Elec onic
Resea ch A chi e
h ps://www.aimsp ess.com/jou nal/e a
ERA, 33(2): 537–555.
DOI: 10.3934/e a.2025025
Recei ed: 19 No embe 2024
Re ised: 03 Janua y 2025
Accep ed: 08 Janua y 2025
Published: 07 Feb ua y 2025
Resea ch a icle
Second g adien he moelas ici y wi h mic o empe a u es
Do in Ies¸an1and Ram´
on Quin anilla2,*
1Al.I. Cuza Uni e si y and Oc a Maye Ins i u e o he Ma hema ics o Romanian Academy, Bd.
Ca ol I, n . 11, 700506 Iasi, Romania
2Depa amen o de Ma em´
a icas, E.S.E.I.A.A.T.-U.P.C., Colom 11, 08222 Te assa, Ba celona, Spain
*Co espondence: Email: [email p o ec ed]; Tel: +34-93-7398162.
Abs ac : This esea ch was conce ned wi h a linea heo y o he moelas ici y wi h mic o empe a-
u es whe e he second he mal displacemen g adien and he second g adien o mic o empe a u es
a e included in he classical se o independen cons i u i e a iables. The mas e balance laws o
mic omo phic con inua, he heo y o he s ain g adien o elas ici y, and G een-Naghdi he mome-
chanics we e used o de i e a second g adien heo y. The semig oup heo y o linea ope a o s al-
lowed us o p o e ha he p oblem o he second g adien he moelas ici y wi h mic o empe a u es
is well-posed. Fo he equa ions o iso opic igids, we p esen ed a na u al ex ension o he Cauchy-
Ko ale ski-Somigliana solu ion o iso he mal heo y. In he case o s a iona y ib a ions, he unda-
men al solu ions o he basic equa ions we e ob ained. Uniqueness and ins abili y o he solu ions we e
ob ained in he case o an iplane shea de o ma ions.
Keywo ds: elas ic solids wi h mic o empe a u es; solids wi h mic os uc u e; second g adien
heo y; cons i u i e equa ions; well-posed p oblems
1. In oduc ion
In ecen yea s, G een-Naghdi he modynamics [1–3] has been used o es ablish some heo ies
o he moelas ici y ha ake in o accoun he second-o de empe a u e g adien [4–6]. On he o he
hand, he balance laws o he con inua wi h mic os uc u e [7–10] led o a heo y o he modynamics o
elas ic ma e ials whe e he inne s uc u e has mic oelemen s wi h mic o empe a u es. A he begin o
his pape , we use he heo y o non-simple elas ic solids [11–13] and esul s om he modynamics o
mul ipola con inua [14] o ob ain a second g adien heo y o he moelas ici y wi h mic o empe a u es.
An in oduc ion o he concep s o he mal displacemen and he mal mic odisplacemen s as well as
he heo y o mul ipola con inua allows us o de i e he local o m o ene gy balance and cons i u i e
equa ions. In [3], he au ho s es ablished a heo y o he moelas ici y cha ac e ized by cons i u i e
538
equa ions ha depend on he i s g adien o he displacemen ec o , on empe a u e, and on he i s
g adien o he mal displacemen . In he p esen wo k, we ha e conside he ollowing new independen
cons i u i e a iables: he second g adien o he mal displacemen , he second g adien o he mal
mic odisplacemen s, as well as mic o empe a u es. To simpli y he w i ing, we limi ou a en ion only
o he in oduc ion o he second-o de spa ial de i a i es o he he mal a iables.
We exp ess he ield equa ions o he linea case in e ms o componen s o he displacemen ec o ,
he mal displacemen , and he mal mic odisplacemen s, and ob ain a ou h-o de sys em o equa ions.
