Modeling and Simulation of Two-Dimensional Consolidation for Thermoset Matrix Composites

Transcription

1 Modeling and Simulation of Two-Dimensional Consolidation for Thermoset Matrix Composites Min Li and Charles L. Tucker III Department of Mechanical and Industrial Engineering University of Illinois at Urbana-Champaign 1206 West Green Street, Urbana, Illinois Submitted to Composites Part A: Applied Science and Manufacturing July 30, 2001 Abstract A finite element method is developed to solve two-dimensional consolidation problems for composites manufacturing. The consolidation governing equations, one for solid stress and one for fluid pressure, are derived using a local volume averaging approach, and the two equations are strongly coupled. A special anisotropic, hyperelastic constitutive equation is developed for the solid stress. This equation matches Gutowski s model for consolidation transverse to the fibers, and has a high stiffness parallel to the fibers. An updated Lagrangian method is used to solve the equations, using implicit time integration and a successive substitution method. The code is applied to several case studies to explore two-dimensional consolidation effects. A free edge affects the thickness profile during consolidation, but the final thickness can still be uniform. This effect is substantial in the region close to the edge, and it propagates progressively from the edge towards the center. Simulations were also performed for laminates that bend to form a corner. The corner is thicker than the flat region after consolidation. Wiggles, similar to fiber buckling, arise at low values of shear modulus when using a male mold. Large values of the solid shear modulus cause the corner effect to extend far into the adjacent flat region. The length of the flat region also affects the consolidation of the corner. Keywords: thermoset cure, consolidation, hyperelasticity, anisotropic materials, finite element analysis, edge effects, corner effects To whom correspondence should be addressed. 1

2 1 Introduction Autoclave curing is a process to produce fiber-reinforced polymeric parts in final shape. The simulation and optimization of autoclave processing have seen widespread application in industry to understand and improve product quality. During processing, the autoclave is heated according to a predetermined temperature cycle and, at the same time, pressurized according to a predetermined pressure cycle. The applied heat increases the temperature in the composite, resulting in changes in the molecular structure of the resin and, correspondingly, in resin viscosity. When the resin viscosity has become sufficiently low, the applied pressure squeezes excess resin from the composite into a bleeder ply, as the laminate consolidates from the top toward the bottom. The resin then cures and cross-links, producing a rigid finished part. Process simulations for composite curing and consolidation use physically-based models to simulate the process, and predict the cure cycle time. The fiber volume fraction, resin pressure, thickness, degree of cure, temperature and other properties in the final part can also be predicted by the simulations. Most consolidation and curing models have been one-dimensional, focusing on variations in the laminate thickness direction. Lin, Ranganathan and Advani (1) review consolidation phenomena with different manufacturing processes and point out the mechanisms and the approaches to address them. Hubert and Poursartip (2) provide a comprehensive review of flow and compaction models for thermoset matrix composites during curing. Barone and Caulk (3) proposed a thermochemical model based on a heat conduction equation with internal heat generated by the exothermic chemical reaction. Springer (4) studied the relationship between the applied pressure and the resin flow during the cure of fiber reinforced composites, where the layers were found to consolidate in a wavelike manner. Loos and Springer (5) developed resin flow and void models of the curing process. The resin velocity was related to the pressure gradient, fiber permeability, and resin viscosity through Darcy s law. Gutowski, Morigaki and Cai (6) developed three-dimensional flow and one-dimensional consolidation models of the composite. The resin flow was modeled using Darcy s law for an anisotropic porous medium. The general case was then solved for compression molding and bleeder ply molding. Gutowski and Dillon (7) reviewed the stresses of an aligned fiber bundle, comparing data for transverse compression, axial extension, and coupling behavior to the three-dimensional fiber deformation model. Li, Zhu, Geubelle and Tucker (8) studied the thermochemical part of autoclave curing process for thermoset composites, and implemented a design sensitivity method with a gradient-based search method to find optimal cure cycles that provide rapid curing without thermal degradation. Li and Tucker (9) further extended this approach and considered both thermochemical and consolidation for one-dimensional curing problems. Time-optimal cure cycles were obtained that meet both cure and consolidation requirements. Two-dimensional aspects of consolidation have received far less attention. The only 2-D consolidation model we know of was proposed by Hubert, Vaziri and Poursartip (10). A numerical 2-D plane strain, flow-compaction finite element model was developed to simulate autoclave processing of fiber-reinforced composite laminates. Their model uses a formulation similar to the one developed here. However, it uses an incremental, quasi-linear elasticity model for the solid bed stress, and the constitutive equation for stress is not fully described. That formulation may give different stresses for the same deformation state, depending on the loading path used to reach that state, and it is not easily adapted to large strain problems. In contrast, here we develop a special hyperelastic model for the solid bed stress, which is valid for large deformations and whose stress is independent of the loading path. In addition, Hubert et al. do not update the mesh geometry or fiber orientation as consolidation proceeds, but here we update the geometry at each time step and account fully for the associated geometric nonlinearities. Finally, our governing equations include some terms (such as the spatial variation in porosity and permeability) that are missing from the model of Hubert et al. Hubert et al. performed a parametric study to investigate the effect of the material properties on the compaction of angle-shaped laminates. The corner region consolidated less than the flat portions of the 2

