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A technique for solving the three-dimensional problem of bending in the theory of elasticity in orthotropic plates of variable thickness is developed in the paper. On the basis of the method of expansion of displacements into an infinite series, the problem has been reduced to the solutions of two independent problems, which are described by two independent systems of two-dimensional infinite equations.

Published papers on the calculation of plates of variable thickness are realized within the framework of classical theory of plates, developed with a number of simplifying hypotheses. It should be noted that in the places where the thickness of a plate of variable thickness is the greatest, it is not advisable to apply any simplifying hypotheses. Scientific researches have shown that in calculations of thick plates, it is necessary to take into account not only the moments and forces, but the bimoments as well. The account of bimoments in the sections of the plate is based on the application of the method of expansion of displacements into an infinite series along one of spatial coordinates directed along the normal.

In published literature, there are a number of papers devoted to the investigation of this problem. The law of thickness variation of plates and shells is taken in the form of linear, quadratic and other functions. Various problems of the behavior of thin-walled structural elements under the action of static and dynamic loads are considered.

A wide survey of papers devoted to plates of variable thickness is given in [

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As an object of investigation, a thick orthotropic elastic plate of variable thickness has been chosen; the plate is located between two asymmetric face surfaces z = h 1 ( x 1 , x 2 ) , z = − h 2 ( x 1 , x 2 ) . Let h 1 ( x 1 , x 2 ) ≥ h 2 ( x 1 , x 2 ) ≥ 0 , then the thickness of the plate is H = h 1 ( x 1 , x 2 ) + h 2 ( x 1 , x 2 ) .

In contrast to classical theory of plates, the components of the displacement vector are defined as functions of three spatial coordinates and times u 1 ( x 1 , x 2 , z , t ) , u 2 ( x 1 , x 2 , z , t ) , u 3 ( x 1 , x 2 , z , t ) . The components of the strain tensor are determined by the Cauchy relations. The plate is considered as a three-dimensional orthotropic body [

σ 11 = E 11 ε 11 + E 12 ε 22 + E 13 ε 33 , σ 22 = E 21 ε 11 + E 22 ε 22 + E 23 ε 33 , σ 12 = 2 G 12 ε 12 , σ 13 = 2 G 13 ε 13 , σ 23 = 2 G 23 ε 23 , (1)

where E 11 , E 12 , ⋯ , E 33 ―are the elastic constants determined by Poisson’s coefficients and elastic moduli, given in [

For the equations of motion of the plate we would use the three-dimensional equations of motion of the theory of elasticity:

∂ σ i 1 ∂ x 1 + ∂ σ i 2 ∂ x 2 + ∂ σ i 3 ∂ z = ρ u ¨ i , ( i = 1 , 2 , 3 ) . (2)

Here ρ ―is the density of plate material.

The boundary conditions on the lower and upper surfaces of the plate z = h 1 ( x 1 , x 2 ) and z = − h 2 ( x 1 , x 2 ) are written relative to the generalized external forces in the form:

σ 33 = q 3 ( + ) , σ 31 = q 1 ( + ) , σ 32 = q 2 ( ( + ) ) , at z = h 1 ( x 1 , x 2 ) , σ 33 = q 3 ( − ) , σ 31 = q 1 ( − ) , σ 32 = q 2 ( − ) , at z = − h 2 ( x 1 , x 2 ) . (3)

The solution of the set problem of the theory of elasticity for thick plates of variable thicknesses (1), (2), and (3) is built by the method of expansion of the components of the displacement vector into the Maclaurin series [

u k = B 0 ( k ) + B 1 ( k ) z h 1 + B 2 ( k ) ( z h 1 ) 2 + B 3 ( k ) ( z h 1 ) 3 + ⋯ + B m ( k ) ( z h 1 ) m + ⋯ , ( k = 1 , 2 ) u 3 = A 0 + A 1 z h 1 + A 2 ( z h 1 ) 2 + A 3 ( z h 1 ) 3 + ⋯ + A m ( z h 1 ) m + ⋯ . (4)

Here B m ( k ) , A m ―are the unknown functions of two spatial coordinates and time:

B m ( k ) = B m ( k ) ( x 1 , x 2 , t ) = h 1 m m ! ( ∂ m u k ∂ z m ) z = 0 , ( k = 1 , 2 ) , A m = A m ( x 1 , x 2 , t ) = h 1 m m ! ( ∂ m u 3 ∂ z m ) z = 0 . (5)

