Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

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Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

Christophe Desceliers, Christian Soize

To cite this version:

Christophe Desceliers, Christian Soize. Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization. International Journal of Non-Linear Mechanics, Elsevier, 2004, 39 (3), pp.343-368. �10.1016/S0020-7462(02)00191-9�. �hal- 00686206�

Nonlinear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization.

C. Desceliers, C. Soize

Laboratory of Engineering Mechanics,University of Marne-La-Vall´ee, France

Full postal address:

Christian Soize

Laboratoire de M´ecanique Universit´e de Marne-la-Vall´ee 5, boulevard Descartes

77454 Marne-La-Vall´ee Cedex 2 France

E-mail address: [email protected]

Abstract

This paper deals with the nonlinear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the nonlinear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to a third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the nonlinear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed.

The model is constructed by the introduction of a reference configuration which is deduced from the nonlinear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.

Keywords :Nonlinear Structural Dynamics, Rotating Structure, Viscoelasticity in Finite Displacement, Sensitivity Analysis.

1. Introduction.

Vibration analysis of rotating structures is an important subject in mechanical engineering. This field has been extensively studied in particular for linear vibration problems (for instance, see the non exhaustive list of references, [ 3, 16, 19, 21, 24, 28, 44, 45, 46, 48, 49]). Sometimes, nonlinear vibration analysis of rotating structures has to be performed because of the presence of finite displacement of the structure.

This can be the case for rotating structures constituted of slender parts, such as for certain helicopter rotor blades. Many publications devoted to nonlinear vibrations of solids are based on the use of nonlinear beam and plate theories (for instance, see [10, 18, 36, 47]). It should be noted that, if the use of such approximated theories are very efficient for isotropic materials having no significant viscoelastic effects, some difficulties can arise for a structure in finite displacement constituted of nonisotropic viscoelastic material (for instance, this is the case of certain advanced helicopter rotor technology made of composite materials). This paper is devoted to nonlinear dynamics of rotating structures constituted of nonisotropic viscoelastic three-dimensional bodies in finite displacement, for which complex loads and complex Dirichlet boundary conditions are considered, and for which a very little exists in the literature. In the other hand, it should be noted that many works have been published concerning viscoelastic constitutive equations (for linearized theory, see for instance [6, 20, 31, 38, 43] and for nonlinear theory with finite displacement, see [13]) and concerning the use of viscoelastic models in vibration problems (see for instance [4, 33, 38, 40, 42, 47]). Concerning the constitutive model, physically nonlinear theory should be mentioned (see for instance [8]). In this theory, a nonlinear relation between the Cauchy stress tensor and the linearized deformation tensor is constructed, preserving invariance under superposed rigid body motions by writing the constitutive equations in the local reference frame of a material point of the structure rather than in a global reference frame. Other invariant theories use this kind of strategy to study invariant infinitesimal motions of elastic body (see for instance [9], [39]). Such theories could be used to study small or moderate vibrations around a given finite deformation (see also [22]). Nevertheless, it should be noted that this paper is not concerned with infinitesimal motion superposed to a given deformation.

This paper presents the nonlinear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement without introducing any approximation in order to allow sensitivity analysis to be numerically performed. On a first part of the boundary of the structure considered, a rigid body displacement field operates to move the structure in a rotation motion. A time dependent Dirichlet

condition concerning the structural displacement field is applied to a second part of the boundary of the structure (for instance, in the context of helicopter rotor blades, this time dependent Dirichlet condition corresponds to the cyclic step which allows the aerodynamic loads to be adapted). A surface force field depending on the time history of the structural displacement field is applied to a third part of the boundary of the structure (for instance, such a surface force field corresponds to general unsteady aerodynamics forces). We present the constitutive equations of the viscoelasticity in finite displacement (deduced from the Coleman and Noll theory [13]) in the rotating frame and with respect to the steady configuration. The important objective of this paper is to deduce the numerical formulation based on the use of the finite element method [2, 12, 14, 23, 50] to analyze nonlinear responses of the dynamical system for small data perturbations to the geometric and the mechanical parameters. In this context, weak formulations [7, 11, 14, 17, 37, 38, 41] of the boundary value problems are systematically introduced in order to deduce the matrix equations resulting from a finite element discretization. Taking into account that we are concerned with three-dimensional structures, several formulations could be constructed (each corresponding to several possible choices of different reference configurations). In this paper, we present a strategy to analyze such a mechanical system. This consists in constructing a steady problem corresponding to the structure in equilibrium under prestresses and steady loads in the rotating frame (for instance, the time average part of unsteady body forces). It should be noted that this configuration does not correspond to a given motion, but has to be constructed. We choose the deformed structure associated with the steady equilibrium as the reference configuration. The nonlinear viscoelastodynamic equations are transported on the reference configuration, because the nonlinear dynamical responses about the steady configuration in the rotating frame are of prime importance. For instance, such a formu- lation allows linearization of the dynamics to be performed in order to study linear instabilities of the structure coupled with unsteady aerodynamic flow but, in the present paper, we are not interested in such a linearized theory. A treatment by translation of the time dependent Dirichlet conditions is presented for this kind of nonlinear viscoelastodynamic system. Finally, a complete algebraic developement is presented which allow sensitivity analysis and random analysis to be numerically performed without any kinematical approximations. As a result of such a strategy, among others and due to the specific treatment used for the time dependent Dirichlet condition, nonlinear integrodifferential matrix equations with periodic coefficients are deduced which can be used for a parametric analysis.