The bounda y-ini ial- alue p oblems a e also o mula ed. The semig oup heo y o linea ope a o s
allows us o p o e ha he p oblem o he second g adien he moelas ici y wi h mic o empe a u es
is well-posed. Fo he equa ions o iso opic igids, we p esen a na u al ex ension o he Cauchy-
Ko ale ski-Somigliana solu ion o he iso he mal heo y. In he case o s a iona y ib a ions, we
es ablish he undamen al solu ions o he basic equa ions. An i-plane shea de o ma ions a e also
conside ed and uniqueness and ins abili y esul s a e ob ained.
The ele ance in in oducing empe a u e g adien e ec s in he momechanics can be ecalled
om [15].
2. Balance equa ions
In his sec ion, we p opose a second g adien heo y o solids wi h mic o empe a u es by using
he basic laws o mechanics o ma e ials wi h mic os uc u e and G een-Naghdi he momechanics.
Th oughou his pape , he mo ion o he body is e e ed o he e e ence con igu a ion B, occupied
by he body a ime 0, and o a ixed sys em o ec angula Ca esian coo dina es Oxj, (j =1, .., 3).
La in subsc ip s ange o e he in ege s (1, 2, 3), and G eek subsc ip s ange o e he in ege s (1,
2). Ca esian enso no a ion is used h oughou . In wha ollows, xja e e e ence coo dina es, yj
a e spa ial coo dina es, a supe posed do deno es ma e ial ime di e en ia ion, and ,jdeno es pa ial
di e en ia ion o wi h espec o xj. We deno e by ∂B he bounda y o B. Following [1, 2], we ge
he ollowing local balance o en opy:
ρ˙η=Si,i+ρ(s+ξ).(2.1)
By using he heo y o con inua wi h mic os uc u e [9], we can ob ain he balance o he i s momen
o en opy in he o m:
ρ˙ηj= Λki,k+Si−Hi+ρ(Qi+ξi),(2.2)
In he ela ions (2.1) and (2.2), we ha e used he ollowing no a ions: ρis he e e ence mass densi y;
ηis he en opy pe uni mass o he body; Siis he en opy lux ec o ; sis he ex e nal a e o supply
o en opy pe uni mass; ξis he in e nal a e o p oduc ion o en opy pe uni mass; ηjis he i s
en opy momen ec o ; Λi j is he i s en opy lux momen enso ; Hiis he mean en opy lux ec o ;
Qiis he i s momen o he ex e nal a e o supply o en opy, and ξjis he i s momen o he in e nal
a e o p oduc ion o en opy. The en opy lux Σand he i s en opy momen lux ec o σja egula
poin s o ∂Ba e gi en by [1,2,8],
Σ = Sjnj, σk= Λjknj,(2.3)
whe e niis he ou wa d uni no mal o ∂B. Le θbe he absolu e empe a u e. We deno e by x he
cen e o mass o a gene ic mic oelemen V. We assume ha o x′∈V, we ha e
θ(x′, )=θ(x, )+Ti(x, )(x′
i−xi).(2.4)
Elec onic Resea ch A chi e Volume 33, Issue 2, 537–555.
