Gâteaux and Fréchet derivatives of the operator of geometrically nonlinear bending problem of sandwich plate

The geometrically nonlinear problem of bending of sandwich plates with a transversally soft core in a one-dimensional statement is considered. Mathematically, the problem is formulated as an integral identity generating an operator equation in the Sobolev space. The Gâteaux derivative of the operator is calculated and it is proved that it coincides with the Fréchet derivative.


Introduction
Layered structures, in particular, sandwich plates and shells (Fig. 1), are used in various fields of technology (aircraft manufacturing, shipbuilding, etc.). Three-layer structures have many qualities that conventional structures made of metal alone do not have. They have high specific stiffness and can withstand high specific loads. Layered plates and shells have good heat and sound insulation qualities, damping vibration-absorbing properties. [1][2][3][4][5][6][7][8]. This paper is devoted to finding the Gâteaux derivative [9] of an operator of a geometrically nonlinear bending problem for a sandwich plate with a transversally soft core, formulated as an operator equation in Sobolev space. It is established that the Gâteaux derivative of an operator is a continuous operator, whence it follows that the Gâteaux derivative coincides with the Fréchet derivative [9, 10]. Generalized statements of physically nonlinear and geometrically linear problems are considered in [11][12][13][14]. The nonlinear problems of the shells and the theory of soft net shells were studied in [15][16][17][18][19][20][21][22][23][24][25]. The numerical solution of geometrically nonlinear problems was carried out in [26][27][28][29].

Problem statement
We study the problems of determining the stress-strain state of infinitely wide sandwich plates with transversally soft core. The plate length is equal a , the thickness of the aggregate is t 2 , the thickness of the supporting layers is ) ( 2 k t , k is the layer number. To describe the stress-strain state in the carrier layers, the Kirchhoff-Love model equations are used; in the core, the equations of the elasticity theory, simplified within the accepted model of the transversally soft layer and integrated over the thickness with satisfaction of the conjugation conditions of the layers by displacements [30][31][32]. We introduce the following notation: (hereinafter we assume that k = 1, 2), , a x  . We consider the geometrically nonlinear case, i.e., is the tensile-compressive stiffness of the k-th layer,   be the Sobolev spaces [33] with inner products . We denote the inner product in V and functional f given on W W  and V are determined by the formulas The form given by (2) is linear and continuous with respect to the second argument, which means it generates an operator defined by the formula where W ) , (   is the inner product in W, and the functional f defined by (3) generates an element defined by the formula Therefore, problem (1) can be written as an operator equation

Gâteaux derivative of an operator of an equation
is a linear operator with respect to  , then the Gâteaux variation is called the Gâteaux differential and is denoted by , and ) (u A is called the Gâteaux derivative of the operator A at the point u . Let's calculate the Gâteaux derivative of the operator A defined by (2), (4). We denote ) , , , ( By the corollary of the Han-Banach theorem (see [34] From here and from the relation (7) it follows that It is easy to see that the operator ) ) , is the Gâteaux derivative of the operator . Thus, the following theorem holds.

Theorem 2. Let an operator
A be generated by relations (2), (4). Then it is differentiable everywhere according to Gâteaux, its Gâteaux derivative is defined by the relation (6).

Fréchet derivative of an operator of an equation
is the linear operator of  and is called the Fréchet differential of the operator is called the Fréchet derivative of the operator A at the point u . We prove that the following theorem is true.
Theorem 3. Let the operator A be generated by relations (2), (4). Then its Gâteaux derivative is a continuous operator.
Proof. For all ) , ( and any unit vector ) , ( y Z from W , using the Sobolev embedding theorem and the generalized Hölder inequality, we obtain By the corollary of the Han-Banach theorem ( [34], Theorem 2.7.4), we can choose a unit vector From here and from the relation (8) it follows that is a continuous operator. The theorem is proved. It follows from Theorem 3 and Theorem 2.1 [9] that the Gâteaux derivative coincides with the Fréchet derivative.

Conclusion
It is proved that the operator of a geometrically nonlinear problem of bending of a three-layer plate with a transversally soft filler is differentiable according to Gâteaux and its Gâteaux derivative is calculated. It is established that the Gâteaux derivative coincides with the Fréchet derivative. In the future, this property will be used in the study of buckling of the plate and finding the critical load at which a buckling occurs. Approximate methods for solving this problem will be developed based on the approaches developed in [35][36][37][38][39][40][41][42][43][44][45][46].