On the basis of expansion (4), the components of the strain and stress tensor also expand into the Maclaurin series in the form:

ε i j = ε i j ( 0 ) + ε i j ( 1 ) z h 1 + ε i j ( 2 ) ( z h 1 ) 2 + ε i j ( 3 ) ( z h 1 ) 3 + ⋯ + ε i j ( m ) ( z h 1 ) m + ⋯ ( i , j = 1 , 2 , 3 ) , (6)

σ i j = σ i j ( 0 ) + σ i j ( 1 ) z h 1 + σ i j ( 2 ) ( z h 1 ) 2 + σ i j ( 3 ) ( z h 1 ) 3 + ⋯ + σ i j ( m ) ( z h 1 ) m + ⋯ ( i , j = 1 , 2 , 3 ) . (7)

Here, the expansion coefficients are defined as:

ε i j ( m ) = ε i j ( m ) ( x 1 , x 2 , t ) = h 1 m m ! ( ∂ m ε i j ∂ z m ) z = 0 , σ i j ( m ) = σ i j ( m ) ( x 1 , x 2 , t ) = h 1 m m ! ( ∂ m σ i j ∂ z m ) z = 0 ( m = 1 , 2 , 3 , ⋯ ) .

Based on the Cauchy relation and expansion (4), we obtain the expressions for the expansion of strain (6) of thick plates of variable thickness. The components of elongation strain are:

ε 11 ( m ) = ∂ B m ( 1 ) ∂ x 1 − B m ( 1 ) m h 1 ∂ h 1 ∂ x 1 , (8a)

ε 22 ( m ) = ∂ B m ( 2 ) ∂ x 2 − B m ( 2 ) m h 1 ∂ h 1 ∂ x 2 , (8b)

ε 33 ( m ) = ( m + 1 ) A m + 1 h 1 . (8c)

Components of angle strain are written in the form:

ε 12 ( m ) = 1 2 ( ∂ B m ( 1 ) ∂ x 2 + ∂ B m ( 2 ) ∂ x 1 − m h 1 ∂ h 1 ∂ x 2 B m ( 1 ) − m h 1 ∂ h 1 ∂ x 1 B m ( 2 ) ) , (9a)

ε 13 ( m ) = 1 2 ( ( m + 1 ) B m + 1 ( 1 ) h 1 + ∂ A m ∂ x 1 − A m h 1 ∂ h 1 ∂ x 1 ) , (9b)

ε 23 ( m ) = 1 2 ( ( m + 1 ) B m + 1 ( 2 ) h 1 + ∂ A m ∂ x 2 − A m h 1 ∂ h 1 ∂ x 2 ) . (9c)

Based on Hooke’s law (1) and expansions (5) and (7), we obtain the expressions for the coefficients of stress expansion (7). The coefficients of expansion of normal stresses are defined as:

σ 11 ( m ) = E 11 ε 11 ( m ) + E 12 ε 22 ( m ) + E 13 ε 33 ( m ) , σ 22 ( m ) = E 21 ε 11 ( m ) + E 22 ε 22 ( m ) + E 23 ε 33 ( m ) , σ 33 ( m ) = E 31 ε 11 ( m ) + E 32 ε 22 ( m ) + E 33 ε 33 ( m ) . (10а)

The coefficients of the expansion of tangential stresses are defined as:

σ 12 ( m ) = 2 G 12 ε 12 ( m ) , σ 13 ( m ) = 2 G 13 ε 13 ( m ) , σ 23 ( m ) = 2 G 23 ε 23 ( m ) , (10b)

where m = 0 , 1 , 2 , 3 , ⋯ .

Based on expansion (4), we would demonstrate that the proposed elasticity problem in plates of variable thickness is described by two unrelated problems, each of which is formulated on the basis of a system of infinite recurrent two-dimensional equations with the corresponding boundary conditions. The first system of recurrent equations has the form:

∂ σ 11 ( 2 m ) ∂ x 1 − 2 m σ 11 ( 2 m ) h 1 ∂ h 1 ∂ x 1 + ∂ σ 12 ( 2 m ) ∂ x 2 − 2 m σ 12 ( 2 m ) h 1 ∂ h 1 ∂ x 2 + ( 2 m + 1 ) σ 13 ( 2 m + 1 ) h 1 = ρ B ¨ 2 m ( 1 ) , (11а)

