Computationally efficient beam elements for accurate stresses in sandwich laminates and laminated composites with delaminations

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Abstract

Laminated composites are prone to delamination failure due to the lack of reinforcement through the thickness. Therefore, during the design process the initiation and propagation of delaminations should be accounted for as early as possible. This paper presents computationally efficient nine degree-of-freedom (dof) and eight-dof shear locking-free beam elements using the mixed form of the refined zigzag theory (RZT(m)). The corresponding nine-dof and eight-dof elements use the anisoparametric and constrained anisoparametric interpolation schemes, respectively, to eliminate shear locking in slender beams. The advantage of the present element over previous RZT beam elements is that no post-processing is required to accurately model the transverse shear stress while maintaining the computational efficiency of a low-order beam element. Comparisons with high-fidelity finite element models and three-dimensional elasticity solutions show that the elements can robustly and accurately predict the displacement field, axial stress and transverse shear stress through the thickness of a sandwich beam or a composite laminate with an embedded delamination. In fact, the accuracy and computational efficiency of predicting stresses in laminates with embedded delaminations make the present elements attractive choices for RZT-based delamination initiation and propagation methodologies available in the literature.

Introduction

Laminated composites are prone to delamination failure due to the lack of reinforcement through the thickness, and this failure mode adversely affects the structural integrity of composite structures. Hence, the initiation and propagation of delaminations should be accounted for at the early stages in the design process. In this respect, tools for accurate stress predictions are an important prerequisite.

Currently, the standard approach in industry is to use three-dimensional finite element (3-D FE) models or layerwise theories to predict accurate 3-D stress fields. At the preliminary design stage, detailed yet computationally expensive 3-D FE solutions are prohibitive for rapid design as meshes with multiple elements per layer are typically required for converged results. Therefore, 3-D layerwise models are often only used on a component-scale level in areas of high stress concentration or for safety-critical components.

For most composite laminates, the thickness dimension is at least an order of magnitude smaller than representative in-plane dimensions, which allows these structures to be modeled as thin beams, plates or shells. This feature facilitates a reduction from a 3-D problem to a 2-D one coincident with a chosen reference axis or surface. The major advantage of this approximation is a significant reduction in the total number of variables and computational effort required.

In multi-layered composite structures, the effects of transverse shear and normal deformations are especially pronounced because the ratios of longitudinal to transverse moduli are approximately one order of magnitude greater than for isotropic materials (ExxisoGxziso=2.6,E11G131405=28 and ExxisoEzziso=1,E11E3315010=15). Second, differences in layerwise transverse shear and normal moduli lead to abrupt changes in the slopes of the three displacement fields ux,uy,uz at layer interfaces. This is known as the zigzag phenomenon (see Fig. 1) and, as shown by Demasi [1], the zigzag form of the displacements ux, uy and uz can be derived directly from interfacial continuity requirements of the through-thickness stresses.

The classical theory of plates (CTP) [2], [3] and its extension to laminated structures, namely classical laminate analysis (CLA) [4], are commonly regarded as inadequate for predicting accurate through-thickness stresses under the conditions described in the previous paragraph. This theory neglects the effects of transverse shear and transverse normal strains, the displacement fields neglect the zigzag effect, and the transverse displacement is assumed to be constant through the thickness.

To overcome these deficiencies a large number of approximate higher-order 2-D theories have been formulated with the aim of predicting accurate 3-D stress fields while maintaining low computational expense. Refinements of CLA along these lines have focused mainly on displacement-based models due to the relatively intuitive physical meaning of the displacement variables that govern the distortion of the plate cross-section. These theories extend from first-order shear deformation theories by Mindlin [5] and Yang, Norris and Stavsky [6] to higher-order Levinson–Reddy-type shear deformation models that enforce vanishing shear strains at the top and bottom surfaces in the displacement field a priori [7], [8], and further to generalized higher-order theories that do not make this initial assumption and may account for transverse normal deformation, i.e. thickness stretching [9], [10]. Finally, starting with the works of Lekhnitskii [11] and Ambartsumyan [12] in the Russian literature, and Di Sciuva [13] and Murakami [14] in the Western literature, attempts were made to incorporate changes in the layerwise slopes of the in-plane displacements ux and uy via unknown zigzag bending rotations multiplied by layup-dependent zigzag functions. Since then, more accurate zigzag functions have been proposed by Tessler et al. [15], [16], [17], [18] and Icardi [19], with the latter work providing the most recent assessment of different zigzag theories.

