Boundary-value problem for the stability of asymmetrically built and loaded multi-layered circular sandwich type plates

The present report gives the formulation of the mechanical/mathematical models of the stability of asymmetrically built and loaded circular multi-layered sandwich-type plates with (constructionally) orthotropic hard and transversally isotropic soft layers. Using tensor formalism the corresponding basic formulas are given and the governing equations and natural boundary conditions are derived


Formulation of the Problem
In the theory of structures with combined materials the sandwich-type ones plays special role because of their good and predictable physical characteristics.Theories of classical three and regularly multi-layered sandwich plates and shells are done well in the literature.From the point of view of practical applications irregularly built multi-layered plate and shell structures are also very important.Our investigations are dealing with such structures, closely the aim of them are the comparative study of the analytical, experimental and numerical solution of the stability for asymmetrically built and loaded multi-sandwich rectangular plates with (constructionally) orthotropic hard and transversally isotropic soft layers, using the common suppositions in the theory of sandwich and sandwich-type structures (Hoff (1950), Bolotin (1965), Sun et al. (1968)).
Our foregoing research dealt with the stability of regularly multi-layered [5] and asymmetrically built and loaded three-and multi-layered rectangular sandwich-type (consisting of alternating each other hard and soft layers) plates [4], [6].In the latter case -continuing the proceedings -to the formulation of the stability task it was supposed, that • material of all layers are elastic and orthotropic (transversally isotropic) or in the case of constructive anisotropy the layout of the hard layers permits to use the "effective stiffness theory", i.e. by "smoothing" (energetically) the stiffness characteristics of the reinforced layers the stiffness of equivalent flat layers can be determined; • the "hard" layers constitute elastic plates obeying the Kirchhoff-Love laws; • in the "soft" layers the antiplane shear stresses and in the "transversally soft" layers moreover the antiplane normal stresses are also characteristics and all these are constant across the soft layer thickness and proportional to the corresponding strains.These mean, that the displacement fields of soft layers are linear and determinate by the adjacent hard layers (Mindlin-type layers) [1].
• according to the loading of the plate it is supposed, that the hard layers are loaded with a constant normal and tangential to the boundary membrane forces only, but this forces could be different -in particular case zero -for different layers and directions.
In the following for the investigations of the stability of circular sandwich plate all these suppositions are saved.
The main differences in the tasks of the stability of the rectangular and circular sandwich plates are in the geometry.Many of authors at formulation of different tasks of circular plates for deriving the corresponding strain components simply are going out from their form in the Descartes coordinates and use transformation rules between the coordinate systems [2], [3].This method can be work well in simply tasks, but in the case of more complex structures some members of the basic equations can be missed.To avoid this problem in the given paper by formulation of the mathematical-mechanical model of the stability of multi-layered circular sandwich-type plate from the beginning the correct tensor formalism in the system of polar coordinates was used.The corresponding formulas -well known in the thin elastic plate theory -are given in the Appendix 1.

Mathematical -Mechanical Modeling 2.1 Initial position
Let us investigate the stability of the asymmetrically built multi-layered circular sandwich-type plate of radius r 0 with n (constructionally) orthotropic hard and n − 1 transversally isotropic soft layers of thickness h λ and s λ (λ = 1, 2) correspondingly, loaded on the boundary of the hard layers by normal N 11 λ and may be tangential N 12  λ distributed in-plane edge forces.Let join to the plate a polar coordinate system such a way, that the basic [r,θ] plane of the system coincide with the middle surface of the first hard layer and in normal direction to this plane the hard layers let have labels λ = 1, 2, ..., n, where ∆λ = 1.(In case of symmetrically to the middle plane built plate there is more advantageous to join the coordinate system to the middle surface of the plate and to use a labeling λ = ±1, ±2, ±3, ..., ±m, if n is even and λ = 0, ±2, ±4, ..., ±m, if n is odd, where in both case m = n − 1 and ∆λ = 2.) The λ-th soft layer takes place between the hard layers with labels λ and λ + ∆λ (Fig. 1).
Our former investigations for the stability of regularly multilayered (Pomázi (1974 and1992)) and asymmetrically built and loaded three-layered plates with constructionally orthotropic hard layers (Pomázi (1980(Pomázi ( , 1985(Pomázi ( and 1990)) give the basic methods, formulas and governing equations of the problem.

Displacement fields
The mechanical-mathematical model of a multi-layered plate with n hard layers is based on the displacement field of the plate, which could be described by 3n displacement functions (as Lagrangean coordinates) in the coordinate directions u α (x,y), v α (x,y), w α (x,y) of the points belonging to the basic surfaces of the hard layers.The displacements in the hard layers are determined by the Kirchhoff-Love law; in the soft layers -based on the suppositions -they are linear functions of local normal coordinates and can be expressed by the displacements of adjacent hard layers.So, using this linearity of displacement functions and the corresponding constitutive equations the strain and stress fields of the hard and soft layers can be obtained (Pomázi (1980 and1992)).
Going out from the aspects of modeling used in the connected previous works [1], [11] the strain and stress field of the plate in this case will be based also on the displacement fields of the points belonging to the middle surface of the hard layers, consequently the model will be continuous by the in-plane coordinates and discrete in normal direction of the plate, expressed by the labels λ of the layers.
Using the common designations of the mathematical and physical quantities and the basic assumptions for the displacement -strain fields mentioned in point 1. we have the displacements (Fig. 2): For hard layers: where u k (k = α, 3 and α = 1, 2) are the vector coordinates and u x k = (u, v, w)the physical ones.Relation of these coordinates is given by the well known formula: For soft layers: where:

