Characterization of Strength of Carbon Fiber Reinforced Polymer Composite Based on Micromechanics

Transcription

1 Characterization of Strength of Carbon Fiber Reinforced Polymer Composite Based on Micromechanics Characterization of Strength of Carbon Fiber Reinforced Polymer Composite Based on Micromechanics Wangnan Li, Hongneng Cai*, and Jie Zheng State Key Laboratory for Mechanical Behavior of Materials, Xi an Jiaotong University, Xi an , China Summary Micro-mechanics of failure (MMF) is a micromechanics-based failure criterion for composites, which can be applied to characterize the composite strength accurately and reliably. The strength and mechanical properties of unidirectional laminate are measured by the static tensile loading and compressive loading tests as well as three points bending test. The micromechanics model is proposed to amplify the detailed mechanical and residual thermal stresses for the key points in the fibers and matrix. The critical MMF parameters are characterized by using the stress amplification factors and the measured strengths of unidirectional laminate. The carbon fibers reinforced plastic UTS50/E51 is used to demonstrate the application of MMF theory. The failure envelopes are constructed and compared on the basis of the MMF theory and Tsai-Wu failure theory. The failure mechanism and static tensile strength of quasi-isotropic laminate of polymer composites is studied based on MMF. The UTS50/E51 CFRP is used to demonstrate the application of theory of MMF. Keywords: Micromechanics of failure, Constituent-based strength characterization, Critical MMF parameters, Non-interactive failure criteria 1. Introduction In recent years, composite materials are used in various advanced structures due to their high specific strength and stiffness, outstanding design ability, long fatigue life, corrosive resistance and other relevant fields. The extent of potential application of composite material in the engineering field depends on whether or not the strength of the composite structure that can be evaluated correctly at various loading condition. The strength prediction and failure mechanism analysis attracted many researchers attention and produced a lot of strength theories for composite materials 1-5. Hinton and Soden 6 indicated that the current lamina-based strength theories for composite material are not accurate enough to guide the engineering * hntsai@mail.xjtu.edu.cn Smithers Rapra Technology, 2014 design. These strength theories were established using the mathematical approximation formulas that satisfy some key points of the results given by experimental tests. Large error occurs during the prediction of strength of the structure under complicate loading conditions. In addition, the specific failure mechanism is not indicated when the conventional lamina-based strength theories are used for practical design. In addition, the classical laminated plate and shell theory-based strength theories of composite materials are not adequate for the three dimensional stress states in the large, thick and curved structures 7. Those theories are essentially used for the twodimensional cases, where the thickness effect is ignored. In order to guarantee the reliability of the structure, a lot of tests from laminate, sub-structure to whole structure need to be carried out. So the scientists 8 suggest a novel physically-based strength theory for composite materials should be developed. The micro-mechanics of failure (MMF) proposed by Ha 9 is a real physics-based strength theory for failure prediction of the structures made of CFRP laminates, in which the initial failure of constituents is analyzed at microlevel. Either square or hexagonal unit cell model loaded with the loads at the boundaries corresponding to the stress state at macro-level is used to look insight into the stresses in fibers and matrix in micro-level. In this way, it becomes possible to predict the failure mode of laminates. Compared with the square model, the hexagonal model includes more fibers and can reflect the interactive effect between the fibers. The result of the research shows that the strength prediction using the hexagonal model agrees well with Polymers & Polymer Composites, Vol. 22, No. 2,