2. Nonlinear boundary value problem of a rotating structure in a fixed reference frame

LetR0be the fixed reference frame whose origin is the pointOwhich is assumed to be located on the axis of rotation of the structure. InR0 and at timet, the deformed configuration occupies a bounded domain which is denoted byΩ(t)e ⊂ ³ with boundary∂Ω(t). It is assumed that boundarye ∂Ω(t)e is sufficiently smooth. In R₀ and at time s, let X(t, s,e ext) ∈ Ω(s)e be the position of a particle which is located inR0and at timet > sat positionex_t∈Ω(t). Lete s7→w(t, s,e ex_t)be the ³-valued displacement field time history of domainΩ(t)e defined on]− ∞, t], such that

e

w(t, s,ext) =X(t, s,e ext)−ext. (1) InR0 and at timet, the rotating structure is subject to an external surface force field eF(t,ext) applied to the partΣ(t)e of boundary∂Ω(t)e and to an external body force fieldρ(t,e ext)ef(t,ext)applied inΩ(t)e in which ρ(t,e ext) is the mass density ofΩ(t). In order to model general loads (for instance unsteadye aerodynamic loads which depend on the time history of the structural displacement field), we write F(t,e ex_t) =R+∞

0 G(τ, t,e ex_t,{w(t, te −τ,ey)}_˜y∈Σ(t)˜ )dτ, in which for allex_t ∈Σ(t)e and for any functions δwe fromΣ(t)e into ³, functionτ 7→ G(τ, t,e ex_t, δw)e with values in ³ is integrable on]0, +∞[. The notation{w(t, te −τ,ey)}_˜y∈Σ(t)˜ denotes the mappingey7→w(t, te −τ,ey)fromΣ(t)e into ³. On a second partΥ(t)e of the boundary∂Ω(t), we impose a rigid-body displacement field defined by the rotatione associated with a direct orthogonal(3×3)real matrix[Q(t)]. On a third partΓ(t)e of boundary∂Ω(t),e a time dependent Dirichlet condition related to the displacement fieldw(t, s,e ext)is applied and defined byw(t, s,e ext) =we^dir(t, s,ext). Lete^!(t,ext)be the Cauchy stress tensor related toΩ(t). Therefore, ine R0

and at timet, the equations of the three-dimensional rotating structure are

∀ext ∈Ω(t),e div˜xte^!(t,ext) +ρ(t,ee xt)ef(t,ext) =ρ(t,e ext) ∂²we

∂s²(t, s,ext)

s=t

, (2)

∀ext ∈Σ(t),e e^!(t,ext)en(t,ext) =eF(t,ext), (3)

∀ex_t ∈Υ(t),e w(t, s,e ex_t) =n

[Q(s)] [Q(t)]^T −[I]o

ex_t, (4)

∀ex_t ∈Γ(t),e w(t, s,e ex_t) =we^dir(t, s,ex_t), (5) with eF(t,ext) =

Z +∞

0

G(τ, t,e ext,{w(t, te −τ,ey)}_y^˜∈˜Σ(t))dτ , (6) and whereen(t,ex_t)∈ ³is the unit outward normal vector to the boundary∂Ω(t), the exponente Tmeans the transpose of a matrix and[I]is the(3×3)identity matrix.

3. Nonlinear dynamical boundary value problem of a rotating structure in a rotating frame.

LetR1be the rotating reference frame attached to the rotating structure and let(e0,1,e0,2,e0,3)denote the canonical basis of ³. Rotating frameR1is defined by the originOand the direct orthonormal basis (e_1,1(t),e1,2(t),e1,3(t)) which is derived from (e_0,1,e0,2,e0,3) by the rotation represented by matrix [Q(t)]and such thate1,p(t) = [Q(t)]e0,pfor anypfixed in{1,2,3}. Any particle located at positioney inR0at timetis located at positiony =X_t(ey)inR1whereX_t(ey) = [Q(t)]^T ey. In R1and at timet, the deformed configuration occupies the bounded domain denoted byΩ(t)⊂ ³with boundary∂Ω(t).

It is straightfoward to show thatΩ(t) =X_t(Ω(t))e and∂Ω(t) =X_t(∂Ω(t)). Lete Σ(t),Υ(t)andΓ(t)be the parts of boundary∂Ω(t)such thatΣ(t) =Xt(Σ(t)),e Υ(t) =Xt(Υ(t))e andΓ(t) =Xt(eΓ(t)).

