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For piezo-laminated structures among the various variables, the evaluation of the transverse normal electric displacement is of particular interest. The Dz is, in fact, closely related to the electrical charge:

Q = Z

Ω DzdΩ

where Ω is the plate/shell surface. The charge consists of a fundamental input/output in a closed-loop control of a smart structure. Faster and accurate evaluation of Q is a key point in the development of an efficient and reliable closed-loop control algorithm. However, Dz , in classical applications, is only given a posteriori via post-processing of the primary variables (the mechanical displacements and the electrical potential). An extended RMVT application, with D_z assumed as primary variable, has been employed in this work, which has been called RM V T − Dz. By considering the variational statement in Eq. 4.15 for the PVD of electro-mechanical problems, the RMVT is obtained modelling a priori the transverse normal electric displacement D_nM (the new subscript M is introduced to remark that the transverse normal electric displacement is now modelled and not obtained via constitutive equations). The added Lagrange multiplier is δDT_nM(EnG− EnC).

Z V h δT_pGσpC + δTnGσnC− δEpGT DpC− δEnGT DnM − δDnMT (EnG− EnC) i dV = δLe−δLin (4.21) Substituing in Eq. 4.21 the rearranged constitutive Eq. 2.31 for the RM V T − D_z, the governing equations for the electro-mechanical case are:

Kuu u + KuΦ Φ + KuDn Dn = Pu −Muuu¨

KΦu u + KΦΦ Φ + KΦDn Dn = PΦ

KDnu u + KDnΦ Φ + KDnDn Dn = 0

CHAPTER 4. VARIATIONAL PRINCIPLES

The matrices K_uu, K_uΦ, K_Φu, K_ΦΦ are completely different from that in Eq. 4.16 for the PVD because of the introduction of a Lagrange multiplier and the consequently rearrange- ment of constitutive equations.

Chapter 5

Weak formulation for shell problems

The Finite Element Methods are based on a weak approximation of the structural problem. In other words, some considerations on the solution error are introduced, the error has a integral meaning over the whole finite element. For this reason the balance of an element is reached only in an integral sense, the element is globally in equilibrium, but locally it should be out of balance. A simple proof is when the stresses are evaluated in a shared point between more than one elements, the stresses should be different depending on the finite element con- sidered for the evaluation postprocess cycle. The formulations that take into account the satisfaction of the balance at both global and local level are called strong formulations.

5.1

Finite Element Method

A Finite Element Method (FEM) [6,112–114] approximation can be formulated independently from the choice of the thickness functions adopted. According to FEM, the generalized displacements can be expressed as a linear combinations of the shape functions,in compact form, as follows:

u = Njuj δu = Niδui with i, j = 1, ..., (n◦nodes) (5.1) where u_j and δu_i are the nodal displacements and their virtual variations. Introducing in Eq. 5.1 the Unified Formulation, Eq. 3.5, with the thickness functions, one has:

us= FsNjusj δuτ = FτNiδuτi

with i, j = 1, ..., (n◦nodes) and τ, s = 1, ..., N (expansion order) (5.2) In this work, a 9-node finite plate/shell element is employed and Lagrangian shape functions Ni, Nj are used to interpolate the primary variables. These lagrangian shape functions are expressed in a local reference system of the finite element (ξ, η), see Figure 5.1, where N_i assume value 1 in i-nodes and value 0 in the other nodes.

CHAPTER 5. WEAK FORMULATION FOR SHELL PROBLEMS

where the lagrangian shape functions for a 9-node element are the following: N1 = 1₄  ξ2− ξη2− η N2 = 1₂  1 − ξ2η2− η N3 = 1₄  ξ2+ ξ  η2− η N4 = 1₂  ξ2+ ξ  1 − η2  N5 = 1₄  ξ2+ ξη2+ η N6 = 1₂  1 − ξ2η2+ η N7 = 1₄  ξ2− ξη2+ η  N8 = 1₂  ξ2− ξ1 − η2  N9 =  1 − ξ21 − η2 (5.3)

In classical FEM techniques, the strain components are computed from displacements by using geometrical relations Eq. 2.58. In particular, by substituting Unified Formulation Eq. 3.5 and FEM approximation Eq. 5.1 into Eq. 2.58, one has:

p =Fτ(Dp+ Ap)(NiI)uτi,

n=Fτ(Dnp− An)(NiI)uτi+ Fτ,zDnz(NiI)uτi,

(5.4)

where I is the 3 × 3 identity matrix. This procedure may result in some numerical problems related to the shear and membrane lockings. The Shear Locking Phenomena, in classical FEM analysis, lead to a “blocking effect” by the shear, in other words, the stiffness due to the shear contributions becomes too relevant. The Shear Locking is a convergence problem, and it is a pure numerical problem. The locking effect grows proportionally to the decreasing of the plate/shell thickness, the structure becomes infinitely stiff to the shear for h → 0. A possible way to contrast the shear locking is to increase the mesh, but if the structure be- comes thinner, the number of elements needed to contrast the locking phenomena increases exponentially. This solution is not practicable for the analysis of thin plate/shell structure. In literature many methods were developed to contrast the shear locking phenomena [115], the most commons are based on a modification of the Gauss quadrature numerical integration method, a brief overview is given below:

Reduced Integration:

If the shell thickness decrease h → 0, the shear strain energy grows, and the finite element becomes too stiff. This locking problem implies that the shear strain conditions αz = 0 and βz = 0 are not satisfied. A possible solution is to calculate the integrals, on the shell surface with the Guass quadrature method, only in the points where it is more probable to satifsy the shear strain conditions _αz = 0 and βz = 0. The number of integration points is less than the nodes element number, this leads to underestimate the shear strain integrals, and it contrast the shear locking phenomena, this is the main idea of the Reduced Integration method [116–118]. For example, for a 4-node element the reduced integration point is only 1, for a 9-node element the reduced integration points are 4.

The Reduced Integration method could lead to some singularities [K_reduced] = 0, the pres- ence of spurius modes, and for dynamical analysis the presence of some negative harmonic solutions ω < 0.

Selective reduced Integration:

The variation of the internal work can be splitted into the bending contribution and the shear contribution as follows:

CHAPTER 5. WEAK FORMULATION FOR SHELL PROBLEMS

to use full integration for the bending contribution, one has: [Kbending] → F ull integration [Kshear] → Reduced integration

Throughout this splitted integration [119, 120], it is possible: to avoid the singularities in the major part of the possible structural cases, and the elimination of the negative harmonic solutions ω < 0 in dynamic analysis. It is not possible to completely avoid the spurius modes that are still present.

A different approach to the shear locking problem is the Assumed Shear Strain Field Concept, or well known Mixed Interpolation of Tensorial Components (MITC) method. The key of the MITC method is the evalution, by a new re-interpolation process, of the shear strains in the element points where the shear strain conditions _αz = 0 and βz= 0 could be probably satisfied.

In this work a 9-nodes shell finite element with the MITC method is employed for the mul- tifield analysis, and its implementation is described below in Sec. 5.2.