Abstract
Abstract This study aims to generalize a previously developed accurate and inexpensive 3-D zig-zag theory up to an arbitrary representation form and to determine which simplifications are yet accurate in determining transverse shear and normal stress/deformation effects on vibrations of soft-core sandwiches with not moving middle/neutral plane (pumping). Natural frequencies are calculated using displacements assumed differently across the thickness, having fixed d.o.f., not yet explored forms of representation and zig-zag functions differently accounting for the transverse normal deformability and that partially or fully fulfill physical constraints. Applications are presented for sandwich plates and beams with length-to-thickness ratios and material properties of faces and core varying within an industrial range, for which layerwise effects are very important and so suited to the evaluation of theories. Analytical solutions are found using the same trial functions and expansion order for all theories, so to evaluate their accuracy under the same conditions. The choice of the representation form and of zig-zag functions is shown immaterial if displacement field coefficients are recomputed across the thickness by enforcing the fulfillment of all physical constraints (using symbolic calculus). Furthermore, it is shown that assigning a specific role to each coefficient is immaterial, as well as exchanging the order of representation of in-plane and transverse displacement components and even that zig-zag functions could be omitted. This no longer occurs for lower-order theories with only a partial fulfillment of constraints. Pumping motions are highlighted as the first modes, which require the theories much accurately accounting for transverse normal deformability.
Highlights
This study aims to generalize a previously developed accurate and inexpensive 3-D zig-zag theory up to an arbitrary representation form and to determine which simplifications are yet accurate in determining transverse shear and normal stress/deformation effects on vibrations of soft-core sandwiches with not moving middle/neutral plane
Note that in the discussion that follows and throughout the paper the appellation of higher-order theories is reserved to ZZA Icardi and Sola (2014), ZZA*, HWZZM, HWZZM* retaken from Icardi and Urraci (2018a), HWZZ Icardi and Urraci (2018b), HSDT_34, ZZA*_43, ZZA-XX, ZZA-XX’, ZZA_RDF, HWZZ_RDF and ZZA_X1, to _X4 introduced in this paper, while the appellation lower-order is attributed to theories HWZZMA, B, B2, C, C2 and 0 and MHR±, MHR4± retaken from Icardi and Urraci (2018a), HRZZ, HRZZ4, MHWZZA, MHWZZA4, MHR, MHR4 retaken from Icardi and Urraci (2018b), PP23, ZZ retaken from Icardi (2001) and HSDT_32, HSD_33 introduced in this paper, in addition to FSDT and HSDT used for comparisons
Free vibrations of flexible soft-core sandwich structures assumed as multilayer structures having strong variations of properties of constituent laminae are studied, with the intended aim to discriminate which theoretical assumptions, meaning which different representation of variables across the thickness and which different zig-zag functions, can be more appropriate to efficiently and accurately capture pumping modes dominated by the transverse normal deformation
Summary
As it is well known, sandwiches find continuously increasing applications in aerospace, naval and terrestrial applications owing to superior specific strength and stiffness than traditional materials. Dynamic studies are still overwhelmingly carried out using ESL, excluding those previously mentioned concerning sandwiches, those carried out using DL and ZZ and CUF have highlight the superior performance of refined theories (see, e.g., Vescovini and Dozio (2016), Boscolo and Banerjee (2014), Khdeir and Aldraihem (2016), Sayyad and Ghugal (2015), Kazancı (2016), Lin and Zhang (2018)) Those accounting for the transverse normal deformability have shown such deformation important even to predict first free-vibration modes of sandwiches with flexible soft-core, as highlighted in the previously cited articles (Cho et al (1991), Barouni and Saravanos (2016), Frostig and Thomsen (2004, 2009a, 2009b), Rao and Desai (2004), Malekzadeh et al (2005), Schwarts-Givli et al (2007), Lopatin and Morozov (2010), Rahmani et al (2010), Yang et al (2017)). To prove all these claims, the theories of this paper are applied to the study of free vibrations of flexible soft-core sandwich plates, for which the transverse deformation is as important as transverse shear one
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