2 Wangnan Li, Hongneng Cai, and Jie Zheng the test data. But the hexagonal model cannot explain the homogeneous characteristics in the transverse y and z directions of material coordinate in the macro-level due to the length difference between the edges of hexagonal unit cell model in the y and z directions. Therefore, a new micromechanics FEM model which can explain the homogeneity of the y and z directions is necessary. In this paper, the face-centered cubic micro-mechanics model is constructed to analyze the stresses in fibers and matrix in micro-level. Also, a simple non-interactive failure criterion is used to characterize the strength of matrix. The strength and mechanical properties of multi-directional laminate of UTS50/E51 are measured. The stresses amplification factors of the key points in the fibers and matrix are calculated. The MMF critical parameters are characterized by using the stress amplification factors and the measured strengths of unidirectional laminate. The biaxial stress failure envelopes for unidirectional laminate of UTS50/E51 are constructed on the basis of the MMF theory. The MMF method is used to demonstrate the prediction of the static tensile strength of quasi-isotropic laminate of polymer composites. 2. The micromechanics of failure (MMF) 2.1 Introduction of MMF The micro-mechanics of failure (MMF) proposed by Ha 9 is a micromechanicsbased failure theory for composites. This theory analyzes the stress of composite by dividing composites into three regions, including the fibers, matrix and fibers-matrix interface separately. of constituents in micro-level and the MMF critical parameters from the tested results of unidirectional laminates are extracted. Then, it is possible to evaluate and predict the strength of various composites structures. Three distributed models of the fibers and the matrix are shown in Figure 1, including the square model, hexagonal model, and face-centered cubic model. The composite material is characterized by using four MMF four critical parameters, including the tensile strength of fibers T f, the compressive strength of fibers C f, the tensile strength of matrix T m, and the compressive strength of matrix C m. T f and C f are used to judge the failure of fibers under tensile and compressive loading, respectively. T m and C m are used to judge the failure of matrix under tensile and compressive loading, respectively. The computing methods of the MMF parameters are shown in Equation (1): T f = σ f 1,(σ f 1 > 0) f C f = σ vm,(σ f 1 < 0) T m = I m 1 = σ 1 + σ 2 + σ 3 C m = σ m vm = 0.5 σ 1 σ 2 ( ) 2 + ( σ 1 σ 3 ) 2 + ( σ 2 σ 3 ) 2 + 6(τ τ τ 2 12 ) where T f and C f are extracted using the maximum stress in fiber direction, and T m and C m are extracted using the function of stress components at the key points of matrix. The superscript and subscript f and m represent the fiber and matrix, respectively. The material axes 123 are used as coordinate direction. The failure index of the fibers and matrix is defined as: k f = max max σ f,i l, max σ f,i vm, max I 1 T f C f m,j T m, max σ m,j vm C m where i= 1,... n 1, j = 1,... n 2, n 1,n 2 represent the numbers of key points in the fiber and matrix, respectively. When the failure index k f reaches 1, the failure occurs either in the fiber or in the matrix. The simple non-interactive failure criterion for matrix is proposed to judge the failure of matrix. When the maximal first stress invariant I 1 reaches T m, or the maximum equivalent stress reaches C m, the crack will occur in the matrix. Here, it could be pointed out that Christensen 10 proposed the interactive failure criterion to judge the failure of the matrix. Though the judgment of matrix failure is performed easily, however, the extraction of T m and C m from the results of the unidirectional laminate tests is very difficult for the engineering application. Figure 1. Fiber distribution model of carbon fiber reinforced plastics. (a) Square (b) Hexagonal (c) Face-centered cubic (1) (2) First, unit cell model is constructed and as many typical key points as possible are designated in different regions of the fiber and matrix, then the FEM is used to analyze the stresses 106 Polymers & Polymer Composites, Vol. 22, No. 2, 2014

