40的英文Optimal Curing for Thermot Matrix Composites:
Thermochemical and Consolidation Considerations
Min Li and Charles L.Tucker III关于 英语
Department of Mechanical and Industrial Engineering
University of Illinois at Urbana-Champaign
Urbana,Illinois61801
Submitted to Polymer Composites
March16,2001
Abstract
A design nsitivity method is ud tofind optimal autoclave temperature and pressure histories for curing of thermot-matrix composite laminates.The method us afinite element simulation of the heat transfer,curing reaction,and consolidation in the laminate.Analytical nsitivities,bad on the direct diff
erentiation method,are ud within thefinite element simulation tofind the design nsitivities,
<,the derivatives of the objective function and the constraints with respect to the design variables. Standard gradient-bad optimization techniques are then ud to systematically improve the design, until an optimal process design is reached.In this study the objective is to minimize the total time of
the cure cycle,while the constraints include a maximum temperature in the laminate(to avoid thermal degradation)and a maximum deviation of thefinalfiber volume fraction from its target value(to achieve proper consolidation).The simulations of curing process are performed for EPON862/W epoxy under
a conventional cure cycle,for both thin and thick parts.Time-optimal cure cycles are found using the optimization program.Simulations of fast-curing cycles are also examined.The optimal cycles are similar in form to conventional cure cycles,but give substantially shorter cure times.The entire scheme works automatically and efficiently,simultaneously adjusting multiple design variables at each iteration. Keywords:thermot cure,consolidation,design nsitivity analysis,optimal curing,thick composites
雾霾英文uzumaki naruto
1Introduction
Autoclave curing is a process to producefiber-reinforced polymeric parts infinal shape.The simulation and optimization of autoclave processing have en widespread application in industry as a means 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 increas the tem-perature 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 ex-cess resin from the composite into a bleeder ply as the laminate consolidates.The resin then cures and cross-links,producing a rigidfinished part.
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This study prents the u of a numerical optimization technique,combined with analytical models to simulate the consolidation and curing process,and demonstrates this approach as a viable way to design time-optimal cure cycles,which lead to enhanced productivity and reduced cost.
Section1introduces the process control issues and gives a literature review of this problem.Section2 describes the details of the one-dimensional model of the process.Section3gives thefinite element e
qua-tions ud to solve the primal problem.Section4discuss the design nsitivity analysis and optimization scheme.Section5prents the results for a range of curing and consolidation problems.Section6summa-rizes this work.
1.1Process Control
The controllable variables in autoclave curing are the thermal cycle and the pressure cycle during the process. Three major conditions must be achieved to have a successful autoclave curing process.First,the process must allow full consolidation of the laminate.This condition is achieved by a balance of consolidating pressure and a viscosity profile that ensures sufficient time for full consolidation.
Second,the resin must be fully cured in all parts of the laminate.The gelation point coincides a rapid rate of cure and relea of heat by the resin as it cross-links.Further heating is then necessary to complete the curing process.The curing reaction is exothermic and the thermal conductivities of the laminate and bleeder are low,therefore,when the composites get cured too quickly,a large amount of heat is relead, but only a small fraction of heat is conducted away.This may cau thermal degradation in the composite, so uncontrolled exotherms must be avoided.
batterypackThird,becau of the high cost of autoclave and press molding equipment,overall process time is a critical economic factor in high volume production,and it must be minimized to decrea the cost of the products.However,the rate at which the laminate may be heated and cooled is limited by the thermal inertia of the autoclave,together with its charge of molds and laminates.This typically means that processing cycles lastfive hours or more.How systematically to obtain a time-optimal process cycle while meeting all of the conditions is the objective of this study.
1.2Literature Review
Process simulations for composite curing u a physically-bad model to simulate the process,and predict the cure cycle time.Thefiber volume fraction,thickness,degree of cure,temperature and other properties in thefinal part can also be predicted by a simulation.
Barone and Caulk(1)studied the influence of the applied heat on the curing process and propod a thermochemical model bad on a two-dimensional heat conduction equation with internal heat generated by the exothermic chemical reaction.Springer(2)studied the relationship between the applied pressure and the resinflow during the cure offiber reinforced composites.The mechanism by which resinflows
through the composite was obrved,and the layers were found to consolidate in a wavelike manner.Loos and Springer(3)further developed resinflow 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(4)developed three-dimensionalflow and one-dimensional consolidation models of the composite.The resinflow was modeled using Darcy’s law for an anisotropic porous medium.The general ca was then solved for compression molding and bleeder ply molding.
Bad on rigorous volume averaging principles,Tucker and Desnberger(5)established governing equations forflow and heat transfer in resin transfer molding.The equations include the balance equa-tions of mass,momentum,energy and chemical reaction forflow through porous media.Although derived originally for stationaryfiber beds,the equations can easily be extended to the ca of a moving,de-forming solid bed,which then leads to the complete governing equations for the consolidation and cure in autoclave process.This is the way the consolidation governing equations were derived in this study.