InR1and at times, letX(t, s,x_t)∈Ω(s)be the position of a particle which is located inR1and at time t > s, at positionx_t∈Ω(t). Lets7→w(t, s,x_t)be the time history of the ³-valued displacement field of domainΩ(t), defined on]− ∞, t], such that

w(t, s,x_t) =X(t, s,x_t)−x_t. (7) It could be shown that [15], for allxtinΥ(t),

w(t, s,xt) = 0, (8)

which means that reference frameR1and all the material points belonging onΥ(t)e move with the same rigid body motion described by the rotation matrix[Q(t)].

The natural configuration is defined as the structure at rest without any pre-stresses. InR₀, the natural configuration occupies a bounded domain denoted asΩ₀⊂ ³with boundary∂Ω₀. InR₀at timet, a particule located atx0∈Ω0in the natural configuration will occupy the positionXe0(t,x0)∈Ω(t)e in the deformed configuration. It should be noted that the natural configuration could be defined as the image of domainΩ0 by the mapping x0 7→ R⁻¹₀ (x0). We then introduce the rotating natural configuration as the image of domain Ω0 by the mapping x0 7→ R⁻¹₁ (x0). Note that, in R0, the rotating natural configuration occupies a domainΩe0(t) which is the image ofΩ0 by the mapping X⁻¹_t justifying the rotating natural configurationterminology introduced. In R1 at time t, a particle located at position x0 in the rotating natural configuration will occupy the position denoted by X0(t,x0) ∈ Ω(t) in the deformed configuration. LetΣ₀,Υ₀andΓ₀be the parts of boundary∂Ω₀such thatΣ(t) = X₀(t,Σ₀), Υ(t) =X0(t,Υ0)andΓ(t) =X0(t,Γ0). Considering Eq. (8), it could be noted thatΥ(t)is a fixed part of boundary∂Ω(t)denoted asΥ. Consequently, the domainΩ0should be chosen such thatΥ0= Υ. Let

u0(t,x0)be the ³-valued displacement field of the deformed configuration inR1and at timet, defined on domainΩ0, such that

u0(t,x0) =X0(t,x0)−x0, ∀x₀∈Ω₀. (9) In general,u0(0,x0) is not zero. Letx0 7→ G0(τ, t,x0, δu0)be the ³-valued function defined onΣ0

and such that for any functiony07→δu0(y0)fromΣ0into ³, we have

G0(τ, t,x0, δu0)dS0(x0) = [Q(t)]G(τ, t,e ex_t, δw)e dSe_t(ex_t), (10) wheredSe_t(ext)is the surface element of the boundary∂Ω(t)e anddS₀(x₀) is the surface element of the boundary∂Ω0withex_t=X0(t,x0)for allx0inΩ0and where

δw(ey) = [Q(t)]n

δu₀(y₀)−u0(t,y0)o +n

[Q(t−τ)]−[Q(t)]o

X0(t,y0), (11) witheyt=Xe0(t,y0)for ally0inΥ0. With respect to the rotating natural configuration inR1, Eqs (2) to (6) of the three-dimensional rotating structure could be rewritten as (see for instance [15])

∀x0∈Ω0, div_x0

!0(t,x0) 0(t,x0)

+ρ0(x0)f0(t,x0) =ρ0(x0) ¨u0(t,x0)+2ρ0(x0) [R] ˙u0(t,x0) +ρ0(x0)n

[ ˙R(t)] + [R(t)]²o n

x0+u0(t,x0)o

, (12)

∀x0∈Σ0, ^!0(t,x0) 0(t,x0)n0(x0) =F0(t,x0), (13)

∀x₀∈Υ₀, u0(t,x0) = 0, (14)

∀x0∈Γ0, u0(t,x0) =u^dir₀ (t,x0), (15) with F0(t,x0) =

Z +∞

0

G0(τ, t,x0,{u0(t−τ,y0)}_y0∈Σ0)dτ . (16) where, in R₁ and at time t, the function u^dir₀ (t,x0) defined on Γ₀ with values in ³, is such that u^dir₀ (t,y₀) =u₀(0,y₀)−[Q(s)]^Twe^dir(0, t,ey_t) +{[Q(t)^T−[Q(s)]^T}ey_twithey_t=Xe₀(t,y₀)for ally₀in Γ0, where the unit outward normal vector to boundary∂Ω0is denoted byn0(x0)∈ ³, where the tensor- valued function ^!0(t,x0) = ∂_x0X0(t,x0), defined on Ω0, is the deformation gradient of the rotating natural configuration, where the mass densityρ0(x0)defined on domainΩ0corresponds to the Lagrangian transport byXe0ofρ(t,ee xt), where the ³-valued functionf0= (det^!0)⁻¹[Q(t)]^Tef◦Xe⁻¹₀ is the density of the applied body forces per unit volume in the rotating natural configuration, where the ³-valued function F0(t,x0), defined onΣ₀, is the density of the applied surface forces per unit area in the rotating natural configuration such thatF0dS₀ = [Q(t)]^T(eFdSe_t)◦Xe0 and where ₀ = (det^!₀)^!−1₀ {^"◦X0} ^!-T 0

denotes the Piola-Kirchhoff stress tensor such that^"(t,x_t) = [Q(t)]^Te^"(t,[Q(t)]x_t)[Q(t)]is the Cauchy stress tensor of the deformed configuration inR1. From Eq. (9), we deduce that^!0= [I] +∂_x0u0. It

should be noted that if the rotational velocity and the axis of rotation are independant oft, then[ ˙R(t)] = 0 and[R(t)]is a constant matrix denoted as[R]. With this assumption, Eq. (12) is rewritten as

divx0

0(t,x0) ₀(t,x0)

