LocaL nonsymmetricaL postbuckLing equiLibrium path of the thin fgm pLate niesymetryczna LokaLna ścieżka równowagi pokrytycznej cienkiej płyty z materiału funkcjonaLnie gradientowego

The influence of the imperfection sign (sense) on local postbuckling equilibrium path of plates made of functionally graded materials (FGMs) has been analyzed. Koiter’s theory has been used to explain this phenomenon. In the case of local buckling, a nonsymmetrical stable equilibrium path has been obtained. The investigations focus on a comparison of the semi-analytical method (SAM) and the finite element method (FEM) applied to the postbuckling nonlinear analysis of thin-walled complex FG plated structures.


Introduction
Since the mid 1980's Functionally Graded Materials (FGMs) have been a relatively new class of composite materials, which have become a very popular research field and have been used in numerous engineering applications. A standard functionally gradient material is an inhomogeneous composite made up of two constituents -typically of metallic and ceramic phases. Within FGMs, different microstructural phases have different functions, and the overall FGMs attain the multistructural status from their property gradation. In most cases, these phases content changes gradually along the thickness of the plate or shell. This eliminates adverse effects between the layers (e.g., shear stress concentrations and/or thermal stress concentrations), typical for layered composites what generally improves material utility properties. The combination of ceramic with a metal component improves the characteristics of FGM structures i.e. a better resistance to high temperature (ceramic) and good mechanical features (metal), reducing further a fracture possibility of the whole gradient structure. These features make high temperature environments the leading application area of FGM structures.
The nonlinear analysis of plates and shells devoted to basic types of loads is covered in the monograph by Hui-Shen [4]. Author considers static bending and thermal bending as an introduction to buckling and postbuckling behaviour of FGM plates and shells. The shear deformation effect is employed in the framework of Reddy's higher order shear deformation theory (HSDT) [20].
In [19], alongside the HSDT for FGM plates, Reddy compares the application of the first order shear deformation theory (FSDT) and the classical laminated plate theory (CLPT) to functionally graded plate analysis. According to the presented results for thin-walled plates, it is obvious that an application of the FSDT gives practically the same results as the HSDT. The discrepancy between both theories is of 2% in the calculated deflections of the plates under analysis.
The buckling problem of functionally graded plates is discussed in the frame of different approaches and for different loads: in [21] -biaxial in-plane compression; thermal loads (constant temperature) with axial compression in [24]; biaxial in-plane compression in [2] and [16], and through the thickness temperature gradient in [23].
Birman and Byrd [1] give a wide review of theories employed for a description of grading material properties and focus on the principal developments in functionally graded materials (FGMs) with an emphasis on the recent works published since 2000 (up to 300 works cited).
In some papers (e.g., [14,27]), the concept of 'physical neutral surface' that allows one to uncouple the in-plane and out-of-plane deformations is introduced.
Due to the complexity of buckling problems of FG plates under compound mechanical and thermal loads, the finite element method (FEM) seams to be the only possible solution in many cases. Therefore, in the literature one can find many papers which present results of a solution to different problems of FG plate buckling, obtained with an application of the FEM, for example [15,17,22].
In current paper in the finite element method solution, FG plates were modelled as multilayered composite structures whose graded material properties in the range of 10-40 isotropic layers were defined. After the convergence analysis the model with twenty layers sciENcE aNd tEchNology was accepted. For meshing, a shell element with four nodes and six degrees of freedom in each node was employed. The rotational DOF in the plane of the element was constrained via the penalty function.
Conducting with the FEM the nonlinear buckling analysis of a rectangular FGM plate, subjected to one-directional compression in its plane, the authors of the present paper have observed some intriguing influence of the imperfection sign (i.e., its direction) on postbuckling equilibrium paths of investigated FGM plates. Therefore, this work is aimed at an explanation of this phenomenon. The general asymptotic Koiter's theory of stability has been assumed as the basis of investigation. Among all versions of the general nonlinear theory, Koiter's theory [6,7,25,26] of conservative systems is the most popular one, owing to its general character and development. Even more, so after Byskov and Hutchinson [3] formulated it in a convenient way. The theory is based on asymptotic expansions of the postbuckling path for potential energy of the system.
The nonlinear stability of thin-walled multilayer structures in the first order approximation of Koiter's theory is solved with the modified analytical-numerical method (ANM) presented in [8]. The analytical-numerical method (ANM) should also consider the second order approximation in the postbuckling analysis of elastic composite structures. The second order postbuckling coefficients were estimated with the semi-analytical method (SAM) [12], modified by the solution method given in [11]. The investigation of stability of equilibrium states requires an application of a nonlinear theory that enables us to estimate an influence of different factors on the structure behaviour. The analysis of postbuckling behaviour of thin-walled composite plate structures using the SAM will be by far faster and more thorough than the FEM.
The initial imperfections were introduced by updating the finite element mesh with the first mode shape of the eigen-buckling solution, with a given magnitude corresponding to the plate thickness and assumed sign (direction). The eigen-buckling analysis, where the critical load was determined despite the eigen-mode, preceded the nonlinear buckling analysis.