3 Characterization of Strength of Carbon Fiber Reinforced Polymer Composite Based on Micromechanics The fiber shows the high rigidity only in longitudinal direction, but in the transverse direction the stiffness is very low. The maximum tensile stress and compressive stress along the fiber direction are used to characterize the tensile strength and compressive strength of the fiber. Gosse et al. 11 proposed the strain invariant failure theory to characterize the strength of matrix, the first strain invariant J 1 and the equivalent strain ε eq being used to characterize the tensile strength and compressive strength of matrix, respectively. Because J 1 is suitable to judge the failure due to the volume increase, while ε eq is suitable to judge the matrix failure due to the shearing deformation. This paper uses the first stress invariant I 1 and the equivalent stress σ eq, which correspond to J 1 and ε eq, to describe the tensile strength and compressive strength of matrix, respectively. value to keep the face flat, as shown in Figures 3a-3c. The procedure is repeated two times in order to obtain stress amplification factors for stresses in the other two orthogonal (2- and 3-) directions. Similarly, for shear deformations, the prescribed shear strain is applied in each of the three directions, as shown in Figures 3d- 3e. The local stresses are extracted from various positions in the unit cell model, 17 key points in fiber and 19 key points in matrix. The extracted stresses are normalized as mechanical amplification factors of the stresses in the lamina at macro-level. One input stress in macro-level in lamina maps 36 sets of states of stresses, i.e., 17 sets for fiber and 19 sets for matrix, and stored as stress amplification factors. The normalized thermal residual stresses are also considered as thermal-mechanical amplification factors. The thermal residual stresses occur from the temperature difference from the curing temperature to room temperature. The stresses of lamina are converted to the stresses of fiber and matrix at microlevel using the stress amplification factors. The output result of macrostresses from the finite element analysis at macro-level or from classical laminated plate theory is amplified through these amplification factors before failure analysis. The critical point among these key points is identified by comparing the values of strength ratio using failure criterion with the MMF critical parameters of the constituents. The stresses modification from the macro-level to the microlevel is carried out with the expression using the obtained stress amplification factors: 2.2 Micromechanics Analysis of Stresses The stresses amplification in fiber and matrix can be analyzed using unit cell model, shown in Figure 2, whereby individual fiber and matrix are modeled by three-dimensional elements. This unit cell model is applied with prescribed loads of average unit stress in one of three normal stresses or one of three shear stresses, as shown in Figure 3. The boundary constraint conditions set for the model are same as described in the literature by Cai et al. 12. For example, to obtain stress amplification factors for prescribed load in the fiber (or 1-) direction for one of the faces, the prescribed load with average unit stress is applied to one of nodes in the face where the normal displacements of all nodes in that face are coupled with same value to keep the face flat, and the model is free in the other five faces except that the normal displacements of all nodes in each face are coupled with same Figure 2. Location of points for the extraction of stress amplification factors within the unit cell model of carbon fiber reinforced plastics Figure 3. (a-c) Prescribed normal stress of average 1, (d-f) Prescribed shear stress of average 1 Polymers & Polymer Composites, Vol. 22, No. 2,

4 Wangnan Li, Hongneng Cai, and Jie Zheng σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 mech M 11 M 12 M 13 M 14 M 15 M 16 M 21 M 22 M 23 M 24 M 25 M 26 M 31 M 32 M 33 M 34 M 35 M 36 = M 41 M 42 M 43 M 44 M 45 M 46 M 51 M 52 M 53 M 54 M 55 M 56 M 61 M 62 M 63 M 64 M 65 M 66 where σ mech is the micro stress in fiber or matrix, σ mech is the macro stress in CFRP lamina, T is the temperature difference between environment temperature and the curing temperature, M σ is the stress amplification factors of point i either in fiber or matrix resin, A σ is the thermal stress amplification factors. 2.3 Implementation Method of MMF Theory Analysis procedure using MMF theory includes five steps: 1. Measure the macro-mechanical properties of the laminate using unidirectional laminate tests. 