Flat problem of the elastic environment reinforced with three families of fibres

Abstract

A6strucZ-Rectilinear reiaforcing structures are used more often for research of tensely deformed state of flat constructions from fibrous composites. It has been shown in a number of works [I-31, that use o f complex curvilinear structures of reinforcing can results in effective designs taking into account the charge of armature rigidity ana durability. In the given work the general approach on a presence of flat designs reinforcing structures which provide required reinforcing fibres effective work is considered. Resolving systems flat problem are constructed for possible combinations of three fibres families of reinforcing. These families can be not extensible and equal streachable families of fibres. Use of algorithms of invariant solutions of partial differential equations construction has allowed to construct and investigate some model solutions. Let reinforcing is executed by fibres of constant crosssection section. Structural model 141 is used for the statement of a composite. Flat problems of elasticity for the environments reinforced w i t h three families of fibres are considered.The model contains the algebraic and differential equations of reinforcement intensity wk(x, y ) , components of deformations tensor Eii(x,y), deformations in fibres of the first, second and third families E k ( x , y ) , pressure in fibres of the first, second and third families sk (x, y ) , averaging pressure sii (x, y ) where x,y -the Cartesian coordinates, g k ( x , y ) -corners of reinforcing, indexes i, j = 1,2, k = 1,2,3.Under condition of a temperatures field constancy T the initial system is:Notations are used:where Lz -are factors of linear expansion of k -family material of fibres (k = 1,2,3). Symbols mean partial differentiation on coordinates x, y accordingly. The right part in (2) takes into account both a case of equally deformed hmilies offires ( E t = const, E: = const + EL) and a case of not extensible (Ek = 0, E: =E;) fibres families, and their possible combinations (one of families of fibres are equally deformable fibres, others -not extensible fibres). The average pressure sB (x, y ) looks like: 3 k=l sij = USE + s k WRZ bl& .a = 1 -q -wz -w3. (4) 1J Ln (4) pressure in binding are determined under formulas: SF. 3 -EU Eii, ( j = 3 -i , i-l,Z) Here EU ,U,,!,, -are accordingly Jung module, Poisson factor and factor of temperature expansion of a binding material, E; -modules Jung k-family of fibres. Pressure should satis@ to the equations of balance: (l+u) ~l j , l C Q ,~ =+bf,(i=1,2). (5) Right sides in (5) 3 k=l bj = ((1 -U)P, + w~Q)F;: are components of the mass distributed loading on directions of rectangular Cartesian system of coordinates; rc,rk -mass density of materials binding and fibres of families; I$ -components of the specific distributed loading working on a mss unit. Boundary conditions on a contour are added to system (1) -(5). The equation of a contour G is set in the parametrical form: x = fi (s), y = f2 (s), S-any parameter. Here is G = Cp U G, . On a contour Gp static conditions with normal ...

Determination of Fiber-Structure for the Effective Reinforcement of Planar Composite-Material Constructions

“…Fibrous reinforcement allows applying new principles in product design and production, the material and product being created simultaneously as a common integrated technological process. As a result, a product with new unique performance qualities is obtained [1,2,3,4,5,6,7]. Until recently, reinforcement has been carried out predominantly using rectilinear fiber-structures.…”