A GENERAL ASYMPTOTIC DYNAMIC MODEL FOR LIPSCHITZIAN ELASTIC CURVED RODS

A GENERAL ASYMPTOTIC DYNAMIC MODEL FOR LIPSCHITZIAN ELASTIC CURVED RODS

ROSTISLAV VOD ´AK

Received 31 June 2005 and in revised form 15 November 2005

We study the asymptotic behaviour of solutions to the linear evolution problem for clamped curved rods with the small thicknessunder minimal regularity assumptions on the geometry. In addition, nonconstant density of the curved rods is considered.

1. Introduction

The main task of this paper is to relax regularity assumptions on a shape of elastic curved rods in a general asymptotic dynamic model and to derive this asymptotic model from a linear evolution equation of three dimensional elasticity by asymptotic technique.

We use the asymptotic approach presented by Aganovic and Tutek [17] for straight rods, which was modified by Jurak and Tambaˇca [8,9] (see also Trabucho and Via˜no [16]) and by Ignat, Sprekels, and Tiba [7] for curved rods generated by a functionΦ∈ C^k([0,l])³,k≥3, andk≥2, respectively. The approach from [8,9] was applied to a dy- namic model of curved rods in [13]. Following an idea from Blouza and Le Dret [3], a new formulation of the equations of elasticity which requiredΦ∈C²([0,l])³was found in Tiba and Vod´ak [14]. In addition, the general asymptotic model obtained in [14] was well-defined for a Jordan unit speed curve with Lipschitzian parameterization which led to a special construction of its approximations with smooth Jordan unit speed curves.

The whole construction was valid forΦ_i>0 a.e. on (0,l) for somei,i=1, 2, 3. This re- strictive condition was excluded by the modification of the construction in [15]. In this paper, we want to extend the theory presented in [14,15] on the dynamic model for the curved rods. Among other papers concerning with the dynamic models for thin curved structures we mention here Raoult [10], Xiao [18], and Sprekels and Tiba [12]. Further we recommend the reader ´Alvarez-Dios and Via˜no [1], Berm ´udez and Via˜no [2].

Finally, we give a brief outline of the paper. InSection 2, we introduce the basic no- tations and notions that will be further needed.Section 3 contains auxiliary proposi- tions, which are used throughout the paper.Section 4is devoted to a weak formulation of the linear elasticity equation and its transformation.Section 5deals with basic esti- mates.Section 6gives us a basic overview about behaviour of the displacements if→0 and about qualitative properties of their limit state. InSection 7, the passage to the limit →0 is performed and the main existence and uniqueness result is proved.

Copyright©2005 Hindawi Publishing Corporation Journal of Applied Mathematics 2005:4 (2005) 425–451 DOI:10.1155/JAM.2005.425

The main result of this paper can be summarized in the following theorem.

Theorem1.1. Let a functionΦ∈W^1,∞(0,l)³be a parametrization of a unit speed curve.

Let, further,F∈L²(0,T;L²(Ω)³),G∈W^1,1(0,T;L²(0,l;L²(∂S)³))andFˇF+Gbe defined as inLemma 7.4. Then, there is a unique pairU,φ ∈L^∞(0,T;ᐂ^t,n,b0 (0,l))such that∂_tU∈ L^∞(0,T;L²(0,l)³)∩C([0,T]; [ᐂ^t,n,b0 (0,l)]), which generates a unique solution to the prob- lem (7.41)–(7.42). Moreover, the constant extension toΩ=(0,l)×SofU,φmay be ap- proximated by solutionsU∈L^∞(0,T;V(Ω)³)∩W^1,∞(0,T;L²(Ω)³)of the problem (4.13)–

(4.14) as follows:

U=lim

→0U∗-weakly inL^∞0,T;V(Ω)³,

∂_tU=lim

→0∂_tU∗-weakly inL^∞0,T;L²(Ω)³, φ=lim

→0

1 2

∂2U,b

−

∂3U,n

∗-weakly inL^∞0,T;L²(Ω).

(1.1)

2. Preliminaries

We denote byR³the usual three-dimensional Euclidean space with scalar product (·,·) and norm| · |. LetS⊂R²be a bounded simply connected domain of classC¹satisfying the “symmetry” condition

Sx2dx2dx3=

Sx3dx2dx3=

Sx2x3dx2dx3=0. (2.1) We denote byΩ=(0,l)×S,Ω=(0,l)×Sopen cylinders inR³, wherel >0 and>0

“small,” are given.

