LOAD CARRYING CAPACITY OF BROKEN ELLIPSOIDAL INHOMOGENEITY AND CRACK EXTENDED IN PARTICLE REINFORCED COMPOSITES

Transcription

1 18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS LOAD CARRYING CAPACITY OF BROKEN ELLIPSOIDAL INHOMOGENEITY AND CRACK EXTENDED IN PARTICLE REINFORCED COMPOSITES Y. Cho 1 * 1 Deartment of Manufacturing & Design Engineering, Jeonju Universit, Wansan-Gu, Jeonju, , Korea * Corresonding author (chot@jj.ac.kr) Kewords: Load Carring Caacit, Ellisoidal Inhomogeneit, Cracking Damage, Crack Exended, Axismmetric Finite Element Method, Elastic Stress Distribution 1. Introduction Comosites, which contain articles or short-fibers in a ductile matrix have alread used or have the otential as engineering material because of their good formabilit and machinabilit as well as imroved mechanical roerties. In the comosites, a variet of damage modes such as fracture of reinforcements, interfacial debonding between reinforcements and matrix, and cracking in matrix develo from earl stage of deformation under monotonic or cclic loads.[1-4] The observed damage modes deend on the combination of the mechanical roerties of the constituents and the insitu interfacial strength between them. In order to extend the alication of the comosites and to develo a new comosite sstem, an understanding of the micromechanism of damage rocess in the comosites is essential. Man theories for article or short-fiber reinforced comosites were established based on the Eshelb's solution (1957) for an ellisoidal inhomogeneit in an infinite bod. [5-11] However, the corresonding solution for a broken ellisoidal inhomogeneit has not been reorted. Therefore, it is imossible to construct the theor of the comosite containing cracking damage in the same scheme. This aer deals with the load carring caacit of intact (Fig.1(b)), broken (Fig.1(a)) and 5%-extended matrix crack roortional to broken (Fig.1(d)) ellisoidal inhomogeneities in an infinite bod and a damage theor of article or short-fiber reinforced comosites. The load carring caacit of the broken inhomogeneit is exressed in terms of the average stress of the intact inhomogeneit and some coefficients. Based on the finite element analses of the intact and broken ellisoidal inhomogeneities, the coefficients are given as functions of an asect ratio for a variet of combinations of the elastic moduli of inhomogeneit and matrix. 2. Load Carring Caacit of an Ellisoidal Inhomogeneit Load carring caacit of an ellisoidal inhomogeneit embedded in an infinite bod can be defined as an average stress in the inhomogeneit. The load carring caacit deends on the elastic moduli of the inhomogeneit and matrix. High average stress comaring with the remote alied stress means high load carring caacit of the inhomogeneit. On the other hand, when the average stress is reduced b the debonding or cracking damage of the inhomogeneit, the load carring caacit is also reduced, and the stress free in a void means that the load carring caacit of the void is equal to ero. Figure 1(b) shows an intact ellisoidal inhomogeneit embedded in an infinite bod under alied stress., (a) (b) (c) (d) Fig. 1 Princile of suerosition for a broken ellisoidal inhomogeneit in infinite bod

2 The elastic stiffness tensor of the infinite bod L (matrix) and the inhomogeneit are denoted b 0 L and 1, resectivel. The stress of the ellisoidal inhomogeneit is uniform and given b the Eshelb's equivalent inclusion method as well known. [1] 1 1 = L0 ( S I)[( L1 L0 ) S + L0 ] L1 ( S I) L0 (1) Where, S is Eshelb's tensor which is exressed as a function of shae of the inhomogeneit and Possino's ratio of the matrix. For an ellisoidal inhomogeneit cracked in the cross section of x-lane as shown in Fig. 1(a), the stress distribution in the inhomogeneit seems to be comlex and its solution have not been reorted as far as the authors traced references. Figure 1 shows the rincile of suerosition for a cracked ellisoidal inhomogeneit in an infinite bod. The average stress in the inhomogeneit reresents its load carring caacit, and the difference between the average stresses of the intact and broken inhomogeneities indicates the loss of load carring caacit due to the cracking damage. As shown in Fig. 1(a), the stress state in the broken c inhomogeneit is given b the sum of the c* stresses and, where the stress is in the intact inhomogeneit under the alied stress and c* is the broken inhomogeneit subjected