3 n0 e0 Composite Laminate Bleeder Tooling e0 n0 Figure 1: Schematic of the geometry for two-dimensional consolidation analysis. Note that e 0 and n 0 change direction as the prepreg conforms to the curved tool. laminate. Also, the fiber bed shear modulus significantly affects the compaction behavior near corners, while the resin viscosity and fiber bed permeability only affect the compaction rate. Here we find qualitatively similar results, but we also observe a numerical behavior that is related to fiber buckling. This study presents a complete model and numerical method to simulate two-dimensional consolidation processes. Section 2 describes the governing equations of the 2-D model and proposes a special anisotropic, hyperelastic constitutive equation for the solid contribution to stress. Section 3 gives the finite element equations and solution procedure. Section 4 uses the model to predict the consolidation behavior of composite laminates in autoclave curing processes and explores two-dimensional consolidation effects. Section 5 summarizes this work. 2 Process Model The geometry of the two-dimensional model is illustrated in Fig. 1. Here e 0 is a unit vector that is locally aligned with the initial fiber orientation, and n 0 is a unit vector normal to the plane of the undeformed sheet. Note that e 0 and n 0 change direction as the prepreg conforms to the curved tool. The process model can be divided into two parts: a thermochemical part and a consolidation part. 2.1 Thermochemical Governing Equations The governing equation for transient heat conduction, including internal heat generation due to the exothermic cure reaction, may be written as ρc p T t = (k T ) + ρ H R c t where T is temperature and t is time. The thermal conductivity tensor k may be anisotropic. The density, specific heat, and heat of reaction of the laminate are ρ, c p, and H R respectively. c is the degree of cure, which is defined as the fraction of the reactive groups in the resin that have reacted. Further, c/ t denotes the rate of the cure reaction, which, together with the heat of reaction, determines the heat release rate during the cure process. (1) 3

4 The heat release due to the cure reaction, which appears as a source term in Eqn. (1), is a function of the reaction kinetics. The kinetic equation for the material considered in this work is ( c = g(c, T ) = A exp E ) c m (1 c) n (2) t RT where A is a frequency factor, E is the activation energy, and R is the universal gas constant. The parameters m and n are constants. The governing equations (1) and (2) are subjected to initial conditions of T = T 0 and c = c 0 at t = 0. Note that c 0 is a very small number, since we must have c 0 = 0 in order to get the curing process started. The boundary conditions are convective heat transfer conditions at the surfaces of the bleeder and tooling. 2.2 Consolidation Governing Equations We derive the consolidation governing equations using the local volume averaging method. The equations of Tucker and Dessenberger (11), derived for stationary fiber beds in resin transfer molding, are extended here to simulate the consolidation process with a moving, deforming solid bed. We use a subscript f to denote the fluid (or resin) phase, while the subscript s denotes the solid (or fiber) phase. φ f and φ s denote the volume fractions of fluid and solid, respectively, and φ s + φ f = 1. Let B represent any quantity that has a value at each point. There are three distinct ways to compute an average value for B. The spatial average is the average value of all phases within a representative volume V, and is denoted by angle brackets as B. A second type of average includes only the points that lie within a single phase, but still averages over the entire volume V. This is called a phase average. Phase averages will also be denoted by angle brackets, but the quantity inside the brackets will carry a subscript denoting the phase over which the average is taken. For example, v f denotes the phase-average velocity for the fluid. The third average considers only points lying within a single phase, and averages their values over just the volume occupied by that phase. This is called an intrinsic phase average, and it is denoted by angle brackets with a superscript labeling the relevant phase. For example, v f denotes the intrinsic phase-average velocity for the fluid. The phase average is related to the intrinsic phase average by relations like v f = φ f v f and v s = φ s v s s. We assume ρ s and ρ f are constant and that inertial and gravitational effects are negligible. We begin with the continuity equations for a constant-density fluid and solid in a consolidating porous medium, which are φ f t φ s t + v f = 0 for the fluid (3) + v s = 0 for the solid (4) We then introduce constitutive equations for the fluid-solid drag force and solid stress. The fluid-solid drag force per unit volume f d is f d = φ2 f µ [ v S f v s s] (5) where µ is the fluid viscosity and S is the permeability tensor. The total solid stress is assumed to be σ s s = p f I + τ s s (6) where I is the identity tensor. The first term, p f I, is part of the fluid-solid interaction, and assumes that 4

5 the fluid pressure is transmitted directly to the solid. The second term, τ s s, is the solid extra stress, and is determined by the deformation of the fiber bed. The above equations are combined with the momentum balance equations, which are φ f p f f d = 0 for the fluid (7) σ s + f d p f φ f = 0 for the solid (8) We substitute the basic relation φ s + φ f = 1 into the continuity expressions, Eqns. (3) and (4), to get φ s t = v f v s = v f (9) (10) The momentum equations (7) and (8) are further manipulated. We introduce the general drag force term from Eqn. (5) into Eqn. (7), and get the generalization of Darcy s law with a moving solid phase as v f v s s = 1 φ f µ S p f (11) The resin permeability tensor S is also anisotropic, and depends on fiber volume fraction φ s. Next we multiply the solid constitutive equation (6) by φ s and use the relation between the phase average and the intrinsic phase average to get a expression for σ s. We then substitute this into the solid momentum equation (8), and combine this result with Eqn. (7). This gives τ s p f = 0 (12) To further reduce the number of unknown variables, we multiply Eqn. (11) by φ f and take the divergence of both sides of this equation. This gives v ( f φ f v s s) ( 1 = µ S ) p f (13) We then substitute Eqn. (10) to eliminate v f. The left-hand side can be further simplified using relations such as φ s + φ f = 1 and v s = φ s v s s. This gives ( 1 v s s = µ S ) p f (14) Equations (12) and (14) are the final governing equations for two-dimensional or three-dimensional consolidation. Equation (12) behaves like a solid-mechanics equilibrium problem, with p f acting like a body force term. Equation (14) has the usual form for flow in a porous medium (e.g., see Tucker and Dessenberger (11)), with an additional source term due to the divergence of v s s. These two equations are connected because p f appears in both equations, and further because the solid velocity equals the material derivative of the solid displacement vector u s, v s s = u s (15) The solid extra stress tensor is determined by the solid deformation state, which in turn is determined by the 5

