Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

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778 FERDINANDO AURICCHIO, GIUSEPPE BALDUZZI AND CARLO LOVADINA 4. Model derivations In this section we develop some beam models. We will start from the HR variational formulations of Section 3.2 (this is Step 1 of the procedure described at the beginning of Section 3), then define the approximated fields (Step 2), and perform the integration along the thickness (Step 3). To simplify further discussion, we will switch to engineering notation, the equivalence between tensor and engineering notation being summarized as follows, where E1 = � 1 0 0� and E2 = � 0 0 0� : 0 0 1 0 1 1 Tensor notation Engineering notation ∇ · σ � d dx E1 + d dy E2 ∇ � Pσ ˆσ Ss � � d d + Ps ˆs dx ET 1 dy ET 2 σ · n (nx E1 + n y E2)Pσ ˆσ 4.1. Profile approximation and notations. We now introduce the notation we use for a generic approximated field γ (x, y) involved in the beam models. We first define a profile vector function pγ (y): pγ : A → � m . We insist that the m components of pγ (y) are a set of linearly independent functions of the cross-section coordinate y. They may be global polynomials, as well as piece-wise polynomials or other shape functions. The approximate field γ (x, y) is expressed as a linear combination of the components of pγ (y), with arbitrary functions of the axial coordinate x as coefficients, collected in a vector ˆγ (x). Therefore, γ (x, y) can be written as a scalar product between pγ (y) and ˆγ (x), that is, γ (x, y) = p T γ (y) ˆγ (x), (11) where superscript T denotes transposition and ˆγ : l → � m contains the functional coefficients as components. We emphasize that, once the profile vector pγ (y) has been assigned, the field γ (x, y) is uniquely determined by the components of ˆγ (x). Such components, for all the involved fields, are indeed the unknowns of the beam models we will develop. We also remark that, due to assumption (11), the computation of partial derivatives is straightforward, since: ∂ ∂ γ = ∂x ∂x ( pT γ ˆγ ) = pT γ d dx ˆγ = pT γ ˆγ ′ , ∂ ∂ γ = ∂y ∂y ( pT γ ˆγ ) = d dy pT γ ˆγ = p′T γ where we use a prime to indicate derivatives along both x and y, as there is no risk of confusion. Adopting the notation introduced in (11) and switching now to an engineering notation we set � � � su(x, y) pT s(x, y) = = u (y) 0 sv(x, y) 0 pT v (y) � � � û(x) = Ps ˆs, (12) ˆv(x) ⎧ ⎫ ⎨σxx(x, y) ⎬ σ (x, y) = σyy(x, y) ⎩ ⎭ τxy(x, y) = ⎡ ⎣ p T σx (y) 0 0 0 pT σy (y) 0 0 0 pT τ (y) ⎤ ⎧ ⎫ ⎨ ˆσx(x) ⎬ ⎦ ˆσy(x) ⎩ ⎭ ˆτ(x) = Pσ ˆσ , (13) where, for Ps and Pσ (ˆs and ˆσ , respectively), we drop the explicit dependence on y (or x, respectively), for notational simplicity. The virtual fields are analogously defined as δs = Ps δˆs and δσ = Pσ δ ˆσ . ˆγ ,
PLANAR BEAMS: MIXED VARIATIONAL DERIVATION AND FE SOLUTION 779 In Section 3, D−1 denotes the fourth-order elastic tensor; from now on, we use the same notation to indicate the corresponding square matrix obtained following engineering notation. Therefore, we have D −1 = 1 ⎡ ⎤ 1 −ν 0 ⎣−ν 1 0 ⎦ . E 0 0 2(1+ν) 4.2. Formulation of the problems. We consider the special case of a beam for which ¯s = 0 on A0 = ∂�s, t �= 0 on Al, f = 0 in �, t = 0 on S0 ∪ Sn. Hence, ∂�t = Al ∪ S0 ∪ Sn; the beam is clamped on the left-hand side A0, and is subjected to a nonvanishing traction field on the right-hand side Al. Furthermore, we suppose that t|Al can be exactly represented using the chosen profiles for σ · n. Recalling (13), and noting that n|Al = (1, 0)T , this means that there exist suitable vectors ˆtx and ˆtτ such that � pT σx t = ˆtx � . (14) p T τ ˆtτ Therefore, the boundary condition σ · n|Al = t may be written as (see (13)) � � � � ˆσx(l) ˆtx = . (15) ˆτ(l) ˆtτ All these assumptions can be modified to cover more general cases; nonetheless, this simple model is already adequate to illustrate the method’s capabilities. 4.2.1. HR grad-grad approach. In the notation of Section 4.1, the HR grad-grad stationarity (5) becomes δ J gg HR = � �� d � dx ET d 1 + dy ET � � � T 2 Psδˆs Pσ ˆσ d� + δ ˆσ � T P T �� d σ dx ET d 1 + dy ET � � 2 Ps ˆs d� � − δ ˆσ T P T σ D−1 � Pσ ˆσ d� − δˆs T P T t dy = 0. (16) Expanding (16) we obtain δ J gg HR = � (δˆs ′T P T � s E1 Pσ ˆσ + δˆs T P ′T s E2 � Pσ ˆσ )d� + � � � − Using the Fubini–Tonelli theorem, (17) can be written as δ J gg HR = � where ∂�t (δ ˆσ T P T σ ET 1 Ps ˆs ′ + δ ˆσ T P T σ ET 2 P′ � δˆs T P T s t dy = 0. (17) δ ˆσ � T P T σ D−1 Pσ ˆσ d� − (δˆs l ′T Gsσ ˆσ + δˆs T Hs ′ σ ˆσ + δ ˆσ T Gσ s ˆs ′ + δ ˆσ T Hσ s ′ ˆs − δ ˆσ T Hσ σ ˆσ )dx − δˆs T Tx Gsσ = G T σ s = � P A T s E1 � Pσ dy, Hσ σ = P A T σ D−1 Pσ dy, Hs ′ σ = H T σ s ′ = � A P ′T s E2 Pσ dy, Tx = � Al P T s t dy. Al s s ˆs)d� � � � = 0, (18) x=l (19)
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