539
We call he unc ions Timic o empe a u es. As in [1], we conside he he mal displacemen αand he
he mal mic odisplacemen s βjby
˙α=θ, ˙
βj=Tj.(2.5)
We now conside a domain Pa ime , bounded by a su ace ∂P, and le Pbe he espec i e domain in
e e ence con igu a ion, wi h he bounda y ∂P. In iew o [1,9, 11,14], we p opose an ene gy balance
in he o m: ZP
ρ(¨ui˙ui+˙e)d =ZP
ρ( i˙ui+sθ+QiTi)d (2.6)
+Z∂P
( i˙ui+ Σθ+σjTj+Gjθ,j+ ΠjiTi,j)da
o e e y egion Po Band e e y ime. He e uiis he displacemen ec o , eis he in e nal ene gy
pe uni mass iis he body o ce pe uni mass, i he s ess ec o associa ed wi h he su ace ∂Pbu
measu ed pe uni a ea o ∂P, and Giand Πi j a e he monopola and dipola en opy lux pe uni a ea,
espec i ely. We impose ha he dipola body o ce and he spin ine ia pe uni mass a e no p esen
(see [14]). F om (2.6), we can de i e he balance o linea momen um so ha , by he well-known
me hod, we ob ain
j= i jni(2.7)
and
ji,j+ρ i=ρ¨ui(2.8)
whe e i j is he s ess enso . A e he use o he di e gence heo em and he equali ies (2.1)–(2.3),
(2.7), and (2.8), he ela ion (2.6) can be w i en in he o m:
ZP
ρ˙ed =ZP
[ ji ˙ui,j+ρ˙ηθ +ρ˙ηjTj+Sjθ,j+ Λk jTj,k−(Si−Hi)Ti−ρξθ −ρξjTj]d (2.9)
+Z∂P
(Gjθ,j+ ΠjiTi,j)da.
Wi h an a gumen simila o ha used o de i e he ela ion (2.7), om (2.9), we ob ain
(Gj−Gk jnk)θ,j+(Πji −Πk jink)Ti,j=0,(2.10)
whe e Gk j and Πk ji a e enso s associa ed o he su ace loads Giand Πji, espec i ely. Wi h he help
o (2.10), we ob ain he local exp ession o m o he ene gy balance
ρ˙e= ji ˙ui,j+ρ˙ηθ +ρ˙ηjTj+Fjθ,j+ Γk jTj,k+(Hi−Si)Ti(2.11)
+Gk jθ,jk + Πk jiT,i jk −ρξθ −ρξjTj,
whe e he ollowing no a ion
Fj=Sj+Gk j,k,Γk j = Λk j + Πmk j,m,(2.12)
is used.
Elec onic Resea ch A chi e Volume 33, Issue 2, 537–555.
540
Following [14], we assume a mo ion o he body ha is di e en om he gi en mo ion by a supe -
posed uni o m igid body angula eloci y, and ha ρ, e, i j, η, θ, ηj,Tj,Sj,Λk j,Hj,Fj,Γk j, ξ, and ξjdo
no change by such mo ion. The equali y (2.11) implies ha
ji = i j.(2.13)
I we conside he Helmhol z ee ene gy ψby
ψ=e−θη −Tjηj,(2.14)
and we see ha he ene gy balance may be w i en in he o m:
ρ(˙
ψ+θ˙η+Tj˙ηj)= i j ˙ei j +Fjθ,j(2.15)
+ Γk jTj,k+(Hi−Si)Ti+Gk jθ,jk + Πki jTi,jk −ρξθ −ρξjTj,
whe e we ha e in oduced he s ain enso
2ei j =ui,j+uj,i.(2.16)
3. Cons i u i e equa ions
F om now on, we de ine he cons i u i e equa ions o ψ, i j, η, ηj,Sj,Hj,Gk j,Fj,Γk j,Πk j, ξ, and
ξj, and we suppose ha hese a e unc ions o he se V=(ei j, θ, Tj, α,j, βk,j, α,i j, βk,i j). To simpli y
he w i ing, we omi he explici dependence o xkand hen he ma e ial should be homogeneous and
assume ha he e is no kinema ical cons ain . In he heo y es ablished in [3], he cons i u i e a iables
a e ei j, θ, and α,j. I we in oduce he no a ion A=ρψ, hen Eq (2.15) becomes
(∂A
∂ei j
− i j)˙ei j +(∂A
∂θ +ρη)˙
θ+(∂A
∂Ti
+ρηi)˙
Ti(3.1)
+(∂A
∂α,i
−Fi)θ,i+(∂A
∂βj,i
−Γi j)Tj,i+(∂A
∂α,i j
−Gji)θ,i j +(∂A
∂βi,jk
−Πk ji)Ti,jk
+ρθξ +(ρξi+Si−Hi)Ti=0.