+ρ₀(x₀)f0(t,x0) =ρ₀(x₀)[R]²n

x0+u0(t,x0)o

+ 2ρ₀(x₀) [R] ˙u0(t,x0) +ρ₀(x₀) ¨u₀(t,x₀).(17)

4. Nonlinear steady boundary value problem in the rotating frame

The external loads applied to the structure and the displacement fieldu^dir₀ can be decomposed:

f0(t,x0) =f^stat₀ (x0) +f₀^dyn(t,x0), ∀x0∈Ω0, (18) F0(t,x0) =F^stat₀ (x0) +F^dyn₀ (t,x0), ∀x0∈Σ0, (19) u^dir₀ (t,x0) =u^stat₀ (x0) +u^dyn₀ (t,x0), ∀x0∈Γ0, (20) wheret7→ρ0(x0)f₀^dyn(t,x0)denotes the time-fluctuation part of body forcet7→ρ0(x0)f0(t,x0)around its steady partρ0(x0)f₀^stat(x0), wheret7→F^dyn₀ (t,x0)denotes the time-fluctuation part of surface force t7→F0(t,x0)around its steady partF^stat₀ (x0)and whereu^dyn₀ (t,x0)denotes the time-fluctuation part of applied displacementu^dir₀ (t,x₀)around its steady partu^stat₀ (x₀), these steady parts being defined below.

If the structure is only subjected to steady partsρ0(x0)f₀^stat,F^stat₀ andu^stat₀ of external loads, then the structure is in equilibrium in a steady configuration which occupies inR1a bounded domain denoted as Ω_r with boundary∂Ω_r. In R1, a particule which is located at positionx0 ∈ Ω0 in the rotating natural configuration will occupy the positionX^ref₀ (x0) ∈Ω_r in the steady configuration. LetΣ_r,Υ_r andΓ_r be the parts of boundary∂Ωr such that Σr = X^ref₀ (Σ0), Υr = X^ref₀ (Υ0) andΓr = X^ref₀ (Γ0). Let u^ref₀ (x0)be the^!3-valued displacement field of the steady configuration inR1, defined fromΩ0, such that

u^ref₀ (x0) =X^ref₀ (x0)−x0, ∀x0∈Ω0. (21) Therefore, ^ref₀ =∂_x0X^ref₀ is the deformation gradient of the steady configuration defined on domain Ω0. Using Eq. (21) we deduce that ^ref₀ = [I] +∂_x0u^ref₀ . The Cauchy stress tensor of the steady configuration is denoted as^"r(xr)and its Piola-Kirchhoff transport [11] by X^ref₀ is the tensor-valued function ^ref₀ = (det ^ref₀ ){ ^ref₀ }⁻¹{^"r◦X^ref₀ }{ ^ref₀ }-^T defined onΩ0. If the functions f0(t,x0), u^dir₀ (t,x0) andG0(τ, t,x0, δu0) are T-periodic in t (which is consistent with the constant rotational

velocity assumption), thenf₀^stat(x0),F^stat₀ (x0)andu^stat₀ (x0)are defined by f₀^stat(x0) = 1

T Z T

0

f0(t,x0)dt , ∀x0∈Ω0, (22) F^stat₀ (x0) = 1

T Z +∞

0

dτ Z T

0

G0(τ, t,x0,{u^ref₀ (y0)}_y0∈Σ0)dt , ∀x0∈Υ0, (23) u^stat₀ (x0) = 1

T Z T

0

u^dir₀ (t,x0)dt , ∀x0∈Σ0. (24) InR₁, the nonlinear steady boundary value problem is deduced from the nonlinear dynamical boundary value problem by replacing{f0,F0,u^dir₀ , u0, 0, 0}with{f₀^stat,F^stat₀ ,u^stat₀ , u^ref₀ , ^ref₀ , ^ref₀ }in Eqs. (12) to (15) and (17), and is written as (see for instance [15])

∀x₀∈Ω₀, divx0

ref

0 (x₀) ^ref₀ (x₀)

+ρ₀(x₀)f₀^stat(x₀) =ρ₀(x₀)[R]²

x0+u^ref₀ (x₀)

(25)

∀x0∈Σ0, ^ref₀ (x0) ^ref₀ (x0)n0(x0) =F^stat₀ (x0), (26)

∀x0∈Υ0, u^ref₀ (x0) = 0, (27)

∀x0∈Γ0, u^ref₀ (x0) =u^stat₀ (x0). (28)

5. Nonlinear dynamic boundary value problem in the rotating frame with respect to the reference configuration

Below, we refer the steady configuration as the reference configuration of the nonlinear dynamical boundary value problem. Consequently, Eqs. (12) to (17), (25) and (26) have to be transported onto the domain Ωr. InR1and at time t, a particule located at positionxr in the reference configuration will occupy the position denoted asXr(t,xr) ∈ Ω(t) in the deformed configuration. Let ur(t,xr) be the

!

