Composite Interfaces, Vol. 14, No 1, pp 49-62(2007) Alsoavailableonlinewww.brillnl/ci Influence of inhomogeneous interphase on thermal stresses in fiber-reinforced composites L.H. YOU* College of Mechanical Engineering, Chongqing University, Chongqing City 400044, China and NCCA, Bournemouth University, Talbot Campus, BH12 5BB, UK Received 14 September 2005: accepted 5 January 2006 Abstract-The effects of an inhomogeneous interphase on thermal response of fiber-reinforced composites are investigated in this paper. The work is based on a linear variation of Youngs modulus and thermal expansion coefficient of the interphase and the assumption of generalized plane strain. An accurate analytical approach is presented to determine thermal stresses in composites reinforced with isotropic fibers containing an inhomogeneous interphase. How the inhomogeneous interphase, and Youngs modulus and thermal expansion coefficient of the matrix affect thermal response of the composites is examined. It is found that the inhomogeneous interphase causes different stress distribution from that of the homogeneous interphase. Raising Youngs modulus and thermal expansion coefficient of the matrix obviously increases the maximum radial, circumferential and axial stresses in all the constituents of the composites Keywords: Thermal stresses: fiber-reinforced composites; varying interphase: Youngs modulus thermal expansion coefficient; generalized plane st 1 INTRODUCTION Due to different material properties of the constituents, different deformations and stresses will occur in these constituents of fiber-reinforced composites even ubjected to the same temperature loads. These thermal deformations and stresse may cause some undesirable outcomes such as fiber breaks, debonds and cracks at interface, and plastic deformations of interfacial layers and matrix [1-3]. Therefore thermal analysis is always an active topic in composite mechanics There have been quite a number of numerical and analytical approaches deal- ing with thermal analysis of fiber-reinforced composites with a homogeneous inter- ase E-mail: Lyou@bournemouth. ac uk
Composite Interfaces, Vol. 14, No. 1, pp. 49–62 (2007) VSP 2007. Also available online - www.brill.nl/ci Influence of inhomogeneous interphase on thermal stresses in fiber-reinforced composites L. H. YOU ∗ College of Mechanical Engineering, Chongqing University, Chongqing City, 400044, China and NCCA, Bournemouth University, Talbot Campus, BH12 5BB, UK Received 14 September 2005; accepted 5 January 2006 Abstract—The effects of an inhomogeneous interphase on thermal response of fiber-reinforced composites are investigated in this paper. The work is based on a linear variation of Young’s modulus and thermal expansion coefficient of the interphase and the assumption of generalized plane strain. An accurate analytical approach is presented to determine thermal stresses in composites reinforced with isotropic fibers containing an inhomogeneous interphase. How the inhomogeneous interphase, and Young’s modulus and thermal expansion coefficient of the matrix affect thermal response of the composites is examined. It is found that the inhomogeneous interphase causes different stress distribution from that of the homogeneous interphase. Raising Young’s modulus and thermal expansion coefficient of the matrix obviously increases the maximum radial, circumferential and axial stresses in all the constituents of the composites. Keywords: Thermal stresses; fiber-reinforced composites; varying interphase; Young’s modulus; thermal expansion coefficient; generalized plane strain. 1. INTRODUCTION Due to different material properties of the constituents, different deformations and stresses will occur in these constituents of fiber-reinforced composites even subjected to the same temperature loads. These thermal deformations and stresses may cause some undesirable outcomes such as fiber breaks, debonds and cracks at interface, and plastic deformations of interfacial layers and matrix [1–3]. Therefore, thermal analysis is always an active topic in composite mechanics. There have been quite a number of numerical and analytical approaches dealing with thermal analysis of fiber-reinforced composites with a homogeneous interphase. ∗E-mail: Lyou@bournemouth.ac.uk
LH. You Ishikawa et al. presented a collocation method to calculate thermal expansion coefficients of unidirectional composites [4]. Using a three-dimensional hexagonal array finite element model, Bowles and Tompkins determined the longitudinal and transverse thermal expansion coefficients of different graphite fiber-reinforced resin, metal, and ceramic matrix composites [5]. With plane stress, plane strain and generalized plane strain finite element models, Muller and Schmauder examined the residual stresses at the fiber-matrix interface of the composites with regular cubic and hexagonal fiber arrangements [6]. Treating the geometrical structure of fiber reinforced composites as perfectly periodic, uniformly spaced fiber arrangements in square and hexagonal cells, as well as different cells in