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Progressive damage arrest mechanism in tailored laminated plates with a cutout
Acta Mechanica Solida Sinica, Vol. 21, No. 1, February, 2008 Published by AMSS Press, Wuhan, China. DOI: 10.1007/s10338-008-0809-2
ISSN 0894-9166
PROGRESSIVE DAMAGE ARREST MECHANISM IN TAILORED LAMINATED PLATES WITH A CUTOUT De Xie1
Sherrill B. Biggers, Jr.
(Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA)
Received 1 March 2007; revision received 13 December 2007
ABSTRACT This study addresses the effectiveness of a simple stiffness tailoring concept to delay damage initiation, control damage progression, and improve residual strength in tensile-loaded composite plates with a central circular cutout. The tailoring concept is to simply reposit all axially oriented (0◦ ) material into regions near the edge of the plate away from the cutout. This tailoring is done in a way so as not to affect the weight of the plate. This accomplishes several beneficial changes in the way that the plate resists loading with no increases in material cost or weight. Lowering the axial stiffness of the laminate surrounding the cutout lowers the stress concentration. Increasing the axial stiffness near edges of the plate attracts loading away from the vicinity of the cutout to further lower stresses in the critical cutout region. This study focuses on in-plane response including damage progression and residual strength as a function of the degree of tailoring and cutout size. Strength and stiffness properties typical of IM7/8551-7 preperg material were assumed and a modified version of the Hashin failure criteria was used to identify the local damage. Results show that tailoring can significantly increase the damage initiation load and the residual strength. In some cases, observed evidence shows that tailoring performs as a damage arrest mechanism.
KEY WORDS progressive failure analysis, tailoring design, damage arrest, finite element analysis
I. INTRODUCTION Damage analysis and residual strength evaluation are important aspects in composite structural design and health monitoring. One major group of modeling damage directly incorporates damage into constitutive equations by introducing a set of internal state variables (IVBs)[1–6] . Another group uses failure criterion to detect the damage at a specific material point and then degrades the material properties at failure points[7–26] . Due to the complex feature of the phenomena, no single criterion could provide accurate and meaningful predictions of failure over all cases other than a very limited range. Therefore, different failure criteria together with various material degradation rules have been proposed and recently a world-wide failure exercise (WWFE) has been carried out to evaluate these methods[20–22] . The nature of fibrous composite materials and structures enables them to be highly adaptable to satisfy specific functions through proper choices of constituent materials and design concepts. In laminated composites, the designer may specify different materials, fiber orientations, and layers at various locations through the thickness or over the plane of the laminate to improve its response to mechanical or thermal loads. Often innovative uses of materials can be achieved with simple, low-cost
Corresponding author. E-mail: [email protected], Tel: +86-27-87540101 Project supported by NASA, the State of South Carolina, and Clemson University through the EPSCoR grant ‘Development and Enhancement of Research Capability for Aircraft Structures and Materials’.
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manufacturing methods. One of them is a simple stiffness tailoring which simply removes all axially oriented 0◦ materials from the central regions of a rectangular plate and relocates the material in the edge of the plate. This uniaxial stiffness tailoring has been shown to improve buckling loads, initial failure loads and residual strength in laminated plates and shells with/without cutouts[23, 24, 27–30] . This paper addresses the effectiveness of the simple stiffness tailoring design concept to delay damage initiation, control damage progress, and improve residual strength in plates with a central circular cutout under uniaxial tensile loading. All improvements on the plate are attributed to the tailoring design since the laminates consist of a single material without additional materials such as skin-stiffeners.
II. TAILORING APPROACH A typical plate evaluated is shown in Fig.1. Loading is uniaxial in the x-direction (vertical) and applied monotonically as a uniform end displacement in tension. The tailoring concept is simply to reposit all axially oriented (0◦ ) material into regions of certain widths near the edges of the plate. The edge regions in tailored plates are of total width b1 while the overall plate width is b. The width ratio, ¯b = b1 /b, defines the extent of the tailoring. When ¯b = 1.0, the plate is uniform. When ¯b < 1.0, the total thickness of the 0◦ material in the edge regions is increased such that the total weight of the material in the plate remains unchanged. The 0◦ material is uniformly distributed in the edge regions and its ply thickness t¯ is 1 − πr2 /(ab) t0 t¯ = (1) ¯b where t0 is the original ply thickness. This is obtained based on the assumption that the volume of 0◦ material remains unchanged before and after tailoring. Therefore, improvement on the plate behavior is attributed to the tailoring design without adding materials.
Fig. 1. Geometry of uniform and tailored plates with a central circular cutout.