3-valued displacement field of the deformed configuration, defined onΩr, such that

ur(t,xr) =Xr(t,xr)−xr. (29) Letx_r 7→ G_r(τ, t,x_r, δu_r)be the mapping fromΣ_r into^!3such that, with x0= {X^ref₀ }⁻¹(x_r)inΣ0

and for any functionδu0fromΣ0into^!3, we have

G_r(τ, t,x_r, δu_r)dS_r(x_r) =G0(τ, t,x0, δu_r◦X^ref₀ +u^ref₀ )dS0(x0), (30) where(δur◦X^ref₀ )(x0)meansδur(X^ref₀ )(x0))and wheredSr(xr)is the surface element of boundary

∂Ω_r. Equations (13) to (17) can be rewritten as (for instance, see [15])

∀xr ∈Ωr, div_xr

r(t,x0) r(t,xr)

+ρr(xr)fr(t,xr) =ρr(xr) ¨ur(t,xr)+2ρr(x0) [R] ˙ur(t,xr) +ρr(xr)[R(t)]²n

x0+u0(t,x0)o

, (31)

∀x_r ∈Σ_r, _r(t,x_r) _r(t,x_r)n_r(x_r) =F_r(t,x_r), (32)

∀x_r ∈Υ_r, u_r(t,x_r) = 0, (33)

∀xr ∈Γr, ur(t,xr) =u^dir_r (t,xr), (34) with Fr(t,xr) =

Z +∞

0

Gr(τ, t,xr,{ur(t−τ,yr)}_yr∈Σr)dτ . (35) where, in R1 and at time t, the function u^dir_r (t,x_r) defined on Γ_r with values in ^!3, is such that u^dir_r (t,x_r) = u^dir₀ (t,{X^ref₀ }⁻¹(x_r))−u^stat₀ ({X^ref₀ }⁻¹(x_r)), the unit outward normal vector to the boundary∂Ωr is denoted asnr(xr)∈^!3, the tensor-valued function r(t,xr) =∂_xrXr(t,xr), defined onΩr, is the deformation gradient of the rotating natural configuration, the mass densityρr(xr)defined on domain Ωr corresponds to the Lagrangian transport by X^ref₀ of ρ0(x0), the ^!3-valued function fr(t,xr)is the density of the applied body forces per unit volume in the rotating natural configuration such thatρ_r(x_r)f_r(t,x_r)dx_r = ρ0(x0)f0(t,x0)dx0, the^!3-valued functionF0(t,x0), defined onΣ0, is the density of the applied surface forces per unit area in the rotating natural configuration such that F0dS0= (F_rdS_r)◦X^ref₀ and _r = (det _r) ⁻¹_r n

"◦X^ref₀ o -^T

r denotes the Piola-Kirchhoff stress tensor. From Eq. (29), we deduce that _r = [I] +∂_x0u_r.

InR₁and at timet, the nonlinear dynamical boundary value problem of the rotating structures with respect to the reference configuration (the steady configuration) is given by Eqs. (31) to (35). Consequently, the transport of Eq. (25) on domainΩ_rand the transport of Eq. (26) onΣ_rgive

divxr^"r(x_r) +ρ_r(x_r)f_r^stat(x_r) =ρ_r(x_r) [R]²xr, ∀x_r ∈Ω_r , (36)

"r(x_r)n_r =F^stat_r (x_r), ∀x_r ∈Σ_r, (37) whereρ_r(x_r)f_r^stat(x_r)defined onΩ_r, is the external body forces applied to the reference configuration (the steady configuration) and corresponds to the Lagrangian transport byX^ref₀ of external body forces ρ0(x0) f₀^stat(x0) defined on domain Ω0 and whereF^stat_r (x_r) is the external surface forces applied to boundaryΣ_rcorresponding to the Lagrangian transport byX^ref₀ of surface forcesF^stat₀ (x0)definedΣ0. It should be noted that

F^stat_r (x_r) = 1 T

Z +∞

0

dτ Z T

0

Gr(τ, t,xr,0)dt , ∀x_r ∈Σ_r . (38)

6. Viscoelasticity in finite displacement.

In this paper, we consider viscoelastic materials and we refer the reader to the theory of linear viscoelasticity in finite displacement developed by B.D. Coleman and W. Noll [13]. In order to deduce a total Lagrangian formulation of the constitutive equation, it is necessary to present a short review of this theory. It is assumed thatdete₀(t,x0) > 0, where e₀(t,x0) = ∂_x0Xe0(t,x0) is the deformation gradient of the natural configuration inR0, at timet. Fors≤t, let^!e(t, s,ex_t) = ¹₂(e^T e−[I])be the Green-Lagrange strain tensor defined on domainΩ(t)e in whiche(t, s,ex_t) = ∂^˜_xtX(t, s,e ex_t). From Eq.