which either 30 or 60 fibers are randomly placed in the ductile matrix, Nakamura and Suresh investigated the combined effects of thermal residual stresses and fiber spatial distribution on the deformation of a 6061 aluminum alloy by using the finite element code ABAQUS [7]. Employing a 3D prism for the hexagonal fiber pattern, Abedian and Szyszkowski discussed stress singularities on the free surface at the fiber/matrix nterface with the ANSYS finite element code [8]. Recently, Choo et al. carried out a finite element analysis of the inelastic relaxation of the time-and path-dependent thermal residual stresses in continuous fiber-reinforced composites [9]. Chen and Liu presented a boundary element method with thin-body capabilities to mod multiple cells of fiber-reinforced composites [10] Numerical methods are effective in coping with complicated situations of thermal analysis of the composites. However, they are expensive. Therefore, some analytical models were developed to simplify the analysis Treating continuous fiber-reinforced composites as a cylinder consisting of a fiber and a concentric binder shell, Hashin and Rosen predicted the elastic moduli of the composites with hexagonal and random arrays of fibers [ll]. Using the rule of mixtures to calculate the properties of the outer cylinder, Mikata and Taya gave a four concentric cylinder model which consists of innermost transversely isotropic fiber, middle isotropic coating and matrix, and outermost transversely isotropic composite, and determined the elastic stress field in coated continuous fiber-reinforced composites subjected to thermomechanical loading [12]. Using the composite cylinder assemblage model of Hashin [13] to determine the properties of the outer cylinder, Kuntz et al. [14] also investigated the effects of fiber coating on the residual stresses in the same composites as those investigated by Mikata and Taya [12]. Applying the assumption of the generalized plane strain, Avery nd Herakovich developed an analytical method to determine the deformations and stresses in the composites reinforced with circumferentially orthotropic, radially orthotropic and transversely isotropic fibers [15]. Considering some composites may contain a number of coatings, Warwick and Clyne extended the model of Mikata and Taya [12] to that consisting of an arbitrary number of component cylinders, and predicted the thermal stresses in SiC monofilament systems [16] Sutcu presented a simple recursive algorithm to determine the material properties of a uniaxially aligned composite which contains an arbitrary number of coatings
50 L. H. You Ishikawa et al. presented a collocation method to calculate thermal expansion coefficients of unidirectional composites [4]. Using a three-dimensional hexagonal array finite element model, Bowles and Tompkins determined the longitudinal and transverse thermal expansion coefficients of different graphite fiber-reinforced resin, metal, and ceramic matrix composites [5]. With plane stress, plane strain and generalized plane strain finite element models, Müller and Schmauder examined the residual stresses at the fiber–matrix interface of the composites with regular cubic and hexagonal fiber arrangements [6]. Treating the geometrical structure of fiberreinforced composites as perfectly periodic, uniformly spaced fiber arrangements in square and hexagonal cells, as well as different cells in which either 30 or 60 fibers are randomly placed in the ductile matrix, Nakamura and Suresh investigated the combined effects of thermal residual stresses and fiber spatial distribution on the deformation of a 6061 aluminum alloy by using the finite element code ABAQUS [7]. Employing a 3D prism for the hexagonal fiber pattern, Abedian and Szyszkowski discussed stress singularities on the free surface at the fiber/matrix interface with the ANSYS finite element code [8]. Recently, Choo et al. carried out a finite element analysis of the inelastic relaxation of the time- and path-dependent thermal residual stresses in continuous fiber-reinforced composites [9]. Chen and Liu presented a boundary element method with thin-body capabilities to model multiple cells of fiber-reinforced composites [10]. Numerical methods are effective in coping with complicated situations of thermal analysis of the composites. However, they are expensive. Therefore, some analytical models were developed to simplify the analysis. Treating continuous fiber-reinforced composites as a cylinder consisting of a fiber and a concentric binder shell, Hashin and Rosen predicted the elastic moduli of the composites with hexagonal and random arrays of fibers [11]. Using the rule of mixtures to calculate the properties of the outer cylinder, Mikata and Taya gave a four concentric cylinder model which consists of innermost transversely isotropic fiber, middle isotropic coating and matrix, and outermost transversely isotropic composite, and determined the elastic stress field in coated continuous fiber-reinforced composites subjected to thermomechanical loading [12]. Using the composite cylinder assemblage