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Computer experiment, Latin hypercube sampling, Design of experiments, Uncertainty analysis, Dependence,

To cite this version:Alexandre Loredo, Alexis Castel. A multilayer anisotropic plate model with warping functions for the study of vibrations reformulated from Woodcock's work. Journal of Sound and Vibration, Elsevier, 2013, 332 (1) AbstractIn this paper, a suitable model for static and dynamic analysis of inhomogeneous anisotropic multilayered plates is described. This model takes into account the variations of the transverse shear strains through the thickness of the plate by means of warping functions. Warping functions are determined by enforcing kinematic and static assumptions at the interfaces. This model leads to: a 10 × 10 stiffness matrix coupling to each other the membrane strains, the bending and torsion curvatures, and the x and y-derivatives of the transverse shear strains; and a classical 2 × 2 transverse shear stiffness matrix. This model has been proven to be very efficient, especially when high ratios between the stiffnesses of layers -up to 10 6 -are present. This work is related to Woodcock's model, so it can be seen as a reformulation of his work. However, it brings several enhancements: the displacement field is made explicit; it is reformulated with commonly used plate notations; laminate equations of motion are fully detailed; the place of this model among other plate models is now easy to see and is discussed; the link between this formulation and the original one is completely written with all necessary proofs; misses and errors have been found in the energy coefficients of the original work and have been corrected; it is now easy to improve or to adapt the model for specific applications with the choice of refined or specific warping functions.Static deflection and natural frequencies for isotropic and anisotropic sandwich plates are given and compared to other models: they show that the present model is very accurate for the simulation of such structures.

A numerical vibroacoustic model that can manage multilayered plates locally covered with damping patches is presented. All the layers can have an on-axis orthotropic viscoelastic behavior. Continuity of displacements and transverse shear stresses at each interface is enforced, which permits to write the entire displacement field in function of the displacements of the--common--first layer, leading to a two-dimensional plate model. The problem is then discretized by Rayleigh-Ritz's method using a trigonometric basis that includes both sine and cosine functions in order to treat various boundary conditions. The excitation can be of mechanical kind (concentrated or distributed forces) or of acoustic kind (plane wave of any incidence, diffuse field, etc.). The model permits to compute different vibroacoustic indicators: the mean square velocity of the plate, the radiation efficiency, and the transmission loss. Comparisons between the present model and numerical results from literature or finite element computations show that the model gives good results in both mechanical and acoustical aspects. Then, a comparison of the effects of different distributions of patches is presented. The role of the surface covering rate is first discussed, followed by a study involving different geometries for the same surface covering rate.

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A multilayered plate theory which uses transverse shear warping functions is presented. Two methods to obtain the transverse shear warping functions from three-dimensional elasticity equations are proposed. The warping functions are issued from the variations of transverse shear stresses computed at specific points of a simply supported plate. The first method considers an exact 3D solution of the problem. The second method uses the solution provided by the model itself: the transverse shear stresses are computed integrating equilibrium equations. Hence, an iterative process is applied, the model is updated with the new warping functions, and so on. Once the sets of warping functions are obtained, the stiffness and mass matrices of the models are computed. These two models are compared to other models and to analytical solutions for the bending of simply supported plates. Four different laminates and a sandwich plate are considered. Their length-to-thickness ratios vary from 2 to 100. An additional analytical solution that simulates the behavior of laminates under the plane stress hypothesis -shared by all the considered models -is computed. Both presented models give results very close to this exact solution, for all laminates and all length-to-thickness ratios.