F om (3.1), we ind ha [3]
i j =∂A
∂ei j
, ρη =−∂A
∂θ , ρηj=−∂A
∂Tj
,(3.2)
Fi=∂A
∂α,i
,Γi j =∂A
∂βj,i
,Gji =∂A
∂α,i j
,Πk ji =∂A
∂βi,jk
,
and
ρξθ +(ρξi+Si−Hi)Ti=0.(3.3)
We in oduce he no a ions
θ(x, 0)=T0,Tj(x, 0)=T0
j, α(x, 0)=α0, βj(x, 0)=β0
j,(3.4)
Elec onic Resea ch A chi e Volume 33, Issue 2, 537–555.
541
whe e 0is a e e ence ime and T0,T0
j, α0, and β0
ja e gi en cons an s. As in [3], we conside new
he mal a iables
T=θ−T0, θi=Ti−T0
i, χ =Z
0
Tds, φi=Z
0
θids.(3.5)
F om (3.4) and (3.5), we ge
α=χ+T0( − 0)+α0, βj=φj+T0
j( − 0)+β0
j, α,i=χ,i, βi,j=φi,j,(3.6)
˙χ=T,˙φi=θi.
F om now on, we es ic ou a en ion o he linea heo y whe e he unc ions ui,T, and θjcan be
w i en as
ui=ϵu′
i,T=ϵT′, θj=ϵθ′
j
whe e ϵis a pa ame e small enough o squa es and highe powe s o be neglec ed, and u′
i,T′,
and θ′
ja e independen o ϵ. As usual, we assume ha Ais a quad a ic o m o he a iables
ei j,T, θj, α,j, βk,j, α,i j,and βk,i j and ha Hi, ξ, and ξja e linea unc ions o he same a iables. We
conside he case o a ma e ial wi h a cen e o symme y. Thus, we ha e
2A=Ai j sei je s −2bi jei jT+2Ci j sei jφ ,s+2Di j sei jχ, s −aT2−2Li jTφi,j(3.7)
−2Ni jTχ,i j −Bi jθiθj−2Ci jθiχ,j−2dipq φp,q θi+Ki jχ,iχ,j+2Mipq χ,iφp,q
+Ei j sφi,jφ ,s+2Hi j sφi,jχ, s +Ui jkpq φi,jkφp,q +Qi j sχ,i jχ, s.
The ollowing symme ies a e sa is ied:
Ai j s =Aji s =A si j,bi j =bji,Ci j s =Cji s,Di j s =Dji s =Djis ,(3.8)
Bi j =Bji,Ni j =Nji,dipq =dip q,Ki j =Kji,Ei j s =E si j,Mipq =Mip q,
Hi j s =Hi js ,Ui jkpq =Upq i jk =Uik jpq ,Qi j s =Q si j =Qji s.
I ollows om (3.2), (3.7), and (3.8) ha
i j =Ai j se s −bi jT+Ci j sφ ,s+Di j sχ, s,
ρη =bi jei j +aT +Li jφi,j+Ni jχ,i j,
ρηi=Bi jθj+Ci jχ,j+dipq φp,q ,(3.9)
Fj=−Ci jθi+Ki jχ,i+Mjpq φp,q ,
Γi j =C s jie s −LjiT+Eji sφ ,s+Hji sχ, s,
Gji =D si je s −Ni jT+Qi j sχ, s +H si jφ ,s,
Πk ji =−d i jkθ +Msi jkχ,s+Ui jkpq φp,q .
Fo iso opic ma e ials, he numbe o cons i u i e coe icien s is d as ically educed (see [12]). F om
(3.3), we see ha he esponse unc ion ξ anishes when he mic o empe a u es Tj anish. In he linea
Elec onic Resea ch A chi e Volume 33, Issue 2, 537–555.