model of Hashin [13] to determine the properties of the outer cylinder, Kuntz et al. [14] also investigated the effects of fiber coating on the residual stresses in the same composites as those investigated by Mikata and Taya [12]. Applying the assumption of the generalized plane strain, Avery and Herakovich developed an analytical method to determine the deformations and stresses in the composites reinforced with circumferentially orthotropic, radially orthotropic and transversely isotropic fibers [15]. Considering some composites may contain a number of coatings, Warwick and Clyne extended the model of Mikata and Taya [12] to that consisting of an arbitrary number of component cylinders, and predicted the thermal stresses in SiC monofilament systems [16]. Sutcu presented a simple recursive algorithm to determine the material properties of a uniaxially aligned composite which contains an arbitrary number of coatings
Influence of inhomogeneous interphase on composites on its fibers [17]. You gave an efficient method to tackle thermal analysis of composites containing anisotropic continuous fibers and an arbitrary number of interfacial layers [18] The above analytical research studies did not consider the plasticity of composites When thermal deformations of composites are large enough, some components of composites may become plastic. In order to address this problem, Zhang et al. used a two concentric cylinder model together with the increment theory of plasticity to derive two partial differential equations governing the deformation behavior of composites and numerically solved these two partial differential equations [19] Adopting Tresca's yield criterion and linear strain-hardening material model, You and Long discussed the effects of material properties of interfacial layer on thermal response of fiber-reinforced composites [20]. This work was further extended to the composites reinforced with anisotropic fibers [21] Compared to the amount of work that has been published on the composites with a homogeneous interphase, there are far fewer publications dealing with thermal or thermomechanical analysis of the composites with an inhomogeneous interphase Taking the variation of Youngs modulus of the interphase to be constant, or sub ject to power variation, reciprocal variation and cubic variation, respectively, Ja yaraman and Reifsnider investigated thermal residual stresses in fiber-reinforced composites with an inhomogeneous interphase [22]. Using the same variations to describe the thermal expansion coefficient of the interphase, they also discussed the effect of the interphase on residual thermal stresses in the composites [23]. In- troducing adhesion parameters to describe the imperfect adhesions between main constituents and taking into account the inhomogeneous interphase, Kakavas et al. determined th e coeficient of thermal expansion of fiber-reinforced compos- The introduction of the continuously varying material properties of the inhomoge- neous interphase greatly increases the complexity to achieve an analytical solution of the thermal or thermomechanical analysis of the composites. In this paper, an effort will be made to address this issue. By employing a linear variation of mater ial properties of the inhomogeneous interphase, a second order ordinary differential equation will be derived from the assumption of the generalized plane strain and the basic equations of axisymmetric problems in elasticity. The equation is further changed into a hypergeometric equation by variable substitution whose accurate analytical solution is represented with the hypergeometric function. The bound ary conditions of the composites and the continuous conditions of deformations and stresses between the constituents are employed to determine all the unknown constants. The obtained analytical solution is used to examine the effects of the inhomogeneous interphase, material properties of the matrix on thermal stresses in the composites
Influence of inhomogeneous interphase on composites 51 on its fibers [17]. You gave an efficient method to tackle thermal analysis of composites containing anisotropic continuous fibers and an arbitrary number of interfacial layers [18]. The above analytical research studies did not consider the plasticity of composites. When thermal deformations of composites are large enough, some components of composites may become plastic. In order to address this problem, Zhang et al. used a two concentric cylinder model together with the increment theory of plasticity to derive two partial differential equations governing the deformation behavior of composites and numerically solved these two partial differential equations [19]. Adopting Tresca’s yield criterion and linear strain-hardening material model, You and Long discussed the effects of material properties of interfacial layer on thermal response of fiber-reinforced composites [20]. This work was further extended to the composites reinforced with anisotropic fibers [21]. Compared to the amount of work that has been published on the composites with a