to internal stress on the crack surface. Therefore, the average stress of the broken inhomogeneit is exressed b c c* = + = + h = ( I + h) = k (2) Where h is a coefficient exressing the reduction of average stress due to the cracking damage of an ellisoidal inhomogeneit, k is the ratio of the average stresses of the broken and intact inhomogeneities and I was unit tensor. Once h is the determined, the load carring caacit of the broken inhomogeneit can be evaluated inclusion method. Based on the axismmetric finite element analses of the intact and broken ellisoidal inhomogeneities, the matrix h is obtained as a function of an asect ratio the inhomogeneit and combination of elastic moduli of the inhomogeneit and matrix. [12] The comonents of the stresses are given c b = x,, x, τ x, c c c c c c = x,, x x and c c c c c c c = x,, x x. The average stress comonents in the broken inhomogeneit shown in c c Fig.1(c) are * * c, x and * due to internal stress c* on the crack surface due to τ, and c* τ x due to τ x. Therefore, we have 1 0 h h k = ( 1+ h) = h (3) h h in the matrix form. Furthermore, in the case of the axismmetric ellisoidal inhomogeneit, h 13 = h23 and h 44 = h 55 relations are obtained. As a result, once the three comonents, h 33, h 13 and h 44 are obtained, the average stress of the broken ellisoidal inhomogeneit can be evaluated b Eqs.(1) and (2). Out of the three comonents, h 33 and h 13 are determined b the analsis under uniaxial tension, and h 44 is b the analsis under ure shear. 3. Numerical Procedure Elastic stress analses of intact and damaged ellisoidal inhomogeneities embedded in an infinite bod under uniaxial tension as shown in Fig.1 were carried out based on the axismmetric finite element method using the quadrilateral 8-node isoarametric elements. Mesh division was carried out for a wide ellisoidal domain including an inhomogeneit in the center, and uniaxial uniform tensile stress in the -axis direction is alied on the surface of the domain as boundar condition. The sie of the inhomogeneit is denoted b 2a and 2b in r (x and ) direction and direction, resectivel. On the same finite element meshes, the node oints on the - and r-axes were fixed for the intact inhomogeneit, and the node oints on the - and r-axes excet crack lane were fixed for the cracked inhomogeneit. In the analses, an asect ratio (b/a) of the inhomogeneit and the combination in the elastic moduli of the inhomogeneit and matrix were widel changed. The average stress in the intact and cracked inhomogeneities were calculated and used as the load carring caacit. [12] 4. Numerical Results and Discussion 4.1 Stress distribution in and around an inhomogeneit Stress distributions of the intact and cracked inhomogeneities are shown for the case in which Young s modulus ratio is E /E m =5.0 and Poisson s

3 PAPER TITLE ratios are ν m =0.3 for the matrix and ν =0.17 for the inhomogeneit. The asect ratios are 0.5 (oblate sheroid), 1.0 (shere) and 3.0 (rolate sheroid). The stress in the intact inhomogeneities is uniform as well known as Eshelb's solution for ellisoidal inclusion. The maximum tensile stress in the matrix is created in the region near the ole of the inhomogeneit. On the other hand, in the case of the cracked inhomogeneit, the stress is released on the region near the crack surface and becomes nonuniform in the inhomogeneit, and it concentrates at the crack ti region and the ole of the inhomogeneit in the surrounding matrix. [12] The stress distributions of along the -axis and r- axis are shown in Fig. 2 for the intact inhomogeneities and in Fig. 3 for the broken and 5%-extended matrix crack roortional to broken inhomogeneities, resectivel. These distributions were obtained b the data at the Gauss oints close to the -axis and r-axis. It is found in Fig. 2 that the uniform stress in the intact inhomogeneit increases with increasing its asect ratio. This means that longer fiber has larger load carring caacit. (a) Stress distribution (b) Stress distribution along the -axis along the r-axis Fig. 2 Stress distribution ( / ) along the - and r- axis for an intact inhomogeneit under uniaxial tension In Fig. 3(a) for the cracked inhomogeneit and it's the crack extended to matrix, the stress is equals to ero at the crack surface and increases with the distance from the crack surface, and then the maximum stress is obtained at the ole of the inhomogeneit. It is found from the above