A fundamental characteristic of purely displacement-based theories is that all strains and stresses are derived from the displacement assumptions using the kinematic and constitutive equations, respectively, and transverse strains and transverse stresses are typically not recovered accurately in this manner  [20]. More accurate transverse stresses can be recovered a posteriori by integrating the in-plane stresses in Cauchy’s 3-D indefinite equilibrium equations [21], and various techniques exist to achieve this within the displacement-based finite element method (FEM) [22], [23], [24], [25]. The disadvantage of this technique is that the post-processed transverse stresses no longer satisfy the underlying equilibrium equations of the theory, in terms of force resultants and moments, and are therefore variationally inconsistent. A second disadvantage of this technique is that higher-order derivatives of the kinematic variables are required, and for C0-continuous finite elements, computing these derivatives leads to oscillations that require smoothing [22].

The aforementioned post-processing operation can be precluded if independent assumptions for the transverse stresses are made. This results in a mixed displacement/stress-based approach, whereby the governing equilibrium equations and boundary conditions are derived by means of a mixed-variational statement. For example, in the Hellinger–Reissner mixed variational principle [26], [27], the strain energy is expressed in complementary form in terms of in-plane and transverse stresses, and Cauchy’s 3-D equilibrium equations are introduced as constraints via Lagrange multipliers. This has the advantage that the six stress fields are always equilibrated and provide very accurate predictions of through-thickness stresses [28], [29].

Forty years after publishing his work on the Hellinger–Reissner principle, Reissner [30] had the insight that it is sufficient to make separate assumptions for the transverse stresses because only these have to be specified independently to guarantee interfacial continuity requirements. This variational statement is known as Reissner’s mixed-variational theorem (RMVT), and makes model assumptions for the three displacements ux,uy,uz and independent assumptions for the transverse shear stresses τxz,τyz and transverse normal stress σz. Compatibility of the transverse strains from kinematic relations, i.e. from ux, uy and uz, and constitutive equations, i.e. from τxz, τyz and σz, is enforced by means of Lagrange multipliers.

Murakami [14] was one of the first authors to use RMVT for composites and simultaneously enhance the axiomatic first-order displacement field of Yang, Norris and Stavsky [6] by including a zigzag function. Murakami made piecewise-parabolic assumptions for the transverse shear stresses that satisfy the interlaminar and surface traction conditions. Even so, Murakami’s transverse shear stress assumptions lead to poor results for laminates with more than three layers [31] because the assumptions do not equilibrate with the axial stresses at each point through the thickness, but rather only in an average sense via the equivalent-single layer equilibrium equations. Thus, the particular choice of the transverse shear stress assumption is of great importance when applying RMVT.

Furthermore, Murakami’s zigzag function suffers from certain limitations for sandwiches with large face-to-core stiffness ratios and arbitrary layups as it is not based on actual transverse shear moduli that drive the underlying physics of the problem [31], [32]. As an alternative, the zigzag function of the refined zigzag theory (RZT) developed by Tessler, Di Sciuva and Gherlone [15], [16], [17], [18] may be used. In this theory, the zigzag slopes βi(k) are defined by the difference between the transverse shear rigidities Giz(k) of layer k, and the effective transverse shear rigidity Gi of the entire layup βi(k)=GiGiz(k)1,whereGi=12hk=1Nl2h(k)Giz(k)1,i=x,ywhere Nl is the total number of layers, and 2h(k) and 2h are the thickness of layer k and total laminate thickness, respectively. Thus, the zigzag slopes βi(k) vanish when the transverse shear modulus of a layer is equal to the effective “spring-in-series” stiffness Gi, and a non-zero value quantifies the normalized difference from Gi.

The early displacement-based versions of RZT require stress recovery steps for accurate transverse stress predictions. To remedy this deficiency, Tessler [33] developed a mixed-variational approach of RZT, known as RZT(m) for 1-D beams using RMVT. The novelty of this work is that the assumption for the transverse shear stresses is based on the equilibrium condition between in-plane stress and the transverse shear stress. As a result of enforcing the critical condition of equilibrated stresses, very accurate through-thickness stresses can be computed directly from the underlying model assumptions. Recently, the formulation was further extended to 2-D plates [34].