6 solid displacement u s. Therefore there are two primary unknown variables in this problem: p f and u s. We also need an additional, constitutive equation for the fiber extra stress τ s. In our 2-D consolidation problem, we assume that the autoclave pressure acts on the laminate through the bleeder in terms of the surface traction, and resin pressure at the surface of the laminate equals the pressure in the bag. Therefore, at the beginning of the process, when vacuum is applied, p b = 0. When vacuum is removed, p b equals atmospheric pressure. Thus, the boundary conditions are n τ s = (p a p b )n on S p (16) p f = p b on S p (17) where p a is the autoclave pressure, p b is the bag pressure, and n is the unit vector normal to the transverse direction. The negative sign on the right-hand side of Eqn. (16) accounts for the fact that the surface traction becomes more negative as pressure increases. Here S p is the surface where the prescribed pressure is applied. There is no resin flow into or out of the surface S q between the laminate and tool. 2.3 Hyperelasticity for Solid Stress In two-dimensional consolidation, the fiber bed is deformed and compressed in the thickness direction. This is a large-strain problem, since the thickness change is comparable to the initial thickness. The fiber longitudinal stiffness is very large compared to the autoclave pressure, so extension in the fiber direction is negligible, but the fiber bed is compressed volumetrically. The stress-strain behavior in compression is highly nonlinear (see Gutowski et al. (6)), which requires a nonlinear constitutive equation. While a simple stress-strain expression is sufficient for a 1-D consolidation model, 2-D and 3-D models requires a complete, large-strain constitutive equation for the solid stress. Here a hyperelastic model is developed to represent the fiber bed stress τ s. Denote the material coordinates in the reference configuration by X and the spatial coordinates in the current configuration by x. Note that x = X + u, where u is the displacement vector. The deformation gradient tensor F describes both shape change and solid-body rotation, and is defined as F = x/ X. The Cauchy deformation tensor C is defined as C = F T F. An elastic (or hyperelastic) material is one that stores all the work done on it as internal energy. This means the material must possess a scalar strain energy function W(F), giving the energy per unit unconstrained volume stored at the deformation state F. In this study, we propose a strain energy function W as W = W 0 ( ) + E sφ 0 4 (L lnl) + G 2 (I 1 3) G(lnJ) + K 2 (lnj)2 (18) where E s is the fiber axial stiffness and φ 0 is the initial fiber volume fraction. G is a shear modulus, K is a bulk modulus, and J is the determinant of F. W 0 ( ) is a function that captures the nonlinear response in transverse compression, and it will be discussed shortly. The two scalars L and that characterize the strain state are defined as L = e 0 C e 0 = n 0 C 1 n 0 (19) Equation (18) is constructed by superposing two existing transversely isotropic models. The first model was presented by Lurie (12), and regards W as a function of C. It has the material symmetry axis e 0 and is suitable for modeling the high axial stiffness of the fibers. The second model was proposed by Pillai, Tucker and Phelan (13), and regards W as a function of C 1. It has the symmetry axis n 0 and is useful for modeling nonlinear compression in the thickness direction. These two models are formally equivalent when e 0 and n 0 coincide (see Li (14)), but for our superposed model, we choose e 0 and n 0 to be orthogonal (see Fig. 1). 6

7 For any hyperelastic material the Cauchy stress is related to the strain energy through τ s = 2 J F W C FT (20) Substituting Eqn. (18) gives, after some manipulation, the stress response τ s = 2 ( ) W0 nn + E sφ 0 J 2J (L 1)ee + G J (B I) + K (lnj)i (21) J where B is the Finger tensor, defined as B = F F T, and I is the unit tensor. e is a unit vector along the fiber direction, and n is a unit vector transverse to the fibers, both in the deformed configuration. They are found from e = e0 F T L n = n0 F 1 (22) Equations (21) and (22) allow the solid bed stress τ s to be calculated for any deformation. 2.4 Constitutive Equation of Fiber Extra Stress Equation (21) gives the fiber bed behavior that we want. First, in the transverse direction the fiber bed is initially soft, but it stiffens progressively as consolidation proceeds. This transverse stress is the term that multiplies nn. Second, in the axial direction the fiber is elastic with a very high stiffness. This is the term multiplying ee. Solid stresses arising from shear parallel to the fibers should be very small. However, a model with no resistance in the axial shear direction will make the numerical solution unstable. The third and fourth terms are equivalent to the compressible neo-hookean stress, and provide some shear stiffness to make the numerical solution stable. The function W 0 ( ), which models the response to transverse compression, is chosen, together with the contribution of the compressible neo-hookean model in the transverse direction, to match the model presented by Gutowski et al. (6). Consider a deformation in which the laminate is compressed transversely and there is no deformation in the fiber axial direction. In this case the deformation relations are x 1 = X 1 x 2 = a X 2 (a < 1) (23) The initial fiber orientation vector is e0 T = [1 0], and thus nt 0 = [0 1]. We then compute all the quantities appearing in Eqn. (21) and find that the transverse fiber stress (assuming K = 0) is τ sn n τ s n = 2 a 3 W 0 + G(a2 1) a Note that the isotropic shear modulus G contributes to this stress component. For this same deformation, Gutowski et al. (6) proposed a model for transverse fiber extra stress as ( ) φs 1 φ τ szz = A 0 s ( ) 1 4 (25) φ s 1 φ a where A s is called the spring constant and φ a is the maximum possible fiber volume fraction. Fibers which are wavy or slightly misaligned will result in a lower φ a. The minus sign on the right-hand side of Eqn. (25) (24) 7