2. Back-calculate material properties of the fiber including the longitudinal and transverse Young modulus of the fiber E fl, E ft, the shear modulus G flt, G ftz, Poisson s ratio µ flt, µ ftz and the thermal expansion coefficient α fl, α ft using the mixture rules for composite material 13, as listed in Equation (4). The properties of the epoxy matrix are set as constants, Young s modulus E m 3.0 GPa, Poisson s ratio µ m 0.38, thermal expansion coefficient α m /k: E L = E fl + E m (1 ) E fl E E T = (1 C) ft E m E m + E ft (1 ) + C{ G V + E (1 V )} E flt f m f ft E Z = E T E fz = E ft G G LT = (1 C) flt G m G m + G flt (1 ) + C{ G V + G (1 V )} G flt f m f flt E G TZ = T 2(1+ ν TZ ) G = E ft ftz 2(1+ ν ftz ) G LZ = G LT G flz = G flt ν LT = (1 C) { ν flt + ν m (1 )}+ C ν E V + ν E (1 V ) flt ft f m m f ν flt E ft + E m (1 ) ν TZ = ν ZT = ν ftz + ν m (1 ) ν ftz ν LZ = ν LT ν flz = ν flt α L = α V E + α (1 V )E fl f fl m f m α fl E fl + (1 )E m α T = α ft (1+ ν ft )+ α m (1 )(1+ ν m ) α L ν LT α ft C = (4) s σ1 σ 2 σ 3 σ 4 σ 5 σ 6 mech + A 1 A 2 A 3 A 4 A 5 A 6 s (3) Δ T 3. Calculate the stress amplification factors and the thermal stress amplification factors of key points in the constituents by FEM using the constructed micromechanical unit cell model. 4. Calculate stresses of the key points of constituents by Equation (3), then the MMF parameters of the key points of constituents using Equation (1), and extract the maximum values of σ 1, -σ 1, I 1, σ eq among the key points as the critical MMF strength parameters of the constituents. 5. Design or predict the strengths of various structures by mechanical analysis, starting from preliminary analysis in macro-level and looking insight in detailed into the constituents in micro-level, and the critical MMF strength parameters being used as the judgment standards. 3. Macro-mechanics Test and Micromechanics Analysis of Laminate 3.1 Materials for Experiment and Experimental Procedures The stacking sequences of tested CFRP laminates are [0] 10, [90] 20, [45/-45] 5s of UTS50/E51, which consists of UTS50 carbon fiber and E51 epoxy resin. The volume fraction of the fiber is approximately All the laminates were made by hot pressing technique. The curing procedure includes 80 C for 30 min, and subsequent 130 C for 60 min under pressure of 5 MPa. The laminates were then cut by a face-centered cubic-grit wheel into the specimen size for the tests. This tensile test of unidirectional laminate was carried out according to the ISO527 standard, and the tensile strain of specimens during the process of loading was measured. The three point bending method was used to test the longitudinal compressive strength, according to ISO14125 standards, and 108 Polymers & Polymer Composites, Vol. 22, No. 2, 2014

5 Characterization of Strength of Carbon Fiber Reinforced Polymer Composite Based on Micromechanics the transverse compression test was implemented according to ISO14126 standard. The test scheme and specimen size are shown in Table 1 and a uniaxial static tensile load or static compressive load is applied with displacement rate 1 mm/min. The average value of test results of five specimens of each group is used to represent the characteristic static strength of unidirectional laminate. The tests were performed by using a universal testing machine INSTRON1195 with a 10 kn loading cell which is controlled by microcomputer. 3.2 Experimental Results The static tensile strength in the longitudinal and transverse direction, the static compressive strengths in the longitudinal and transverse directions, and the static shear strength of [45/ 45] 5s laminate are listed in Table 2. On the basis of the experimentally tested stress-strain relationships from UD laminates, the material properties of the UD laminates are obtained, the results being listed in Table 3. Table 4 shows the material properties of the fibers which are back-calculated by Equation (4) using the laminate properties listed in Table Micromechanics Finite Element Analysis The face-centered cubic unit cell model is constructed to analyze the stresses of fiber and matrix in microlevel by FEM using ANSYS code. Figure 4 shows the FEM meshes of the face-centered cubic model. The mechanical load and boundary constraints conditions are applied to the models as described in Section 2.2. Then the stress amplification factors and the thermal stress amplification factors of the key points in the fibers and matrix are obtained under prescribed mechanical loads with average unit