DSpace at VNU: Determining the plane strain bulk modulus of the composite material reinforced by aligned fibre

VNU JO URNAL OF SCIENCE, M athem atics - Physics, T X X II, N q - 2006 D E T E R M IN IN G TH E PL A N E S T R A IN B U L K M O D U L U S OF T H E C O M P O SIT E M A T E R IA L R E IN F O R C E D B Y A L IG N E D F IB R E N g u y e n D in h D u c^ ’\ N g u y e n T ie n D ac*^ ^ Vietnam National University, Hanoi ^ Department o f Mathematics-Mechanics and Informatics, College o f Science- VNU The composite material reinforced by aligned fibre is an important type of material because of its practical applications in many components of structures This material is a tranversely isotropic medium having five independent elastic constants Many researchers are involved in determining these elastic constants By using com­ plex function method for solving the plane strain problem in elasticity theory, [4] has given a relatively precise method for determining the stiffnesses of transversely isotropic medium [5] has introduced a method for defining the engineering constants of the composite material reinforced by fibre basing on physical experiments In this paper, we would give a derivation method to determine the plane strain bulk modulus of the composite material reinforced by aligned fibre basing on a geomet­ ric approximation of representative volume element Our result is identical to some other authors’ This is a new method to obtain the plane strain bulk modulus of the composite material reinforced by aligned fibre A b s tra c t T h e p r o b l e m Assuming a cyclic structure of the composite material reinforced by aligned fibre, its representative volume element is in form of a rectangular parallelepiped containing a cylinder By using a geometric approximation of outer rectangular parallelepiped, we obtain a new model for representative volume element called composite cylinders model Specifically, T he fibre phase is taken to compose of a number of long circular cylinders embedded by a continuous m atrix phase W ith each individual fibre of radius a, there is an association with an annulus of matrix material of radius b Each individual cylinder combination of this type is referred to a composite cylinder [6] As a result, our problem can be posed as below Figure T he representative volume element of the composite material reinforced by fibre and composite cylinder model T ypeset by 4^ S - T e X N g u y e n D in h D ue, N g u y e n T ie n D a c Let us consider a heterogeneous cylinder composed of two isotropic elastic materials Inner solid cylinder (0 ■< r •< a) and outer cylindrical shell (a ■< r ■< b) made of elastic materials having properties (A1 , Hi) and (A2 , /^2)1 respectively A heterogeneous cylinder is subjected to a hydrostatic stress p2 on its outer boundary r — b Our objective is to obtain the plane strain bulk modulus of two phase composite as a function of elastic properties and volume concentrations of each phase G o v e r n in g r e l a ti o n s h i p s By setting the above problem, the root of the problem is found in form of Ur — u r (r) , Ug = u z = (1) We have the equilibrium as follow dơrr dr _ ơớỡ) - (2) r Taking Eq.(l) into Eq.(2) we have d2u r dur + d r2 r Jdr - 7 Ur _ = °- ^ ) The solution of Eq.(3) is given by ur = A r H r (4) The solution Eq.(4) implies specific forms in the separate fibre and m atrix phases So lving m e t h o d The problem is solved by applying elasticity theory for the plane strain case Specif­ ically, the problem about cylinderical shell is subjected to the hydrostatic stresses boundaries We will consider the displacement and stress states inthe separate fibre on and matrix phases of composite cylinder before examining equivalent homogeneous cylinder 3.1 T h e m a t r i x p h a s e o f c o m p o site c y lin d e r In the area of the m atrix phase, the displacement and stress states in forms u r = A l t H —, (5) r — 2(Ả2 + ^ 2)^2 ~ ^ 2 (đ) D e t e r m i n i n g the p la n e s tr a in bulk m o d u lu s o f In tro d u c in g ơrV into the boundary and interface conditions r(2 ) "=6 = P2 , r=a p (7) (where p is the interaction stress between the fiber and matrix phases), we define inte­ gration constants as well as the displacement and stress fields in the m atrix phase as follows u (2) = Pa2 - P b 2(A2 + ^ )(o2 - b2) (p - p2)a2b2 I 2^2(o2 ~ b2) r ’ (2) _ pq2 - P2b2 _ (p - P2)a2b2 J_ rr a —62 a —ỉ>2 r ( 8) (9) T h e fib r e p h a s e o f the c o m p o site c y lin d e r In this p art (0 •< r ■< a), the displacement and stress fields have forms —A ự , ( 10) ơrr^ = 2(Ai + ụ.