III. DAMAGE MODEL To evaluate the tailoring design concept proposed in the previous section, the progressive damage analysis is performed. The damage evolution model used is a ply-discount method in which stress and stiffness are reduced locally and instantaneously to zero once failure occurs at a point in a ply. These changes due to the damage were introduced at individual integration points as they occurred during loading. A progressive damage model, combining the Hashin[7] and Chang[10] failure criteria, was used to indicate damage initiation. The fiber failure index and matrix failure index are defined as
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ACTA MECHANICA SOLIDA SINICA
– Fiber failure index
ef =
σ11 Xt
2 +
2 σ11 − , ef = Xc – Matrix failure index
σ11 > 0
(2a)
σ11 < 0
(2b)
2
F1 , σ22 > 0 F2
2 σ22 2 Y σ22 F1 c = + −1 + , 2S 2S Yc F2
em =
em
σ22 Yt
F1 , F2
2008
+
(2c)
σ22 < 0
(2d)
where Xt and Xc are the longitudinal tensile strength and compressive strength, respectively. Yt and Yc are the transverse tensile strength and compressive strength, respectively. S is the shear strength. F1 and F2 are the nonlinear correctors obtained by Chang[10] as 2 2τ12 4 + 3ατ12 G12 2S 2 F2 = + 3αS 4 G12
F1 =
(3a) (3b)
where G12 is the shear modulus and α is a material non-linearity parameter measured experimentally[10] . Before any failure occurs, the material behavior at a typical point in the laminar is described by a linear 2D orthotropic stiffness matrix C: ⎧ ⎫ ⎡ ⎫ ⎤⎧ C11 C12 0 ⎨ σ11 ⎬ ⎨ ε11 ⎬ σ22 = ⎣ C12 C22 0 ⎦ ε22 (4) ⎩τ ⎭ 0 0 C66 ⎩ γ12 ⎭ 12 where C11 =
E11 E22 ν12 E22 , C22 = , C12 = , and C66 = G12 . 1 − ν12 ν21 1 − ν12 ν21 1 − ν12 ν21
Material degradation within the damaged area was evaluated based on the mode of failure predicted by the failure criteria of equation (2). Therefore, the residual strength and stiffness of composites strongly depend on the mode of failure in each layer. Here a sudden reduction model with uncoupled longitudinal and transverse failure is used. The property degradation models for each layer are proposed as followings. For fiber failure in a layer (ef > 1), the longitudinal modulus E11 is set to zero so that the laminate can no longer carry any load in the longitudinal direction. Poisson’s ratio ν12 is reduced to zero to uncouple the longitudinal failure from the transverse property. For matrix failure in a layer (em > 1), the transverse modulus E22 is set to zero so that the laminate can no longer carry any load in the transverse direction. Poisson’s ratio ν12 is reduced to zero for uncoupling. In actual practice, a small number a little higher than zero was used for the value of reduced stiffness instead of zero to avoid numerical problem such as divergence. For both fiber and matrix failures, ν21 = ν12 (E22 /E11 ) is enforced to fully uncouple the longitudinal and transverse failure from each other. Therefore, damage modes are summarized as – Fiber failure: C11 = C12 = 0, C22 = E22 – Matrix failure: C11 = E11 , C22 = C12 = 0 This uncoupling failure model indicates that the laminate having failures in one direction can still support load in the other direction. However, this load capacity should be reduced to some degree. Therefore, the shear modulus G12 is degraded gradually as[10] G12 = (1 − d)G012
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where the damage index d has the expression as d=
2 3 − 2ατ12 /γ12 3αG012 τ12 0 2 1 + 3αG12 τ12
where G012 is the original value of the shear modulus and G12 is the updated value during loading. R finite element code[31, 32] . The UMAT routine The analysis was conducted using the ABAQUS R was used to account for the present damage model. The stress was updated based within ABAQUS on the total strain. In calculation, no severe convergence problems were encountered.
IV. RESULTS AND DISCUSSIONS The baseline panel is a square (152.4×152.4 mm2) and has a uniform quasi-isotropic layup [±45/0/90]s with mechanical properties of typical IM7/8551-7 prepreg material (as shown in Table 1). The value of the material non-linearity parameter α is chosen to be 1.2 × 107 MPa−3[10] . The convergence study on both mesh refinement and load increment refinement was performed as a first step. Then four cutout sizes combined with various tailoring extent were investigated for parametric study. Table 1. Mechanical properties of IM7/8551-7
Stiffness (GPa) E11 = 130.3 E22 = 9.38 ν12 = 0.33 G12 = 4.5
Strength (MPa) Xt = 2526 (axial tension) Xc = 916 (axial compression) Yt = 69.4 (transverse tension) Yc = 69.4 (transverse compression) S = 120 (in-plane shear)
4.1. Convergence Study The convergence studies were performed on a plate with 12.7 mm diameter cutout (2r/b = 0.33). Grid refinement was investigated first. The ultimate load of a uniform plate and tailored plates with several width ratios are plotted in Fig.2 as a function of the number of elements. It is clear that the ultimate load decreases rapidly as coarse meshes are refined and tends to approach a constant after the number of element reaches a certain value, i.e. mesh with 736 elements for this specific case. This shows the plates modeled here are converged with respect to the mesh refinement at sufficient big number of elements. Another important convergence investigation Fig. 2 Convergence study on grid refinement. concerns the displacement increment used in the nonlinear solution procedure. If the increment is too big, some slight changes of edge reaction vs. displacement curves will be likely to be omitted and thus degrades the ultimate load prediction. The predicted value usually overshoots its real value. The smaller the displacement increment is, the more accurate the predicted ultimate load is. But this may results in excessive computational time. Therefore a reasonable value of displacement increment is needed to achieve both sufficiently accurate prediction of the ultimate load and an acceptable computational expense. In Fig.3, the ultimate load is plotted against the reciprocal of the displacement increment, 1/(ΔU ). When 1/(ΔU ) reaches 3.9 × 103 mm−1 , the plot becomes almost a horizontal line corresponding to a displacement increment of 0.00254 mm and a strain increment of 3.33 × 10−5 mm/mm. Based on these studies, the problem investigated here is reasonably converged with respect to both grid refinement and displacement increment refinement with a 736 element mesh together with a 0.00254 mm displacement increment. These parameters are the baseline for the other cases to be discussed later. A typical mesh for 2r/b = 0.3 is shown in Fig.4. Due to the symmetry of the geometry, boundary conditions, and the stacking sequence, only one quarter of the plate is modeled using four-node plane stress element (CPS4 of ABAQUS).