homogeneous interphase, there are far fewer publications dealing with thermal or thermomechanical analysis of the composites with an inhomogeneous interphase. Taking the variation of Young’s modulus of the interphase to be constant, or subject to power variation, reciprocal variation and cubic variation, respectively, Jayaraman and Reifsnider investigated thermal residual stresses in fiber-reinforced composites with an inhomogeneous interphase [22]. Using the same variations to describe the thermal expansion coefficient of the interphase, they also discussed the effect of the interphase on residual thermal stresses in the composites [23]. Introducing adhesion parameters to describe the imperfect adhesions between main constituents and taking into account the inhomogeneous interphase, Kakavas et al. determined the coefficients of thermal expansion of fiber-reinforced composites [24]. The introduction of the continuously varying material properties of the inhomogeneous interphase greatly increases the complexity to achieve an analytical solution of the thermal or thermomechanical analysis of the composites. In this paper, an effort will be made to address this issue. By employing a linear variation of material properties of the inhomogeneous interphase, a second order ordinary differential equation will be derived from the assumption of the generalized plane strain and the basic equations of axisymmetric problems in elasticity. The equation is further changed into a hypergeometric equation by variable substitution whose accurate analytical solution is represented with the hypergeometric function. The boundary conditions of the composites and the continuous conditions of deformations and stresses between the constituents are employed to determine all the unknown constants. The obtained analytical solution is used to examine the effects of the inhomogeneous interphase, material properties of the matrix on thermal stresses in the composites
LH. You 2. ANALYTICAL FORMULAE OF DEFORMATIONS AND STRESSES Fiber-reinforced composites can be simplified as a three-phase model consisting of the inner fiber, middle interphase and outer matrix whose outer radii are rf ri and rm, respectively. In order to obtain an accurate analytical solution, all the constituents of the composites are taken to be isotropic. The variations of Youngs modulus and thermal expansion coefficient of the interphase are taken to be linear Since the Poissons ratio of the interphase has a weak influence on thermal stresses in the composites, it is assumed to be constant, i. e. does not vary along the radial direction. Using the subscript i to indicate the interphase, the variations of Youngs modulus and thermal expansion coefficient of the interphase can be written as Ei= eo+ eir Mi=ao +ar According to the following continuous conditions of material properties at the interfaces between the fiber and interphase and between the interphase and matrix r, E= et f Ei= em, ai=a the unknown constants in equation(1) are found to be r E ri E Subjected to a uniform temperature change, the deformations of the composites can be decomposed into elastic and thermal ones, respectively. Using Hooke's law in elasticity to describe the elastic deformations, the total radial, circumferential and axial strains can be formulated below E[t,-w(+0)+a△T [o-v(o+a)+a△T a2-v1(+a)]+a1△T According to the assumption of the generalized plane strain, the axial strain in all the constituents of the composites is equal to a uniform strain Ez0. Substituting it and
52 L. H. You 2. ANALYTICAL FORMULAE OF DEFORMATIONS AND STRESSES Fiber-reinforced composites can be simplified as a three-phase model consisting of the inner fiber, middle interphase and outer matrix whose outer radii are rf , ri and rm, respectively. In order to obtain an accurate analytical solution, all the constituents of the composites are taken to be isotropic. The variations of Young’s modulus and thermal expansion coefficient of the interphase are taken to be linear. Since the Poisson’s ratio of the interphase has a weak influence on thermal stresses in the composites, it is assumed to be constant, i.e. does not vary along the radial direction. Using the subscript i to indicate the interphase, the variations of Young’s modulus and thermal expansion coefficient of the interphase can be written as Ei = E0 + E1r, αi = α0 + α1r. (1) According to the following continuous conditions of material properties at the interfaces between the fiber and interphase and between the interphase and matrix r = rf , Ei = Ef , αi = αf , r = ri, Ei = Em, αi = αm, (2) the unknown constants in equation (1) are found to be E0 = riEf − rf Em ri − rf , E1 = Em − Ef ri − rf , (3) α0 = riαf − rf αm ri − rf , α1 = αm − αf ri − rf . Subjected to a uniform temperature change, the deformations of the composites can be decomposed into elastic and thermal ones, respectively. Using Hooke’s law in elasticity to describe the elastic deformations, the total radial, circumferential and axial strains can be formulated below εr = 1 Ei σr − νi(σθ + σz) + αiT, εθ = 1 Ei σθ − νi(σr + σz) + αiT, (4) εz = 1 Ei σz − νi(σθ + σr) + αiT. According to the assumption of the generalized plane strain, the axial strain in all the constituents of the composites is equal to a uniform strain εz0. Substituting it and