\)A \ ( 11) By the continuity conditions of displacement and stress on interface r — a Ơ ( ) r= a r=a ); uur(1) r=a ( 12) and interaction stress p we obtain the following relations of U (1) = ' p = u 2(Ai + a*i ) ’ (Ai *f j ) (A2 + 2/i2)f2^2 (A2 + // ) [(Ai + Hi) + ụ iịb + [(Ai + fix) - (A2 + Ai2)]a2 (13) (14) 3.3 T h e e q u iv a le n t h o m o g e n e o u s c y lin d e r We consider equivalent homogeneous solid cylinder having radius r — b made of isotropic elastic material of properties The homogeneous cylinder is subjected to the hydrostatic stress P2 on the boundary r — b It is clear th a t solved forms are similar to those of the fibre phase u r = A r, (15) ơrr — 2(A + (16) N g u y e n D in h D u e , N g u y e n T ien D a c By defining integration constant from boundary condition, we obtain as follow U r — P (17) - 2(A + ỊÌ) In fact, the displacements at the outer boundaries of the composite cylinder and the equiv­ alent homogeneous cylinder are equated to provide the same average sta te of dilatation within each of them 2) r—b = (18) U r r =b Introducting the relationships Eq.(8), Eq.(17) into Eq.(18) taking into account Eq.(14) gives us the following relationship N (A2 + ^ ) [ ( ^ + Hi) — ( ^ + ^ )]^ (A + M) = (A*+ m ) + (Ai + ; i - ; 2) - [(Ai + m ) z (A s+ W )j • Where £ = a2/b is volume fraction of the fibre phase In fact K 23 = K + ụ ,/3 = A+/J is plane-strain bulk modulus of transversely isotropic medium [7] We carry out some transformations at Eq.(19) and obtain as follows r, T, ^ where K i Hi ^ (i = ,0 , (# + 4/i2/ ) [ K i - # + (/^1 - / ^ ) / ] £ + {K i - W ) ; (1 - [K - n J 1,2) are the bulk and shear moduli of the fibre and m atrix isotropic phases, respectively Expression Eq.(20) is a formula for determining the plane strain bulk modulus of the composite material reinforced by aligned fibre which is a transversely isotropic medium It coincides with the relationship derived by Christensen [1] C o n c lu sio n s By using a geometric approximation of the representative volume element of the composite material reinforced by aligned fibre, this paper presents an alternative derivation method in order to obtain the plane strain bulk modulus of this material The governing ideas of this approach base on the composite cylinder model and applying elasticity theory for the plane strain problem Our result is identical to Christensen’s [1] We believe that this derivation method can be widenned to determine the remaining elastic moduli of the composite material reinforced by aligned fibre, which is a transversely isotropic medium The results of researching presented in this paper have been supported by the National Council for National Sciences D e t e r m i n i n g th e p l a n e s t r a i n bulk m o d u lu s o f References R M Christensen, Mechanics of Composite Materials, John Wiley and Sons Inc New York, 1979 G A Vanin, Micro - mechanics of composite Materials, ” Naukova dumka” , Kiev 1985 Dao Huy Bich, The theory of elasticity, The Vietnam National University Publisher Hanoi, 2001 Nguyen Hoa Thinh, Nguyen Dinh Due, The Composite Material - Mechanics and technology, T he Science and Technique Publisher, Hanoi, 2002 TYan Ich Thinh, Composite material-Mechanics and structural calculations, The Education Publisher, Hanoi, 1994 Nguyen Dinh Due, Hoang Van Tung, Do Thanh Hang, A method for determining the module K of composite material with sphere pad seeds, Journal of Science Mathemathics-Physics, VNƯ, T.XXII, No2, 2006 D K Hale, Review the physical properties of composite materials, Journal of m a­ terials science 11 (1976) pp 2105-2141  ... element of the composite material reinforced by aligned fibre, this paper presents an alternative derivation method in order to obtain the plane strain bulk modulus of this material The governing... are the bulk and shear moduli of the fibre and m atrix isotropic phases, respectively Expression Eq.(20) is a formula for determining the plane strain bulk modulus of the composite material reinforced. .. for determining the module K of composite material with sphere pad seeds, Journal of Science Mathemathics-Physics, VNƯ, T.XXII, No2, 2006 D K Hale, Review the physical properties of composite materials,