Calculation of a three-layer plate by the finite element method taking into account the creep of the filler

In the article the derivation of the resolving equations for the calculation of a three-layer plate taking into account the creep of the middle layer by the finite element method is given. Rectangular finite elements are used. The problem reduces to a system of linear algebraic equations. An example of calculating a three-layer plate hinged on the contour and loaded with a uniformly distributed load is considered. A comparison of the results with the solution based on the finite difference method is presented.


Introduction
Three-layer structures are widely used in many industries, including aircraft construction, shipbuilding, civil and industrial engineering, etc. Such structures, as a rule, consist of two outer layers with high mechanical characteristics (steel, aluminium, fiberglass) and a lightweight filler located between them. Porous polymers (foams) are widely used as a filler. For polymers in addition to elastic properties, a pronounced rheology is characteristic. The calculation of threelayered structures taking the creep into account is considered in [1][2][3]. Resolving equations for a triangular finite element of a three-layer plate and shell are given in [1][2]. In the present paper rectangular finite elements with higher accuracy will be considered.

Derivation of resolving equations
The rectangular finite element of the three-layer plate is shown in Fig. 1   For the displacement field within the element, we take the following approximation: where a, b -dimensions of the finite element.
The coordinates x and y in formulas (2) are measured from the centre of the finite element.
Expression (4) can be rewritten as: The relationship between deformations and internal forces has the form: where ] [Dblock matrix of elastic constants. Substituting (6) in the expression for the potential energy, we get: The total energy represents the difference between the potential energy of deformation and the potential of external forces: , Differentiating the total energy with respect to the vector of nodal displacements, we obtain: The stiffness matrix is not presented here due to its bulkiness.

Results and discussion
A three-layer rectangular hinged plate (Fig. 3) was calculated with the following initial data: plate thickness h = 8 cm, modulus of elasticity of sheaths E = 2 • 10 5 MPa, Poisson's coefficient of sheaths ν = 0.3, sheath The core of creep was assumed to be exponential: . hour With an exponential kernel, the creep law (6) is easily represented in a differential form:  (6) and (7) contain the values of the total shear deformation of the filler, which represent the sum of elastic deformations and creep strains: Using (8), we can express from (7) the growth rates of creep strains: The calculation was carried out by the step method, the creep strains at the time t t ∆ + were determined as follows: This method is also used in [4][5][6][7][8][9]. The following boundary conditions were assumed:  Fig. 4 shows the curve of growth of the deflection in the center of the plate. The solid line corresponds to the solution obtained by the authors using the finite element method, the dashed line is the solution by the finite difference method according to the method described in [10]. When 0 = t the results are the same and at ∞ → t the difference is 1.83%.
Stresses in the skins and filler do not change during creep. The stress + σ x and + τ xy distribution in the lower skin is shown in Fig. 5 -6. The stresses in the upper skin under the boundary conditions (10) in absolute value coincide with the stresses in the lower skin. The distribution of tangential stresses in the filler is shown in Fig. 7.

Conclusions
The equations obtained are applicable for arbitrary creep laws of the filler, including nonlinear ones. The correctness of the equations and the reliability of the results are confirmed by a comparison with the solution on the basis of the finite difference method. It is established that under linear creep law the stresses in the shells and filler do not change during the creep process.