Numerical Investigation of Physically Nonlinear Problem of ... - Core

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 150 (2016) 1050 – 1055

International Conference on Industrial Engineering, ICIE 2016

Numerical Investigation of Physically Nonlinear Problem of Sandwich Plate Bending I.B. Badrieva,*, M.V. Makarova,b, V.N. Paimushina,b a

b

Kazan Federal University, 18 Kremlyovskaya Street, Kazan 420008, The Russian Federation Kazan National Research Technical University, 10 Karl Marx Street, Kazan 420111, The Russian Federation

Abstract The present work is devoted to the numerical investigation of geometrically linear problem of bending of sandwich plate with transversal-soft core for the physically non-linear case. The generalized statement of the problem consists in finding a saddle point of some functional. The existence and uniqueness theorem solutions are proved. To solve the problem, we use an iterative process previously proposed by the authors, each step of which is reduced to solving a linear problem of the elasticity theory and finding the projection onto convex closed set. A Matlab software package was developed, numerical experiments for the model problems are performed. The results of numerical experiments show the effectiveness of the proposed iterative method. © 2016 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: sandwich plate; transversely soft core; the physical nonlinearity; saddle problem; iterative method; numerical experiment

1. Introduction In order to facilitate construction, while not reducing its carrying capacity the thin-walled membranes in the form of shells are used. Such membranes are widespread in engineering structures, mechanical engineering, shipbuilding, aircraft industry and rocket technology [1-5]. Development, implementation and continual expansion of the use of structural elements a three-layer structure stimulate the development of research on the construction methods of calculating their strength. A three-layer structure have, in particular, the glazing elements (triplexes) of aircraft and shipbuilding products in which the materials of the outer layers have elastic characteristics, by several orders greater than the elastic characteristics of the middle layer (core) material. When creating methods of strength analysis of

* Corresponding author. Tel.: +7-919-630-7659. E-mail address: [email protected]

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016

doi:10.1016/j.proeng.2016.07.213

1051

I.B. Badriev et al. / Procedia Engineering 150 (2016) 1050 – 1055

such constructions real deformation diagram of the middle layer to a practically acceptable accuracy can be described by the model of an ideal elastic-plastic material, characterized by a first-order elasticity module E3 ,

V*

Poisson's coefficient Q 3 and the limit stress 3 at which achievement the plastic deformation starts. As part of a three-layer structure its middle layer generally can be classified as a transversely soft given therein a compression strain and transverse shear stresses is constant in thickness. When loading of these structural elements by surface effort, smoothly changing to the coordinates, the main in the middle layer are shear stresses, the relationship between them and the corresponding shear deformations can be described by a model of ideal elastic-plastic deformation under shear strains conditions. At the same time the characteristics of the middle layer material is a first-order elastic modulus, a second-order elastic modulus (shear modulus), limit values of tangential stresses and shear strains, at achievement of which the process of the plastic shear deformation of the middle layer starts. In this paper, a numerical investigation of a mathematical model of a three-layer plate deformation with transversely soft core for the physically nonlinear case is carried out. Generalized statement of the problem is formulated as a problem of finding a saddle point of some functional. In [6] a solvability theorem by using the general results of the existence of saddle points is proved. To solve the considered problem previously suggested in [7] method has been used. The calculation results for the model problems are presented. Note that a physically nonlinear problem of determining the equilibrium position of a soft network shells and methods of solving them have been studied in [8–12]. Geometrically nonlinear problems were considered in [13, 14]. 2. Statement of the problem We consider, in one-dimensional spatial coordinates, the stress-strain state of a sandwich plate consisting of two load carrying layers and transversely soft core adhesively bonded to them. Load carrying layers are described by refined Kirchhoff-Love hypothesis as in [15–17] and for the core layer, equations of elasticity theory simplified in accordance with the accepted model of the transversely soft material and integrated over the thickness satisfying conjugate conditions of the layers for the displacements in the transverse direction. Let in accordance with introduced in [16, 17] notations, a be the length the plate, 2 h , 2 h( k ) be the thickness of the filler and the k-th layer, (here and below we assume that k = 1, 2), H ( k )

h  h( k ) , X (1k ) , X (3k ) be the components of the surface load given to

the middle surface of k-th layer, w( k ) and u ( k ) be the bending and axial displacement the surface of the middle points of k-th layer, X (11k ) and M (11k ) be the membrane forces and inner bending moments in the k-th layer, respectively. The edges of the plate carrier layers we assume rigidly fixed so that the conditions hold u ( k ) ( x) (k )

w ( x)