8 accounts for the fact that the stress on the fiber bed is compressive, and becomes more negative as φ s increases. Equating Eqns. (24) and (25), and using the relations J = a, = a 2, and = (φ s /φ 0 ) 2 = J 2 in this deformation, we get ( W 0 = A ) s ( 2 ) 4 + G ( ) (26) 2 φ 0 φ a The advantage of using this expression for W 0 ( ) is that the value of G does not affect the compression behavior revealed in the usual consolidation experiment. The net stress in the transverse direction, including contributions from the neo-hookean model and from Eqn. (26), will exactly match Eqn. (25). This is what we desire, since the purpose of adding the compressible neo-hookean model (represented by G) is to stabilize, but not to change, the solution. However, we found that computationally Eqn. (26) is not the best choice. For a problem with shear deformation, this equation sometimes gives, at the very early stages, highly-deformed elements, which can make the whole simulation unstable. Therefore, we need to modify Eqn. (26) to gives the fiber bed some stiffness at the beginning, but this modification should not change the solution very much, so that the behavior revealed in the usual consolidation experiments still holds. To do so, we introduce a parameter G 0, and modify Eqn. (26) to read ( W 0 = A s 2 ) ( 1 φ 0 φ a ) 4 + (G G 0) 2 ( ) (27) Of course, setting G 0 = 0 recovers Eqn. (26). Typically we make G 0 small compared to G. The transverse consolidation behavior is slightly affected by the value of G 0. In fact, the difference of fiber volume fraction between our model and Eqn. (25) is about 1% when G 0 = 0.5σ, whth the applied stress σ ranging from 100 kpa to 1000 kpa. Equation (27) is the final form for W 0 ( ) used in this study. One could integrate Eqn. (27) to find the expression for W 0. However, it turns out that W 0 / is all that we need in this study. 2.5 Other Material Properties The transverse permeability of the fiber bed is modeled using a modified Carman-Kozeny equation, as proposed by Gutowski et al. (15): 3 S t = r 2 f 4k t For axial permeability we use the model of Gebart (16): φ a 1 φ s ( ) (28) φ a + 1 φ s S a = 8r 2 f C a (1 φ s ) 3 φ 2 s (29) 8

9 Here φ a is the maximum possible volume fraction and C a is called the shape factor; these quantities depend on the fiber arrangement (quadratic or hexagonal packing), but not on the fiber radius or the fiber volume fraction. For quadratic packing, C a = 57. The permeability is much larger along the fiber axis than transverse to the fibers, and along the fiber axis the permeability decreases more slowly with consolidation than that transverse to the fibers. The permeability tensor S is two-by-two for our 2-D problem. For arbitrary fiber orientation e, S can be represented as S = S t I + (S a S t )ee = S a ee + S t (I ee) (30) where the unit vector e corresponds to the instantaneous fiber axis direction. The resin viscosity µ changes with position and time during processing. Generally µ can be expressed as a function of temperature and degree of cure, i.e., µ = µ(t, c). In this study EPON 862/W resin is used and its viscosity was characterized as (17) [ ( c ) ] A+Bc µ = min µ 0, µ c (31) c where c is the critical degree of cure when resin gels, and c = 0.71 for this material. µ is an arbitrarily chosen constant that is much greater than µ 0. The fact that the consolidation stops after gelation is implemented numerically by assigning a very large viscosity µ to the resin after the gel point, i.e., when c approaches c. µ 0, A and B are functions of T as well. Note that both T and c are functions of time and position, so the viscosity is also a function of time and position. 3 Finite Element Analysis The governing equations are solved numerically using the finite element method. The finite element formulation for the thermochemical part of this two-dimensional problem is the same as the analysis in the one-dimensional case, except that now the elements and shape functions are two-dimensional. Li et al. (8) provide a detailed discussion of this part of the solution, and we omit it here. Note that temperatures and degrees of cure are treated in a single set of finite element equations and solved simultaneously at each time step, by the Newton-Raphson iteration. A special lumping technique is used to ensure numerical stability and consistency between the temperature and the cure equations. For the consolidation equations, an updated Lagrangian approach is used where, at the end of each time step, the nodal coordinates and the fiber orientation are updated. This is a large deformation problem, and one must account for both geometric and material nonlinearity. Compared to the elastic method, the hyperelastic analysis has the advantage that it correctly solves problems with any amount of deformation. The coupled thermochemical and consolidation problem is solved using an alternating step approach. First we solve one time step of the thermochemical subproblem (temperature and cure), then we solve one step of the consolidation subproblem (fluid pressure and solid displacement). This procedure is repeated at each time step. The governing equations and numerical solution in fact generalize easily to three dimensions, though we do not undertake any 3-D calculations here. 3.1 Formulation of the Flow Equation For conciseness we use u to denote the vector of nodal values of solid displacement u s, and p to denote the vector of nodal values of fluid pressure p f. As a first step, we choose finite element interpolation (shape) functions N u for displacement solution vectors u, and N p for pressure solution vectors p. Therefore, the 9