results that the stress distributions in and around an ellisoidal inhomogeneit in an infinite bod under uniaxial tension become more comlex b the cracking damage, and that the cracked inhomogeneit still maintains a large amount of load carring caacit. In the case of the cracked ellisoidal inhomogeneit, a singular stress field is dominant around the crack ti and it is ver imortant on the discussion of the fracture behavior from the crack ti. In the resent aer, however, the stress singularit at the crack ti is not referred an more but the load carring caacit will be intensel discussed Load Carring Caacit As mentioned reviousl, the load carring caacit of an ellisoidal inhomogeneit embedded in a matrix is defined b its average stress. In this section, the average stress in the intact and broken inhomogeneities in an infinite bod under uniaxial tension is calculated based on the result of the finite element analses, and the load carring caacit of these inhomogeneities is discussed. In the axismmetric finite element analsis the stress comonents in -axis, radial and circumferential directions,, and r, are obtained. Here, the θ stress comonents in the rectangular coordinate sstem and = transferred from, and r θ are used for the discussion. (a) Stress distribution (b) Stress distribution along the -axis along the r-axis Fig. 3 Stress distribution ( / ) along the - and r-axis for an broken inhomogeneit and 5%- extended matrix crack roortional to broken inhomogeneit to under uniaxial tension (a) Stress in -axis (b) Stress in x-axis direction ( / ) direction ( / x ) Fig. 4 Stress of an intact inhomogeneit in infinite bod under uniaxial tension as a function of an asect ratio 3

4 Figure 4 (a) and (b) show the stresses and x of the intact inhomogeneit in an infinite bod under uniaxial tension as a function of an asect ratio for various combinations in the elastic moduli of inhomogeneit and matrix. The solid lines and lots indicate the results of Eshelb's equivalent inclusion method (Eq. (1)) and the results of the finite element analses, resectivel. Good agreement between both results shows that the resent finite element analses of the ellisoidal domain containing the inhomogeneit in the center well simulate an inhomogeneit in an infinite bod. It is found from Fig. 4(a) that with increasing the asect ratio, the stress in tensile direction increases for the Young's modulus ratio of E /E m >1, and decreases for E /E m <1. The stress erendicular to tensile direction x shown in Fig. 4(b) is alwas negative, and it deends on the Young's modulus ratio at the region of low asect ratio but converges to around with increasing the asect ratio. As shown in the revious section, the stress in the inhomogeneit is released b the cracking damage, but the cracked inhomogeneit still has a large amount of load carring caacit. From Eq. (2), relations between the average stresses of the broken and intact inhomogeneities under multi-axial tension are exressed b c c x c = ( 1+ h33) (4) = x + h13 (5) = + h (6) 13 Using equations (4) and (5), the coefficients and are determined from the numerical results of the average stresses of intact and broken ellisoidal inhomogeneit. (a) Intact and broken Inhomogeneities (b) Intact and 5%-extended matrix crack roortional to broken inhomogeneit Fig. 5 Coefficient (h 33 ) and ratio of load carring caacit as a function of an asect ratio. (a) Intact and broken (b) Intact and 5%- extended inhomogeneities matrix crack roortional to broken inhomogeneit Fig. 6 Coefficient (h 13 ) as a function of an asect ratio Figures 5, 6 show relationshi between the coefficients h 33, h 13, and the asect ratio for various combinations in the elastic moduli. Since the load carring caacit of the cracked inhomogeneit under multi-axial tension can be easil evaluated b the above equations from the average stress of the intact inhomogeneit obtained b the Eshelb's equivalent inclusion method, these coefficients are ver imortant. Hereafter, we discuss in more detail the change of the load carring caacit due to cracking damage of the ellisoidal inhomogeneit under uniaxial tension. Figure 5 also exhibits a ratio of the load carring c caacit / which is defined b the ratio of the average stress in the broken and 5%-extended matrix c crack roortional to broken inhomogeneit ( ) to the stress in the intact inhomogeneit ( ), when it is observed for a scale of the right hand side. As shown in