stresses in first six cases, and one more case with thermal load of unit temperature as body force in the micromechanics models. Table 1. Experimental scheme and specimen size of the laminate UTS50/E51 Test type Stacking sequence Specimen size (mm) (L W H) In-plane shear [45/-45] 5s Longitudinal tensile [0] Longitudinal compressive [0] Transverse tensile [90] Transverse compressive [90] Table 2. Material strength of the UD laminate for UTS50/51 Strength parameters value Longitudinal tensile strength X (MPa) 2100 Longitudinal compressive strength X (MPa) 1932 Transverse tensile strength Y (MPa) 54 Transverse compressive strength Y (MPa) 179 In-plane S (MPa) 140 Table 3. Material properties of the UD laminate for UTS50/51 Material properties value E XX (GPa) 136 E YY (GPa) 10 E ZZ (GPa) E YY G XY (GPa) 4.7 G XZ (GPa) G XY G YZ (GPa) 3.2 µ XY 0.35 µ YZ 0.56 µ XZ µ XY α XX (1/k) α YY (1/k) α ZZ (1/k) α YY Where E ij (i,j=x,y,z) is Young s modulus of the ply; G ij (i,j=x,y,z) shear modulus of the ply; α ij (i,j=x,y,z) thermal expansion coefficient of the ply; µ ij (i,j=x,y,z) Poisson s ratio of laminate. Figure 5 shows the stress amplification factors and the thermal stress amplification factors of the key points in the fibers and matrix under prescribed mechanical loads with average unit stresses in six cases, and one case with thermal load of unit temperature as body force in the face-centered cubic micro-mechanics model. Table 4. Material properties of the fiber for UTS50 Material properties value E fxx (GPa) 240 E fyy (GPa) 42 E fzz (GPa) E YY G fxy (GPa) 23 G fxz (GPa) G fxy G fyz (GPa) 12 µ fxy 0.33 µ fyz 0.71 µ fxz µ fxy α fxx (1/k) α fyy (1/k) α fzz (1/k) α fyy Where E fij (i,j=x,y,z) is Young s modulus of the fibers; G fij (i,j=x,y,z) shear modulus of the fibers; α fij (i,j=x,y,z) thermal expansion coefficient of the fibers; µ i (i,j=x,y,z) Poisson s ratio of fibers. j Figure 4. The FEM meshes of the facecentered cubic model Polymers & Polymer Composites, Vol. 22, No. 2,

6 Wangnan Li, Hongneng Cai, and Jie Zheng Figure 5. Amplification factors of key points of fiber and matrix under different loading (a) Amplification factors of fiber, M 11 (b) Amplification factors of matrix, M 11 (c) Amplification factors of fiber, M 22 (d) Amplification factors of matrix, M 22 (e) Amplification factors of fiber, M 33 (f) Amplification factors of matrix, M 33 (g) Amplification factors of fiber, M 44 (h) Amplification factors of matrix, M 44 Amplification factors of fiber, M 55 (j) Amplification factors of matrix, M 55 (k) Amplification factors of fiber, M 66 (l) Amplification factors of matrix, M 66 (m) Equivalent thermal stress amplification factors of fiber (n) Equivalent thermal stress amplification factors of matrix 110 Polymers & Polymer Composites, Vol. 22, No. 2, 2014

7 Characterization of Strength of Carbon Fiber Reinforced Polymer Composite Based on Micromechanics It s important to note that Figure 5 only gives out the amplification factors of normal stresses of key points in the fibers and matrix under loads with average unit stresses in different kinds of stresses in the face-centered cubic model. The amplification factors of normal stresses of key points are not listed owing to the space limitation. Practically, all stress components amplification factors will be included in the macro-micro stress conversion. Meanwhile, the different stress amplification factors of key points in fibers and matrix are observed when different kinds of loading are applied in the face-centered cubic model, and stress amplification factors of fibers are same in the y and z directions as well as those of matrix being also same. The obtained amplification factors avoid the difference between the y and z directions using the conventional hexagonal model. Therefore, the facecentered cubic model can provide the homogeneities characteristics in the y and z directions in macro-level. 