Let a functionΦ: [0,l]→R³,Φ∈W^1,∞(0,l)³, be a parametrization of a Jordan unit speed curveᏯ inR³ and lett,n,b denote its tangent, normal and binormal vectors.

LetΦ: [0,l]→R³be a smoothing ofΦsuch that it remains a Jordan unit speed curve (i.e.,|Φ(y1)| =1,∀y1∈[0,l]) andt,n,b be the associated local frame. Alternative ways, how to construct local frames under low regularity assumptions, may be found in [12]. The whole construction of the local frame associated with the functionΦand its smoothing can be found in [14,15]. Here we mention only the needed properties of the approximation:{Φ}∈(0,1),{t}∈(0,1),{n}∈(0,1),{b}∈(0,1)⊂C^∞([0,l])³,

t−→t, n−→n, b−→b in measure in (0,l), (2.2) Φ(·)=

_·

0t(z)dz+Φ(0)−→Φ(·) inC[0,l]³ (2.3) for→0,

t_∞+n_∞+b_∞≤ C

^r, t_∞+n_∞+b_∞≤ C ^2r, r∈

0,1

3

. (2.4) The whole construction can be found in [14,15].

The orthogonality properties (t,t)=0, (n,n)=0, (b,b)=0 lead to so called

“laws of motion” of the local frame

t=αb+βn, n= −βt−γb, b= −αt+γn.

(2.5)

From (2.4), it follows that α_∞+β_∞+γ_∞≤ C

^r, α_∞+β∞+γ_∞≤ C ^2r, r∈

0,1

3

. (2.6) We adopt the usual notation for the standard Sobolev and Lebesgue spaces, that is,H¹(Ω), H₀¹(Ω) andL^p(Ω),p∈[1,∞] for the spaces and · 1,2, · pfor their norms. We will use the same notation of the norms also for vector or tensor functions in the case that all their components belong to the above mentioned Sobolev or Lebesgue spaces.H⁻¹(Ω) and [X]stand for the dual space toH₀¹(Ω) orX, respectively. The notationC^m(Ω), with m∈N0, means the usual spaces of continuous functions whose derivatives up to the ordermare continuous inΩ, and we denote byC^∞₀(Ω) the space of all functions which have derivatives of any order onΩand whose supports are compact subsets ofΩ. The symbolsL^p(I;X),p∈[1,∞] andC(I;X), whereXis a Banach space andI is a bounded interval, stand for the Bochner spaces endowed with the normsvL^p(I;X)andv_C(I;X), respectively. We say thatv∈C([0,T];L²_weak(Ω)) if the function_Ωv(t)w dxis continuous on [0,T] for an arbitrary functionw∈L²(Ω).

Except for the standard definition of the weak convergence inXorL^p(I;X),p∈(1,∞), and∗-weak convergence inL^∞(I, [X]), we say without danger of confusion thatv_nv inL²(0,l;H⁻¹(S)), if

_l

0H⁻¹(S)

vn

x1

−vx1

,ψx1 H0¹(S)dx1−→0 for anyψ∈L²0,l;H₀¹(S), (2.7)

where[X]·,·Xdenotes the dual pairing of [X]andX, andvn^∗ vinL^∞(I,L²(Ω)) if

I

Ω

vn−vψ dx dz−→0 (2.8)

for anyψ∈L¹(I,L²(Ω)). Further, we introduce the space ᐂ^t,n,b0 (0,l)=

V,ψ∈H₀¹(0,l)³×L²(0,l) : (V,t)=0,

V_∗= −ψt+ (V,b)n−(V,n)b∈H₀¹(0,l)³. (2.9) We refer the reader to [14] for the proof thatᐂ^t,n,b0 (0,l) is a nontrivial Hilbert space endowed with the norm

V,ψ²= V²1,2+ψ²2+V_∗²_1,2. (2.10)

Letv∈L¹_loc(0,T) andϕ∈C^∞0(0,T). Then we denotev^ϕ=_T

0 v(t)ϕ(t)dt. Now, we in- troduce the mappingsRand ¯P,

R:Ω−→Ω, Rx1,x2,x3

=

x1,x2,x3

, (2.11)

P¯:Ω−→R³, P¯(y)=Φ y1

+y2ny1

+y3by1

, (2.12)

