Modeling hypothesis

An inspection of the two-dimensional modeling hypothesis is essential for the study of this dissertation. Plates are an important class of two-dimensional structures. They are flat structures, and the thickness of plates ts is much smaller than the other two dimensions (length aand widthb). They can be subdivided into three classes according to the ratiob/ts[122]:

1. Thick plates have a ratio b/ts 6 10. Therefore, a three-dimensional mechanical assumption is necessary. All the components of stress, strain, and displacement fields are involved in the analysis of this type of structures.

2. Membranes have a ratio b/t_s >100. A membrane mainly sustains the in-plane loads called membrane forces on its mid-surface so that they are devoid of bending behaviors.

3. Thin plates have a b/tsratio between 10 and 100. They form the most extensive class. Depending on the ratio of maximum deflection of the plate to its thickness w/ts, the contributions of flexural and membrane forces may be different. Thereby, this category can be subdivided into two groups:

MODELING OF MFC TRANSDUCERS INTEGRATED INTO A THIN HOST STRUCTURE 27

Stiff plates have a ratio w/ts 6 0.2. They sustain external loads mostly by internal bending, twisting moments and shear forces. The deformation of mid-surface and the membrane forces can be neglected.

Flexible plates have a ratio w/t_s > 0.3. The plates’ deflection is followed by a stretching of the mid-surface. Hence, a flexible plate behaves like a combination of stiff plates and membranes. They carry external loads by the combined action of bending moments, shear forces and membrane forces.

It is mentioned in [122] that the above classification is conditional because the reference of a plate to one or another group depends on the accuracy of analysis and boundary conditions, etc. Restricted by the small strain variation of MFC transducers, only thin plates are studied in this dissertation. Thereby, either the Kirchhoff plate theory or the Mindlin-Reissner plate hypothesis and their variations are applicable in the study. The displacement field of a plate is given in Figure 2.16, in which,u,v andware the translational components in x,y andz directions, respectively. β_x andβ_y are the rotational components iny andxdirections, respectively.

w

βy

x y

z

w

βx

y x z

x

y z

w

βy βx

v u

t_s

Figure 2.16: Displacement field of a plate

The Kirchhoff plate theory assumes that the thickness of the plate is remained during a deformation; the normal to the mid-surface remains straight and perpendicular to the mid-surface after the deformation too [123]. Then, βx

andβy can be approximately expressed by the derivatives of the transverse displacement w. As a result, the Kirchhoff plate hypothesis significantly simplifies the strain-displacement relations. In the Mindlin-Reissner theory, the normal to the mid-surface remains straight but it is not always necessary to be perpendicular to the mid-surface of the plate [123]. The Mindlin-Reissner hypothesis [123] considers the rotational componentsβ_xandβ_y as independent variables in its strain-displacement relations. We can recognize that both of them decouple in-plane and bending motions of the plates in a similar way

but with different physical variables for the bending behavior of the plate.

Significant potential by using the Kirchhoff theory in this study is to create a straightforward coupling between the piezoelectric effect and the transverse displacement of a host plate, which has not been characterized in the literature.

Furthermore, there is also a class of modified plate theory, called Refined Plate Theories (RPT). It is like a combination of the Kirchhoff plate theory and Mindlin-Reissner theory. Huffington (1963) [124] firstly proposed that the deflection of a platewcan be split into two components: bending component wband shear componentws. Subsequently, the RPT is adopted in many studies of composite structures and orthotropic plates [125–129]. Free vibrations of an orthotropic plate have been studied by using the RPT in [128]. A detailed check of the RPT is performed here to evaluate its potential for the modeling of plates with integrated MFC transducers. The displacement field of a plate that takes the high-order shear deformation into account in RPT is expressed as follows: where,uandv are the in-plane displacement components inxandydirections, respectively. wbandwsare the bending and shear components of the transverse displacement w. The strain field can be written as follows according to the strain-displacement relations in [128]:



The stresses can be obtained through the Hooke’s law [130]. Then, the bending momentsM and shear force taucan be described as follows by integrating the stress components in the thickness-wise of the plate:



Hence, we can observe from Equation (2.10) that the bending effect of the plate only depends onwb. Also, the transverse shear forces depend onws. RPT uses

MODELING OF MFC TRANSDUCERS INTEGRATED INTO A THIN HOST STRUCTURE 29

a similar way according to the Kirchhoff plate theory to describe the bending effect on a plate. It uses a single variablews to express the transverse shear effect, which does not exist in the Kirchhoff plate theory. The transverse shear effect is included in the rotational variablesβ_xandβ_y in the Mindlin-Reissner plate theory. Since the MFC-d31 and MFC-d33 transducers do not generate transverse shear deformations on a plate, and the Kirchhoff plate theory is sufficient to analytically investigate the piezoelectric coupling of the two types of MFC transducers on a host plate. This is also consistent with it the observation in [22]. However, the Mindlin-Reissner plate theory or RPT should be used to model MFC-d15 transducer, which generates transverse shear deformations on a host structure. It is also important to understand that a refined plate theory certainly improves the accuracy on mechanical modeling of a plate with integrated piezoelectric transducers.