4. Strength of material characterization and failure envelopes analysis 4.1 Extraction of Critical MMF Strength Parameters of Constituents For the tested specimen of unidirectional laminates, the stress at the 17 key points of fiber and the 19 key points of matrix in the unit cell model can be calculated, using the stress amplification factors of key points in fiber and matrix, and the macroscopic strength shown in Table 2. For the UD laminate tension, the maximal σ vm of key points is taken as MMF critical parameters of fibers, longitudinal tensile strength, being shown in Figure 6a. The maximal σ 1 among these key points is symbolized using T f. Similarly, for the UD laminate compression, the maximal -σ 1 of key points is taken as MMF critical parameters of fibers, longitudinal compressive strength, being shown in Figure 6b. The maximal -σ vm among these key points is taken as C f. The first stress invariant I 1 of the key points in the matrix are taken as MMF critical parameters of the tensile strength of the matrix, being shown in Figure 6c. The maximal I 1 among these key points is taken as T m. The equivalent stress σ eq of the key points is taken as MMF critical parameters of compressive strength of the matrix, being shown in Figure 6d. The maximal equivalent stress σ eq among these key points is taken as C m. Then we can get the critical MMF strength parameters of constituents of UTS50/E51, being shown in Table Construction of Failure Stress Envelopes for Unidirectional Laminate In order to draw the biaxial failure stress envelopes for unidirectional laminate, the polar coordinates parameters rcosθ and rsinθ are used to represent stress Figure 6. The MMF parameters of key points of constituents of UTS50/E51, (a) tensile MMF parameters of fiber, (b) compressive MMF parameters of fiber, (c) tensile MMF parameters of matrix, (d) compressive MMF parameters of matrix (a) (b) (c) (d) Polymers & Polymer Composites, Vol. 22, No. 2,

8 Wangnan Li, Hongneng Cai, and Jie Zheng Table 5. Critical MMF strength parameters of constituents of UTS50/E51 Strength parameters Values Fiber tensile strength T f (MPa) 3710 Fiber compressive strength C f (MPa) 3440 Matrix tensile strength T m (MPa) 155 Matrix compressive strength C m (MPa) 207 components σ 1 and σ 2 in the off-axis laminate, respectively. For any angle θ, substitute the rcosθ and rsinθ to Equation (3) to calculate the micro-stress of key points in fibers and matrix, and calculate the MMF parameters of fibers and matrix with Equation (1). For any given value of θ, MMF parameters of constituents are function of r. By verifying Equation (5) by combining Equation (2) and MMF parameters of constituents as function of r, the value of r, corresponding to the initial failure of laminate, can be calculated. Excel VBA macro is used to calculate the r i value, which is implied in the four functions in Equation (5). Finally, the stress components σ 1 and σ 2 of laminate can be calculated by turning the r value into the rcosθ and rsinθ at the designated angle θ. The points as many as possible in biaxial failure stress envelopes can be obtained for unidirectional laminate, and finally the smooth failure stress envelope can be joined together. The drawing of σ 1 σ 6 and σ 2 σ 6 failure stress envelopes for unidirectional laminate is the same as described above. 4.3 Failure Envelopes Analysis and Comparison for the UD Laminate Biaxial failure stress envelopes are constructed for unidirectional laminate using the MMF theory, and compared to Tsai-Wu failure envelopes. First, the biaxial stress (σ 1 σ 2 ) failure envelope of laminate is constructed using the MMF theory and compared with the Tsai-Wu envelope, being shown in Figure 7. The failure stress envelope obtained from the face-centered cubic micro-mechanics model using the MMF theory is trapezoidal, while the failure stress envelope predicted by the Tsai-Wu criterion forms an oval shape. Both MMF theory and Tsai-Wu failure criterion give the same prediction for the X, X, Y, and Y. The MMF theory allows greater stress components when the laminate bears a biaxial tensile-compressive loading, corresponding to the second and fourth quadrants in the σ 1 σ 2 plane, while the Tsai-Wu failure criterion allows greater stress components when the laminate is subject to biaxial compression. Using the MMF theory, the failure envelopes can be divided into fiber failure area, which locates within the shallow area in the σ 1 σ 2 plane, and matrix failure area. Figure 8 shows the biaxial stress failure envelopes of the same lamina based on MMF and Tsai-Wu criterion in the longitudinal and longitudinal shear (σ 1 σ 6 ) (5) plane, being shown in Figure 8a, and the transverse and longitudinal shear stress (σ 2 σ 6 ) plane, being shown in Figure 8b. Generally, the shear stress predicted by MMF theory is lower than that predicted by Tsai-Wu failure criterion. Since each point in the MMF failure envelope is obtained on the basis of failure mechanism each fiber failure or matrix failure, the predicted value is more reliable. 