Influence of inhomogeneous interphase on composites the following equilibrium equation +σr-0=0 into the third of the equations(4), the axial stress can be written as 0=2,她+E1(-a△7) The substitution of equations (5)and(6) into the first two of equation(4)leads to the total radial and circumferential strains as follows ,=(--2)0-(1+的 vE20+(1+)a;△T, (1-u-202)0,+(1-v2)r d-/-le:0+(1+)x;△T Substituting the radial and circumferential strains in the above equation into the following deformation compatibility equation and making use of equation (1), a second order ordinary differential equation describing the deformation of the composites is achieved below (Eoter)r+(3Eo+2Er) E1 (E0+E1r)2=0.(9) The general solution of the above equation consists of the solution of its homoge neous equation and the particular solution of equation(9). Dividing equation(9)by Eo and denoting no= E1/Eo, the homogeneous form of equation(9)can be written (1+m0d2+(3+2nor) noo =0 (10) If we introduce a new variable z= -nor, equation(10) can be changed into the following hypergeometric equation -0, The solution of the above hypergeometric equation can be represented with the hypergeometric function which has the form of [25]
Influence of inhomogeneous interphase on composites 53 the following equilibrium equation r dσr dr + σr − σθ = 0, (5) into the third of the equations (4), the axial stress can be written as σz = 2νiσr + νir dσr dr + Ei(εz0 − αiT ). (6) The substitution of equations (5) and (6) into the first two of equation (4) leads to the total radial and circumferential strains as follows εr = 1 Ei (1 − νi − 2ν2 i )σr − νi(1 + νi)r dσr dr − νiεz0 + (1 + νi)αiT, (7) εθ = 1 Ei (1 − νi − 2ν2 i )σr + (1 − ν2 i )r dσr dr − νiεz0 + (1 + νi)αiT. Substituting the radial and circumferential strains in the above equation into the following deformation compatibility equation r dεθ dr + εθ − εr = 0, (8) and making use of equation (1), a second order ordinary differential equation describing the deformation of the composites is achieved below (E0+E1r)r d2σr dr2 +(3E0+2E1r) dσr dr −1 − 2νi 1 − νi E1σr+ α1T 1 − νi (E0+E1r)2 = 0. (9) The general solution of the above equation consists of the solution of its homogeneous equation and the particular solution of equation (9). Dividing equation (9) by E0 and denoting n0 = E1/E0, the homogeneous form of equation (9) can be written as (1 + n0r)r d2σr dr2 + (3 + 2n0r) dσr dr − 1 − 2νi 1 − νi n0σr = 0. (10) If we introduce a new variable z = −n0r, equation (10) can be changed into the following hypergeometric equation (1 − z)z d2σr dz2 + (3 − 2z) dσr dz + 1 − 2νi 1 − νi σr = 0. (11) The solution of the above hypergeometric equation can be represented with the hypergeometric function which has the form of [25]
LH. You G=A1F(a,b,C,z)+B1(1-x)-a-bF(c-a,c-b,1-a-b+c,1-x),(12) where Ai and bi are unknown constants, and 0.5 5-9 b=0.5(1+ C=3 and F(a, b, c, z)is called the hypergeometric function of variable z with parameters a, b and c whose mathematical representation is ab a(a+1b(b+1) F(a,b,c, z=1+z+ 2!c(Cc+1) n(a+1)(a+2)b(b+1)(b+2) 3!c(c+1)(c+2) According to equation(9), the particular solution can be taken as the following form r =co+ Cir +car Substituting equation (15) into(9), the unknown constants co, CI and c2 in the above equation are found to be Ec1△T (5-4v)E1 (16) E1a1△T Substituting equation (16) back into(15), then superimposing the particular solution (15) upon the homogeneous solution(12), and making use of the relationshil z =-nor, the general solution of the second order ordinary differential equation (9)is obtained as o= A, F(a, b, c, -nor)+B, (1 +nor)c-a-b F(c b.1 b+c, 1+nor) c1△T (5-4)E1(E0+E1r)2 (17) The circumferential stress can be obtained as follows by substituting the above radial stress and its first derivative into the equilibrium equation (5)
54 L. H. You σ¯r = AiF(a, b, c, z) + Bi(1 − z)c−a−b F(c − a,c − b, 1 − a − b + c, 1 − z), (12) where Ai and Bi are unknown constants, and a = 0.5 1 − 5 − 9νi 1 − νi , b = 0.5 1 + 5 − 9νi 1 − νi , (13) c = 3, and F(a, b, c, z) is called the hypergeometric function of variable z with parameters a,b and c whose mathematical representation is F(a, b, c, z) = 1 + ab 1!c z + a(a + 1)b(b + 1) 2!c(c + 1) z2 + a(a + 1)(a + 2)b(b + 1)(b + 2) 3!c(c + 1)(c + 2) z3 +··· . (14) According to equation (9), the particular solution can be taken as the following form ¯ σ¯ r = c0 + c1r + c2r2 . (15) Substituting equation (15) into (9), the unknown constants c0, c1 and c2 in the above equation are found to be c0 = − E2 0α1T (5 − 4νi)E1 , c1 = −2E0α1T 5 − 4νi , (16) c2 = −E1α1T 5 − 4νi . Substituting equation (16) back into (15), then superimposing the particular solution (15) upon the homogeneous solution (12), and making use of the relationship z = −n0r, the general solution of the second order ordinary differential equation (9) is obtained as σr = AiF(a, b, c, −n0r) + Bi(1 + n0r)c−a−b × F(c − a,c − b, 1 − a − b + c, 1 + n0r) − α1T (5 − 4νi)E1 (E0 + E1r)2 . (17) The circumferential stress can be obtained as follows by substituting the above radial stress and its first derivative into the equilibrium equation (5)