(k )

0, d w

/ dx 11 (k )

we assume that T

0 , at x B( k ) d u

(k )

0, x

0,

a . The problem is considered in the geometrically linear statement, i.e.,

/ dx , M

11 (k )

D( k ) d 2 u ( k ) / dx 2 , where B( k )

2h ( k ) E ( k ) / (1 Q 12( k ) )(1 Q 21( k ) ) is the

stiffness of the k-th layer on the tension-compression, E ( k ) and Q 12( k ) , Q 21( k ) are the first kind modulus and Poisson's coefficients of the k-th carrying layer material, D( k )

U

B( k ) h(2k ) / 3 is the flexural rigidity of the k-th layer. Let

( w(1) , w(2) , u (1) w(2) ) be the displacement vector points of the middle surface of the k-th layer, and q1 be

tangential stresses in the core. For q1 we assume that the boundary conditions q1 (0) accordance with [15, 16] take into consideration the functional L P (U , q1 )

q1 (a)

0 are satisfied. Let in

P  A  Aq , where

2 1a 2 { ¦ ª¬ B( k ) (d u ( k ) / dx)2  D( k ) (d 2 w( k ) / dx 2 )2 º¼  c1 (q1 )2  c2 (d q1 / dx)2  c3 ( w(2)  w(1) ) } dx ³ 20 k 1

is the strain potential, c1 modules of a core,

2h / G13 , c2

h3 / 3 E3 , c3

(1)

E3 / (2h) , G 13 , E3 are the transverse shear and compression

1052

I.B. Badriev et al. / Procedia Engineering 150 (2016) 1050 – 1055 a

2

A(U , q1 ) ³ ¦ [ X (1k ) u ( k )  M (1k ) d w( k ) / dx X (3k ) w( k ) ] dx

(2)

0 k 1

is the work of given external forces and moments, M (1k ) is the surface moment of external forces, reduced to the middle surface of the k-th layer, and a

2

0

k 1

(1) (2) (k ) 1 2 1 2 1 ³ [( u  u )  ¦ H ( k ) d w / dx  c1q  c2 d q / dx ] q dx

Aq (U , q1 )

(3)

is the work of unknown tangential stresses on corresponding displacements. As shown in [15, 16] a solution of the problem is a saddle point of the functional L. Assuming that the connection between tangential stress q1 and shear deformations corresponds to the ideal elastic-plastic model, we consider the problem under the constraint | q1 ( x) |d q* , 0  x  a , where q* is the given value of limiting stress in core. Let Vk

o (k )

W 2 (0, a) and V be Sobolev spaces [18] with inner products a

(u ,K ) k

k k k k ³ d u / dx d K / dx dx, 0

(4)

a

³ > c2 y ( x) z ( x)  c3 d y / dx d z / dx @ dx

( y, z ) 

0

We denote V

V2 u V2 u V1 u V1 , K 1

{ y  V : | y ( x) |d q* , 0  x  a} . For the sake of easiness let rewrite the

functional L : V u V o R in the form L(U , q1 ) ) (U )  ) 3 (U , q1 )  ) 4 (q1 ) , )

a 1a ª¬ B( k ) (d u ( k ) / dx) 2  D( k ) (d 2 w( k ) / dx 2 ) 2 º¼ dx  ³ ( X (1k ) u ( k )  X (3k ) w( k )  M (1k ) d w( k ) / dx), (5) ³ 20 0

) ( k ) (U ) a

ª

0

¬k

2

) 3 (U , q1 ) ³ « ¦ H ( k )

) 0 (U )

) 0  )1  ) 2 , where

1

º 1 a ª 2h 1 2 h3 d q1 2 º d w( k ) (q )  ( ) » dx,  ( u (2)  u (1) ) » q1dx, ) 4 (q1 ) ³ « dx 2 0 ¬« G13 3 E3 dx ¼» ¼

2 1a c3 ( w(2)  w(1) ) dx ³ 20

(6)

(7)

It is easy to see that the functional ) is well defined on V, the functionals L , ) 3 are well defined on V u V , and functional ) 4 is well defined on V . The inner product in V is denoted by (˜, ˜)V . Note that by virtue of the Riesz-Fisher theorem, there exists a continuous linear operator C : V o V such that ) 3 (U , q1 ) (CU , q1 ) . It is easy to check that the operator C is Lipschitz continuous with some constant J ! 0 . By a generalized solution of the problem of determining the stress-strain state of the sandwich plate with transversal-soft filler we mean a function (Uˆ , qˆ1 )  V u K such that L(Uˆ , qˆ1 )

inf sup L(U , q1 ). U V

q1K

(8)

1053

I.B. Badriev et al. / Procedia Engineering 150 (2016) 1050 – 1055

In [6] on the basis of the common results of the existence of saddle points [19, 20] the following theorem is proved Theorem 1. The problem (8) has an unique saddle point. 3. Method of solution.