(2), we deduce thate = [I] +∂_x^˜_tw. The theory of linear viscoelaticity in finite displacement states thate Cauchy stress tensore^"(t,ext)is such that

e

#0(t,x0)⁻¹e^"(t,Xe0(t,x0))^#e₀(t,x0) =^$0(x0,e^!₀(t,x0)) +

Z +∞

0

Γ0(τ,x0,e^!₀(t,x0)) :{^#e₀(t,x0)^Te^!(t, t−τ,Xe0(t,x0))^#e₀(t,x0)}dτ , (39) where^!e₀ = ¹₂{e^T₀e₀−[I]} is the Green-Lagrange strain tensor of the deformed configuration inR0

with respect to domain Ω0 and where the orthogonal tensor ^#e₀(t,x0) is constructed using the polar decomposition of positive-definite tensore₀(t,x0)and is written as^#e₀=e₀{e^T₀e₀}⁻¹².

In Eq. (39), x0 7→ ^$₀(x₀,^!e₀(t,x0))is a function fromΩ₀into the set of all symmetric second-order tensors andt 7→ Γ0(t,x0,^%) is a function from ^&+ into the set of all symmetric fourth-order tensors.

Futhermore, for all symmetric second-order tensors ^%, for all x0 in Ω0 and for all i, j, k, h fixed in {1,2,3}, functionst7→ {Γ0(t,x0,^%)}_ijkh belongs toL²(^&+, γ⁻²dτ)whereγis a positive real-value function defined on^&+such that, for a given positive integerr,lim_τ7→+∞τ^r γ(τ) = 0.

In a first step, the constitutive equation (39) could be rewritten in rotating reference frameR1in which a Lagrangian formulation on domainΩ0for the linear viscoelasticity in finite displacement theory could be deduced. In a last step, this Lagrangian formulation is transported onto domainΩ_r.

Thus, Eq. (39) could be rewritten as (see for instance [15])

0(t,x0) =^'0(x0,^!0(t,x0)) + Φ0(0,x0,^!0(t,x0)) :^!0(t,x0) +

Z +∞

0

˙Φ0(τ,x0,^!₀(t,x0)) :^!₀(t−τ,x0)dτ , (40) in which^!0= ¹₂{ ^T₀ 0−[I]}is the Green-Lagrange strain tensor of the deformed configuration inR0

with respect to domainΩ0, and where the derivative of functionΦ0(τ,x0,^%)with respect toτ is written

as n

Φ0(s,x0, )o

ijkh= det

2 + [I]¹₂ X³

a,b,p,q=1

2 + [I]⁻¹₂

ai

2 + [I]⁻¹₂

bj

×

2 + [I]⁻¹₂

pk

2 + [I]⁻¹₂

qh

nΓ₀(s,x0, )o

abpq , (41)

where^!0(x0, )is such that, for all symmetric second-order tensors ,

!0(x0, ) = det

2 + [I]¹₂

2 + [I]−¹₂

"0(x0, )

2 + [I]−¹₂

. (42)

In addition, we have lim_τ→+∞Φ₀(τ,x0, ) = 0. If ^#₀(τ,x0) is a bounded function in τ and if τ 7→ Φ0(τ,x0, ) belongs to L¹(^$+, dτ) then 0(t,x0) exists. With these assumptions and because τ 7→ Γ0(τ,x0, ))belongs to L²(^$+, γ⁻²dτ), we deduce that functionτ 7→ Φ0(τ,x0, ) belongs to L²(^$+, γ⁻²dτ)∩L¹(^$+, dτ).

In this paper, only finite displacements are taken into account and material nonlinearities are not considered. Consequently, substituting ^!0(x0,^#0(t,x0)) = ^%0(x0) : ^#0(t,x0) into Eq. (40) and introducing the^&0(τ,x0) = Φ0(τ,x0,0) +^%0(x0)yield

0(t,x0) =^&₀(0,x0) :^#₀(t,x0) + Z +∞

0

˙

&0(τ,x0) :^#₀(t−τ,x0)dτ . (43) It can be proven that

{^&0(t,x0)}_ijkh={^&0(t,x0)}_khij ={^&0(t,x0)}_ijhk={^&0(t,x0)}_jikh, (44) and that there existsc >0such that, for all symmetric second-order tensors^',

n

&0(0,x0)o

ijkh {^'}kh{^'}ij ≥c{^'}ij{^'}ij . (45) It should be noted that the linearization of Eq. (43) with respect tou0yields the usual linear viscoelasticity theory [20, 31, 43]