∂ σ 21 ( 2 m ) ∂ x 1 − 2 m σ 21 ( 2 m ) h 1 ∂ h 1 ∂ x 1 + ∂ σ 22 ( 2 m ) ∂ x 2 − 2 m σ 22 ( 2 m ) h 1 ∂ h 1 ∂ x 2 + ( 2 m + 1 ) σ 23 ( 2 m + 1 ) h 1 = ρ B ¨ 2 m ( 2 ) , (11b)

∂ σ 31 ( 2 m + 1 ) ∂ x 1 − ( 2 m + 1 ) σ 31 ( 2 m + 1 ) h 1 ∂ h 1 ∂ x 1 + ∂ σ 32 ( 2 m + 1 ) ∂ x 2 − ( 2 m + 1 ) σ 32 ( 2 m + 1 ) h 1 ∂ h 1 ∂ x 2 + ( 2 m + 2 ) σ 33 ( 2 m + 2 ) h 1 = ρ A ¨ 2 m + 1 . (11c)

Here m = 0 , 1 , 2 , 3 , ⋯ .

The second system of recurrent equations has the form:

∂ σ 11 ( 2 m + 1 ) ∂ x 1 − ( 2 m + 1 ) σ 11 ( 2 m + 1 ) h 1 ∂ h 1 ∂ x 1 + ∂ σ 12 ( 2 m + 1 ) ∂ x 2 − ( 2 m + 1 ) σ 12 ( 2 m + 1 ) h 1 ∂ h 1 ∂ x 2 + ( 2 m + 2 ) σ 13 ( 2 m + 2 ) h 1 = ρ B ¨ 2 m + 1 ( 1 ) , (12а)

∂ σ 12 ( 2 m + 1 ) ∂ x 1 − ( 2 m + 1 ) σ 12 ( 2 m + 1 ) h 1 ∂ h 1 ∂ x 1 + ∂ σ 22 ( 2 m + 1 ) ∂ x 2 − ( 2 m + 1 ) σ 22 ( 2 m + 1 ) h 1 ∂ h 1 ∂ x 2 + ( 2 m + 2 ) σ 23 ( 2 m + 2 ) h 1 = ρ B ¨ 2 m + 1 ( 2 ) , (12b)

∂ σ 31 ( 2 m ) ∂ x 1 − 2 m σ 31 ( 2 m ) h 1 ∂ h 1 ∂ x 1 + ∂ σ 32 ( 2 m ) ∂ x 2 − 2 m σ 32 ( 2 m ) h 1 ∂ h 1 ∂ x 2 + ( 2 m + 1 ) σ 33 ( 2 m + 1 ) h 1 = ρ A ¨ 2 m . (12c)

Here m = 0 , 1 , 2 , 3 , ⋯ .

On the basis of expansion (7), the boundary conditions on the surface of the plate z = + h 1 ( x 1 , x 2 ) (3a) are rewritten as:

σ 31 ( + ) = σ 31 ( 0 ) + σ 31 ( 1 ) + σ 31 ( 2 ) + σ 31 ( 3 ) + σ 31 ( 4 ) + σ 31 ( 5 ) + ⋯ = q 1 ( + ) (13а)

σ 32 ( + ) = σ 32 ( 0 ) + σ 32 ( 1 ) + σ 32 ( 2 ) + σ 32 ( 3 ) + σ 32 ( 4 ) + σ 32 ( 5 ) + ⋯ = q 2 ( + ) (13b)

σ 33 ( + ) = σ 33 ( 0 ) + σ 33 ( 1 ) + σ 33 ( 2 ) + σ 33 ( 3 ) + σ 3 k ( 4 ) + σ 33 ( 5 ) + ⋯ = q 3 ( + ) (13c)

On the basis of expansion (7), the boundary conditions on the surface of the plate z = − h 2 ( x 1 , x 2 ) (3b) are rewritten as:

σ 31 ( − ) = σ 31 ( 0 ) − α σ 31 ( 1 ) + α 2 σ 31 ( 2 ) − α 3 α 31 ( 3 ) + α 4 σ 31 ( 4 ) − α 5 σ 31 ( 5 ) + ⋯ = q 1 ( − ) , (14а)

σ 32 ( − ) = σ 32 ( 0 ) − α σ 32 ( 1 ) + α 2 σ 32 ( 2 ) − α 3 α 32 ( 3 ) + α 4 σ 32 ( 4 ) − α 5 σ 32 ( 5 ) + ⋯ = q 2 ( − ) , (14b)

σ 33 ( − ) = σ 33 ( 0 ) − α σ 33 ( 1 ) + α 2 σ 33 ( 2 ) − α 3 α 33 ( 3 ) + α 4 σ 33 ( 4 ) − α 5 σ 33 ( 5 ) + ⋯ = q 3 ( − ) , (14c)

where α = h 2 ( x 1 , x 2 ) h 1 ( x 1 , x 2 ) .