10 approximate solutions are N T p u = N u u p = N p p (32) Applying a Galerkin weighted residual method to the flow equation (14) on the whole domain gives [ )] u d = 0 (33) N T p ( S µ p Substituting the approximate solutions and integrating the second term by parts, we get ( Nu x u x + N ) u z u z d + N T S [ v p µ B pp d = f v s s] φ f d S q (34) Here B p is the usual matrix of derivatives of the pressure shape functions. In our consolidation problems there is no resin injected into the laminate, so the right-hand side of Eqn. (34) equals zero. The finite element formulation of this equation is written as S q N T p C pu u + K pp p = 0 (35) where, for a problem with n nodes, C pu is n 2n and K pp is n n, with the matrices given by C pu = N T p B up d K pp = B T S p µ B p d (36) 3.2 Displacement Equation Equation (12) describes the balance between displacement-induced solid stresses and solid/fluid drag force. The discretized form of this spatial equilibrium equation is given by Bonet and Wood (18) as Nu T τ s d + Nu S T p d Nu T t d S p = 0 (37) p From the boundary condition (16), the surface traction is t = (p a p b )n. In the above equation, the first term can be viewed as the internal virtual work done by the stress τ s, and the other two terms can be viewed as the work done by fluid drag force and surface tractions. The finite element formulation of the equation above can be written, as K up p = F τ + F t (38) where the stiffness matrix K up and the load vectors F σ and F t are K up = Nu T B p d F τ = B T τ s d F t = Nu T (p a p b )ˆn d S p (39) S p Note that F t is evaluated only at the boundaries where autoclave pressure, i.e., surface traction, is applied. ˆn is the direction normal to that boundary, and it can be computed from the nodal coordinates associated with that boundary. Since the boundary is a straight line in the two-dimensional case, we find F t by onedimensional Gaussian integration. Also note that F τ cannot be written as something like K uu u. For this hyperelasticity problem, F τ is a very complicated function of u and we cannot factor u out from this expression. Therefore, for the moment 10

11 we use F τ, the complete load vector due to the internal stress. The complete finite element equations for consolidation can be written by combining Eqns. (35) and (38), to get [ ] { 0 0 u C pu 0 ṗ } [ 0 Kup + 0 K pp ] { u p } = { Fσ + F t 0 For convenience of discussion, the above equation is written compactly as } (40) C v + Kv = F (41) Here v = [u p] T represents the complete solution vector for the consolidation subproblem. 3.3 Time Integration For the two-dimensional consolidation subproblem, each node has three degrees of freedoms: x-direction nodal displacement u x, z-direction nodal displacement u z, and intrinsic fluid pressure p f. Let v n denote the solution at time step n. An implicit time scheme, in which v n (v n v n 1 )/ t, is used to solve Eqn. (41). Since this is a nonlinear equation, successive substitution is used to solve for each time increment of displacement and pressure. To do this, we use k to denote the iteration number and transform Eqn. (41) to ( ) ( ) Ck t + Kn k v n k+1 = Ck v n 1 + F n k (42) t To help the successive substitution method to converge, we compute the tangent stiffness matrix of the stress equation, Eqn. (38). Start with a first-order Taylor series expansion, (F σ + F t ) u n k+1 (F σ + F t ) u n k + (F σ + F t ) u n k u and denote K ut (F σ + F t )/ u. Then, in place of Eqn. (40), we have [ ] { } [ ] { } { 0 0 u n k+1 Kut K C pu 0 ṗ n + up u n k+1 u n k F n k+1 0 K pp p n = σ k + F n t k k+1 0 (u n k+1 un k ) (43) Note that we are solving for the incremental displacement, u n k+1 un k. Also we write un k+1 in terms of the incremental displacement as u n k+1 = un k+1 un 1 t = (un k+1 un k ) (un 1 u n k ) t Substituting Eqn. (45) into Eqn. (44) and rearranging, we get the final finite element formulation as K ut K up { } C pu u n k+1 u n F n σ k + Fn t k k K pp p n = C k+1 pu t t (un 1 u n k ) The benefit of reaching convergence of Eqn. (46) quickly justifies the extra work of computing K ut. Also, by correctly evaluating K ut, the solution method approaches a full Newton-Raphson method. As for the physical meaning, K ut is the tangent stiffness matrix of the solid stress equation, K up represents the drag force by the fluid on the solid, and C pu reflects the fluid flow generated by consolidating the solid. Since the diagonal entries of C are zero, and no non-zero entries exist in the top half of C (refer to } (44) (45) (46) 11

12 Eqn. (40)), C is a singular matrix and we cannot find C 1. Therefore, an explicit time integration scheme cannot be used to solve Eqn. (41). 3.4 Linearized Equilibrium Equations So far we know the expressions for every term in Eqn. (46) except K ut. Bonet and Wood (18) showed that, by linearizing the equilibrium equations, K ut can be evaluated as where the components relating node a to node b in element e are [K e c,ab ] ij = [K e σ,ab ] ij = K ut = K c + K σ + K p (47) 2 v e k,l=1 2 v e k,l=1 N a c sym N b ikjl dv for i, j = 1, 2 (48) x k x l N a x k σ kl N b x l δ ij dv for i, j = 1, 2 (49) Here c sym ikjl = (c ikjl +c iklj +c kijl +c kilj )/4, where c is the so-called Eulerian or spatial elasticity tensor, which acts as the material stiffness. To calculate c ijkl, we introduce the second Piola-Kirchoff stress tensor S and another fourth-order tensor, the Lagrangian elasticity tensor. S is defined as S = 2 W/ C. Substituting Eqn. (18) gives while I J K L is given by S = 2 W 0 C + E ( sφ ) e 0 e 0 + G(I C 1 ) + K(lnJ)C 1 (50) 2 L I J K L = S I J E K L = Again W is the strain energy function. The two fourth-order tensors 4 2 W C I J C K L = K L I J (51) and c are related by c ijkl = J 1 F i I F j J F kk F ll I J K L for I, J, K, L = 1, 2 (52) The complete form of c ijkl is (for details see Li (14)) c ijkl =4 ( ) W0 J 1 2 n i n j n k n l + 2 W 0 J 1 + E s φ 0 J 1 L 2 e i e j e k e l + K J 1 δ ij δ kl + (G KlnJ)J 1 (δ ik δ jl + δ il δ jk ) q (53) where q ijkl = n i n k δ jl + n i n l δ jk + n j n k δ il + n j n l δ ik, and δ ij is the Kronecker delta. Also, if (ξ, η) are the local (isoparametric) coordinates on the face of an element, Bonet and Wood (18) show that the third term in Eqn. (47) is K p,ab = 1 2 A ξ (p a p b ) [ x ξ ( Na η N b N ) b η N a + x ( Na η ξ N b N )] b ξ N a dξ dη (54) This is simplified in our two-dimensional problem by setting the vector x/ η to [0, 0, 1] T, so that when the cross product with x/ ξ is taken, the correct normal vector is obtained. The finite element solution and software were verified using several problems which were selected to 12