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Fig. 3. Convergence study on displacement increment refinement. Fig. 4. Finite element mesh of a plate with 2r/b = 0.3.
4.2. Parametric Study Four plates with different cutout sizes (2r/b = 0.2, 0.3, 0.4, 0.5) were investigated for parametric study. Figures 5 and 6 shows plots of average running load versus applied end displacement. Results are shown for uniform (¯b = 1.0) plates and tailored plates with ¯b ranging from 0.10 to 0.76 for four different cutout sizes. Tailoring offers significant advantages in increasing ultimate loads and ultimate end displacement. Increases in stiffness due to tailoring are also observed. However the most important observation is that, in the more tailored cases (¯b = 0.10 and other smaller values), there is an initial peak load followed by the unloading and the reloading till the ultimate load. This shows that the tailored region can arrest, or at least alleviate, a temporarily unstable progress of the damage. This very clear warning far prior to the final failure could be highly useful in many situations. To help understand this behavior, damaged regions in some selected plies and at selected end displacement values are shown in Figs.7 and 8. Zones in dark color represent the damaged areas. Here we first consider a flat panel with the width ratio ¯b = 0.10 and the cutout size 2r/b = 0.30. In this case, the initial peak load occurs at end displacement U = 0.8103 mm, and the reloading starts at U = 0.8636 mm, and the final failure occurs at U = 1.125 mm. Matrix damage initiates in the 90◦ plies at the root of the cutout at U = 0.3048 mm. This damage grows in a stable way both horizontally and vertically until it reaches the tailored edge region at
Fig. 5. Avg. running load vs. end displacement of plates with different cutout size and tailoring degree.
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Fig. 6. Avg. running load vs. end displacement of plates with different cutout size and tailoring degree (Fig.5 continued).
U = 0.5817 mm. This damage then grows vertically and is accompanied by new matrix damage at the upper corner of the panel. The damage regions continue to grow until they meet each other and a final saturation level develops at about U = 0.7925 mm. A narrow band of fiber and matrix damage starts to develop in this ply at the top of the cutout at U = 0.8509 mm, and grows only slightly in the vertical direction until final failure at U = 1.125 mm. Matrix damage initiates in the −45◦ plies at U = 0.5207 mm and in the +45◦ plies at U = 0.6858 mm, both are at the root of the cutout and grow in a narrow horizontal band toward the panel side. These damage events can be seen as slight stiffness reductions in the load-displacement curve prior to the initial load peak. At the initial load peak (U = 0.8103 mm), fiber damage starts to develop at the root of the cutout in 90◦ and 45◦ plies. This damage grows in an unstable way in a narrow horizontal band toward the
Fig. 7. Damage progression in most highly tailored plate (¯b = 0.10, 2r/b = 0.3).
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2008
Fig. 8. Damage progression in most highly tailored plate (¯b = 0.10, 2r/b = 0.3) (Fig.7 continued).
panel side until it is arrested as it reaches the tailored edge regions at U = 0.8636 mm. This unstable growth corresponds to the region of the decreasing load in the load-displacement profile. Once the arrest occurs, reloading commences with little new damage (except in the 90◦ plies at the top of the cutout as mentioned above) until the ultimate load is reached. Final failure is quite abrupt as fiber damage grows rapidly in the 0◦ and other plies through the tailored edge regions at U = 1.125 mm. This final damage develops horizontally, continuing the fiber damage along the net section in the other plies. Figure 9 shows how damage grows in the uniform plate with the same cutout size of 2r/b = 0.30. From these pictures, one can observe that the matrix failure of the 90◦ layer initiates at a rather low load level (U = 0.254 mm) at the edge of the cutout. The damage grows in a controlled manner as loading increases. At approximately twice the damage initiation load level (U = 0.5309 mm) matrix
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damage initiates at two remotely separate locations in the 90◦ layer and both damage zones progress at an increased pace until they meet. The fiber failure of both 90◦ and 0◦ layers only occurs when the edge reaction reaches its ultimate value and damage progresses unimpededly even when the load begins to drop. Therefore, fiber failure takes place very rapidly and always represents catastrophic failure. One can see that the fiber damage zones look like a slot and can be considered as a crack that separates the plate at centerline. The matrix failure of the 0◦ material is quite different from that of the 90◦ material and progresses in the same way as the fiber failure of the 0◦ layer. In other words, the matrix damage accompanies the fiber damage in the 0◦ layer. Since the damage pattern among the fiber failure of 90◦ and both fiber and matrix failure of 0◦ layer are very similar, only that for 90◦ is shown in Fig.9. Both the fiber and matrix failure of +45◦ layer and fiber failure of −45◦ present very similar damage propagation patterns to the fiber failure of 90◦ and only those of the latter layer are shown in Fig.9 as well. It initiates soon after the matrix failure in 90◦ layer occurs and progresses across the centerline to the unloaded side as the plate totally fails. No damage propagation was found for the fiber failure of −45◦ layer and the matrix failure of +45◦ layer. Figures 10 and 11 shows the damage growth in the most gently tailored plate with ¯b = 0.64. In this case, the tailoring layer mostly approaches the edge of the cutout. Patterns of progressive damages are observed more like that of the uniform plate discussed earlier. Progressive damages are observed in both 90◦ and −45◦ layers in the form of the matrix failure and the matrix failure in 90◦ layer dominates most of the damage progression. The fiber failure in all layers (90◦ , 0◦ and ±45◦) and matrix failure
Fig. 9. Damage progression in uniform plate (¯b = 1.00, 2r/b = 0.3).