Two independent systems of boundary conditions relative to the expansion coefficients (4) and (7) correspond to each boundary condition at the edges of the plate.

If the displacements at the edge of the plate are equal to zero, then we get:

B 0 ( 1 ) = 0 , B 0 ( 2 ) = 0 , B 2 ( 1 ) = 0 , B 2 ( 2 ) = 0 , B 4 ( 1 ) = 0 , B 4 ( 2 ) = 0 , ⋯ A 1 = 0 , A 3 = 0 , A 5 = 0 , ⋯ , (15а)

B 1 ( 1 ) = 0 , B 1 ( 2 ) = 0 , B 3 ( 1 ) = 0 , B 3 ( 2 ) = 0 , B 5 ( 1 ) = 0 , B 5 ( 2 ) = 0 , ⋯ A 0 = 0 , A 2 = 0 , A 4 = 0 , ⋯ . (15b)

If the edge of the plate is free of supports, then the boundary conditions have the form:

σ 11 ( 0 ) = 0 , σ 12 ( 0 ) = 0 , σ 13 ( 1 ) = 0 , σ 11 ( 2 ) = 0 , σ 12 ( 2 ) = 0 , ⋯ σ 13 ( 3 ) = 0 , σ 11 ( 4 ) = 0 , σ 12 ( 4 ) = 0 , σ 13 ( 5 ) = 0 , ⋯ , (16а)

σ 11 ( 1 ) = 0 , σ 12 ( 1 ) = 0 , σ 13 ( 0 ) = 0 , σ 11 ( 3 ) = 0 , σ 12 ( 3 ) = 0 , ⋯ σ 13 ( 2 ) = 0 , σ 11 ( 5 ) = 0 , σ 12 ( 5 ) = 0 , σ 13 ( 4 ) = 0 , ⋯ . (16b)

If the edge of the plate is supported, then the boundary conditions have the form:

σ 11 ( 0 ) = 0 , σ 12 ( 0 ) = 0 , A 1 = 0 , σ 11 ( 2 ) = 0 , σ 12 ( 2 ) = 0 , ⋯ A 3 = 0 , σ 11 ( 4 ) = 0 , σ 12 ( 4 ) = 0 , A 5 = 0 , ⋯ , (17а)

σ 11 ( 1 ) = 0 , σ 12 ( 1 ) = 0 , A 0 = 0 , σ 11 ( 3 ) = 0 , σ 12 ( 3 ) = 0 , ... A 2 = 0 , σ 11 ( 5 ) = 0 , σ 12 ( 5 ) = 0 , A 4 = 0 , ... . (17b)

If the edge of the plate is supported and there is no displacement along the tangential direction to the plate contour, then the boundary conditions have the form:

σ 11 ( 0 ) = 0 , B 0 ( 2 ) = 0 , A 1 = 0 , σ 11 ( 2 ) = 0 , B 2 ( 2 ) = 0 , A 3 = 0 , ⋯ σ 11 ( 4 ) = 0 , B 4 ( 2 ) = 0 , A 5 = 0 , ⋯ , (18а)

σ 11 ( 1 ) = 0 , B 1 ( 1 ) = 0 , A 0 = 0 , σ 11 ( 3 ) = 0 , B 3 ( 1 ) = 0 , A 2 = 0 , ⋯ σ 11 ( 5 ) = 0 , B 5 ( 1 ) = 0 , A 4 = 0 , ⋯ . (18b)

Thus, a three-dimensional bending and elasticity problem for thick plates of variable thickness is set and reduced to the solutions of two independent two-dimensional problems, which are described by two dependent systems of infinite recurrent differential equations in partial derivatives. It should be noted that the equations of motion for the bimoment theory of plates of constant thickness developed in [

Usarov, М.К. (2017) To the Solution of Three-Dimensional Problems of Oscillations in the Theory of Elasticity in Thick Plates of Variable Thickness. Journal of Applied Mathematics and Physics, 5, 2044-2050. https://doi.org/10.4236/jamp.2017.510170