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Research Papers

# On the Use of Zigzag Functions in Equivalent Single Layer Theories for Laminated Composite and Sandwich Beams: A Comparative Study and Some Observations on External Weak Layers

[+] Author and Article Information
Marco Gherlone

Department of Mechanical
and Aerospace Engineering,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
Torino 10129, Italy
e-mail: marco.gherlone@polito.it

Vidal and Polit, have presented a zigzag theory on the basis of the “sine” model by Touratier [21], in which the higher-order term for in-plane displacements is a sine function of the thickness coordinate: According to Di Sciuva's original works, continuity of the transverse shear stresses is guaranteed by adding a through-the-thickness C0-continuous zigzag term, and the so-called “sin-c” models are, thus, obtained [53-55]. The same authors, in Ref. [100], compared the results of the sin-c model for beams with those of the “sinzz” model, in which the zigzag effect is modeled using Murakami's zigzag function. However, the sinzz model has one kinematic variable more than the sin-c model and, as a consequence, the comparison might not be considered completely representative of the two approaches adopted to include zigzag kinematics.

A penalty term $Iλ≡(λ/2)∫xaxbη2(x)dx$ is added to the strain energy; λ is a penalty parameter that is set to a large value in order to enforce $η(x)→0$.

Once Eq. (13) is satisfied, the other two conditions needed to completely define the zigzag function interfacial values may be chosen without any effect on the final results. In fact, the numerical results obtained with the refined zigzag function, Eq. (13) and Eq. (14), may be obtained with a different zigzag function, based on Eq. (13) plus any two further conditions [73]. The latter ones can be designed in order to give a particular physical meaning to the kinematic variables of the first-order zigzag theory, Eq. (2). The choice made in the framework of the RZT is represented by Eq. (15). However, the results would be exactly the same, in terms of displacement components, strains, and stresses, with any other choice of the other two conditions.

The four-layer beam will also be considered in the numerical results section and denoted as laminate F.

The equivalence between the results obtained with the two zigzag functions occurs in terms of displacement components ($ux(k)(x,z)$ and $w(x)$), strains ($ɛxG(k)(x,z)$ and $γxzG(k)(x,z)$), and stresses ($σxH(k)(x,z)$ and $τxzH(k)(x,z)$). Similarly, in the mixed formulation, $τxzM(k)(x,z)$ and $γxzH(k)(x,z)$ are also equivalent. The amplitudes of $u(x)$, $θ(x)$, and $ψ(x)$ are different.

EDZ1 is equivalent to Mur-Dis, except for the transverse displacement, which is modeled as the in-plane displacements, i.e., through-the-thickness linear plus Murakami's zigzag contribution.

Manuscript received June 13, 2012; final manuscript received February 6, 2013; accepted manuscript posted February 19, 2013; published online August 19, 2013. Assoc. Editor: George Kardomateas.

J. Appl. Mech 80(6), 061004 (Aug 19, 2013) (19 pages) Paper No: JAM-12-1229; doi: 10.1115/1.4023690 History: Received June 13, 2012; Accepted February 06, 2013; Revised February 06, 2013

## Abstract

The paper presents a comparison between two existing zigzag functions that are used to improve equivalent single layer (ESL) theories for the analysis of multilayered composite and sandwich beams. ESL theories are easy to implement and computationally affordable but, in order to correctly describe the mechanical behavior of laminated structures (especially those exhibiting high transverse anisotropy or high thickness-to-side length ratios), the displacement field needs to be enriched by a through-the-thickness piecewise linear contribution denoted as “zigzag.” The zigzag term of the displacement field is used to model the local distortion of the cross section in each lamina of multilayered structures and is related to the continuity of transverse stresses. The paper considers two zigzag functions that have been proposed in the open literature (namely Murakami's zigzag function and the refined zigzag function) and compares their performances when they are used to improve the classical Timoshenko beam theory; both displacement-based and mixed formulations are considered. To the best of the author's knowledge, such a comparative study has never been published. The problem of a simply supported beam subjected to a transverse distributed load is considered as a test case. Several stacking sequences, ranging from monolithic to sandwich-like and from symmetric to arbitrary, are considered. The special case of laminates with external weak layers is also investigated and the effects of these lay-ups on the derivation of the refined zigzag function are analyzed for the first time. The capability of the tested zigzag functions to help evaluate the overall deflection and model the through-the-thickness distribution of the axial displacement and stress is investigated. It has been recognized that the refined zigzag function is more accurate, especially for unsymmetric and arbitrary lay-ups and can be adopted to efficiently introduce zigzag kinematics into any ESL theory.

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## Figures

Fig. 1

Fig. 2

Layer and interface notations (for a three-layer beam). (a) Thickness and interfaces of each layer. (b) Local thickness coordinate (for the second layer).

Fig. 3

Zigzag function with interfacial values (for a three-layer beam)

Fig. 7

Through-the-thickness distribution of the normalized axial displacement (five-layer (90/0/90/0/90) beam, L/2h = 8)

Fig. 6

Through-the-thickness distribution of the normalized axial displacement (four-layer (90/0/90/0) beam, L/2h = 8)

Fig. 5

Refined zigzag function (for a three-layer beam)

Fig. 8

Simply supported beam with sinusoidal transverse load. The supports placed at the beam ends graphically represent the physical origins of the geometric boundary conditions, i.e., w(0)=w(L)=0. Constraints on the other kinematic variables, u, θ, and ψ, are not shown, but they are expressed by the trigonometric functions of the exact solution of the problem, Eqs. (25).

Fig. 11

Through-the-thickness distribution of the normalized axial displacement (laminate H, L/2h = 8)

Fig. 20

Through-the-thickness distribution of the normalized axial displacement (laminate N, L/2h = 8)

Fig. 21

Through-the-thickness distribution of the normalized axial stress (laminate N, L/2h = 8)

Fig. 12

Through-the-thickness distribution of the normalized axial displacement (laminate I, L/2h = 8)

Fig. 13

Through-the-thickness distribution of the normalized axial displacement (laminate J, L/2h = 8)

Fig. 14

Normalized maximum deflection versus length-to-thickness ratio (laminate K)

Fig. 15

Through-the-thickness distribution of the normalized axial displacement (laminate K, L/2h = 8)

Fig. 17

Normalized maximum deflection versus length-to-thickness ratio (laminate M)

Fig. 18

Through-the-thickness distribution of the normalized axial stress (laminate M, L/2h = 8)

Fig. 19

Normalized maximum deflection versus length-to-thickness ratio (laminate N)

Fig. 9

Normalized maximum deflection versus length-to-thickness ratio (laminate F)

Fig. 10

Through-the-thickness distribution of the normalized axial displacement (laminate F, L/2h = 8)

Fig. 16

Through-the-thickness distribution of the normalized axial stress (laminate L, L/2h = 8)

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