10
2 Literature Review
uation of transverse shear strains is distributing nonlinearly through the thickness
and disappearing at the outer surfaces. To model thick structures in a precise way,
Reddy [36, 37] proposed a third-order shear deformation (TOSD), one of the higherorder shear deformation (HOSD) hypotheses, for composite laminated structures.
The TOSD hypothesis assumes that the through-thickness displacement function is
a third-order function of the position in the thickness direction. This yields secondorder of the transverse shear strains, with the maximum shear strain at the mid-surface
and zero shear strain at the outer surfaces. Afterwards, the theory was further applied
to composite structures by Hanna and Leissa [38] and extended to model smart structures by Correia et al. [39, 40], Moita et al. [41], Selim et al. [42]. In addition, Loja et
al. [43] and Soares et al. [44] proposed higher-order B-spline finite element models
for composite structures laminated with piezoelectric patches.
2.1.4 Zigzag Hypothesis
Considering a laminated structure with different material properties, the interlayer shear stresses are discontinuous when applying aforementioned plate or shell
hypotheses. To avoid the inter-layer shear stress discontinuity, zigzag hypothesis
or layerwise hypothesis was introduced. The hypothesis assumes that the displacement distribution function is different for each substrate layer, either with first-order
or higher-order, in such a way the inter-layer shear stress continuity can be satisfied. A first-order zigzag shear deformation (or layerwise first-order shear deformation) theory was developed for smart structure by Ray and Reddy [45], Vasques and
Rodrigues [46]. A third-order zigzag shear deformation theory was implemented
into analysis of smart structures by Kapuria [47], Kapuria et al. [48]. Furthermore,
Polit et al. [49] developed Murakami’s zigzag formulation for modeling of laminated
piezoelectric smart structures, while Carrera and Demasi [50] applied the theory for
composite structures.
2.1.5 Bernoulli and Timoshenko Beam Hypotheses
Regarding to beam- or arch-shaped one-dimensional structures, they can be shrunk
to a line for simplicity using specific beam hypothesis. Bernoulli and Timoshenko
hypotheses are the most frequently used ones for mathematical modeling of beamshaped structures. These two beam hypotheses were proposed earlier than plate and
shell hypotheses. Therefore, the Kirchhoff-Love plate/shell hypothesis can be understood as an extension of the Bernoulli beam hypothesis. Analogously, the ReissnerMindlin plate/shell hypothesis was extended from the Timoshenko beam hypothesis.
Neglecting the transverse shear strains, Crawley and Luis [51] proposed an analytical model for beam-like structures embedded with piezoelectric layer. Afterwards,
Tzou and Chai [52], Kucuk et al. [53] developed linear models based on the Bernoulli
2 Literature Review
uation of transverse shear strains is distributing nonlinearly through the thickness
and disappearing at the outer surfaces. To model thick structures in a precise way,
Reddy [36, 37] proposed a third-order shear deformation (TOSD), one of the higherorder shear deformation (HOSD) hypotheses, for composite laminated structures.
The TOSD hypothesis assumes that the through-thickness displacement function is
a third-order function of the position in the thickness direction. This yields secondorder of the transverse shear strains, with the maximum shear strain at the mid-surface
and zero shear strain at the outer surfaces. Afterwards, the theory was further applied
to composite structures by Hanna and Leissa [38] and extended to model smart structures by Correia et al. [39, 40], Moita et al. [41], Selim et al. [42]. In addition, Loja et
al. [43] and Soares et al. [44] proposed higher-order B-spline finite element models
for composite structures laminated with piezoelectric patches.
2.1.4 Zigzag Hypothesis
Considering a laminated structure with different material properties, the interlayer shear stresses are discontinuous when applying aforementioned plate or shell
hypotheses. To avoid the inter-layer shear stress discontinuity, zigzag hypothesis
or layerwise hypothesis was introduced. The hypothesis assumes that the displacement distribution function is different for each substrate layer, either with first-order
or higher-order, in such a way the inter-layer shear stress continuity can be satisfied. A first-order zigzag shear deformation (or layerwise first-order shear deformation) theory was developed for smart structure by Ray and Reddy [45], Vasques and
Rodrigues [46]. A third-order zigzag shear deformation theory was implemented
into analysis of smart structures by Kapuria [47], Kapuria et al. [48]. Furthermore,
Polit et al. [49] developed Murakami’s zigzag formulation for modeling of laminated
piezoelectric smart structures, while Carrera and Demasi [50] applied the theory for
composite structures.
2.1.5 Bernoulli and Timoshenko Beam Hypotheses
Regarding to beam- or arch-shaped one-dimensional structures, they can be shrunk
to a line for simplicity using specific beam hypothesis. Bernoulli and Timoshenko
hypotheses are the most frequently used ones for mathematical modeling of beamshaped structures. These two beam hypotheses were proposed earlier than plate and
shell hypotheses. Therefore, the Kirchhoff-Love plate/shell hypothesis can be understood as an extension of the Bernoulli beam hypothesis. Analogously, the ReissnerMindlin plate/shell hypothesis was extended from the Timoshenko beam hypothesis.
Neglecting the transverse shear strains, Crawley and Luis [51] proposed an analytical model for beam-like structures embedded with piezoelectric layer. Afterwards,
Tzou and Chai [52], Kucuk et al. [53] developed linear models based on the Bernoulli