((t,X0(t,x0)) =^&0(0,x0) : ⁾ 0(t,x0) + Z +∞

0

˙

&0(τ,x0) : ⁾ 0(t−τ,x0)dτ , (46) where ⁾ ₀ = ¹₂{∂x0u0+∂x0u^T₀} denotes the linearized strain tensor. Substituting ^&₀(τ,x0) dτ =

%0(x₀)1 ⁺(τ)dτ+^*₀(x₀)δ₀(τ)into the right-hand side of Eq. (43) yields the constitutive equation of instantaneous viscoelasticity in finite displacement which is written as

0(t,x0) =^%0(x0) :^#0(t,x0) +^*0(0,x0) : ˙^#0(t,x0). (47)

Let ^ref₀ = ¹₂(^!ref₀ ^{T !ref}₀ −[I]) be the Green-Lagrange strain tensor of the reference configuration defined onΩ0. Substituting 0(t,x0) = ^ref₀ (x0)into Eq. (43) yields

ref

0 (x0) =^"0(x0) : ^ref₀ (x0), ∀x0∈Ω0. (48) In section 4, we introduced ^ref₀ as the Piola-Kirchhoff transport byX^ref₀ of tensor-valued function^#_r. Consequently, we deduce from Eq. (48)

#r(x_r) = 1 det^!ref₀ (x0)

!

ref 0 (x0)n

"0(x0) : ^ref₀ (x0)o

!

ref

0 (x0)^T, (49)

with x0 = {X^ref₀ }⁻¹(xr). Let r(t,xr) = ¹₂(^!T_r ^!r −[I]) be the Green-Lagrange strain tensor of the deformed configuration defined onΩ_r. In section 5, we have introduced _r as the Piola-Kirchhoff transport byXrof Cauchy stress tensor^#. In section 3, ₀is also defined as the Piola-Kirchhoff transport byX0of Cauchy stress tensor^#. Therefore, it could be proved [15] that _r is also the Piola-Kirchhoff transport byX^ref_o of tensor-valued function 0. Consequently, Eq. (47) could be rewritten as

r(t,xr) =^#_r(x_r) +^$_r(0,xr) : _r(t,xr) + Z +∞

0

˙

$r(τ,xr) : _r(t−τ,xr)dτ , (50) where^$r(τ,xr)is such that

{^$_r(τ,x_r)}_abpq= {^$0(τ,x0)}_klmn

det^!ref₀ (x0) {^!ref₀ (x0)}_ak{^!ref₀ (x0)}_bl{^!ref₀ (x0)}_pm{^!ref₀ (x0)}_qn , (51) in whichx0 = {X^ref₀ }⁻¹(x_r). It is easy to prove that, for any{a, b, p, q} fixed in{1,2,3}, functions τ 7→ {^$_r(τ,xr)}_abpqbelong toL²(^%+, γ⁻²dτ)∩L¹(^%+, dτ)with

{^$r(t,xr)}abpq ={^$r(t,xr)}pqab ={^$r(t,xr)}abqp ={^$r(t,xr)}bapq. (52) Tensor^$0(0,x0)is positive-definite but concerning tensor^$_r(0,x_r), we have only proved that there is a positive constantcsuch that

{^$_r(0,x_r)}_abpq{^&}_ab{^&}_pq≥c{^&ref(x0)}_ab{^&ref(x0)}_ab, (53) in which x0 = {X^ref₀ }⁻¹(x_r), ^&ref = p ¹

det ^ref₀

!

ref 0

T

& !

ref

0 and where ^& is any real symmetric second-order tensor. Equation (53) does not mean that tensor^$_r(0,xr)is positive-definite.

7. Weak formulation of the nonlinear steady boundary value problem.

Let C(Ω0) be the set of sufficiently differentiable functions v fromΩ0 into ^%3. LetC_ad(Ω0) be the admissible function space constituted of all functionsvinC(Ω0)such thatv= 0onΥ0and letCad,0(Ω0)

be the subset of all the functions inC(Ω0)such thatv = 0onΥ0∪Γ0. The weak formulation of the nonlinear steady boundary value problem defined in Section 4 is written as follows : find the displacement fieldu^ref₀ ∈ C_ad(Ω0)withu^ref₀ =u^stat₀ onΓ0such that, for allvinCad,0(Ω0),

k_e,0(u^ref₀ , v) +k_c,0(u^ref₀ , v) +k_2,0(u^ref₀ , v) +k_3,0(u^ref₀ , v)−l_a,0(u^ref₀ ,v) =l₀(v), (54) where the symmetric positive-definite bilinear formke,0, skew symmetric negative bilinear formkc,0, nonlinear formsk2,0,k3,0,la,0, defined onCad(Ω0)× Cad(Ω0), and linear forml0defined onCad(Ω0), are such that

ke,0(u, v) = Z

Ω0

0: ∂u

∂x0

: ∂v

∂x0

dx0, (55)

k_c,0(u, v) = Z

Ω0

ρ₀n

[R]²uo

·vdx₀ , (56)

k_2,0(u,v) = 1 2

Z

Ω0

(

0: ∂u

∂x₀

T ∂u

∂x₀

!) : ∂v

∂x₀ dx₀+ Z

Ω0

0: ∂u

∂x₀

: (∂u

∂x₀

T ∂v

∂x₀ )

dx₀, (57)

k3,0(u,v) = 1 2

Z

Ω0

(

0: ∂u

∂x₀

T ∂u

∂x₀

!) :