Formulation of the problem
The square plate is supported at all their edges. It is assumed that the FG plate obeys Hooke's law. The material properties are assumed to be temperature independent.
In strain-displacement relations -in order to enable the consideration of both out-of-plane and in-plane bending of the plate, all nonlinear terms are present [8,9,11]: and κ κ κ where: , u v, w -are components of the displacement vector of the plate in the , , x y z axis direction, respectively, and the plane x y − overlaps the midplane before its buckling.
It should be highlighted that in the majority of publications devoted to stability of structures, the terms 2 2 , , ( ) x y x y u u v v + in strain tensor components (1) are neglected. However, the main limitation of the assumed theory lies in an assumption of linear relationships between curvatures (2) and second derivatives of the displacement w . In such an approach, finite displacements and small or moderate rotations are considered [11].
In thin-walled FG structures -plates or shells, usually the ceramic volume fraction V c and metal fraction V m distribution throughout the structure thickness t are described by a simple power law of: and 0 q ≥ is the volume fraction exponent (i.e., for 0 q = -plate is full ceramic and for q = ∞ -plate is metallic -see Fig. 1).
According to the rule of mixture, the properties of the functionally graded material (E -Young's modulus, ν -Poisson's ratio etc.) can be expressed as follows: In the present study, the classical plate theory is employed to obtain the governing equations of the thin FG plate equilibrium. Using the classical laminated plate theory (CLPT), the stress and moment resultants ( N , M ) are defined as [5,8,9]: where: A , B , D -are extensional, coupling and bending stiffness matrices, respectively. For the FG plate their components are listed below: Due to the presence of the nontrivial submatrix B , the coupling between extensional and bending deformations exists as it is in the case of unsymmetrical laminated plates [5,8,9]. An extensional force results not only in extensional deformations, but also bending of the FG plate. Moreover, such a plate cannot be subjected to the moment without suffering simultaneously from extension of the middle surface. Coupling between extension and bending is a result of a combination of the geometry and FGM properties in the structures. The stretching-bending coupling affects strongly the constitutive equations and the boundary conditions that have a complex form and the solution procedures become difficult.
The equations of stability of thin-walled structures have been derived using a variational method [8,9,11]. After expanding the fields of displacements U and the fields of sectional forces N into a power series with respect to the mode amplitudes ζ 1 (the dimensionless amplitude of the buckling mode), Koiter's asymptotic theory has been employed [3,6,7,8,9,10,13,18,25,26]: where: λ -load parameter, The postbuckling equilibrium path within an imperfect structure with the amplitude ζ 1 * for the single mode (i.e., an uncoupled mode), buckling mode has the following form [3,8,9,13,26]: a a a a cr cr where: σ cr -critical (bifurcational) value of σ (instead of λ ). The coefficients in equilibrium equation (9) are given in papers [3,8,9,13,26]. It can be easily seen that the amplitude ζ 1 * is a small quantity (i.e., only linear terms with respect to ζ 1 * have been accounted for) and the linear pre-buckling state is assumed. The corresponding expression for the total elastic potential energy of the structure has the following form:  [3,8,9,13,26] -to be precisely determined.
The considerations performed in [12] and the results of [11] allow for concluding that the approximated values of the 1111 a coefficients correspond to an application of the following simple supported boundary conditions of the plate at both edges (i.e. x = 0; ) The first condition in (11) means that the external loading is not subjected to any additional increment.
At the critical point, the dependence describing the relationship for the ideal structure (i.e., without the imperfection ζ 1 0 * = ) is subject to bifurcation between the external loading and the displacement amplitude ζ 1 . Equilibrium path equation (9) can be treated as the first variation of the system potential energy, that is to say, as the condition necessary for the system equilibrium.  aNd tEchNology equilibrium state, the second variation of energy was calculated Then the intersection points of the equilibrium path and the structure stability limit of the ideal structure were determined. Finally, the coordinates of two points 1, 2 p p were arrived at: A sample equilibrium path for the ideal square FGM plate is presented in Fig. 2. There it is visible that in the case of the FG plate, an nonsymmetrical stable equilibrium path exists. The unstable configuration corresponds to the postbifurcational equilibrium path of the plate without imperfection in the range < In the linear problem, the critical stress σ cr is the characteristic quantity, whereas in the nonlinear first order problem, the magnitude of the coefficient 111 a determining the sensitivity to imperfections should be accounted for.

