Documentos obtenidos sobre lentes de Fresnel para uso solar

Considering Eq. (4.57) a cross sectional damage index, ˇDA(S), can be constructed re- stricting the integrations to the cross sectional area as

ˇ DA(S) = 1 − R A ¡ P i|P1im| ¢ dA R A ¡ P i|P1i0m| ¢ dA ∀S ∈ [0, L]. (4.58)

In this way, Eq. (4.57) can be rewritten as ˇ D = Z _L 0 ˇ DA(S)dS. (4.59)

The cross sectional damage index has the virtue of being a dimensionally reduced quantity that capture in a scalar the degradation level of the rod at the arch–length coordinate S ∈ [0, L].

Linearization of the virtual work

principle

As stated by Marsden (see [201] Ch. 5), nonlinear problems in continuum mechanics are invariably solved by linearizing an appropriated form of nonlinear equilibrium equa- tions and iteratively solving the resulting linear systems until a solution to the nonlinear problem is found. The Newton-Raphson method is the most popular example of such a technique [29]. Correct linearization of the nonlinear equations is fundamental for the success of such techniques.

As it has been demonstrated in §3.6 the virtual work principle is an equivalent represen- tation of the equilibrium equations. For prescribed material and loading conditions, its solution is given by a deformed configuration fulfilling the equilibrium equations and the boundary conditions. Normally, the development of an iterative step-by-step procedure, such as the Newton-Raphson solution procedure, can be obtained based on the lineariza- tion, using the general directional derivative (see Def. A.22 in §A.5), of the virtual work functional, which is nonlinear with respect to the kinematic and kinetic variables, the loading and the constitutive behavior of the materials (see §4). Two approaches are avail- able: (i) To discretize the equilibrium equations and then linearize with respect to the nodal positions or (ii) To linearize the virtual work statement and then discretize [50]. Here the later approach is adopted in Chapters 6 and 7 due to the fact that it is a more suitable for the solution of problems in solid mechanics.

This chapter is concerned with the linearization of the virtual work principle, in a manner consistent with the geometry of the configurational manifold where the involved kinetic and kinematical quantities belongs. The procedure requires an understanding of the directional derivative. The linearization procedure is carried out using the directional (Gˆateaux) derivative considering it provides the change in an item due to a small change in something upon which item depends. For example, the item could be the determinant of a matrix, in which case the small change would be in the matrix itself.

The fact that the rotational part of the displacement field can be updated using two alter- natively but equivalent rules, the material and the spatial one (see Appendix A), implies that two sets of linearized kinetic and kinematical quantities can be obtained, according to the selected updating rule. It is possible to show that both sets are also equivalent by mean of the replacement of the identities summarized in Eqs. (A.65a) to (A.65c) of §A.4.

5.1. Consistent linearization: admissible variations 100 In any case and by completeness, both set of linearized expressions are obtained in the following sections of this chapter.

5.1

Consistent linearization: admissible variations

At it has been explained in Section 3.1.3 the current configuration manifolds of the rod at time t is specified by the position of its line of centroid and the corresponding field of orientation tensors, Eq. (3.23), explicitly Ct:= {( ˆϕ, Λ) : [0, L] → R3× SO(3)} which is a nonlinear differentiable manifold. Following the procedure presented in [278], where Simo and Vu-Quoc, according to the standard practice, carry out the linearization procedure based on using the Gˆateaux differential (see Appendix A) as a way to approximate to the more rigorous Fr´echet differential1_{, it is possible to construct a perturbed configuration}

onto Ct as follows:

(i) Let β > 0 ∈ R be a scalar and δ ˆϕ(S) = δϕi(S)ˆei be a vector field (see Def. A.26 of Appendix A) considered as a superimposed infinitesimal displacement onto the line of centroid defined by ˆϕ.

(ii) Let δeθθθ = δeθijˆei⊗ ˆej (= δeθijeˆi∧ ˆej (i < j)) be the spatial version of a skew–symmetric tensor field interpreted, for β > 0, as a superimposed infinitesimal rotation onto Λ, Eqs. (3.19) and (3.21), with axial vector δ ˆθ ∈ T_Λspa (see §A.4.4).