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2008
of 0◦ and 45◦ appear only at the end damage propagation and represents complete failure of the plate. Due to the similarity to its fiber failure, the matrix failure of 0◦ layer is not shown in Figs.10 and 11. Peak values taken from each load displacement curve in Figs.5 and 6 are plotted against the tailoring width ratio ¯b in Fig.12. Here another important difference from the postbuckling problem[23, 24] can be seen. In former cases, the most extreme width ratio (smallest value of ¯b) produces the highest ultimate load and the least extreme tailoring produces the lowest ultimate load, whereas both of these results are reversed in the current case of tensile loading. Furthermore, there is an intermediate value of the width ratio that yields the lowest ultimate load in tension. To reveal the relative increases in the ultimate load in tailored panels compared to uniform panels, the data in Fig.12 are normalized by ultimate loads of the uniform panel with the same cutout size and these ultimate load ratios R are plotted in Fig.13. Improvements range from 50% to 100%. However, the damage arrest feature observed in the most highly tailored panels does not occur in the most gently tailored panels. As the cutout size becomes smaller, there is a wider range of tailoring width ratios for which damage arrest occurs. So, there is a design trade-off between having the highest ultimate load and having a slightly lower ultimate load, which can increase the ability to arrest damage.
Fig. 10. Damage progression in most gently tailored plate (¯b = 0.64, 2r/b = 0.3)).
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Fig. 11. Damage progression in most gently tailored plate (¯b = 0.64, 2r/b = 0.3)) (Fig.10 continued).
Fig. 12. Ultimate loads, Nx .
Fig. 13. Ultimate load ratios, R.
V. CONCLUSIONS This paper pointed out some complex effects that tailoring concepts can have on the initiation and progression of the in-plane damage in composite plates with central cutouts under tensile loading. The tailoring concept considered was shown to be effective in improving the average stiffness, the ultimate load, and the ultimate average axial strain in plates. Compared to uniform panels, ultimate loads can be increased by as much as 110% for the particular material studied here. Observation of the damage progression showed that tailoring can arrest unstable damage growth in tensile loading and greatly extended the range of the applied end displacement between damage initiation and final failure.
References [1] Schapery,R.A., Nonlinear viscoelastic and viscoplastic constitutive equations with growing damage. International Journal of Fracture, 1999, 97: 33-66. [2] Talereja,R., A continuum-mechanics characterization of damage in composite materials. Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, 1985, 399: 195-216. [3] Kumar,R.S. and Talreja,R., A continuum damage model for linear viscoelastic composite materials. Mechanics of Materials, 2003, 35: 463-480.
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[4] Allen,D.H., Harris,C.E. and Groves,S.E., A thermomechanical constitutive theory for elastic composites with distributed damage: 1 Theoretical development. International Journal of Solids and Structures, 1987, 23: 1301-1318. [5] Harris,C.E., Coats,T.W., Allen,D.H. and Lo,D.C., A progressive damage model and analysis methodology for predicting the residual strength of composite laminates. Journal of Composites Technology and Research, 1997, 19: 3-9. [6] Basu,S., Waas,A.M. and Ambur,D.R., Computational modeling of damage growth in composite laminates. AIAA Journal, 2003, 41: 1158-1166. [7] Hashin,Z., Failure criteria for unidirectional fiber composites. Journal of Applied Mechanics, 1980, 47: 329334. [8] Minnetyan,L., Gotsis,P.K. and Chamis,C.C., Progressive damage and fracture of unstiffened and stiffened composite pressure vessels. Journal of Reinforced Plastics and Composites, 1997, 16: 1711-1724. [9] Gotsis,P.K., Chamis,C.C. and Minnetyan,L., Progressive fracture of blade containment composite structures. Journal of Reinforced Plastics and Composites, 1997, 16: 1407-1424. [10] Chang,K.Y., Liu,S. and Chang,F.K., Damage Tolerance of laminated composites containing an open hole and subjected to tensile loadings. Journal of Composite Materials, 1991, 25: 274-301. [11] Qing,X.L., Chang,F.K. and Starnes,J., Damage tolerance of notched composite laminates with reinforcing strips. Journal of Composite Materials, 2003, 37: 111-128. [12] Lee,H.H. and Hyer,M.W., Postbuckling failure of composite plates with holes. AIAA Journal, 1993, 31: 1293-1298. [13] McMurray,J.M. and Hyer,M.W., Response and failure characteristics of internally pressurized elliptical composite cylinders. AIAA Journal, 2002, 40: 117-125. [14] Sleight,D.W., Norman,F., Knight,Jr N.F. and Wang,J.T., Evaluation of a progressive failure methodology for laminated composite structures, AIAA Paper 97-1187, Aril 1997. [15] Sleight,D.W., Progressive failure analysis methodology for laminated composite structures. NASA/TP1999-209107. [16] Knight,N.F., Rankin,C.C. and Brogan,F.A., STAGS computational procedure for progressive failure analysis of laminated composite structures. International Journal of Non-Linear Mechanics, 2002, 37: 833-849. [17] Davila,C.G., Ambur,D.R. and McGowan,D.M., Analytical prediction of damage growth in notched composite panels loaded in compression. Journal of Aircraft, 2000, 37: 898-905. [18] Goyal,V.K., Jaunky,N.R., Johnson,E.R. and Ambur,D.R., Intralaminar and interlaminar progressive failure analyses of composite panels with