To solve the problem (1) in analogy to [21–25] in [7] following iteration process has been proposed. Let q10  K be any element. For n 0,1, 2,! we find U n as a solution of variational inequality

() c(U n ),U  U n )V  (CU  CU n , q1n ) t 0

U  V .

(9)

Then we set

q1n 1

PK (q1n  W (q1n  CU n )

(10)

where W ! 0 is an iterative parameter, PK is the projection in V onto closed convex set K. Note that variational inequality (2) is equivalent to the equation ) c(U n )  C * q1n

0 , which is a linear elasticity theory problem, where

*

C : V o V is conjugate to C operator. In paper [7] it is established that the following result holds. Theorem 2. Suppose that 0  W  2D / (2D  J ) , where D max{B(1) , B(2) , D(1) , D(2) } , (Uˆ , qˆ1 )  V u K is the solution of problem (8), and {U n , q1n }f n 0 is the iterative sequence constructed according to (9) and (10). Then U n converges to Uˆ as n o f . 4. Results of the numerical experiments.

In the software environment Matlab was developed software package for numerical implementation of the offered iterative method. For the model problem, numerical experiments were carried out. Iteration parameter was chosen empirically.

Fig. 1. (a) axial displacements (non-linear case); (b) axial displacements (linear case).

The calculations were performed for the following characteristics: a 1 , h1 MPa, E3 M (1k )

25 MPa, X

0, k

3 (1)

{ 1.0 MPa, X

3 (2)

0 , E

(k )

4

7 ˜10 MPa, Q

(k ) 12

Q

(k ) 21

h2

0.005 , h

0.05 , G13

0.3 , q* =15 MPa, X

1 (k )

15

0 ,

1, 2 . The initial approximation q10 be zero. Calculations according to (9), (10) were carried out as

long as the residual norm remained greater than given accuracy H

5 ˜106 . Optimal (by the number of iterations)

1054

I.B. Badriev et al. / Procedia Engineering 150 (2016) 1050 – 1055

value of iteration parameter was W 1 , the number of iterations in this case was equal to 17. The results of numerical experiments for both physically nonlinear and linear cases are shown in Figures 1–4. Figures 1–3 show the graphs of changing the axial displacements, bending and membrane forces of the outer layers and the Figure 4 shows transverse of tangential stresses in the middle layer formed in the structure at the same level of external load 3 X (1) { 1.0 . Of these, Figures 1 (b), 2 (b), 3 (b) correspond to a physically linear statement, and Figures 1 (a), 2 (a), 3 (a) to physically non-linear one.

Fig. 2. (a) bending (non-linear case); (b) bending (linear case).

Fig. 3. (a) membrane forces (non-linear case); (b) membrane forces (linear case).

Fig. 4. Tangential stresses in core for physically linear (the solid line) and non-linear (the dot line) cases.

As expected, when | q1 ( x) | reaches a value of q* , then begins the process of reversible plastic deformation of the middle layer at two sites along the length of the structure is accompanied by a significant increase in the axial

I.B. Badriev et al. / Procedia Engineering 150 (2016) 1050 – 1055

1055

displacements and bending of the outer layers without a change in these layers the quality picture of membrane forces T(11 k ) ( x ) . In addition, from Figure 4 follows implementation of both linear and nonlinear case of the equalities 11 11 T(1)  T(2)

const , meaning that the equilibrium conditions are satisfied.