Analysis of the results
Detailed numerical computations were conducted only for a square FGM plate. The plate is subjected to uniform compression in the direction of x axis. All plate edges are assumed to be simply supported. Although, in the subsection devoted to determination of critical stress, some other boundary conditions along the unloaded edges are considered as well.
The following geometrical dimensions of the square plate (Fig. 3) and the material constants for Al-TiC are assumed: where: indices m and c refer to the metal (Al) and ceramic material (TiC), respectively. In [8], an unbending, prebuckling state, i.e., a distribution field of the zero state according to (A1) has been assumed. According to assumed (A1) and (8) displacements, for the zero state (i.e., prebuckling) the force and moment dependence (5) takes the form: Then for the zero state, it results in an occurrence of nonzero inner sectional forces (A2) (0) , is presented. In legends to these figures the descriptions mean: F -free edges, for which the following has been assumed:  (Fig. 4) for the boundary conditions under consideration increase with a decrease in the values of volume fractions q -as can be expected. Differences in the values of critical loads for cases S1 and S2 become visible for 0.5 q > . They are relatively inconsiderable (below 10%) and larger for case S2. a for case C are by approximately 60% lower than for conditions S1 and S2. For condition S2, the values of 1111 a are higher than for S1, and these differences become visible for 0.5 q > . However, they do not exceed 5% for the range of variability 0.1 10 q ≤ ≤ under consideration. As it has been discussed in detail in [8], sectional moments of the zero state ⋅ l U (according to the notation introduced by Byskov and Hutchinson [3,26]). An appearance of first order internal forces (i.e., related to the forces (1) (1) (1) , , = ) -respectively for cases F, S1, S2, and C; one obtains different distributions of inner forces of the first order, i.e., (1) (1) , , , , x y xy M M M . These inner forces values are determined with accuracy up to a constant, as it takes place for eigenproblems. The conditions on loaded edges (11a) enforce a generation of a self-balancing system of forces (1) x N . Below, few exemplary diagrams for S2 boundary conditions on longitudinal edges, for 0.5 q = are shown.
In Fig.7, distributions of first-order inner membrane forces , , , , To verify the proposed SAM solution, finite element computations were performed for the FG plate under axial loading. The commercial ANSYS software was applied for the numerical calculations. The numerical model was created with an application of a shell finite element. It was a multi-layered four-node element with six degrees of freedom at each node (three translations in the directions of local coordinate axes and three rotations around these axes). The rotational DOF around the normal to the plate midplane was constrained via the penalty function to relate this independent rotation with the in-plane components of displacements. This element is dedicated for modelling multi-layered structures and is equipped with the section option which allows for easy tailoring the lay-ups of the modelled plate. The sensitivity to shear strains in this element is governed by the first-order shear deformation theory, whereas the element formulation is based on the logarithmic strain measure. According to the current analysis requirements, the applied finite element was associated with linear elastic material properties. To discretize the model, a uniform mesh of elements was generated. The boundary conditions on loaded plate edges, which followed from S1 type analytical simple support, were introduced by displacement constrains in appropriate directions as well as coupling of edge node displacements to keep the edges straight.
The initial imperfection was introduced by updating the finite element mesh with the local mode shape of the eigen-buckling solution, with a given magnitude corresponding to the plate thickness. The eigen-buckling analysis, where the critical load was determined despite the eigen-mode, preceded the nonlinear analysis. Therefore, the numerical model employed large displacement formulation. The load was applied to the plate edges in the form of uniformly distributed node forces.
For the square plate under analysis and for 0.5 q = , the results of calculations obtained from the SAM and the FEM were compared. A comparison of the results is presented in Fig. 9. were determined approximately. It is followed by visible differences in the results obtained with both the methods (SAM and FEM) for the same value of imperfection. It can be seen that the sign of imperfection exerts an influence on the postbuckling equilibrium path. The initial deflection ζ 1 * along ceramic yields higher values of total deflections for the given value of load / cr N N than the initial deflection along the direction of metal.
In both cases the assumed absolute value of imperfection ζ 1 * was equal. Thus, as it was discussed above, the application of Koiter's theory through the semi-analytical method enables an explanation of the phenomenon of various postbuckling equilibrium paths for the functionally graded plate for different signs of imperfection with the same absolute magnitude ζ 1 * . In particular, it can be seen for

Conclusions
The analytical and numerical investigations on FGM -a relatively novel material, applications in plate and shell structures are presented. The effect of gradually varying volume fraction of constituent materials leads to continuous change from one surface to another eliminating interface problems and gives smooth material properties of final composite which is especially import in thermal environment applications.
An influence of imperfection values on various postbuckling equilibrium paths of the FG plate has been analyzed. The basis to explain the discussed behaviour is the nonlinear Koiter's theory of conservative systems. In the case of the FG plate, nonzero first-order sectional inner forces that cause an occurrence of nonzero postbuckling coefficients are responsible for the system sensitivity to imperfection. It results in the fact that postbuckling equilibrium paths of plate structures made of FGMs are unsymmetrically stable. This explains the observed differences in plate response dependence on imperfections sign (sense). where Δ is the actual loading. This loading of the zero state is specified as a product of the unit loading and the scalar load factor. Taking into account relationship (5), inner sectional forces of the prebuckling (i.e., unbending) state for the assumed homogeneous field of displacements (A1) are expressed by the following relationships before the redistribution of forces in the plate due to plate deformations: The assumed displacement field and the field of inner forces, corresponding to it for the prebuckling state, fulfil equilibrium equations for the zero state as an identity.
The omission of the displacements of the fundamental state implies that we ignore the difference between the configuration of the non-deformed state and the fundamental state and we may consequently regard the previously defined displacements u v ( ) ( ) , 0 0 as the additional ones from the fundamental state to the adjacent state.
The first order approximation, being the linear problem of stability, allows for determination of values of critical loads, buckling modes, and initial postbuckling equilibrium paths.