542
case, he unc ion ξ ha sa is ies his equi emen mus be ξ=cjTj, whe e cja e cons an s. Since he
body has a cen e o symme y, we ge ξ=0. Thus, om (2.12), (3.3), and (3.9), we ind ha
ρξi=Hi−Si.(3.10)
I we use hese esul s, hen Eqs (2.1) and (2.2) ake he o m
ρη =Sk,k+ρs, ρηi= Λki,k+ρQi.(3.11)
The equa ions o he linea heo y consis o he equa ions o mo ion (2.8), he ene gy equa ions
(3.11), he cons i u i e equa ions (3.9), and he geome ical equa ions (2.16). In iew o (2.12), we can
w i e he Eqs (3.11) in he o m
Fk,k−Gk j,k j −ρ˙η=−ρs,Γi j,i−Π k j, k −ρ˙ηj=−ρQj.(3.12)
Equa ions (2.8) and (3.11) can be exp essed in e ms o he unknowns uj,χ, and φi. Thus, we ob ain
he equa ions
Ai j su ,s j −bi j ˙χ,j+Dji kχ, k j +Ci j sφ ,s j +ρ i=ρ¨ui,(3.13)
−D pqku ,pqk −bi j ˙ui,j+Ki jχ,i j −Qi j sχ,i j s −a¨χ−Rpq j φp,q j
−pi j ˙φi,j=−ρs,
C s jku ,sk −pjk ˙χ,k+ζp jq ˙φp,q +Rjk sχ, sk
+Ejk sφ ,sk −Uj spqmφp,qms −Bjk ¨φk=−ρQj,
whe e
Rjk s =Hjk s −M jks, ζp jq =dp jq −djpq ,pjk =Ljk +Cjk.(3.14)
To he sys em (3.13), we ha e o adjoin bounda y and ini ial condi ions.
4. Bounda y-ini ial- alue p oblems
Now, we s udy he bounda y condi ions and o mula e he basic bounda y-ini ial- alue p oblems.
We assume ha he bounda y o Bconsis s o he union o a ini e numbe o smoo h su aces, smoo h
cu es (edges), and poin s (co ne s). Le Cbe he union o he edges. As in [11, 12], o ob ain he
o m o he bounda y condi ions, we mus s udy he su ace in eg al in (2.5). By using (2.3), (2.7), and
(2.10), we ind ha Z∂P
( i˙ui+ Σθ+σjTj+Gjθ,j+ ΠjiTi,j)da (4.1)
=Z∂P
[ ki ˙ui+(Fk−G k, )θ+(Γk j −Πmk j,m)Tj+Gk jθ,j+ Πk jiTi,j]nkda.
We will use he no a ions
D = ,jnj,Di=(δi j −ninj)∂
∂xj
,(4.2)
whe e δi j is he K onecke del a. Then we ob ain
Gk jθ,jnk=Gk j nknlDθ−θ(Gk jnk)+Dj(Gk jnkθ),(4.3)
Elec onic Resea ch A chi e Volume 33, Issue 2, 537–555.
543
Πk jiTi,jnk= Πk jinknjDTi−TiDj(Πk jink)+Dj(Πk jinkTi).
As in [11,12], om (4.1) and (4.3), we ge
Z∂P
( i˙ui+ Σθ+σjTj+Gjθ,j+ ΠjiTi,j)da (4.4)
=Z∂P
( i˙ui+ Φ1θ+ Φ2Dθ+ ΨjTj+WiDTi)da
+ZC
(Yθ+ ΩiTi)ds
whe e
Φ1=(Fk−G k, )nk−Dj(nsGs j)+(Djnj)nsnpGsp,Φ2=G sn ns,(4.5)
Ψi=(Γki −Π ki, )nk−Dj(nsΠs ji)+(Djnj)nsnpΠspi,Wi= Π sin ns,
Y=<G sn ys>, Ωi=<Π sin ys>, yi=ϵi k s nk.
