2015 # Refined zigzag theory for homogeneous, laminated composite, and sandwich beams derived from Reissner’s mixed variational principle

**Abstract:** A highly accurate and computationally attractive shear-deformation theory for homogeneous, laminated composite, and sandwich laminates is developed for the linearly elastic analysis of planar beams. The theory is derived using the kinematic assumptions of Refined Zigzag Theory (RZT) and a two-step procedure that implements Reissner's Mixed Variational Theorem (RMVT). The basic expression for the transverse-shear stress that satisfies a priori the equlibrium conditions along the layer interfaces is obtained fro…

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“…These observations allow to substantiate the limitations of most "conventional" ZZ models, such as those that may be formulated within the general procedure proposed in this paper. On the one hand, warping functions should depend on the material properties in a far more complex manner than through the linear coefficients a αβ and b αβ as identified by equation (12). More specifically, the warping functions should depend on the ratio between the longitudinal and the transverse shear moduli: ZZ models whose warping functions depend only on the transverse shear moduli will always suffer a certain limitation with respect to highly constrained configurations such as laminated plates with low number of layers.…”

confidence: 99%

“…These observations allow to substantiate the limitations of most "conventional" ZZ models, such as those that may be formulated within the general procedure proposed in this paper. On the one hand, warping functions should depend on the material properties in a far more complex manner than through the linear coefficients a αβ and b αβ as identified by equation (12). More specifically, the warping functions should depend on the ratio between the longitudinal and the transverse shear moduli: ZZ models whose warping functions depend only on the transverse shear moduli will always suffer a certain limitation with respect to highly constrained configurations such as laminated plates with low number of layers.…”

confidence: 99%

“…While the approaches by Ambartsumyan and Lekhnitskii rely on the exact verification of the constitutive equation connecting the transverse shear stress and the kinematic fields, Murakami [8] formulated a ZZ theory by postulating these two fields in an independent manner thanks to Reissner's mixed variational theorem (RMVT) dedicated to multilayered plates [9]. For more recent developments of RMVT-based ZZ theories, the interested reader may refer to papers by Carrera [10], Demasi [11] and Tessler [12,13]. Murakami's zig-zag function (MZZF) has been also extensively applied to classical displacement-based variable kinematics approaches, see, e.g., [14,15].…”

confidence: 99%

“…To remedy this deficiency, Tessler [41] recently developed a mixed-variational approach for 1D beams using RMVT in a novel way, by splitting the variation of the full RMVT functional into two separate operations. First, the variation of the Lagrange multiplier functional that enforces transverse shear strain compatibility is evaluated, and second, the variation of the strain energy functional is computed.…”

confidence: 99%

“…membrane forces and bending moments. Following the work of previous authors [30][31][32]41], the in-plane stress field is then integrated in Cauchy's equilibrium equations to find expressions for the transverse shear (Section 4) and transverse normal stresses (Section 5), which inherently satisfy all interlaminar and surface traction equilibrium conditions. A new set of governing differential field equations and boundary conditions is then derived from the HR mixed-variational statement in Section 6.…”

confidence: 99%

“…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.…”

confidence: 99%