields and equi alen mechanical s esses ( on
Mises) wi hin he Ea h’s c us we e ob ained.
The Sun causes ela i ely la ge displacemen s in he Ea h’s c us du ing annual
pe iods (and, o a lesse ex en , o e a daily pe iod), eaching up o 40.07 m in he angen ial
di ec ion and 3.99 m in he adial di ec ion a he c us al su ace. Howe e , i is a slow
p ocess ha is almos impe cep ible o humans, compa ed o ea hquakes, whe e he c us al
displacemen is isible o he naked eye.
The nume ical analysis u he e ealed s esses ha eached up o 52 MPa. These
s esses, om empe a u e alone, a e no mal o he Ea h and a e no e y damaging o i
bu can cause ec onic mo emen s. Howe e , in combina ion wi h o he geomechanical
phenomena, e.g., idal e ec s, he Ea h’s o a ion, o con ec ion cu en s wi hin he Ea h,
e c., i can cause aul s in he Ea h’s c us .
Ou indings, wi h espec o he impo an in luence o sola he mal hea ing on he
Ea h’s ec onics, a e new as well as p o en by di e en models, bu consis en wi h he
chosen e e ences. We ha e p o ed ha he Sun can ‘mo e’ he Ea h’s c us .
Au ho Con ibu ions:
Me hodology, K.F.; Fo mal analysis, I.W.; W i ing – o iginal d a , D. ˇ
C., K.F.
and I.W. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This a icle was suppo ed by Speci ic Reasea ch SP2023/027 “Applica ion o Mode n
Compu a ional and Expe imen al App oaches in Applied Mechanics”; Speci ic Resea ch SP2022/26
“Compu a ional and expe imen al modeling in he asks o applied mechanics and biomechanics” and
by in e na ional p ojec s CZ.02.1.01/0.0/0.0/17_049/0008441 “Inno a i e The apeu ic Me hods o
Musculoskele al Sys em in Acciden Su ge y” and CZ.02.1.01/0.0/0.0/17_049/0008407 “Inno a i e
and addi i e manu ac u ing echnology—new echnological solu ions o 3D p in ing o me als and
composi e ma e ials” wi hin he Ope a ional P og amme Resea ch, De elopmen and Educa ion
inanced by he Eu opean Union and om he s a e budge o he Czech Republic.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Nomencla u e
Name Physical Uni Explana ion
ALA Moni o ing sys em o measu ing
c Jkg−1K−1Speci ic hea o Ea h’s c us
E Pa Young’s modulus o Ea h’s c us
Ea hcylinde Simpli ied model o geoid o Ea h
e o Mon eCa lo 1 Possible e o o Mon e Ca lo Me hod

Appl. Sci. 2023,13, 4367 22 o 24
Name Physical Uni Explana ion
Func ion
FE Fini e elemen
FEA Fini e Elemen Analysis
FEM Fini e Elemen Me hod
K Nm−3Modulus o he ounda ion
ma e ial1, ma e ial2, . . . Name o ma e ials in Ea h’s c us
ma e ial24
Max Maximum alue
Mean Mean alue
Median Median alue
Min Minimum
MSC.Ma c Men a Fini e elemen so wa e
Node Node o ini e elemen mesh
R m Radius o Ea h’s c us
R1m Inne adius o Ea h’s c us
R2m Ou e adius o Ea h’s c us
S De ia ion S anda d de ia ion
S eps Numbe o Mon e Ca lo simula ions
S ess HMH Equi alen on Mises s ess
T K Tempe a u e
T1, T2, . . . T24 K Tempe a u e loading on sec ion 1, 2, . . . , 24
o Ea h’s c us , see Figu e 3
TcK Ini ial empe a u e on inne adius o
Ea h’s c us , see Figu e 5
Tini ial K Ini ial empe a u e o Ea h0s c us om
in e al Tc;Tp
TpK Tempe a u e on inne adius o Ea h’s c us
s Time
um To al displacemen in poin o Ea h’s c us
u 1 m Tangen ial componen o c us al su ace
displacemen o node 1
u 7094 m Tangen ial componen o c us al su ace
displacemen o node 7094
u 12044 m Tangen ial componen o c us al su ace
displacemen o node 12044
u To m Tangen ial componen o o al c us al
su ace displacemen
u 1 m Radial componen o c us al su ace
displacemen o node 1
u 7094 m Radial componen o c us al su ace
displacemen o node 7094
u 12044 m Radial componen o c us al su ace
displacemen o node 12044
u To m Radial componen o o al c us al
su ace displacemen
uxm Displacemen in X axis di ec ion
uym Displacemen in Y axis di ec ion
X m Axis o coo dina e sys em
Y m Axis o coo dina e sys em
Z m Axis o coo dina e sys em
αK−1The mal expansion coe icien o Ea h’s c us
ϕdeg Angle o Ea h’s c us
λWm−1K−1Conduc i i y o Ea h’s c us
µ1 Poisson numbe o Ea h’s c us
ρKgm−3Densi y o Ea h’s c us
σHMH Pa Equi alen on Mises S ess
σ1,σ2,σ3Pa P incipal s ess
Appl. Sci. 2023,13, 4367 23 o 24
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