He e, ska e he componen s o he uni ec o angen o C,< >deno es he di e ence o he limi s
o om bo h sides o C, and ϵj k is he al e na ing symbol. The i s bounda y-ini ial- alue p oblem
is cha ac e ized by he bounda y condi ions
ui=u∗
i, χ =χ∗, φi=φ∗
i,Dχ=ζ∗,Dφi=γ∗
ion ∂B×I,(4.6)
whe e u∗
i, χ∗, φ∗
i, ζ∗, and γ∗
ia e p esc ibed unc ions.
Fo he second bounda y-ini ial- alue p oblem, he bounda y condi ions a e [11]
i= ∗
i,Φ1= Φ∗
1,Φ2= Φ∗
2,Ψi= Ψ∗
i,Wi=W∗
ion ∂B×I,(4.7)
Y=Y∗,Ωi= Ω∗
ion C×I,
whe e ∗
i,Φ∗
1,Φ∗
2,Ψ∗
i,W∗
i,Y∗, and Ω∗
ia e gi en. The ini ial condi ions a e
ui(x,0) =u0
i(x),˙ui(x,0) = 0
i(x), χ(x,0) =χ0(x),˙χ(x,0) =χ1(x),(4.8)
φ(x,0) =φ0
i(x),˙φi(x,0) =ν0
i(x),x∈B,
whe e u0
i, 0
i, χ0, χ1, φ0
i, and ν0
ia e gi en.
5. An exis ence esul
Now, we p o ide an exis ence and uniqueness esul o he p oblem de e mined by he sys em o
Eqs (3.13), wi h he ini ial condi ion (4.8) and he homogeneous e sion o he bounda y condi ions
(4.6). We will use he heo y o con ac i e linea semig oups [16].
We assume once and o all ha :
(i) The mass densi y ρand he he mal capaci y aa e s ic ly posi i e.
(ii) The ma ix Bi j is posi i e de ini e.
(iii) The quad a ic o m
W(ei j, κi jk, χ,i, χ,ji, φ,i, φ,i j)=
Elec onic Resea ch A chi e Volume 33, Issue 2, 537–555.
544
Ai j sei je s +2Ci j sei jφ ,s+2Di j sei jχ, s +Ki jχ,iχ,j+2Mipq χ,iφp,q
+Ei j sφi,jφ ,s+2Hi j sφi,jχ, s +Ui jkpq φi,jkφp,q +Qi j sχ,i jχ, s,
is posi i e de ini e, i.e., he e exis s a posi i e cons an Csuch ha :
W≥C(ei jei j +χ, χ, +χ, sχ, s +φ ,sφ ,s+φp,q φp,q ).
Le us o p opose he p oblem as an abs ac p oblem in a sui able Hilbe space. We will wo k on he
space
H=W1,2
0(B)×L2(B)×W2,2
0(B)×L2(B)×W2,2
0(B)×L2(B),
whe e W1,2
0,W2,2
0, and L2a e he usual Sobole spaces, W2,2
0=[W2,2
0]3and L2=[L2]3. The elemen s
in his space can be deno ed by U=(u, , χ, θ, φ,ϕ).
We conside he scala p oduc associa ed o he no m
||(u, , χ, θ, φ,ϕ)||2(5.1)
=ZB
[ρ i i+aθ2+Bi jϕiϕj+W(ei j, χ,i, χ,ji, φ,i, φ,i j)]d .
Now, we wan o see ou p oblem as a Cauchy p oblem in H. We de ine he ope a o
A
u
χ
θ
φ
ϕ
=
M
θ
υ
ϕ
Ω
(5.2)
whe e
M=(Mi),Ω=(ωi),
and
Mi=ρ−1(Ai j su ,s j −bi jθ,j+Ci j sφ ,s j +Dji kχ, k j),
ωi=Fi j(C s jku ,sk −pjkθ,k+ζp jq ϕp,q +Rjk sχ, sk
+Ejk sφ ,sk −Uj spqmφp,qms ),
υ=a−1(−D pqku ,pqk −bi j i,j+Ki jχ,i j −Qi j sχ,i j s −Rpq j φp,q j
−pi jϕi,j),
whe e Fi jBjk =δik.