A driving factor in the development of RZT and RZT(m) is that the theories be amenable to the development of C0-continuous finite elements. A nine degree-of-freedom (dof) and eight-dof beam element based on RZT were developed by Gherlone et al. [35] using the anisoparametric interpolation scheme proposed by Tessler and Dong [36] where the transverse displacement variable w is interpolated with one polynomial order greater than the bending rotation θ to prevent shear locking. These lower order two- and three-noded elements were then extended to an entire class of higher-order elements by Di Sciuva et al. [37]. The same interpolation scheme was also used in a version of RZT(m) with cubic in-plane displacement field and quadratic transverse displacement [38].

The aim of this paper is to develop robust beam elements based on Tessler’s RZT(m) that provide computationally efficient in-plane stress and transverse shear stress predictions for laminates and sandwich beams with embedded delaminations which can be modeled explicitly within RZT by means of thin resin-rich layers. The motivation for this work is that accurate stress field predictions within these interply resin-rich zones are critical for predicting accurately the onset and propagation of delaminations using the damage frameworks developed for RZT [39], [40].

The rest of the paper is structured as follows. Section 2 provides a background to RZT which is then extended to RZT(m) in Section 3. An eight-dof constrained anisoparametric and nine-dof anisoparametric beam element based on RZT(m) are then developed in Section 4. Detailed comparisons of through-thickness stresses with benchmark 3-D elasticity and high-fidelity FE results are presented in Section 5 and a discussion of the accuracy of the different interpolation schemes is provided. Finally, conclusions are drawn in Section 6.

Section snippets

Refined zigzag theory from the principle of virtual displacements

Following the standard definition of RZT, and using the notation introduced in Fig. 1, the displacement field through the thickness of a 1-D beam is assumed as ux(k)(x,z)=u(x)+zθ(x)+ϕ(k)(z)ψ(x)uz(x)=w(x)where u and w are the axial displacement and average transverse displacement of the reference plane, respectively, θ is the average bending rotation, ψ is the zigzag rotation and ϕ(k) is the layerwise zigzag function of RZT. The zigzag function is defined by ϕ(k)(z)=z+hGGxz(k)1+i=2k2h(i1)GGxz(

Refined zigzag theory based on Reissner’s variational principle

The transverse shear stresses of the displacement-based RZT in Eq. (8) are layerwise constant functions that do not satisfy the equilibrium of interlaminar and surface tractions. Moreover, from Cauchy’s equilibrium equations we know that the z-wise linear axial stresses of Eq. (7) are equilibrated by z-wise quadratic transverse shear stresses. For these reasons, a more accurate assumption for the transverse shear stress is needed, and is briefly described herein according to [33].

Stiffness matrix and external force vector

Having obtained an expression for the assumed transverse shear stress τxz(a)(k) from the Lagrange multiplier portion of Reissner’s mixed variational statement, the internal and external virtual work expressions of Eq. (27) are now used to develop the eight- and nine-dof, shear locking-free beam elements. Using Eq. (27) and substituting the RZT strain expressions from Eqs. (7), (8) gives δWi=Vδϵxx(k)γxz(k)σxx(k)τxz(a)(k)dV=xaxbAδu,xθ,xψ,xw,xθψ10z0ϕ(k)001010β(k)σxx(k)τxz(a)(k)dAdx=xaxbδωRdx

Results

The accuracy and robustness of the two eight-dof beam elements, i.e. for c=r (V constant, denoted by RZT2V(m)) and c=1 (η constant, denoted by RZT2η(m)) in Eq. (52), and the nine-dof beam element (denoted by RZT3(m)) are examined by means of a number of benchmark tests. Section 5.1 shows the convergence of the tip deflection and midspan axial stress of a cantilevered beam with increasing mesh density.

In Section 5.2, through-thickness distributions of the axial displacement, axial stress and

Conclusions

Laminated composites are prone to delamination failure due to the lack of reinforcement through the thickness, and hence the initiation and propagation of delaminations should be accounted for as early as possible in the design process. In this paper, computationally efficient eight degree-of-freedom (dof) and nine-dof elements, that are free of shear locking, were developed based on the mixed form of the refined zigzag theory, RZT(m).

As shown in this study, the beam elements can robustly and

Acknowledgments

The authors would like to thank NASA Langley Research Center for hosting the first author (R.M.J. Groh) under the Advanced Composites Project in the summer of 2016. Furthermore, the first author would like to acknowledge funding by the Engineering and Physical Sciences Research Council under the grant number EP/MO13170/1 at the University of Bristol.

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