(y1,y2,y3)∈Ω=(0,l)×S, where the second one gives the parametrization of the curved rodΩ=P¯(Ω). Furthermore,

d¯(y)=det_∇¯P¯(y)=1−βy1

y2−αy1

y3 ∀y∈Ω. (2.13)

We can suppose that ¯d(y)>0 for all y∈Ωand for“small” (see (2.6) and the defini- tion ofΩ). Then ¯P:Ω→Ωis aC¹-diffeomorphism according to Ciarlet [4, Theorem 3.1-1]. In the sequel, we will write∂i=∂/∂yi, wherey=(y1,y2,y3)∈Ω, ¯∂i=∂/∂yi, for y=(y1,y2,y3)∈Ω,∂i=∂/∂xi, wherex=(x1,x2,x3)∈Ω,∂t =∂/∂t and ∂tt=∂²/∂t². Thus, in (2.13), ¯∇ =( ¯∂1, ¯∂2, ¯∂3). In the case that a functionvdepends only ontorx1(or y1), we denote its first (second) derivation by ˙v(¨v) andv(v), respectively. Sometimes it is more convenient to use the notation (d/dt)vinstead of ˙v.

The definition of the domainΩenables us to introduce the following spaces:

V Ω

=

V ∈H¹ Ω

:VP¯(_{0_}×S)=VP¯(_{l}×S)=0, (2.14) V(Ω)=

V∈H¹(Ω) :V_({0}×S)=V_({l}×S)=0. (2.15) In an analogous way as above, we denote byV a function defined onΩ, ¯Va function defined onΩ, andVa function defined onΩ.

The covariant and the contravariant basis at the point ¯P(y), y∈Ω, of the curved rod are defined by ¯gi,(y)=∂¯_iP¯(y) and (¯gi,, ¯g^j,)=δ^{i j}, and (using (2.5)) these vectors are given by

¯

g1,(y)=

1−y2βy1

−y3αy1

ty1

+y3γy1

ny1

−y2γy1

by1

,

¯

g2,(y)=ny1

, g¯3,(y)=by1

,

(2.16)

¯

g^1,(y)=ty1

d¯(y), g¯^2,(y)= −y3γy1

ty1

d¯(y) +ny1 ,

¯

g^3,(y)= y2γy1

ty1

d¯(y) +by1

.

(2.17)

Further, we define the covariant and contravariant metric tensors ( ¯gi j,)³_i,j=1, ( ¯g^{i j,})³_i,j=1, where

¯ gi j,=

¯ gi,, ¯gj,

, g¯^{i j,}=

¯

g^i,, ¯g^j,. (2.18)

After substitutiony=R(x), we adopt the notation g^{i j,}(x)=g¯^{i j,}R(x), gi j,(x)=g¯i j,

R(x), gi,(x)=g¯i,

R(x), (2.19) g^j,(x)=g¯^j,R(x), d(x)=d¯R(x), (2.20) wherex∈Ω. Analogously as above, we can find the contravariant tensoro^{i j,} for the mapping ¯P◦Rhaving the form

o^{i j,3}_i,j=1=











 1

d^{2 −} x3γ

d²

x2γ d²

−x3γ d²

1 ²+x3²γ²

d^{2 −} x2x3γ²

d² x2γ

d^{2 −} x2x3γ²

d² 1 ²+x2²γ²

d²













. (2.21)

By “×” we will denote the Cartesian product of two spaces and by·,·any ordered pair. In the text, the symbol|A| will also denote the Lebesgue measure of some mea- surable setA, without danger of confusion. The summation convention with respect to repeated indices will be also used, if not otherwise explicitly stated. We use for constants the symbolsCorC_i, fori∈N0= {0, 1, 2,. . .}.

3. Auxiliary propositions

Proposition3.1 [14]. Lett,n,bsatisfy (2.2)–(2.6) and let the spaceᐂ^t0^,n,b(0,l)be de- fined by (2.9) using the functionst,n,binstead oft,n,b. Let, further,V,ψ ∈ᐂ^t,n,b0 (0,l) be an arbitrary but fixed couple. Then there exist the couples

V,ψ∈ᐂ^t0^,n,b(0,l) (3.1) generating the functionsV_∗,such that{V}∈(0,1),{V_∗,}∈(0,1)⊂C^∞₀(0,l)³,{ψ}∈(0,1)⊂ C^∞0(0,l),

V−→V, V_∗,−→V_∗ inH₀¹(0,l)³, (3.2)

ψ−→ψ in measure on(0,l), (3.3)

for→0, and

V₂+ψ₂≤ C ^r, r∈

0,1

3

. (3.4)

Proposition3.2 [14]. Letλ≥0,µ >0and

A^{i jkl} =λg^{i j,}g^kl,+µg^ik,g^jl,+g^il,g^jk,. (3.5) Then there exists a constantC3>0such that the estimate

3 i,j=1

t_{i j}²≤C3A^{i jkl} (x)t_klt_{i j} (3.6)

holds for allx∈Ω, all∈[0, 1]and all symmetric matrices(t_{i j})³_i,j=1, with the constantC3

being independent ofandx.