circular cutouts. Composite Structures, 2004, 64: 91-105. [19] Spottswood,S.M. and Palazotto,A.N., Progressive failure analysis of a composite shell. Composite Structures, 2001, 53: 117-131. [20] Hinton,M.J. and Soden,P.D., Predicting failure in composite laminates: the background to the exercise. Composites Science and Technology, 1998, 56: 1001-1010. [21] Hinton,M.J., Kaddour,A.S. and Soden,P.D., Evaluation of failure prediction in composite laminates: background to ‘Part B’ of the exercise. Composites Science and Technology, 2002, 62: 1481-1488. [22] Hinton,M.J., Kaddour,A.S. and Soden,P.D., Evaluation of failure prediction in composite laminates: background to ‘Part C’ of the exercise. Composites Science and Technology, 2004, 64: 321-327. [23] Baranski,A.T. and Biggers,Jr S.B., Post-buckling analysis of tailored composite plates with progressive damage. Composite Structures, 1999, 46: 245-255. [24] Xie,D. and Biggers,Jr S.B., Postbuckling analysis with progressive damage modeling in tailored laminated plates and shells with a cutout. Composite Structures, 2003, 59: 199-216. [25] Zhang,Y.F., Tang,C.A., Zhang,Y.B. and Liang,Z.Z., Fractual process and toughening mechanism of laminated ceramic composites. Acta Mechanica Solida Sinica, 2007, 20(2): 141-148. [26] Zhang,J.Q. and Wei,Y.Q., A predictive approach to the in-plane mechanical properties of stitched composite laminates. Acta Mechanica Solida Sinica, 2007, 20(2): 130-140. [27] Biggers,S.B. and Srinivasan,S., Compression buckling response of tailored composite plates. AIAA Journal, 1993, 31: 590-596. [28] Biggers,S.B. and Pageau,S., Shear buckling response of tailored composite plates. AIAA Journal, 1994, 32: 1100-1103. [29] Biggers,Jr S.B. and Browder,Jr T.M., Buckling-load interaction in tailored composite plates. Composites Part B: Engineering, 1994, 4: 745-761. [30] Biggers,S.B. and Srinivasan,S., Postbuckling response of piece-wise uniform tailored composite plates in compression. Journal of Reinforced Plastics and Composites, 1994, 13: 803-821. [31] Xie,D., Damage progression in tailored Laminated panels with a cutout and delamination growth in sandwich panels with tailored face sheets. Ph.D. dissertation. Clemson University. December 2002. [32] HKS, Inc. ABAQUS Theory and User’s Manual, version 6.2. HKS, Pawtucket, RI.
ISSN 0894-9166
PROGRESSIVE DAMAGE ARREST MECHANISM IN TAILORED LAMINATED PLATES WITH A CUTOUT De Xie1
Sherrill B. Biggers, Jr.
(Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA)
Received 1 March 2007; revision received 13 December 2007
ABSTRACT This study addresses the effectiveness of a simple stiffness tailoring concept to delay damage initiation, control damage progression, and improve residual strength in tensile-loaded composite plates with a central circular cutout. The tailoring concept is to simply reposit all axially oriented (0◦ ) material into regions near the edge of the plate away from the cutout. This tailoring is done in a way so as not to affect the weight of the plate. This accomplishes several beneficial changes in the way that the plate resists loading with no increases in material cost or weight. Lowering the axial stiffness of the laminate surrounding the cutout lowers the stress concentration. Increasing the axial stiffness near edges of the plate attracts loading away from the vicinity of the cutout to further lower stresses in the critical cutout region. This study focuses on in-plane response including damage progression and residual strength as a function of the degree of tailoring and cutout size. Strength and stiffness properties typical of IM7/8551-7 preperg material were assumed and a modified version of the Hashin failure criteria was used to identify the local damage. Results show that tailoring can significantly increase the damage initiation load and the residual strength. In some cases, observed evidence shows that tailoring performs as a damage arrest mechanism.
KEY WORDS progressive failure analysis, tailoring design, damage arrest, finite element analysis
I. INTRODUCTION Damage analysis and residual strength evaluation are important aspects in composite structural design and health monitoring. One major group of modeling damage directly incorporates damage into constitutive equations by introducing a set of internal state variables (IVBs)[1–6] . Another group uses failure criterion to detect the damage at a specific material point and then degrades the material properties at failure points[7–26] . Due to the complex feature of the phenomena, no single criterion could provide accurate and meaningful predictions of failure over all cases other than a very limited range. Therefore, different failure criteria together with various material degradation rules have been proposed and recently a world-wide failure exercise (WWFE) has been carried out to evaluate these methods[20–22] . The nature of fibrous composite materials and structures enables them to be highly adaptable to satisfy specific functions through proper choices of constituent materials and design concepts. In laminated composites, the designer may specify different materials, fiber orientations, and layers at various locations through the thickness or over the plane of the laminate to improve its response to mechanical or thermal loads. Often innovative uses of materials can be achieved with simple, low-cost
Corresponding author. E-mail: [email protected], Tel: +86-27-87540101 Project supported by NASA, the State of South Carolina, and Clemson University through the EPSCoR grant ‘Development and Enhancement of Research Capability for Aircraft Structures and Materials’.