Influence of inhomogeneous interphase on composites 0e= Ail F(a, b, c, -nor)- F(a+1,b+1 +B:(1+n)=-1+(c-a-b),"0 x F(c-a,c-b,I-a-b+c, 1+nor (c-a(c-b)nor F(c-a+1,c-b+1,2-a-b+c,1+nor) a-b+ △T xIA E+4E0EIr+3Eir) (18) q Substituting the radial stress and its first derivative with respect to the radial ordinate into equation(6), the axial stress in the interphase is found to be abn 0,=V: A 2F(a, b, F(a+1,b+1,c+1,-nor) Vi Bi( nor) x FCc-a, c-b,1-a-b+c, 1+ nor (c-a(c-bnor F(c-a+l,c-b+1, 2-a-b+c, 1+nor via (E2+3E0E1r+2E2n2)+(E0+E [E20-(ao+a1r)△T] The relationship between the circumferential strain and the radial displacement can be described with the following geometric equation The substitution of the radial circumferential and axial stresses into the second of equation(4)gives the circumferential strain. Then from the above equation, the radial displacement can be written as v 2vi)F(a, b, c, -nor) Eo+ Er F C +B:(1+nor) + no x F(c b+c,1+nor)+(1-v) (c-a)(c-b)nor b+ F(c-a+1,c-b+1,2 b+c, I+ nor)
Influence of inhomogeneous interphase on composites 55 σθ = Ai F(a, b, c, −n0r) − abn0r c F(a + 1, b + 1, c + 1, −n0r) + Bi(1 + n0r)c−a−b 1 + (c − a − b) n0r 1 + n0r × F(c − a,c − b, 1 − a − b + c, 1 + n0r) + (c − a)(c − b)n0r 1 − a − b + c F(c − a + 1, c − b + 1, 2 − a − b + c, 1 + n0r) − α1T (5 − 4νi)E1 (E2 0 + 4E0E1r + 3E2 1 r2 ). (18) Substituting the radial stress and its first derivative with respect to the radial coordinate into equation (6), the axial stress in the interphase is found to be σz = νiAi 2F(a, b, c, −n0r) − abn0r c F(a + 1, b + 1, c + 1, −n0r) + νiBi(1 + n0r)c−a−b 2 + (c − a − b) n0r 1 + n0r × F(c − a,c − b, 1 − a − b + c, 1 + n0r) + (c − a)(c − b)n0r 1 − a − b + c F(c − a + 1, c − b + 1, 2 − a − b + c, 1 + n0r) − 2νiα1T (5 − 4νi)E1 (E2 0 + 3E0E1r + 2E2 1 r2 ) + (E0 + E1r) × [εz0 − (α0 + α1r)T ]. (19) The relationship between the circumferential strain and the radial displacement can be described with the following geometric equation εθ = u r . (20) The substitution of the radial, circumferential and axial stresses into the second of equation (4) gives the circumferential strain. Then from the above equation, the radial displacement can be written as u = 1 + νi E0 + E1r r Ai (1 − 2νi)F(a, b, c, −n0r) − (1 − νi) abn0r c F(a + 1, b + 1, c + 1, −n0r) + Bi(1 + n0r)c−a−b 1 − 2νi + (1 − νi)(c − a − b) n0r 1 + n0r × F(c − a,c − b, 1 − a − b + c, 1 + n0r) + (1 − νi) (c − a)(c − b)n0r 1 − a − b + c × F(c − a + 1, c − b + 1, 2 − a − b + c, 1 + n0r)
LH. You (1+v)a1△Tr (1-2u)E+2(2-3u)E0E1r (5-4v2)E1(E0+E1r) +(3-4)E2]一0r+(1+v)(a+a1r)△T Since the fiber and matrix is treated as homogeneous, isotropic and elastic solids, the material properties in equation(9)keep constants, i. e, Eo= Et, ao= a, and El=aI =0(where t =f stands for the fiber, and I= m for the matrix). Substituting them into equation(9), the following equation is obtained. The resolution of the above equation gives the radial stress. From the equilibrium equation (5), the circumferential stress is determined. The axial stress is achieved from equation(). Substituting these stresses into the second of equation (4), ircumferential strain is reached. The substitution of the circumferential strain equation(20)leads to the radial displacement. These quantities have the forms of r,=A1+B1r-2, A1-B1 a2=2A1+En(e20-a△T), 1+v F[(-2v)A r-B,/], or+(1+v,)a,ATr, (t=f,m), cre A and B, are the unknown constants Making use of the boundary conditions and continuous conditions of the radial displacement and stress at the interfaces between the fiber and the interphase, and between the interphase and matrix, all the unknown constants in the above equation can be determined. These conditions can be formulated below 0 uf=li, at r=r r atr=r ofr dr=0 where the subscripts r and z represent the radial and axial directions, the subscript f, i and m stand for the fiber, interphase and matrix, respectively There are 7 unknown constants E20, A/, B, Ai, Bi, Am and Bm in the analytical expressions of the deformations and stresses. They can be determined by the 7 linear equations given by equation(24)
56 L. H. You − (1 + νi)α1T r (5 − 4νi)E1(E0 + E1r) (1 − 2νi)E2 0 + 2(2 − 3νi)E0E1r + (3 − 4νi)E2 1 r2 − νiεz0r + (1 + νi)(α0 + α1r)T r. (21) Since the fiber and matrix is treated as homogeneous, isotropic and elastic solids, the material properties in equation (9) keep constants, i.e., E0 = Et , α0 = αt and E1 = α1 = 0 (where t = f stands for the fiber, and t = m for the matrix). Substituting them into equation (9), the following equation is obtained. r d2σr dr2 + 3 dσr dr = 0. (22) The resolution of the above equation gives the radial stress. From the equilibrium equation (5), the circumferential stress is determined. The axial stress is achieved from equation (6). Substituting these stresses into the second of equation (4), the circumferential strain is