(iii) Let δ eΘ = δ eΘijEˆi ⊗ ˆEj (= δ eΘijeˆi ∧ ˆej (i < j)) be the material version of a skew– symmetric tensor field interpreted, for β > 0, as a superimposed infinitesimal rota-

tion onto Λ, Eqs. (3.19) and (3.21), with axial vector δ ˆΘ ∈ Tmat

Λ .

(iv) Then, the perturbed configuration

Ctβ , {( ˆϕβ, Λβ) : [0, L] → R3× SO(3)} is obtained by setting2 ˆ ϕβ(S) = ˆϕ(S) + βδ ˆϕ(S) ∈ R3 (5.1a) Λβ(S) = exp £ βδeθθθ(S)¤Λ(S) ∈ SO(3). (5.1b)

The term Λβ defined in Eq. (5.1b) is also a rotation tensor, due to the fact that it is

obtained by means of the exponential map acting on the skew–symmetric tensor βδeθθθ ∈

so(3) and, therefore, the perturbed configuration Ctβ belongs to R3 × SO(3) as well as

the current configuration Ct does (Ctβ ⊂ Ct). It should be noted that the perturbed

configuration also constitute a possible current configuration of the rod.

Note that in Eq. (5.1b) the spatial updating rule for compound rotations has been 1_{In reference [292] a rigorous foundation for this procedure can be found.}

2_{Note that as it has been explained in §A, finite rotations are defined by orthogonal transformations,}

whereas infinitesimal rotations are obtained through skew–symmetric transformations. The exponentia- tion map (see §A.2.4) allows to obtain the finite rotation for a given skew–symmetric tensor.

chosen for the superimposed infinitesimal rotation, i.e. βδeθθθ ∈ T_ΛspaSO(3). If the material updating rule is preferred, Eq. (5.1b) has to be rewritten as

Λβ(S) = Λ(S)exp

£

βδ eΘΘΘ(S)¤ ∈ SO(3) (5.2)

where βδ eΘΘΘ(S) ∈ Tmat

Λ SO(3).

As it has been explained in Appendix A.4, both skew–symmetric tensors δeθθθ and δ eΘΘΘ have associated the corresponding axial vectors δ ˆθ and δ ˆΘ ∈ R3_{, respectively. Alternatively,}

it is possible to work with the field defined by the pair ˆη(S) , (δ ˆϕ(S), δ ˆθ(S)) ∈ T Ct ≈ R3_{× R}3 _{and in this case the definition for admissible variation given in §A.5.1 and §3.1.5}

is recovered. The meaning for the two component of ˆη(S) is analogous to those given for

(δ ˆϕ, δ eΘ) if the material updating rule of rotations is used3_.

Due to attention is focused on the boundary value problem in which displacements and rotations are the prescribed boundary data and starting from the previous definition for

ˆ

η, it follows that the linear space of kinematically admissible variations is ηs_{= {ˆη}s _{= (δ ˆϕ, δ ˆθ) ∈ R}3_{× R}3 _{| ˆη}s_|

∂Φϕˆ = 0} ⊂ T Ct (5.3)

if the spatial updating rule for rotations is used; if the material rule is preferred one has that

ηm = {ˆηm = (δ ˆϕ, δ ˆΘ) ∈ R3× R3 | ˆηs|∂Φϕˆ = 0} ⊂ T Ct. (5.4)

The above definitions allows to construct the expression given in Eq. (3.27) for the tangent space in the spatial form TxˆBt, which was originally developed following Ref. [192], Eqs. (A.83), (A.84), (A.85) and (A.86). Employing a slight abuse in the notation it is possible to write ˆηs_{(S) ∈ T}

ΦCt i.e. the kinematically admissible variation belong to the tangent

space to the current configuration Ct at the material point Φ = ( ˆϕ, Λ) ∈ Ct.