Proposition3.3 [14]. There exists a constantC4>0independent ofsuch that V1,2≤C4

ω(V)₂, ∀V∈V(Ω)³,∀∈(0, 1). (3.7) Proposition3.4 [9]. Let {v_n}^∞_n=1⊂L²(0,l;L²(S)),{∂1v_n}^∞_n=1⊂L²(0,l;H⁻¹(S))be such thatv_n|x1=0=v_n|x1=l=0, for alln∈N, in the sense of the spaceC([0,l];H⁻¹(S)). Assume, in addition, that these sequences satisfy

∂1v_n ξ, ∂_jv_n 0, inL²0,l;H⁻¹(S), j=2, 3, (3.8) whereξ∈L²(0,l;H⁻¹(S)). Thenξ∈L²(0,l), and there exists a unique functionv∈H0¹(0,l) such thatv=ξand

vn v inL²0,l;L²(S), (3.9)

vn−→v inC[0,l];H⁻¹(S). (3.10) If the convergences in (3.8) are strong then the convergence (3.9) is also strong.

Proposition3.5 [14]. We have

d−→1 inCΩ, (3.11)

d(x)νi(x)o^{i j,}(x)νj(x)−→1 inC(0,l)×∂S, (3.12) for→0, whereνi,i=1, 2, 3, are components of a unit outward normal to(0,l)×∂S. Thus, there exist constantsC_j, j=0, 1, 2, such that0< C0≤d(x)≤C1 for allx∈Ω, and0≤ d(x)

νi(x)o^{i j,}(x)νj(x)≤C2for allx∈(0,l)×∂Sand∈(0, 1).

4. Weak formulation of an evolution equation for the curved rods and its transformation

We considerΩ defined by mapping ¯P◦R (see (2.11)–(2.12)) for∈(0, 1) arbitrary but fixed as a three-dimensional homogeneous and isotropic elastic body with the Lam´e constantsλ≥0,µ >0 and with mass density ρ. Let F be the body force andG the surface traction acting on the curved rodΩ such thatF∈L²(0,T;L²(Ω)³) andG∈ W^1,1(0,T;L²(( ¯P◦R)((0,l)×∂S))³), for∈(0, 1). LetΩ be clamped on both bases P¯({0} ×S) and ¯P({l} ×S). The equilibrium displacementUis a (weak) solution of the equation

_T

0

Ω

−

ρ∂tU(t),∂tV(t) +A^{i jkl}ekl U(t)ei jV(t)dy dt

= _T

0

Ω

F(t),V(t) dy dt+ _T

0

S

G(t),V(t) dSdy1dt

(4.1)

for allV∈C^∞₀(0,T;V(Ω)³), whereS=( ¯P◦R)((0,l)×∂S),A^{i jkl}=λδ^{i j}δ^kl+µ(δ^ikδ^jl+ δ^ilδ^jk) and (ei j(V)) ³_i,j=1stands for the symmetric part of the gradient of the functionV.

The solutionUsatisfies the initial state

U|t=0=Q0,, ∂tU|t=0=Q1,. (4.2) Using the fact that the functionst,n,b∈C^∞([0,l])³together with (2.11), (2.12), it is easy to see that the mapping ¯P◦R is the parametrization of the smooth three- dimensional curved rod.

