This article deals with the dynamic response of thin circular clamped GLARE fiber—metal laminates subjected to low velocity impact by a lateral hemispherical impactor. Using a spring-mass model, the differential equations of motion corresponding to loading and unloading stages of impact are derived and solved numerically. Internal damage due to delamination is taken into account. Previously published analytical formulas1,2 concerning the indentation of circular GLARE plates are used during the loading stages of impact. In this study, an equation for the unloading path is derived and used during the unloading impact stage. The load—time, position—time, velocity—time, and kinetic energy—time history responses are calculated. In this regard, the position where delamination occurs, the maximum plate deformation and the position where the impact load becomes zero are predicted. Also, the maximum impact load and the total impact duration are determined. The derived differential equations of motion are applied for GLARE 4-3/2 and GLARE 5-2/1 circular plates subjected to low velocity impact. The predicted load—time history response is compared with published experimental data and a good agreement is found. No other solution of this problem is known to the authors.

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INTRODUCTION GLARE is a Fibre-Metal Laminated material used in aerospace structures which are frequently subjected to various impact damages [1-5]. A high percentage of the total energy absorbed by GLARE plates during impacts is due to the static deformation of the plate [1, 6-7]. Hence, response of GLARE plates subjected to lateral indentation is very important as far as their overall impact behaviour is concerned. This paper deals with the static response of thin circular clamped GLARE fibre-metal laminated plates under the action of a lateral hemispherical indentor located at the centre of the plate. In reference [1] Vlot used an elastic-plastic impact model to solve this problem numerically assuming a deformation profile based on experimental data. Hoo Fatt et al. [6] used the principle of minimum potential energy to model analytically the response of fully clamped square GLARE panels assuming a deformation profile which resembles that of a stretched membrane. They also calculated the first failure load due to glass-epoxy tensile fracture. The objective of this paper is to develop an analytical model for the calculation of static load-indentation curve and the first failure due to glass-ep

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Mixed weak formulations, with two or three main (tensor) variables, are stated and theoretically analyzed for general multi-dimensional dipolar Gradient Elasticity (biharmonic) boundary value problems. The general structure of constitutive equations is considered (with and without coupling terms). The mixed formulations are based on various generalizations of the so-called Ciarlet-Raviart technique. Hence, C 0 continuity conforming basis functions may be employed in the finite element approximations (or even, C À1 basis functions for the Cauchy stress variable). All the complicated boundary conditions, especially in the multi-dimensional scenario, are naturally considered. The main variables are the displacement vector, the double stress tensor and the Cauchy stress tensor. The latter variable may be eliminated in some of the formulations, depending on the structure of the constitutive equations. The standard continuous and discrete Babuška-Brezzi inf-sup conditions for the constraint equation, as well as, solution uniqueness for both the continuous statements and discrete approximations, are established in all cases. For the purpose of completeness, two one-dimensional mixed formulations are also analyzed. The respective constitutive equations possess general structure (with coupling terms). For the 1-D formulations, all the inf-sup conditions are satisfied, for both the continuous and discrete statements (assuming proper selection of the polynomial spaces for the main variables). Hence, the general Babuška-Brezzi theory results in quasi-optimality and stability. For multi-dimensional problems, the difficulty of deducing the inf-sup condition on the kernel is examined. Certain aspects of methodologies employed to theoretically by-pass this problem, are also discussed.

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