Vol. 21, No. 1
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· 63 ·
manufacturing methods. One of them is a simple stiffness tailoring which simply removes all axially oriented 0◦ materials from the central regions of a rectangular plate and relocates the material in the edge of the plate. This uniaxial stiffness tailoring has been shown to improve buckling loads, initial failure loads and residual strength in laminated plates and shells with/without cutouts[23, 24, 27–30] . This paper addresses the effectiveness of the simple stiffness tailoring design concept to delay damage initiation, control damage progress, and improve residual strength in plates with a central circular cutout under uniaxial tensile loading. All improvements on the plate are attributed to the tailoring design since the laminates consist of a single material without additional materials such as skin-stiffeners.
II. TAILORING APPROACH A typical plate evaluated is shown in Fig.1. Loading is uniaxial in the x-direction (vertical) and applied monotonically as a uniform end displacement in tension. The tailoring concept is simply to reposit all axially oriented (0◦ ) material into regions of certain widths near the edges of the plate. The edge regions in tailored plates are of total width b1 while the overall plate width is b. The width ratio, ¯b = b1 /b, defines the extent of the tailoring. When ¯b = 1.0, the plate is uniform. When ¯b < 1.0, the total thickness of the 0◦ material in the edge regions is increased such that the total weight of the material in the plate remains unchanged. The 0◦ material is uniformly distributed in the edge regions and its ply thickness t¯ is 1 − πr2 /(ab) t0 t¯ = (1) ¯b where t0 is the original ply thickness. This is obtained based on the assumption that the volume of 0◦ material remains unchanged before and after tailoring. Therefore, improvement on the plate behavior is attributed to the tailoring design without adding materials.
Fig. 1. Geometry of uniform and tailored plates with a central circular cutout.
III. DAMAGE MODEL To evaluate the tailoring design concept proposed in the previous section, the progressive damage analysis is performed. The damage evolution model used is a ply-discount method in which stress and stiffness are reduced locally and instantaneously to zero once failure occurs at a point in a ply. These changes due to the damage were introduced at individual integration points as they occurred during loading. A progressive damage model, combining the Hashin[7] and Chang[10] failure criteria, was used to indicate damage initiation. The fiber failure index and matrix failure index are defined as
· 64 ·
ACTA MECHANICA SOLIDA SINICA
– Fiber failure index
ef =
σ11 Xt
2 +
2 σ11 − , ef = Xc – Matrix failure index
σ11 > 0
(2a)
σ11 < 0
(2b)
2
F1 , σ22 > 0 F2
2 σ22 2 Y σ22 F1 c = + −1 + , 2S 2S Yc F2
em =
em
σ22 Yt
F1 , F2
2008
+
(2c)
σ22 < 0
(2d)
where Xt and Xc are the longitudinal tensile strength and compressive strength, respectively. Yt and Yc are the transverse tensile strength and compressive strength, respectively. S is the shear strength. F1 and F2 are the nonlinear correctors obtained by Chang[10] as 2 2τ12 4 + 3ατ12 G12 2S 2 F2 = + 3αS 4 G12
F1 =
(3a) (3b)
where G12 is the shear modulus and α is a material non-linearity parameter measured experimentally[10] . Before any failure occurs, the material behavior at a typical point in the laminar is described by a linear 2D orthotropic stiffness matrix C: ⎧ ⎫ ⎡ ⎫ ⎤⎧ C11 C12 0 ⎨ σ11 ⎬ ⎨ ε11 ⎬ σ22 = ⎣ C12 C22 0 ⎦ ε22 (4) ⎩τ ⎭ 0 0 C66 ⎩ γ12 ⎭ 12 where C11 =
E11 E22 ν12 E22 , C22 = , C12 = , and C66 = G12 . 1 − ν12 ν21 1 − ν12 ν21 1 − ν12 ν21
Material degradation within the damaged area was evaluated based on the mode of failure predicted by the failure criteria of equation (2). Therefore, the residual strength and stiffness of composites strongly depend on the mode of failure in each layer. Here a sudden reduction model with uncoupled longitudinal and transverse failure is used. The property degradation models for each layer are proposed as followings. For fiber failure in a layer (ef > 1), the longitudinal modulus E11 is set to zero so that the laminate can no longer carry any load in the longitudinal direction. Poisson’s ratio ν12 is reduced to zero to uncouple the longitudinal failure from the transverse property. For matrix failure in a layer (em > 1), the transverse modulus E22 is set to zero so that the laminate can no longer carry any load in the transverse direction. Poisson’s ratio ν12 is reduced to zero for uncoupling. In actual practice, a small number a little higher than zero was used for the value of reduced stiffness instead of zero to avoid numerical problem such as divergence. For both fiber and matrix failures, ν21 = ν12 (E22 /E11 ) is enforced to fully uncouple the longitudinal and transverse failure from each other. Therefore, damage modes are summarized as – Fiber failure: C11 = C12 = 0, C22 = E22 – Matrix failure: C11 = E11 , C22 = C12 = 0 This uncoupling failure model indicates that the laminate having failures in one direction can still support load in the other direction. However, this load capacity should be reduced to some degree. Therefore, the shear modulus G12 is degraded gradually as[10] G12 = (1 − d)G012
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where the damage index d has the expression as d=
2 3 − 2ατ12 /γ12 3αG012 τ12 0 2 1 + 3αG12 τ12
where G012 is the original value of the shear modulus and G12 is the updated value during loading. R finite element code[31, 32] . The UMAT routine The analysis was conducted using the ABAQUS R was used to account for the present damage model. The stress was updated based within ABAQUS on the total strain. In calculation, no severe convergence problems were encountered.