We transform now (4.1). Denoting U=U( ¯P◦R), ρ=ρ( ¯P◦R) and V= V( ¯ P◦R), we get

_T

0

Ω

ρ∂_tU(t),∂_tV(t) dy dt= _T

0

Ω

ρ∂_tU(t),∂_tV(t)²ddx dt. (4.3) For the transformation of other terms we refer the reader to [14]. It is easy to see that if V∈C₀^∞(0,T;V(Ω)³), thenV∈C₀^∞(0,T;V(Ω)³). DenotingQ0,=Q0,( ¯P◦R),Q1,= Q1,( ¯P◦R), F=F( ¯P◦R) andG=G( ¯P◦R), we can rewrite the model (4.1)- (4.2) using (4.3) and the transformation from [14] as

_T

0

Ω−

ρ∂tU(t),∂tV(t)d+A^{i jkl} ω_klU(t)ω_{i j}V(t)ddx dt

= _T

0

Ω

F(t),V(t)ddx dt+ _T

0

_l

0

∂S

G(t),V(t)dνio^{i j,}νjdS dx1dt (4.4) for allV∈C^∞₀(0,T;V(Ω)³), where the solutionUsatisfies the initial state

U|t=0=Q0,, ∂_tU|t=0=Q1,, (4.5) νi,i=1, 2, 3, are the components of the unit outward normal to (0,l)×∂S, (o^{i j,})³_i,j=1was introduced in (2.21), and where the symmetric tensorω(V) has the form

ω(V)=1

θ(V) +κ(V). (4.6)

The individual nonzero components of the symmetric tensorsθandκare defined by θ₁₂(V)=1

2

∂2V,g1,

, θ₂₂(V)=

∂2V,n, θ₃₃(V)=

∂3V,b, (4.7) θ₁₃(V)=1

2

∂3V,g1,

, θ₂₃(V)=1 2

∂2V,b

+∂3V,n

, (4.8)

κ₁₁ (V)=

∂1V,g1,

, κ₁₂(V)=1 2

∂1V,n, κ₁₃(V)=1 2

∂1V,b. (4.9) Assumptions. The following assumptions will be needed throughout the paper:

(1)ρ=²ρ, whereρ∈L^∞(Ω) and

0< C5≤ρ≤C6 a.e. inΩ; (4.10)

(2)F=²F,F∈L²(0,T;L²(Ω)³),G=³G,G∈W^1,1(0,T;L²(0,l;L²(∂S)³));

(3){Q0,}∈(0,1)⊂V(Ω)³,{Q1,}∈(0,1)⊂L²(Ω)³, 1

ωQ0,

2≤C, ∀∈(0, 1), (4.11)

where the constantCis independent of, and

Q0, Q0 inV(Ω)³, Q1, Q1 inL²(Ω)³ (4.12) for→0, whereQ0∈H₀¹(0,l)³andQ1∈L²(0,l)³, that is, the functionsQ0,Q1

are the constant functions in the second and third variable.

The reason for the choice of these scalings can be found in the inequalities (3.7) and (5.1). We are not able to guarantee boundedness of the functionsUin appropriate spaces without the scalings, which means that the curved rod can be broken when the diameter converges to zero.

After substitution of the above assumptions to (4.4)-(4.5), we get _T

0

Ω

−

ρ∂tU(t),∂tV(t)d+A^{i jkl} 1

ω_klU(t)1

ω_{i j}V(t)ddx dt

= _T

0

Ω

F(t),V(t)ddx dt+ _T

0

_l

0

∂S

G(t),V(t)d

νio^{i j,}νjdS dx1dt (4.13)

for allV∈C^∞₀(0,T;V(Ω)³), and

U|t=0=Q0,, ∂tU|t=0=Q1,. (4.14) 5. Basic estimates of a solution to (4.13)-(4.14)

Proposition5.1. Under the assumptions ofSection 4, there exists a unique weak solution Uto the problem (4.13)–(4.14) such thatU∈L^∞(0,T;V(Ω)³),∂tU∈L^∞(0,T;L²(Ω)³), ρ∂ttU∈L²(0,T; [V(Ω)³]), where the initial conditions in (4.14) are fulfilled in the sense of the spaceC([0,T];L²(Ω)³)orC([0,T];L²_weak(Ω)³), respectively. In addition, this solution satisfies for all∈(0, 1)the estimates

∂_tU²

L^∞(0,T;L²(Ω)³)+1

ωU

2

L^∞(0,T;L²(Ω)⁹)

≤C

Q1,²

2+1

ωQ0,

2 2

+F²_L²_(0,T;L²_(Ω)³₎+G²_W^1,1_(0,T;L²_(0,l;L²_(∂S)³₎₎

, (5.1) ρ∂_ttU_L2(0,T;[V(Ω)³])

≤C

FL²(0,T;L²(Ω)³)+GL²(0,T;L²(0,l;L²(∂S)³))+ 1

²ωU_L2(0,T;L²(Ω)⁹)

, (5.2) where the constantCis independent of.