Strain fields:
For hard layers -because of it's plane-shell-type -the strain field can be describe by the covariant derivatives on the middle surface of the displacement field (see Appendix 1.) and so -in general -the strain and curvature tensors are: where by the Kirchhoff -Love law: ε α3 = 0, and so the rotation of the layer's normal: Consequently in the given case the strain field of the hard layers (missing the labels of λ) can be expressed by the following vector/physical coordinates of strain and curvature tensor: For soft layers we have to take into consideration, that displacements of the λ-th hard layeru λ , v λ , w λ playing roll later at using variational theorem -will appear in the strains of adjacent λ-th and λ − ∆λ-th soft layers, therefore the strains for both of soft layers is describing as follows:

Constitutive laws:
Taking into account that by the basic suppositions material of the hard layers are orthotropic and it is transversally isotropic for the soft layers, for all the layers the Hooke's law is valid:

and
For the hard layers: For the transversally soft layers: and the Hooke's law: 2.5 Generalized constitutive laws: a) For hard layers: Hard layers of sandwich-type plates very often are stiffened by stringers or by bending common thin plates.In such "constructionally anisotropic" cases, if on the half-wave length of the solution -functions, characterizing the strain/stress field of the layer there are more stiffeners, then it is possible to formulate the problem for a layer with constant thickness by determining the equivalent stiffness characteristics due to use the effective stiffness theory [11].The σ and τ stresses in the "generalized constitutive law" usually are coupled except if the stiffened layer's material is isotropic and the layer itself has symmetric characteristics, when the model of the layer will be orthotropic plane-shell.
The strain/stress field of such an equivalent orthotropic hard layer with constant h λ thickness -on the basic of the Kirchhoff-Love law -can be determined by the ε strain and κ curvature vectors and by the N internal force and Mbending/torsional moments vectors, getting by integration of the stresses and their static moments on the layer's thickness.Taking ς for local coordinate from the middle plane of the hard layer for these vectors we have: The torsion: H = 2D 66 .χ. Taking into account the Hooke's law for the stress/train relation by point 2.4 after integration we get the constitutive law for the internal forces and moments in form: where C, K, D are the stiffness matrices of the hard layers having forms: Components of this matrices due to determination of the internal forces/moments are integrals on the thickness of the hard layers of the b ik parameters of the constitutive law of the material of the layers: Matrix Kexpress the coupling effect between the stretching and bending of the hard layers, which effect is usual for the (constructionally) orthotropic layer [4], [6].

b) For transversally soft layers:
On the basis of suppositions for the transversally soft layers of constant thickness s, that the strain/stress state of these layers are constant by the normal to the layer, the generalized constitutive law for these layers will be: where stiffnesses of the layers: where G and Ẽz are the average of the shear and Young moduli in normal direction of the layer: 6 Relations between the physical and tensor coordinates:

Formulation of the Boundary-value Problem
Following the method given in [1], [4] the governing equations and natural boundary conditions of the given task in the frame of our suppositions can be derived by using the Trefftz variational principle for the functional of total potential energy of the sandwich plate.The corresponding formulas are as follows:

Energy densities and the total potential energy:
Following the formalism of Bolotin (1965) the strain energy densities of hard and transversally soft layers (missing the label of the hard layers: λ) are respectively : where parameter δ is of second order terms is determined by the quadrate of relations of the full thickness (H) to the characteristic half-wave length of the deformation of the plate (l), or by the relation of the characteristic Young moduli of the soft (E)and the hard Ē layers.
Here the term σ33 ε33 corresponds to the effect of anti-plane tension/compression of the transversally soft layer.If this term is negligibly small compared with others, then can be missed and the layer is as usual simple soft.More sophisticated analysis of the characters of the layers is given by Bolotin (1965), Pomázi and Moskalenko (1967).
Neglecting the second order terms, the energy densities are: After integration of these densities by thickness of the layer we get the surface densities in forms: and calculating also the surface potential energy (work) of external forces Nαβ by the formula: Taking into account the strain expressions for the soft layers by point 2.3 we have: where coefficients of the displacement's variations (writing up by the physical coordinates of the internal forces using relations of point 2.6): In integral-expression of δ Ũλ λ−∆λ the last two member have derivative of the normal displacement's variation δw α therefore -similar to the method used at the hard layers abovethese members after partial integration and using the Gauss-Ostrogradsky theorem: where the surface covariant derivatives: T 13 w 1 = T 13 w , 1 and T 23 w 2 = T 23 w , 2 + 1 r T 13 .
Returning to the physical coordinates of the internal forces and to the derivatives by the coordinates using equalities ( ) , 1 = ( ) , r , ( ) , 2 = ( ) , θ , finally we have:

Fig. 2 .
Fig. 2. The displacement field of the layers