reached. The substitution of the circumferential strain into equation (20) leads to the radial displacement. These quantities have the forms of σr = At + Btr−2 , σθ = At − Btr−2 , σz = 2νtAt + Et(εz0 − αtT ), (23) u = 1 + νt Et (1 − 2νt)Atr − Btr−1 − νtεz0r + (1 + νt)αtT r, (t = f, m), where At and Bt are the unknown constants. Making use of the boundary conditions and continuous conditions of the radial displacement and stress at the interfaces between the fiber and the interphase, and between the interphase and matrix, all the unknown constants in the above equation can be determined. These conditions can be formulated below. uf = 0, atr=0, uf = ui, σrf = σri, at r = rf , ui = um, σri = σrm, at r = ri, σrm = 0, at r = rm, (24) rf 0 σzf r dr + ri rf σzir dr + rm ri σzmr dr = 0, where the subscripts r and z represent the radial and axial directions, the subscripts f , i and m stand for the fiber, interphase and matrix, respectively. There are 7 unknown constants εz0, Af , Bf , Ai, Bi, Am and Bm in the analytical expressions of the deformations and stresses. They can be determined by the 7 linear equations given by equation (24).
Influence of inhomogeneous interphase on composites 3. APPLICATIONs Using the above-developed analytical method, we firstly discuss how the inhomoge- neous interphase affects the stress distribution in the composites. Then, the effects of the Youngs modulus and thermal expansion coefficient of the matrix on the ther nal stresses in the composites will be investigated. The Youngs modulus, Poissons ratio and thermal expansion coefficient of the fiber are taken to be Er= 524.6 GPa, v=0.19 and a =2 x 10-o/ C. Those of the matrix are taken as Em= 115 GPa, Vm=0.36 and am=9 x 10-6/C.The Poissons ratio of the interphase is taken to be the average of that of the fiber and matrix,i.e Vi=0. 275, and its Youngs modulus and thermal expansion coefficient are determined according to those of the fiber and matrix by means of equation (1) The outer radii of the fiber, interphase and matrix are taken to be 5, 6 and 10 um, espectively. The temperature load is a uniform temperature drop of -600oC. The calculated stresses for the composites with an inhomogeneous and homoge- neous interphase are depicted in Fig. I where or, ag and o, indicate the radial, circumferential and axial stresses in the composites with an inhomogeneous inter phase, and or, af and of stand for those in the composites with a homogeneous For the composites with a homogeneous interphase, only the radial stress at the interfaces between the fiber and interphase, and between the interphase and matrix is continuous. The circumferential and axial stresses at these interfaces change abruptly. From the fiber to the matrix, they jump from -355 MPa and 1034 MPa to 559 MPa and 182 MPa. Then. the circumferential stress decreases ontinuously from the inner radius of the interphase to its outer radius, whereas the axial stress stays unchanged. From the interphase to the matrix, another smal ump occurs. The circumferential and axial stresses jump from 427 MPa and 182 MPa to 450 MPa and 373 MPa. After taking into account the inhomogeneous MPa 600 200 0 3 -600 -800 1000 Figure 1. Radial, circumferential and axial stresses in composites with an inor hom
Influence of inhomogeneous interphase on composites 57 3. APPLICATIONS Using the above-developed analytical method, we firstly discuss how the inhomogeneous interphase affects the stress distribution in the composites. Then, the effects of the Young’s modulus and thermal expansion coefficient of the matrix on the thermal stresses in the composites will be investigated. The Young’s modulus, Poisson’s ratio and thermal expansion coefficient of the fiber are taken to be Ef = 524.6 GPa, νf = 0.19 and αf = 2 × 10−6/ ◦ C. Those of the matrix are taken as Em = 115 GPa, νm = 0.36 and αm = 9 × 10−6/ ◦ C. The Poisson’s ratio of the interphase is taken to be the average of that of the fiber and matrix, i.e. νi = 0.275, and its Young’s modulus and thermal expansion coefficient are determined according to those of the fiber and matrix by means of equation (1). The outer radii of the fiber, interphase and matrix are taken to be 5, 6 and 10 µm, respectively. The temperature load is a uniform temperature drop of −600◦C. The calculated stresses for the composites with an inhomogeneous and homogeneous interphase are depicted in Fig. 1 where σν r , σν θ and σν z indicate the radial, circumferential and axial stresses in the composites with an inhomogeneous interphase, and σc r , σc θ and σc z stand for those in the composites with a homogeneous interphase. For the composites with a homogeneous interphase, only the radial stress at the interfaces between the fiber and interphase, and between the interphase and matrix is continuous. The circumferential and axial stresses at these interfaces change abruptly. From the fiber to the matrix, they jump from −355 MPa and −1034 MPa to 559 MPa and 182 MPa. Then, the circumferential stress decreases continuously from the inner radius of the interphase to its outer radius, whereas the axial stress stays unchanged. From the interphase to the matrix, another small jump occurs. The circumferential and axial stresses jump from 427 MPa and 182 MPa to 450 MPa and 373 MPa. After taking into account the inhomogeneous Figure 1. Radial, circumferential and axial stresses in composites with an inhomogeneous and homogeneous interphase
