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

Gherlone, Marco

doi:10.1115/1.4023690

Cited by 71 publications

(67 citation statements)

References 95 publications

(201 reference statements)

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“…Recently, Gherlone [35] and the present authors [31] have shown that MZZF suffers from certain limitations for sandwiches with large face-to-core stiffness ratios and arbitrary layups, as the MZZF is not based on actual transverse shear moduli that drive the underlying physics of the problem. As an alternative ZZ function, the Refined Zigzag Theory (RZT) developed by Tessler, Di Sciuva and Gherlone [36][37][38][39] may be used.…”

Section: Mixed-variational Theoriesmentioning

confidence: 84%

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

Groh

2016

Composite Structures

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Current consensus suggests a trade-off for predicting accurate 3D stress fields in multilayered plates. Relatively expensive layerwise models facilitate accurate stress predictions from underlying models assumptions, whereas more efficient equivalent single-layer theories often rely on variationally inconsistent transverse stresses from recovery steps. In contrast, we show that variationally consistent 3D stress fields are predicted from an equivalent single-layer model using a contracted form of the Hellinger-Reissner functional. Notably, this procedure facilitates computationally efficient analysis of 3D heterogeneous plates, i.e. laminates comprising layers with material properties that differ by multiple orders of magnitude and that can also vary continuously in-plane. The model includes effects of higher-order transverse shearing and zig-zag deformations by expressing the in-plane stress field as a Taylor series expansion of global and local higher-order stress resultants. Equilibrated transverse stress assumptions are derived from Cauchy's 3D equilibrium equations, which identically satisfy the interfacial and surface traction equilibrium conditions when applied within a variational statement that enforces the 3D equilibrium equations as constraints, hence the Hellinger-Reissner mixed-variational statement. By using inherently equilibrated stress fields as model assumptions, it is shown that only the classical membrane and bending equilibrium equations have to be enforced explicitly. As a result, a contracted Hellinger-Reissner functional emerges that requires fewer displacement Lagrange multipliers than when independent stress fields are assumed. The governing equations are derived in a generalised framework such that the order of the model, and hence its accuracy, increases automatically when implemented in a computer code. By refining the order of the stress assumptions, the model is adaptable to plate-like structures ranging from thin engineering laminates to highly heterogeneous, thick laminates comprising straight-fibre or tow-steered variable-stiffness laminates, foam, honeycomb or other compliant layers.

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Section: Mixed-variational Theoriesmentioning

confidence: 84%

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

Groh

2016

Composite Structures

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“…It is reminded that in [55] this model has been already proven to be accurate and computationally efficient when closed form solutions are considered, since it achieves the accuracy of layerwise models with just five d.o.f. Therefore, the aim of present numerical results is neither that of: a) discussing the advantages of the zigzag modeling approach here used, nor discussing the advantages offered by its variable kinematics, as both aspects have been already comprehensively overviewed in [43,[51][52][53][54][55][56]; b) nor discussing the relative merits of a physically-based zigzag representation like the present one over the Murakami's zigzag function just based upon kinematic assumptions, because both modeling options have been already compared and extensively discussed in [17]; c) finally, nor comparing the available displacement-based and mixed theories incorporating zigzag functions to other existing models and nor to discuss their fidelity to 3D exact elasticity or finite element solutions, since assessments were already given among many others in the references quoted in [17] that have shown their value.…”

Section: Numerical Applications and Discussionmentioning

confidence: 99%

“…As discussed by Gherlone [17], the Murakami's zigzag function that is just based upon kinematic assumptions is much easy to implement and requires a less effort than the physically consistent zigzag functions obtained through enforcement of physical conditions at the interfaces, but it is not always equally accurate. On the contrary, in the physically-based zigzag functions the slope is not forced to reverse at each interface, because as shown by exact solutions for undamaged and damaged sandwiches this neither occurs for the through-the-thickness distribution of the transverse displacement and it does not always necessarily occur for the in-plane displacement components.…”

Section: Introductionmentioning

confidence: 99%

“…On the contrary, in the physically-based zigzag functions the slope is not forced to reverse at each interface, because as shown by exact solutions for undamaged and damaged sandwiches this neither occurs for the through-the-thickness distribution of the transverse displacement and it does not always necessarily occur for the in-plane displacement components. The assessments by Gherlone [17] carried out for thin to thick monolithic, sandwich-like, symmetric, asymmetrical and arbitrary multilayered beams proven that Murakami's zigzag function leads to the same results of physically based functions for periodical stack-ups, while less accurate results are obtained for arbitrary stacking sequences. In particular, better results are obtained by zigzag models with physically based functions when a weak layer is placed on the top or bottom and for asymmetrical sandwiches with high face-to-core stiffness ratios.…”

Section: Introductionmentioning

confidence: 99%

“…The readers can find a complete list of publications with the most relevant contributions, a comprehensive discussion of their features and how these features reflect on their performances and computational effort in the papers by Chen and Wu [15], Kreja [16] and Gherlone [17]. The advantage of mixed/hybrid models over displacements-based ones is the possibility of more easily enforce all the necessary physical interfacial and boundary constraint conditions, the stresses being treated as primary variables separately from displacements.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

C0 Layerwise Model with Fixed Degrees of Freedom and Variable In- and Out-of-Plane Kinematics by Strain Energy Updating Technique

Icardi

Sola

2015

Aerospace

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Abstract:Physically based zigzag models have the merit of giving accurate stress predictions for laminates and sandwiches keeping fixed the functional degrees of freedom, though at the expense of the introduction of their derivatives. In the present paper, a technique that enables deleting these derivatives is developed. The objective is finding a priori corrections of displacements, which make the energy of the model with all the derivatives neglected equivalent to that of its initial counterpart model containing all the derivatives. Numerical applications show that this technique can obtain accurate results, even for strongly asymmetrical lay-ups, keeping low the computational cost.

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A new formulation of continuous transverse shear stress field for static and dynamic analysis of sandwich beams with soft core

Ren

Zhao

2019

Numerical Meth Engineering

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Summary This article presents a novel formulation for sandwich beams with soft core based on a layerwise description of continuous transverse shear stress field. The transverse shear stress field is assumed to follow a quadratic distribution in thick face layers and a uniform distribution in soft core. Based on stress‐strain relations, transverse shear strain is obtained, and in‐plane displacement field is determined by taking the integral of the transverse shear stain. By enforcing interlaminar displacement continuity, an explicit displacement field expression containing only displacement variables is formulated. In this formulation, interlaminar continuity conditions of displacement and transverse shear stress are satisfied simultaneously, and transverse shear and crosssectional warping effects in thick face layers are accurately taken account. To show characteristics and advantages of the present model, some classical theoretical models are further developed under different kinematic assumptions. The governing equations are derived using Hamilton principle, and finite element solutions for static and dynamic analysis of sandwich beams are presented. Comparisons with exact solution and other well‐known solutions, well demonstrate high accuracy of the present model for sandwich beams with different features. Besides, numerical discussions further reflect accuracy advantages of the present model in static and dynamic analysis of sandwich beams.

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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

Cited by 71 publications

References 95 publications

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

A computationally efficient 2D model for inherently equilibrated 3D stress predictions in heterogeneous laminated plates. Part I: Model formulation

C0 Layerwise Model with Fixed Degrees of Freedom and Variable In- and Out-of-Plane Kinematics by Strain Energy Updating Technique

A new formulation of continuous transverse shear stress field for static and dynamic analysis of sandwich beams with soft core

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