IV. RESULTS AND DISCUSSIONS The baseline panel is a square (152.4×152.4 mm2) and has a uniform quasi-isotropic layup [±45/0/90]s with mechanical properties of typical IM7/8551-7 prepreg material (as shown in Table 1). The value of the material non-linearity parameter α is chosen to be 1.2 × 107 MPa−3[10] . The convergence study on both mesh refinement and load increment refinement was performed as a first step. Then four cutout sizes combined with various tailoring extent were investigated for parametric study. Table 1. Mechanical properties of IM7/8551-7
Stiffness (GPa) E11 = 130.3 E22 = 9.38 ν12 = 0.33 G12 = 4.5
Strength (MPa) Xt = 2526 (axial tension) Xc = 916 (axial compression) Yt = 69.4 (transverse tension) Yc = 69.4 (transverse compression) S = 120 (in-plane shear)
4.1. Convergence Study The convergence studies were performed on a plate with 12.7 mm diameter cutout (2r/b = 0.33). Grid refinement was investigated first. The ultimate load of a uniform plate and tailored plates with several width ratios are plotted in Fig.2 as a function of the number of elements. It is clear that the ultimate load decreases rapidly as coarse meshes are refined and tends to approach a constant after the number of element reaches a certain value, i.e. mesh with 736 elements for this specific case. This shows the plates modeled here are converged with respect to the mesh refinement at sufficient big number of elements. Another important convergence investigation Fig. 2 Convergence study on grid refinement. concerns the displacement increment used in the nonlinear solution procedure. If the increment is too big, some slight changes of edge reaction vs. displacement curves will be likely to be omitted and thus degrades the ultimate load prediction. The predicted value usually overshoots its real value. The smaller the displacement increment is, the more accurate the predicted ultimate load is. But this may results in excessive computational time. Therefore a reasonable value of displacement increment is needed to achieve both sufficiently accurate prediction of the ultimate load and an acceptable computational expense. In Fig.3, the ultimate load is plotted against the reciprocal of the displacement increment, 1/(ΔU ). When 1/(ΔU ) reaches 3.9 × 103 mm−1 , the plot becomes almost a horizontal line corresponding to a displacement increment of 0.00254 mm and a strain increment of 3.33 × 10−5 mm/mm. Based on these studies, the problem investigated here is reasonably converged with respect to both grid refinement and displacement increment refinement with a 736 element mesh together with a 0.00254 mm displacement increment. These parameters are the baseline for the other cases to be discussed later. A typical mesh for 2r/b = 0.3 is shown in Fig.4. Due to the symmetry of the geometry, boundary conditions, and the stacking sequence, only one quarter of the plate is modeled using four-node plane stress element (CPS4 of ABAQUS).
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Fig. 3. Convergence study on displacement increment refinement. Fig. 4. Finite element mesh of a plate with 2r/b = 0.3.
4.2. Parametric Study Four plates with different cutout sizes (2r/b = 0.2, 0.3, 0.4, 0.5) were investigated for parametric study. Figures 5 and 6 shows plots of average running load versus applied end displacement. Results are shown for uniform (¯b = 1.0) plates and tailored plates with ¯b ranging from 0.10 to 0.76 for four different cutout sizes. Tailoring offers significant advantages in increasing ultimate loads and ultimate end displacement. Increases in stiffness due to tailoring are also observed. However the most important observation is that, in the more tailored cases (¯b = 0.10 and other smaller values), there is an initial peak load followed by the unloading and the reloading till the ultimate load. This shows that the tailored region can arrest, or at least alleviate, a temporarily unstable progress of the damage. This very clear warning far prior to the final failure could be highly useful in many situations. To help understand this behavior, damaged regions in some selected plies and at selected end displacement values are shown in Figs.7 and 8. Zones in dark color represent the damaged areas. Here we first consider a flat panel with the width ratio ¯b = 0.10 and the cutout size 2r/b = 0.30. In this case, the initial peak load occurs at end displacement U = 0.8103 mm, and the reloading starts at U = 0.8636 mm, and the final failure occurs at U = 1.125 mm. Matrix damage initiates in the 90◦ plies at the root of the cutout at U = 0.3048 mm. This damage grows in a stable way both horizontally and vertically until it reaches the tailored edge region at
Fig. 5. Avg. running load vs. end displacement of plates with different cutout size and tailoring degree.
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Fig. 6. Avg. running load vs. end displacement of plates with different cutout size and tailoring degree (Fig.5 continued).
U = 0.5817 mm. This damage then grows vertically and is accompanied by new matrix damage at the upper corner of the panel. The damage regions continue to grow until they meet each other and a final saturation level develops at about U = 0.7925 mm. A narrow band of fiber and matrix damage starts to develop in this ply at the top of the cutout at U = 0.8509 mm, and grows only slightly in the vertical direction until final failure at U = 1.125 mm. Matrix damage initiates in the −45◦ plies at U = 0.5207 mm and in the +45◦ plies at U = 0.6858 mm, both are at the root of the cutout and grow in a narrow horizontal band toward the panel side. These damage events can be seen as slight stiffness reductions in the load-displacement curve prior to the initial load peak. At the initial load peak (U = 0.8103 mm), fiber damage starts to develop at the root of the cutout in 90◦ and 45◦ plies. This damage grows in an unstable way in a narrow horizontal band toward the
Fig. 7. Damage progression in most highly tailored plate (¯b = 0.10, 2r/b = 0.3).