5. The predicted strength and failure mechanism analysis of quasi-isotropic laminate 5.1 The Predicted Strength of Static Tensile of Quasi-isotropic Laminate The tensile strength and failure mechanism of quasi-isotropic laminate is analyzed and compared. The failure of multi-directional laminate usually is a progressive process, so the progressive failure algorithm is used. Gradual damage degradation model has been proposed by the Camanho et al. 14. This method is extended to progressive analysis combined with the threedimensional (3D) finite element analysis and MMF theory. Because the Tsai-Wu theory cannot analyzes constituent failure, the mechanical properties of the failure layer are degraded to one percent of the initial value. The load-displacement curves of static tensile of quasi-isotropic laminate obtained from MMF analysis and Tsai-Wu analysis are compared. The quasi-isotropic laminate was made of multidirectional laminate with a stacking sequence of [45/0/-45/90] 2s. The specimen geometry and boundary conditions are shown in Figure 9. The analysis is implemented using the UMAT written with the Abaqus scripts, and the FEM element type is threedimensional solid elements C3D8R. The predicted load-displacement curves for the quasi-isotropic laminate 112 Polymers & Polymer Composites, Vol. 22, No. 2, 2014

9 Characterization of Strength of Carbon Fiber Reinforced Polymer Composite Based on Micromechanics bearing the static tensile loading are obtained using the MMF analysis and Tsai-Wu analysis, as shown in Figure 10. The figure shows that the static tensile strength of quasi-isotropic laminate obtained from MMF analysis is 790MPa, being greater than the result 670MPa obtained from Tsai- Wu analysis. At the same time, the predicted load-displacement curve has two times of load jumps before the final failure. This shows that the initial failure and subsequent failure occurs in the laminate before final failure. The detailed analysis method is described in next section. Figure 7. Biaxial failure stress envelopes for UTS50/E51 unidirectional laminate under combined longitudinal and transverse loading Figure 8. Biaxial failure stress envelopes for UTS50/E51 unidirectional laminate under combined (a) longitudinal and longitudinal shear loading and (b) transverse and longitudinal shear loading Figure 9. The quasi-isotropic laminate specimen with stacking sequence [45/0/-45/90] 2s and boundary conditions Polymers & Polymer Composites, Vol. 22, No. 2,

10 Wangnan Li, Hongneng Cai, and Jie Zheng 5.2 The Failure Mechanism Analysis of Static Tension of Quasi-isotropic Laminate The MMF is used to analyze the failure mechanism of static tensile of quasiisotropic laminate. The stresses of fiber and matrix in micro-level and failure index distribution of all layers of the quasi-isotropic laminate under static tensile loading are analyzed using the MMF. The distribution of four kinds of the failure indices k for each layer is analyzed. The maximum value among the four kinds of the failure index, which reaches 1, is used to judge the constituent failure. The extreme values of four kinds of the initial and final failure index k of each layer of quasi-isotropic laminate is plotted and compared. Figure 11 shows the result for the first four layers. The MMF analysis shows the initial failure of quasiisotropic laminate under static tensile loading is triggered by matrix failure in the 90 degree layer, being shown in Figure 11a. The second failure of quasi-isotropic laminate under static tensile loading is triggered by matrix failure in the ±45 degree layer, as shown in Figure 11b. The final failure of quasi-isotropic laminate under static tensile loading is triggered by fiber failure in the 0 degree layer, being shown in Figure 11c. Figure 10. The predicted load-displacement curves of the static tensile failure strength of quasi-isotropic laminate of UTS50/E51 with stacking sequence [45/0/- 45/90] 2s under static tensile loading 6. The static tensile test of quasi-isotropic laminate The tested specimen sizes of quasiisotropic laminate are shown in Figure 9, the results being shown in Figure 12. The static tensile tests are performed according to the standard of ISO527, being loaded at a displacement rate of 1 mm/min, using