Acknowledgements

This work was supported by the Russian Science Foundation (project 16-11-10299). References [1] N.I. Akishev, I.I. Zakirov, V.N. Paimushin, M.A. Shishov, Theoretical-experimental method for determining the averaged elastic and strength characteristics of a honeycomb core of sandwich designs, Mechanics of Composite Materials. 47 (2011) 377௅386. [2] I.B. Badriev, M.V. Makarov, V.N. Paimushin, On the interaction of composite plate having a vibration-absorbing covering with incident acoustic wave, Russian Mathematics. 3 (2015) 66–71. [3] D.V. Berezhnoi, V.N. Paimushin, V.I. Shalashilin, Studies of quality of geometrically nonlinear elasticity theory for small strains and arbitrary displacements, Mechanics of Solids. 44 (2010) 837௅851. [4] L.U. Sultanov, Mathematical modeling of deformations of hyperelastic solids, Applied Mathematical Sciences. 143 (2014) 7117௅7124. [5] E.I. Starovoitov, V.D. Kubenko, D.V. Tarlakovskii, Vibrations of circular sandwich plates connected with an elastic foundation, Russian Aeronautics. 2 (2009) 151௅157. [6] I.B. Badriev, V.V. Banderov, G.Z. Garipova, M.V. Makarov, R.R. Shagidullin, On the solvability of geometrically nonlinear problem of sandwich plate theory, Applied Mathematical Sciences. 9 (2015) 4095௅4102. [7] I.B. Badriev, G.Z. Garipova, M.V. Makarov, V.N. Paimushin, R.F. Khabibullin, Solving Physically Nonlinear Equilibrium Problems for Sandwich Plates with a Transversally Soft Core, Lobachevskii Journal of Mathematics. 4 (2015) 474௅481. [8] I.B. Badriev, V.V. Banderov, Iterative methods for solving variational inequalities of the theory of soft shells, Lobachevskii Journal of Mathematics. 4 (2014) 354–365. [9] I.B. Badriev, R.R. Shagidullin, A study of the convergence of a recursive process for solving a stationary problem of the theory of soft shells, Journal of Mathematical Sciences. 5 (1995) 519–525. [10] I.B. Badriev, V.V. Banderov, O.A. Zadvornov, On the solving of equilibrium problem for the soft network shell with a load concentrated at the point, PNRPU Mechanics Bulletin. 3 (2013) 17–35. [11] I.B. Badriev, O.A. Zadvornov, A.M. Saddek, Convergence Analysis of Iterative Methods for Some Variational Inequalities with Pseudomonotone Operators, Differential Equations. 7 (2001) 934௅942. [12] I.B. Badriev, O.A. Zadvornov, A.D. Lyashko, A study of variable step iterative methods for variational inequalities of the second kind, Differential Equations. 7 (2004) 971௅983. [13] I.B. Badriev, V.V. Banderov, M.V. Makarov, V.N. Paimushin, Determination of stress௅strain state of geometrically nonlinear sandwich plate, Applied Mathematical Sciences. 78 (2015) 3887–3895. [14] I.B. Badriev, M.V. Makarov, V.N. Paimushin, Solvability of a physically and geometrically nonlinear problem of the theory of sandwich plates with transversal௅soft core, Russian Mathematics. 59 (2015) 57௅60. [15] V.N. Paimushin, Variational methods for solving non௅linear spatial problems of the joining of deformable bodies, Dokl. Akad. Nauk SSSR. 5 (1983) 1083௅1086. [16] V.N. Paimushin, Nonlinear theory of the central bending of three௅layer shells with defects in the form of sections of bonding failure, Soviet Applied Mechanics. 11 (1987) 1038௅1043. [17] V.N. Paimushin, S.N. Bobrov, Refined geometric nonlinear theory of sandwich shells with a transversely soft core of medium thickness for investigation of mixed buckling forms, Mechanics of composite materials. 1 (2000) 59௅66. [18] R.A. Adams, Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975. [19] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, North௅Holland, Amsterdam, 1976. [20] I.B. Badriev, Application of duality methods to the analysis of stationary seepage problems with a discontinuous seepage law, Journal of Soviet Mathematics. 4 (1989) 1310௅1314. [21] I.B. Badriev, On the Solving of Variational Inequalities of Stationary Problems of Two௅Phase Flow in Porous Media, Applied Mechanics and Materials. 392 (2013) 183௅187. [22] R. Glowinski, J.௅L. Lions, R. Tremolieres, Numerical analysis of variational inequalities, Dunod, Paris, 1976. [23] I.B. Badriev, M.M. Karchevskii, Convergence of the iterative Uzawa method for the solution of the stationary problem of seepage theory with a limit gradient, Journal of Soviet Mathematics. 4(1989) 1302௅1309. [24] M. Fortin, R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary௅Value Problems, North௅ Holland, Amsterdam, 1983. [25] I.B. Badriev, O.A. Zadvornov, Analysis of the stationary filtration problem with a multivalued law in the presence of a point source, Differential Equations. 7 (2005) 915–922.