We no e ha ou p oblem can be w i en as
dU
d =AU +F( ),U(0) =(u0, 0, χ0, χ1,φ0,ν0),(5.3)
whe e
F( )=(0, ( ),0, ρa−1s,0, ρFi jQj).
Elec onic Resea ch A chi e Volume 33, Issue 2, 537–555.
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The domain o he ope a o Ais he subspace o elemen s o ou Hilbe space such ha
∈W1,2
0,ϕ∈W2,2
0, θ ∈W2,2,
Ai j su ,s j +Dji kχ, k j −bi jθ,j∈L2,
−D pqku ,pqk −Qi j sχ,i j s −Rpq j φp,q j ∈L2,
and
C s jku ,sk +Rjk sχ, sk −Uj spqmφp,qms ∈L2.
This domain is a dense subse o ou space.
A e an easy bu labo ious calcula ion, we can see ha
<AU,U>=0,
o e e y elemen Ua he domain o he ope a o .
The nex s ep in ou app oach is o show ha ze o belongs o he esol en o he ope a o . Le U∗
be in H. We mus p o e ha he equa ion
AU =U∗
admi s a solu ion. Tha is
=u∗, θ =χ∗,ϕ=φ∗,
M= ∗, υ =θ∗,Ω=ϕ∗.
We subs i u e he i s h ee equa ions in o he o he s o ind ha
Ai j su ,s j +Ci j sφ ,s j +Dji kχ, k j =ρ ∗
i+bi jχ∗
,j,
−D pqku ,pqk +Ki jχ,i j −Qi j sχ,i j s −Rpq j φp,q j
=aθ∗+bi ju∗
i,j+pi jφ∗
i,j,
C s jku ,sk +Rjk sχ, sk +Ejk sφ ,sk
−Uj spqmφp,qms =Bjkϕ∗
k+pjkχ∗
,k−ζp jq φ∗
p,q .
I we deno e
αi1=ρ ∗
i+bi jχ∗
,j, α2=aθ∗+bi ju∗
i,j+pi jφ∗
i,j,
α3j=Bjkϕ∗
k+pjkχ∗
,k−ζp jq φ∗
p,q ,A1i=Ai j su ,s j +Ci j sφ ,s j +Dji kχ, k j,
A2=−D pqku ,pqk +Ki jχ,i j −Qi j sχ,i j s −Rpq j φp,q j,
A3j=C s jku ,sk +Rjk sχ, sk +Ejk sφ ,sk −Uj spqmφp,qms ,
hen ou sys em can be w i en as
Ai1=αi1,A2=α2,Ai3=αi3.
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552
We do no assume any condi ion on he coe icien s µ, C2,E2, and b1, bu we a e going o ob ain a
couple o quali a i e esul s o ou sys em. Ne e heless we will need o suppose ha ρand ba e wo
posi i e eal numbe s. To do his, i will be con enien o wo k wi h a unc ion ha will allow us o
conclude he desi ed esul s. We de ine his unc ion in he o m:
H( )=ZD
(ρu2+bφ2)dx+ω∗( + 0)2,(9.2)
whe e ω∗and 0a e wo non-nega i e eal numbe s o be selec ed. We ha e
˙
H( )=2ZD
(ρu˙u+bφ˙φ)dx+2ω∗( + 0),
and
¨
H( )=2ZD
(ρu¨u+bφ¨φ)dx+2ZD
(ρ|˙u|2+b|˙φ|2)dx+2ω∗.