a universal testing machine INSTRON1195. The tested average value of three specimens is used to represent the static tensile strength of the quasi-isotropic laminate. Figure 13 provides the comparison between the predicted value and the tested value. The figure shows that the predicted result obtained from the MMF method is closer to the experimental result, the relative error being 4.4%; however, the predicted result obtained from the Tsai-Wu method leads to larger relative error, being 18.9%. It means that the MMF approach provides the accurate prediction. Meanwhile no obvious load jump occurred in the tested load-displacement curve. This might be due to the fact that matrix failure of the quasi-isotropic laminate under static tensile loading does not make a great difference for the tested load-displacement curve. 7. Conclusions 1. The non-interactive micromechanics of failure approach to characterize the strength of Figure 11. The extreme values of four kinds of failure indices of first four layers of quasi-isotropic laminate made of UTS50/ E51 laminate with stacking sequence [45/0/-45/90] 2s under static tensile, (a) the initial failure, (b) the second failure, and (c) the final failure 114 Polymers & Polymer Composites, Vol. 22, No. 2, 2014

11 Characterization of Strength of Carbon Fiber Reinforced Polymer Composite Based on Micromechanics composite constituent was established. This method includes theory of micro-mechanics of failure (MMF), micro-mechanics analysis methodology and macro-mechanical test, extraction of critical MMF strength parameters of constituents. 2. The critical MMF strength parameters of constituents of UTS50/E51 were characterized. The MMF parameters were calculated using the stress amplification factors and the measured strengths of unidirectional laminate, and the critical MMF strength parameters of constituents were extracted. For UTS50/E51, the fiber tensile strength T f is 3710 MPa, fiber compression strength C f 3440 MPa, matrix tensile strength T m 156 MPa, and matrix compression strength C m 205 MPa. Figure 12. The test result of the static tensile failure strength of quasi-isotropic laminate of UTS50/E51 with stacking sequence [45/0/-45/90] 2s Figure 13. Comparison between the predicted value and the tested result of quasiisotropic laminate UTS50/E51 with stacking sequence [45/0/-45/90] 2s 3. Failure stress envelopes were constructed using the MMF theory for unidirectional laminate. 4. Prediction of the static tensile test of quasi-isotropic laminate on the basis of macro-level and micromechanical failure analysis was carried out. The predicted failure mode agrees with the experimental result. Acknowledgement This work is supported by National Science Foundation of China and Civil Aviation Administration of China (No ), and supported by the Fundamental Research Funds for the Central Universities. References 1. Tsai S.W., Strength Characteristics of Composite Materials, NASA Contractor Report, CR-224. National Aeronautics and Space Administration, Tsai S.W. and Wu E.M., A General Theory of Strength for Anisotropic Materials. Journal of Composite Materials, 5 (1971) Harlow D.G. and Phoenix S.L., Bounds on the probability of failure of composites. International Journal of Fracture, 15(4) (1979) Echaabi J., Trochu F., and Gauvin, Review of failure criteria of fibrous composite materials. Polymer Composites, 56 (1996) Chiang M.Y.M., Wang X.F., Schultheisz C.R., and He J.M., Prediction and three-dimensional Monte-Carlo simulation for tensile properties of unidirectional hybrid composites. Composites Science and Technology, 65(11-12) (2005) Hinton M.J. and Soden P.D., Predicting failure in composite laminates: the background to the exercise. Composites Science and Technology, 58(7) (1998) Puck A. and Schürmann H., Failure Analysis of FRP Laminates by Means of Physically Based Phenomenological Models, Composites Science and Technology, 58(7) (1998) Committee on New Materials for Advanced Civil Aircraft National Materials Advisory Board. New materials for next-generation commercial transports. Washington, D.C. National academy press, 1996, Ha S.K., Micro-Mechanical Failure Theory Continuous Fiber Reinforced Composites, Proceedings of the 12th Composites Durability Workshop (CDW-12), St. Maximin, France, March (CD-Rom), Polymers & Polymer Composites, Vol. 22, No. 2,

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