13 Pa x/l=0 x/l=0.5 x/l=0.8 x/l=1 Pb x L Figure 2: The geometry for a laminate with a free edge, which exposes the resin at the right edge to the bag pressure p b. Traction caused by the autoclave pressure p a is applied to the solid at the top surface. The left side is a symmetry boundary with no x displacement. Four different locations are examined. They are marked as x/l = 0, 0.5, 0.8 and 1, respectively, where L is the length of the laminate. test various parts of the finite element governing equations (14). 4 Two-Dimensional Consolidation Case Studies In this section, the two-dimensional model is used to examine the consolidation behavior of composite laminates in the autoclave curing process. Problems are constructed to explore two-dimensional effects, including the behavior near the edge of the laminate, and the behavior near a corner. For all the simulations in this section, the material is EPON 862/W epoxy, with material properties of H R = 128 kj/kg, K = 0, E s = 200 GPa, and G 0 = 0.3 MPa. Also, conventional temperature and pressure cycles are used (see Fig. 3). With such processing conditions, an equilibrium condition between the applied pressure and the solid stress is reached before the resin gels (within 10 minutes). Then, when the consolidation is completed, the solid carries all the applied load, and the resin flow is unimportant. 4.1 Edge Effects In the autoclave curing process, the laminate is placed inside a bag, a vacuum is pulled on the bag, and pressure is applied to the top surface. A dam is usually placed along the edges. In this section, the effect of a free edge on two-dimensional consolidation is examined. The geometry of the analysis is illustrated in Fig. 2. The resin at the edge with the dam is exposed to the bag pressure p b during consolidation, but there is no traction on the solid at the edge. The left side is assigned a symmetry boundary condition, so it has only vertical displacement but no horizontal displacement, and there is no resin flow into or out of the left side. The 2-D finite element code is used to simulate the consolidation process for a laminate with a initial thickness of 10 mm and an initial fiber volume fraction of φ 0 = 50%. The bleeder at the top has a thickness of 6 mm and the tooling at the bottom has a thickness of 4 mm. The length L is 100 mm. The laminate is discretized with four-node isoparametric quadrilateral elements, ten in the thickness direction and ten in the length direction. The simulation is performed for G = 1 MPa and the time step is t = 5 seconds. The calculated profiles of laminate thickness versus time are illustrated in Fig. 4. Since the right edge has the same resin pressure as the top surface, equal to the bag pressure, extra resin can flow out both upward 13

14 Temperature ( o C) Temperature Pressure Time (minute) Pressure (psi) Figure 3: Conventional cure cycle for EPON 862/W epoxy, used in all calculations. Laminate thickness (mm) t=0 t=1 t=2 t=3 t=4 t=5 t= x/l Figure 4: Laminate thickness versus position at various times with a free edge. The initial thickness for the laminate is 10 mm, and the length is 100 mm. Thickness at regions closer to the edge (at x/l = 1) decreases more rapidly than at the center (at x/l = 0). The final thickness is uniform. G = 1 MPa and times are in minutes. 14

15 Laminate thickness (mm) t=0 t=1 t=2 t=3 t=4 7 t=5 t= x/l Figure 5: The edge effect in a longer laminate. The initial thickness for the laminate is 10 mm, and the length is 400 mm. The plateau near the center (at x/l = 0) is apparent. The final thickness is uniform. G = 1 MPa and times are in minutes. and from the right edge during consolidation. Therefore we expect that the right edge will consolidate more quickly than the center of the laminate (the left side of the mesh). Figure 4 shows that after one minute the left side has consolidated to 8.7 mm, while the right edge consolidates more quickly, to 7.4 mm. As time increases, the edge effect extends farther to the left. The center continues to consolidate, with thickness decreasing, but the right edge thickness does not change much. The whole laminate is fully consolidated at 10 minutes, when the thickness reaches 7 mm at all locations. Other results from this simulation (not shown here) reveal that the fiber volume fraction increases most rapidly at the right edge, and most slowly at the left side. This corresponds to the way the thickness decreases as the laminate is compacted, as expected. The right edge, which is exposed to the bag pressure, has zero resin pressure as a boundary condition. At other locations the resin pressure initially follows the applied autoclave pressure. As consolidation continues, the resin pressures all decrease, but at different rates. The resin pressure at the left side decreases the slowest, and resin pressure decreases more quickly at locations closer to the right edge. The presence of the edge should not affect the consolidation far away from the edge. To see how wide the edge effect is, a simulation was performed for a much longer laminate, with a thickness of 10 mm and length of 400 mm. The laminate is discretized with four-node quadrilateral elements, ten in the thickness direction and forty in the length direction. The profiles of laminate thickness versus time for this longer laminate are illustrated in Fig. 5. The thickness closer to the edge (at x/l = 1) decreases more rapidly than in the center (at x/l = 0), for the reasons discussed above. However, in this case the edge effect is limited to a well-defined region, and there is a plateau of uniform thickness in the center of the laminate. The edge effect extends about 40 mm inward, to the position x/l = 0.9, at t = 1 minute. Though the edge effect extends farther inward as time increases, the effect is much weaker and there is always a plateau clearly visible at the center of the laminate. The whole laminate consolidates at about 10 minutes, when the thickness reaches 7 mm at all locations. From Figs. 4 and 5, we conclude that the longer and the thinner the laminate is, the less visible of the edge effect is in areas far away from the edge. 15