We can no ice ha
ZD
(ρu¨u+bφ¨φ)dx=−ZD
(µ|∇u|2+2C2∇φ∇φ+E2|∇φ|2+b1|∆φ|2)dx
ZD
(ρ|˙u|2+b|˙φ|2)dx−2E(0).
The e o e
¨
H( )=4ZD
(ρ|˙u|2+b|˙φ|2)dx+2(ω∗−2E(0)).
A simple use o Holde ’s inequali y allows us o conclude
H( )¨
H( )−(˙
H( ))2≥ −2(ω∗+2E(0))H( ).
I we pu homogeneous ini ial condi ions, we ha e ha E(0) =0 and i we ake ω∗=0, we can
conclude ha
H( )¨
H( )−(˙
H)2≥0.(9.3)
This inequali y allows us o es ablish (see [19], p. 19) ha
H( )≤ H(0)1− /T∗H(T∗) /T∗
o all be ween 0 and T∗. Thus, in he case whe e we impose null ini ial da a, we ob ain ha H( )=0
o all in he in e al and ,consequen ly we ob ain he null solu ion. This allows us o conclude he
uniqueness o he solu ions.
I we now go back o he gene al case and suppose ha he ini ial ene gy is nega i e, we can ake
ω∗=−2E(0) and again conclude he p e ious inequali y. We can also ge (see [19], p. 20)
H( )≥ H(0) exp ˙
H(0)
H(0) .(9.4)
We no e ha we can always selec 0la ge enough o gua an ee ha ˙
H(0) >0. When E(0) =0 and
˙
H(0) >0, we also conclude he g ow h es ima o .
Elec onic Resea ch A chi e Volume 33, Issue 2, 537–555.
553
Theo em 4. Le us o suppose ha ρand ba e posi i e. Then:
(i) The ini ial-bounda y- alue p oblem o he an i-plane shea de o ma ions has a unique solu ion.
(ii) When E(0) <0 o (E(0) =0,˙
H(0) >0), hen he solu ion is exponen ially uns able.
A simila a gumen could p o e he uniqueness and ins abili y o he solu ions o he empe a u e
equa ion in he case ha we only assume ha ais s ic ly posi i e.
10. Conclusions
The esul s ob ained in his pape can be summa ized as ollows:
(a) We p esen a linea heo y o he moelas ici y wi h mic o empe a u es whe e he second he mal
displacemen g adien and he second g adien o mic o empe a u es a e included in he classical se o
independen cons i u i e a iables.
(b) We exp ess he ield equa ions o he linea heo y in e ms o componen s o he displacemen
ec o , he mal displacemen , and he mal mic odisplacemen ,s and ob ain a ou h-o de sys em o
equa ions. The bounda y-ini ial- alue p oblems a e also o mula ed.
(c) The semig oup heo y o linea ope a o s is used o p o e ha he p oblem o he second g adien
he moelas ici y wi h mic o empe a u es is well-posed.
(d) We es ablish a coun e pa o he Cauchy-Ko ale ski-Somigliana solu ion o he iso he mal
heo y.
(e) In he case o s a iona y ib a ions, we es ablish he undamen al solu ions o he ield equa ions.
( ) Uniqueness and ins abili y o he solu ions a e ob ained in he case o an i-plane shea de o ma-
ions.
Use o AI ools decla a ion
The au ho s decla e hey ha e no used A i icial In elligence (AI) ools in he c ea ion o his a icle.
Acknowledgmen s
The au ho s hank he e e ees o hei help ul sugges ions. This esea ch did no ecei e any
speci ic g an om unding agencies in he public, comme cial, o no - o -p o i sec o s.
Con lic o in e es
Ram´
on Quin anilla is an edi o ial boa d membe o Elec onic Resea ch A chi e and was no in-
ol ed in he edi o ial e iew o he decision o publish his a icle. The au ho s decla e ha hey ha e
no known compe ing inancial in e es s o pe sonal ela ionships ha could ha e appea ed o in luence
he wo k epo ed in his pape .
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