( ∂u

∂x₀

T ∂v

∂x₀ )

dx0, (58)

l_a,0(u,v) = 1 T

Z +∞

0

dτ Z T

0

ds Z

Σ0

G0(τ, s,x0,u)·vdS₀(x₀), (59) l0(v) =

Z

Ω0

ρ0f₀^stat·vdx0− Z

Ω0

ρ0

n [R]²x0

o

·vdx0. (60)

8. Finite element discretization of the nonlinear steady boundary value problem.

LetC_ad,h(Ω0)be the subset of C_ad(Ω0)corresponding to a finite element discretization of domainΩ0. Let[N0(x0)]be the3×N_hreal interpolation matrix such that the corresponding approximation ofu^ref₀ andv(always denoted asu^ref₀ andv) are written as[N0(x0)]Uand[N0(x0)]V, in whichU=

"

U^d_stat U^f

#

andV= 0

V^f

belong to^!Nh, whereU^d_stat is the vector of the degrees-of-freedom related to the mesh ofΓ0. The finite element approximation of Eq. (54) is written as

V^T h

[Ke,0] + [Kc,0] + [K2,0(U)] + [K3,0(U)]i

U=V^TL0+V^TL_a(U), (61) where the symmetric positive-definite matrices[Ke,0],[K3,0(U)], the symmetric negative-definite matrix [Kc,0], the nonsymmetric matrix [K2,0(U)] and the vectors La,0(U), L0 are defined in Appendix A.

Introducing the matrices



[A^dd₀ (U^d_stat,U^f)] [A^df₀ (U^d_stat,U^f)]

[A^{f d}₀ (U^d_stat,U^f)] [A^{f f}₀ (U^d_stat,U^f)]



= [Ke,0] + [Kc,0] + [K2,0(U)] + [K3,0(U)], (62)





[F₀^d(U^d_stat,U^f)]

[F₀^f(U^d_stat,U^f)]



=La,0(U) +L0, (63)

equation (61) yields

[A^{f d}₀ (U^d_stat,U^f)]U^d_stat+ [A^{f f}₀ (U^d_stat,U^f)]U^f =F^f₀(U^d_stat,U^f). (64) Moreover, a method based on Lagrange multipliers could be used in order to take into account the Dirichlet conditionu^ref₀ =u^stat₀ onΓ0[15].

9. Weak formulation of the nonlinear dynamic boundary value problem in the rotating frame with respect to the reference configuration.

Let C(Ω_r) be the set of sufficiently differentiable functions v from Ω_r into ³. LetC_ad(Ω_r) be the admissible function space constituted of all functionsvinC(Ω_r)such thatv= 0onΥ_rand letCad,0(Ω_r) be the subset of all the functions in C(Ω_r) such that v = 0 on Υ_r ∪Γ_r. The weak formulation of the nonlinear dynamic boundary value problem defined in Section 5 is written as follows : find the displacement fieldu_r(t,x_r) ∈ C_ad(Ω_r) with u_r(t,x_r) = u^dir_r (t,x_r) on Γ_r such that, for all vin Cad,0(Ωr),

m_r(¨u_r, v) +c_r( ˙u_r, v) +k_g,r(u_r, v) +k_c,r(u_r, v) +k_e,r(u_r, v) +1

2k2,r(u_r, u_r, v) +k2,r(v, u_r, u_r) +k_3,r(u_r, ur, ur, v) +

Z +∞

0

g_r(τ, ur(t−τ),v)dτ+ Z +∞

0

g_2,r(τ, v,ur(t),ur(t−τ))dτ +1

2 Z +∞

0

g_2,r(τ,ur(t−τ),ur(t−τ),v)dτ+ Z +∞

0

g_3,r(τ, ur(t−τ),ur(t−τ),ur(t),v)dτ

− Z +∞

0

l_a,r(τ, t,u_r(t−τ),v)dτ =l_r(t, v), (65) where, for allu,v,wandrinCad(Ωr), we have

k_g,r(u, v) = Z

Ωr

!r: ∂u^T

∂x_r

∂v

∂x_r

dx_r, (66)

k_e,r(u, v) = Z

Ωr

"r(0,x_r) : ∂u

∂x_r

: ∂v

∂x_r dx_r, (67)

k_c,r(u, v) = Z

Ωr

ρ_rn [R]²u

o·vdx_r, (68)

gr(t, u, v) = Z

Ωr

˙

"r(t,xr) : ∂u

∂xr

: ∂v

∂xr

dxr, (69)

c_r(u, v) = 2 Z

Ωr

ρ_r n [R]u

o

·vdx_r, (70)

mr(u, v) = Z

Ωr

ρru·vdxr, (71)