16 40 35 No traction F E 30 Autoclave pressure z (mm) Male mold Laminate 10 No-slip boundary D 5 C Symmetry boundary 0 B 0 A x (mm) Figure 6: The geometry for a laminate with a 90 degree corner, using a male mold. The initial thickness is 5 mm, and the length of the flat region is 50 mm. Only half of the geometry is modeled, based on symmetry along curve AB. Though it affects the thickness profile during consolidation, the edge does not affect the final thickness profile, provided that full consolidation is achieved before the resin gels. If so, the thickness at all locations in the laminate will reach a uniform value. However, in the situations where the consolidation process is not completed, i.e., consolidation is stopped by resin gelation, the edge effect can cause a non-uniform thickness profile. 4.2 Corner Effects on a Long Part In this section, the consolidation behavior of laminates that bend to form a corner are examined. The geometry for a laminate with a 90 degree corner is illustrated in Fig. 6. The initial thickness is 5 mm, and the length of the flat region is 50 mm. The initial radius of the inner corner (curve AC) is 5 mm, and the radius of the outer corner (curve BD) is 10 mm. In this figure only half of the geometry is meshed, based on a symmetry boundary (curve AB). Other boundary conditions include: a no-slip boundary at the inner side (curve ACE), autoclave pressure applied at the outer side (curve BDF), a symmetry boundary (curve AB), and no traction at the end of the flat region. The no-slip condition imposes zero-displacement constraints at that boundary. Since this part is thin, the temperature and degree of cure profiles are nearly uniform through the thickness, and they are not affected by the bleeder ply and the tool. So in this problem the bleeder ply and the tool are not meshed. The laminate is discretized using 559 nodes and 504 four-node quadrilateral elements. A simulation is first performed with G = 1 MPa and a time step t = 2 seconds. The initial mesh (with dotted lines), deformed shape (with solid lines), and the distribution of final fiber volume fraction of the laminate are illustrated in Fig. 7. We see that most of the flat region has consolidated uniformly, with the fiber volume fraction reaching 71.6%. However, the corner does not consolidate much, and this corner effect extends into 16

17 the flat region next to the corner. A interesting point is that the final thickness at the corner is larger than the flat region, as also reported by Hubert et al. (10). Moreover, wiggles occur along the outside of the corner, an effect that has not been reported before. We believe that these wiggles are representative of a real physical effect. In the next section we present evidence that they are not a numerical instability. The wiggles occur because, when using a male mold, the fibers along the outside of the corner move inward, and thus the length of these fiber needs to decrease to accommodate the geometry. This places the fibers in axial compression. However, due to the large fiber axial stiffness (200 GPa), the resistance to this fiber axial compression is very strong. To accommodate this length decrease, fiber buckling occurs. Therefore, the thickness change at the corner is smaller than in the flat region, and wiggles occur at the corner. Because of the wiggles, the outer side of the corner has a high fiber volume fraction, which is close to the value in the flat region. However, the inner side of the corner has a smaller change of fiber length, and thus retains a lower fiber volume fraction. To study the effect of shear modulus G on the consolidation process, a simulation was performed with G = 10 MPa. The results are illustrated in Fig. 8. Wiggles still exists in this case, but they are much weaker than in Fig. 7. Also, because of the higher value of G, the less-consolidated area in the inner corner becomes larger, and the outer side of the corner, which has wiggles but with a higher fiber volume fraction, becomes even thinner. The thickness at the corner decreases slightly from 5 mm to 4.69 mm, while the thickness in the flat region decreases to 3.49 mm. Another simulation was also performed for G = 1 GPa, with results shown in Fig. 9. Wiggles do not occur in this case, while the whole corner does not consolidate much at all. Apparently the larger value of G stabilizes the material and suppress the wiggles. With this high value of G, the corner effect extends far out into the flat region. Only about one-third of the flat region, near the end, has consolidated uniformly. Note also the difference in fiber volume fraction profiles, especially in the flat region close to the corner, between Figs. 7 and 9. The combination of very stiff fibers and a large shear modulus strongly couples the behavior of nearby portions of the laminate. This allows the corner to influence consolidation in the flat region, far away from the corner. With a lower shear modulus the corner effect is more localized. Simulations were also performed for the consolidation of this part using a female mold (see Li (14)). Although the thickness at the corner is still larger than the flat region, wiggles do not occur when using female mold. This is because, when using a female mold, the fibers along the inside of the corner move outward, and thus the length of these fiber needs to increase to accommodate the geometry. This places these fibers in tension instead of compression. Table 1 gives the thickness changes at the corner and in the flat region for these cases, in both the male and the female molds. Note that, with a female mold, the change of thickness at the corner is the horizontal displacement at point A, while with a male mold, the change of thickness at the corner is the horizontal displacement at point B (see Fig. 6). From Table 1 we see that the thickness reduction in the flat region is about 1.51 mm for all cases, independent of the values of G and independent of whether the mold is male or female. However, with the male mold, changes in thickness are different at the corner for different values of G. The larger the shear modulus, the stronger the resistance of changing fiber length at the corner, and the smaller the thickness reduction at the corner. For the female mold, the thickness decrease at G = 1 MPa is much smaller than for the corresponding male mold case. In fact, as G increases, the thickness at the corner actually increases. Because of their strong resistance to axial extension, the fibers deform in this way to accommodate the geometry change. 17