LH. You interphase, both stresses vary consecutively from the fiber to the interphase and from the interphase to the matrix. Along the radial coordinate, they rise from the maximum compressive stresses at the inner radius of the interphase and reach their maximum tensile stresses. Then, they drop until the outer radius of the interphase is reached. Due to the variety of the stresses in the interphase, the stresses in the fiber are also changed. After considering the inhomogeneous interphase, the compressive radial, circumferential and axial stresses in the fiber are decreased, among which the reduction of the axial stress is the largest. Unlike the obvious changes in the fiber, he effect of the inhomogeneous interphase on the stresses in the matrix is very weak. The radial and circumferential stresses basically keep the same. Only the axial stress shows a very slight increase. By changing the Youngs modulus of the matrix from 80 GPa to 320 GPa, the relationships between the maximum stresses and the Youngs modulus of the matrix are given in Figs 2a, 2b and 2c where or!, a0 and at indicate the maximum radial, circumferential and axial stresses in the fiber, ori, aei and o zi stand for those in the interphase, and orm oom and a-m represent those in the matrix All the maximum stresses in the fiber are compressive. Among them, the maximum radial and circumferential stresses in the fiber are the same. Along with the increase of the Youngs modulus, they increase linearly. In contrast, the variation of the maximum axial stress is nonlinear. At first, it increases quickly; then its increase becomes slower and slower. The variations of the maximum radial and axial stresses in the interphase are similar to those in the fiber. However, the maximum circumferential stress in the interphase is tensile. It goes up when the Youngs modulus of the matrix grows. Among the three stresses, only the maximum radial stress changes linearly; the other two stresses vary nonlinearly. In the region near the inner radius of the interphase, the increase of the maximum circumferential stress in the interphase is the smallest. Accompanying the rising of Youngs modulus of the matrix, it becomes a little quicker and quicker. Oppositely, the MPa 800 1200 1800 170 200 260 Figure 2a. The effect of matrix Youngs modulus on maximum radial, circumferential and axial
58 L. H. You interphase, both stresses vary consecutively from the fiber to the interphase and from the interphase to the matrix. Along the radial coordinate, they rise from the maximum compressive stresses at the inner radius of the interphase and reach their maximum tensile stresses. Then, they drop until the outer radius of the interphase is reached. Due to the variety of the stresses in the interphase, the stresses in the fiber are also changed. After considering the inhomogeneous interphase, the compressive radial, circumferential and axial stresses in the fiber are decreased, among which the reduction of the axial stress is the largest. Unlike the obvious changes in the fiber, the effect of the inhomogeneous interphase on the stresses in the matrix is very weak. The radial and circumferential stresses basically keep the same. Only the axial stress shows a very slight increase. By changing the Young’s modulus of the matrix from 80 GPa to 320 GPa, the relationships between the maximum stresses and the Young’s modulus of the matrix are given in Figs 2a, 2b and 2c where σ max rf , σ max θf and σ max zf indicate the maximum radial, circumferential and axial stresses in the fiber, σ max ri , σ max θi and σ max zi stand for those in the interphase, and σ max rm , σ max θm and σ max zm represent those in the matrix. All the maximum stresses in the fiber are compressive. Among them, the maximum radial and circumferential stresses in the fiber are the same. Along with the increase of the Young’s modulus, they increase linearly. In contrast, the variation of the maximum axial stress is nonlinear. At first, it increases quickly; then its increase becomes slower and slower. The variations of the maximum radial and axial stresses in the interphase are similar to those in the fiber. However, the maximum circumferential stress in the interphase is tensile. It goes up when the Young’s modulus of the matrix grows. Among the three stresses, only the maximum radial stress changes linearly; the other two stresses vary nonlinearly. In the region near the inner radius of the interphase, the increase of the maximum circumferential stress in the interphase is the smallest. Accompanying the rising of Young’s modulus of the matrix, it becomes a little quicker and quicker. Oppositely, the Figure 2a. The effect of matrix Young’s modulus on maximum radial, circumferential and axial stresses in fiber