Fig. 6, the load carring caacit ratio is alwas smaller than one, i.e. the inhomogeneit reduces its load carring caacit b the cracking damage and crack extended to matrix. It is found that the load carring caacit ratio increases with increasing the asect ratio and deends on the combination of the elastic moduli. The load carring caacit ratio is equal to ero for a enn shae inhomogeneit (b/a=0), while it aroaches to unit for a continuous long fiber (b/a= ). This means that a enn shae inhomogeneit loses comletel the load carring caacit b the cracking damage while the infinitel long fiber in an infinite bod never loses it. With increasing Young's modulus ratio E /E m, the load carring caacit ratio increases in the region of low asect ratio and decreases in the region of high asect ratio. Particularl, the

5 PAPER TITLE influence of E /E m to the load carring caacit is drastic for the inhomogeneit with low asect ratio. This suggests that in a comosite reinforced with ellisoidal inhomogeneit with low asect ratio a stiffer inhomogeneit maintains higher load carring caacit after cracking damage. 5. Conclusions Axismmetric finite element analses have been carried out on the elastic stress distribution and load carring caacit of intact and broken ellisoidal inhomogeneities embedded in an infinite bod under uniaxial tension. Conclusions obtained from the numerical results are summaried as follows: (1) For the intact inhomogeneit, the stress distribution is uniform in the inhomogeneit and nonuniform in the surrounding matrix, as well known as Eshelb's solution. On the other hand, for the broken inhomogeneit and its crack extended to matrix, the stress in the region near the crack surface in the inhomogeneit is considerabl released and the stress distribution becomes more comlex. (2) The average stress in the broken inhomogeneit is exressed b the average stress of the intact inhomogeneit. Based on the numerical results, the coefficients in the exression describing this relationshi under multi-axial tension are obtained as a function of the asect ratio of the inhomogeneit. (3) The inhomogeneit in an infinite bod loses load carring caacit b cracking damage, but the broken inhomogeneit and its crack extended to matrix still maintains it to some extent. The ratio of the load carring caacit of the intact, broken and crack extended to matrix roortional to broken inhomogeneities is given as a function of the asect ratio of the inhomogeneit. It is found that the broken inhomogeneit with higher asect ratio maintains higher load carring caacit than one with low asect ratio. [3] Whitehouse, A.F. and Clne, T.W., Cavit Formation during Tensile Straining of Particulate and Short Fibre Metal Matrix Comosites, Acta Metall. Mater., Vol. 41, , [4] Tohgo, K., Mochiuki, K., Takahashi, H. and Ishii, H., Alication of Incremental Damage Theor to Glass Particle Reinforced Nlon 66 Comosites, Localied Damage IV, Comuter-Aided Assessment and Control, Comutational Mechanics Publications, , [5] Tohgo, K. and Chou, T.-W., Incremental Theor of Particulate-Reinforced Comosites Including Debonding Damage, JSME Int. J., Vol. 39, , [6] Tohgo, K. and Weng, G.J., A Progressive Damage Mechanics in Particle-Reinforced Metal-Matrix Comosites under High Triaxial Tension, Trans. ASME, J. Eng. Mater. Technol., Vol. 116, , [7] Eshelb, J.D., The Determination of the Elastic Field of an Ellisoidal Inclusion, and Related Problems, Proceedings of the Roal Societ, London, Vol. A241, , [8] Mori, T. and Tanaka, K., Average Stress in Matrix and Average Elastic Energ of Materials with Misfitting Inclusions, Acta Metall., Vol. 21, , [9] Mura, T., Micromechanics of Defects in Solids, Martinus Nijhoff, The hague, [10] Arsenault, R.J. and Taa, M., Thermal Residual Stress in Metal Matrix Comosite, Acta Metall., Vol. 35, , [11] Tandon, G.P. and Weng, G.J., A Theor of Particle-Reinforced Plasticit, Trans. ASME, J. Al. Mech., Vol. 55, , [12] Cho, Y.-T., Tohgo, k. and Ishii, I., Finite Element Analsis of a cracked Ellisoidal Inhomogeneit in an Infinite Bod and Its Load Carring Caacit, JSME Int. J., Vol.40, No.3, , References [1] Loretto, M.H. and Koniter, D.G., The Effect of Matrix Reinforcement Reaction on Fracture in Ti- 6Al-4V-Base Comosites, Metall. Trans. A, Vol. 21A, , [2] Llorca, J., Martin, A., Rui, J. and Elices, M., Particulate Fracture during Deformation of a Sra Formed Metal-Matrix Comosite, Metall. Trans. A, Vol. 24A, ,