18 z (mm) Fiber volume fraction Male mold G=1MPa, E=200GPa G0=0.3MPa x (mm) (a) Whole laminate. z (mm) Fiber volume fraction Male mold G=1MPa, E=200GPa G0=0.3MPa x (mm) (b) At corner. Figure 7: The final fiber volume fraction distribution of a long laminate with a male mold. G = 1 MPa. 18

19 z (mm) 40 Fiber volume fraction Male mold G=10MPa, E=200GPa G0=0.3MPa x (mm) Figure 8: The final fiber volume fraction distribution of a long laminate with a male mold. G = 10 MPa. z (mm) 40 Fiber volume fraction Male mold G=1GPa, E=200GPa G0=0.3MPa x (mm) Figure 9: The final fiber volume fraction distribution of a long laminate with a male mold. G = 1 GPa. 19

20 Table 1: Initial laminate thickness minus final laminate thickness for the long part, at the corner and in the flat region, versus shear modulus. Mold Shear modulus Thickness change Thickness change type G (MPa) at corner (mm) in flat region (mm) Male Male Male Female Female Female Corner Effects on a Short Part In the previous section the length of the flat region was 50 mm, ten times the inner radius of the corner. Due to the long flat region, the resistance to shearing along the fiber axis was large, thus the shear deformation was small. In this section we examine a short laminate, with a flat region only 5 mm long. This is comparable to the inner radius and much shorter than the long part. The initial thickness and the radius of the inner and outer corners remain unchanged. The laminate is discretized in 756 nodes and 704 four-node quadrilateral elements, with mesh refinement around the corner. Simulations were performed using G = 1 MPa and t = 2 seconds. The deformed shape (with solid lines) of the laminate, as well as the distribution of final fiber volume fraction, are illustrated in Fig. 10. Again the initial mesh is denoted by dotted lines. In this case, both the corner and the flat part are very well consolidated, with the fiber volume fraction reaching 71.6%. Since this is a short part, the resistance of fiber axial deformation is not as strong as in the long part. Therefore the corner consolidates much better than the long part, and shear deformation is clearly visible in this short part. The tip of the flat region, due to this shearing effect, was pushed forward. Wiggles again occur in this male mold, for the same reason described in Section 4.2. To determine whether the wiggles are caused by physics or by the numerical method, two checks were made. First, since this result was initially computed using 2 2 Gaussian integration, another simulation was performed using 3 3 integration. The same results, with wiggles, were obtained. So the wiggles are not a mesh artifact like hourglass modes. Second, this result was found to have achieved mesh convergence. We also found that, with the refined mesh, the wiggles becomes smaller, but still occur. We can expect that as the mesh gets finer and finer, the wiggles will become smaller and smaller (while maintaining the fiber length), but wiggles will always exist. This is because, in our current model, the fiber has no bending stiffness, so the natural wavelength of the buckling is always smaller than the mesh size. Thus, the numerical buckling wavelength is controlled by the element size. If the fiber diameter were taken into account, the fiber would have some bending stiffness, and there would be a natural (but very small) wavelength for fiber buckling. Note that our 2-D simulation allows this buckling type deformation only in the plane of the problem. A real laminate might accommodate the consolidation by out-of-plane buckling, which would be visible as fiber waviness at the outside of the corner. The stresses profiles are illustrated in Figs. 11 and 12. Instead of computing the stresses in the x-z coordinates, here we compute them in the directions parallel and normal to the local fiber orientation. The 20

21 z (mm) Fiber volume fraction Male mold G=1MPa, E=200GPa G0=0.3MPa x (mm) Figure 10: Under refined mesh, the final fiber volume fraction distribution of a long laminate with a male mold. G = 1 MPa. normal stress in the fiber direction is evaluated as σ e = e τ s e, while the shear stress parallel to the fibers is τ en = e τ s n. Along the fiber direction the stress σ e is negative, which shows that fibers are in compression. The shear deformation is small in the corner, but in the flat region the shear stress τ en is large. Besides the stress distribution, the code is also capable of computing the resin pressure, the relative resin velocity (the difference between resin velocity and solid velocity), and the fiber orientation at any time step. Simulations were also performed for larger values of G. In this short part, wiggles also occur at the corner when G increases to 10 MPa, but they are much smaller than Fig. 10. The flat region consolidates uniformly, but most of the corner does not consolidate much. No wiggles occur when G increases to 1 GPa, where the whole corner does not consolidate much at all, and the corner effect extends well into the flat region. Usually the fiber longitudinal stiffness E s is much larger than both G and the autoclave pressure p a. Therefore we expect that changes in E s will not affect the consolidation much. A simulation was performed for G = 1 MPa and E s = 2 GPa, and the result were very close to Fig. 10. Table 2 gives the thickness changes at the corner and in the flat region for short parts. With this short part, the final thickness at the flat region is about 3.48 mm, very close to all the cases in the long part (both the female and the male mold). However, for a given G, the change of thickness at the corner in this short part is larger than for the long part (compare to Table 1). This is what we expect, since the influence of the resistance to fiber axial deformation is not as strong in this short part as in the long part. To summarize the corner effect on consolidation, first, for the same thickness change during consolidation, the geometry requires a change of length of the fibers at the corner. However, the resistance to this change is very strong due to the large fiber axial stiffness. Therefore, the corner is thicker than the flat region in the final part, and the corner has a lower fiber volume fraction. Second, the fibers at the corner experience axial compression in a male mold. For sufficiently small values of shear modulus G, wiggles occur at the corner. The wiggles can be eliminated from the simulation by using large value of G. The female mold does not generate wiggles, because fibers at the inside of the corner experience axial tension. Third, though the 21