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Fig. 8. Damage progression in most highly tailored plate (¯b = 0.10, 2r/b = 0.3) (Fig.7 continued).
panel side until it is arrested as it reaches the tailored edge regions at U = 0.8636 mm. This unstable growth corresponds to the region of the decreasing load in the load-displacement profile. Once the arrest occurs, reloading commences with little new damage (except in the 90◦ plies at the top of the cutout as mentioned above) until the ultimate load is reached. Final failure is quite abrupt as fiber damage grows rapidly in the 0◦ and other plies through the tailored edge regions at U = 1.125 mm. This final damage develops horizontally, continuing the fiber damage along the net section in the other plies. Figure 9 shows how damage grows in the uniform plate with the same cutout size of 2r/b = 0.30. From these pictures, one can observe that the matrix failure of the 90◦ layer initiates at a rather low load level (U = 0.254 mm) at the edge of the cutout. The damage grows in a controlled manner as loading increases. At approximately twice the damage initiation load level (U = 0.5309 mm) matrix
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damage initiates at two remotely separate locations in the 90◦ layer and both damage zones progress at an increased pace until they meet. The fiber failure of both 90◦ and 0◦ layers only occurs when the edge reaction reaches its ultimate value and damage progresses unimpededly even when the load begins to drop. Therefore, fiber failure takes place very rapidly and always represents catastrophic failure. One can see that the fiber damage zones look like a slot and can be considered as a crack that separates the plate at centerline. The matrix failure of the 0◦ material is quite different from that of the 90◦ material and progresses in the same way as the fiber failure of the 0◦ layer. In other words, the matrix damage accompanies the fiber damage in the 0◦ layer. Since the damage pattern among the fiber failure of 90◦ and both fiber and matrix failure of 0◦ layer are very similar, only that for 90◦ is shown in Fig.9. Both the fiber and matrix failure of +45◦ layer and fiber failure of −45◦ present very similar damage propagation patterns to the fiber failure of 90◦ and only those of the latter layer are shown in Fig.9 as well. It initiates soon after the matrix failure in 90◦ layer occurs and progresses across the centerline to the unloaded side as the plate totally fails. No damage propagation was found for the fiber failure of −45◦ layer and the matrix failure of +45◦ layer. Figures 10 and 11 shows the damage growth in the most gently tailored plate with ¯b = 0.64. In this case, the tailoring layer mostly approaches the edge of the cutout. Patterns of progressive damages are observed more like that of the uniform plate discussed earlier. Progressive damages are observed in both 90◦ and −45◦ layers in the form of the matrix failure and the matrix failure in 90◦ layer dominates most of the damage progression. The fiber failure in all layers (90◦ , 0◦ and ±45◦) and matrix failure
Fig. 9. Damage progression in uniform plate (¯b = 1.00, 2r/b = 0.3).
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of 0◦ and 45◦ appear only at the end damage propagation and represents complete failure of the plate. Due to the similarity to its fiber failure, the matrix failure of 0◦ layer is not shown in Figs.10 and 11. Peak values taken from each load displacement curve in Figs.5 and 6 are plotted against the tailoring width ratio ¯b in Fig.12. Here another important difference from the postbuckling problem[23, 24] can be seen. In former cases, the most extreme width ratio (smallest value of ¯b) produces the highest ultimate load and the least extreme tailoring produces the lowest ultimate load, whereas both of these results are reversed in the current case of tensile loading. Furthermore, there is an intermediate value of the width ratio that yields the lowest ultimate load in tension. To reveal the relative increases in the ultimate load in tailored panels compared to uniform panels, the data in Fig.12 are normalized by ultimate loads of the uniform panel with the same cutout size and these ultimate load ratios R are plotted in Fig.13. Improvements range from 50% to 100%. However, the damage arrest feature observed in the most highly tailored panels does not occur in the most gently tailored panels. As the cutout size becomes smaller, there is a wider range of tailoring width ratios for which damage arrest occurs. So, there is a design trade-off between having the highest ultimate load and having a slightly lower ultimate load, which can increase the ability to arrest damage.
Fig. 10. Damage progression in most gently tailored plate (¯b = 0.64, 2r/b = 0.3)).
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Fig. 11. Damage progression in most gently tailored plate (¯b = 0.64, 2r/b = 0.3)) (Fig.10 continued).
Fig. 12. Ultimate loads, Nx .
Fig. 13. Ultimate load ratios, R.
V. CONCLUSIONS This paper pointed out some complex effects that tailoring concepts can have on the initiation and progression of the in-plane damage in composite plates with central cutouts under tensile loading. The tailoring concept considered was shown to be effective in improving the average stiffness, the ultimate load, and the ultimate average axial strain in plates. Compared to uniform panels, ultimate loads can be increased by as much as 110% for the particular material studied here. Observation of the damage progression showed that tailoring can arrest unstable damage growth in tensile loading and greatly extended the range of the applied end displacement between damage initiation and final failure.
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