Historically,cure cycles have been developed using trial-and-error methods.The results of some of the trial-and-error methods have led to the development of various rule-bad heuristic expert syste
ms to guide process development(6;7).However,the applicability of heuristic approaches is often limited by the specific material system,geometry,and practical restrictions.Many other approaches have been developed to obtain the optimal cure cycles,such as process contour maps(8),optimization diagrams(9),artificial neural networks(10),and genetic algorithms(11;12).The methods are reasonably accurate and effective, but they cannot obtain an optimal design,especially where there are large numbers of design parameters and constraints involved.
In order to systematically improve the design,an optimization scheme needs to know how the objective function and constraint functions will change as the design variables are altered.The derivatives of the objective and constraints with respect to the design variables are called the design nsitivities.Tortorelli, Haber and Lu(13)ud a Lagrangian multiplier method and convolution theory,and formulated design nsitivities in an adjoint approach for non-linear transient thermal systems.The nsitivities were ud to design a better mold for a metal casting to lesn the porosity in thefinished product.Michaleris,Tor-torelli and Vidal(14)derived the tangent operator and formulated the design nsitivities via both the direct differentiation and adjoint methods for a one-dimensional single degree-of-freedom,two-bar elastoplastic system.Tortorelli,Tiller and Dantzig(15)prent a general framework for the optimal design of non-linear parabolic systems.Furth
er shape variations were accounted for,and a casting solidification problem was prented in which the casting design is optimized to attain a desirable solidification pattern(16).
In some other applications,a L-shaped metal casting geometry was systematically modified to minimize the gate and rir volume while ensuring that no porosity appeared in the product(17),polymer sheeting dies were designed with minimum pressure drop and reduced velocity variation across the die exit(18),and injection molds were designed to minimize moldfill time and satisfy constraints on injectionflow rate and mold clamp force(19).
In a work cloly related to this paper,Li,Zhu,Geubelle and Tucker(20)applied this method and studied the thermochemical part of autoclave curing process for thermot composites.The nsitivities were ud with a gradient-bad numerical optimization tool to systematically improve the curing process.That work considered only the heat transfer and curing reaction,ignoring the consolidation aspect of the process,and produced temperature cycles for rapid curing without thermal degradation.The temperature cycles,which we will refer to as fast-curing cycles,provide short cure times,but are very different from conventional cycles.In this paper we consider consolidation effects,as well as thermochemical phenomena,and jointly optimize the pressure and the temperature cycles.The resulting optimal cycle are found to be quite different from the fast-curing
cycles.
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Figure1:Schematic of the geometry for analysis.
2Process Model
In this study we assume the temperature andfiber volume fraction vary only with time and location in the thickness direction,and consider the simple geometry shown in Fig.1,where denotes the half-thickness of the laminate.The process model can be divided into two parts:a thermochemical part and a consolidation part.
2.1Thermochemical Governing Equations
The governing equation for one-dimensional transient heat conduction,including internal heat generation due to the exothermic cure reaction,may be written as
(1) where is the temperature,is the time and is the location in the thickness direction.The density,specific heat,thermal conductivity,and heat of reaction of the laminate are denoted as,,,and respectively. is the degree of cure,which is defined as the fraction of the reactive groups in the resin that have reacted. Further,denotes the rate of the cure reaction,which,together with the heat of reaction,determines the heat relea rate during the cure process.
The heat relea 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
(2) where is a frequency factor,is the activation energy,and is the universal gas constant.The parameters and are rate constants.
The governing equations(1)and(2)are subjected to initial conditions of and at. The boundary conditions are convective heat transfer conditions at the surfaces of the bleeder and tooling.
Scaling analysis gives the dimensionless groups of the thermochemical problem as
(4)
A cond type of average includes only the points that lie within a single pha,but still averages over the
entire volume.This is called a pha average.Pha averages will also be denoted by angle brackets,but
医学英语电子词典the quantity inside the brackets will carry a subscript denoting the pha over which the average is taken.
For example,denotes the pha-average velocity for thefluid.The third average considers only points lying within a single pha,and averages their values over just the volume occupied by that pha.This is
called an intrinsic pha average,and it is denoted by angle brackets with a superscript labeling the relevant
pha.For example,denotes the intrinsic pha-average velocity for thefluid.
The pha average is related to the intrinsic pha average by relations and最新大学排名
.We assume and are constant and that inertial and gravitational effects are negligible.
We begin with the continuity equations for a constant-densityfluid and solid in a consolidating porous
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medium,which are
(6) where is thefluid viscosity and is the permeability tensor.The total solid stress is
(7)
where is the identity tensor.Thefirst term,,is part of thefluid-solid interaction,and we assume that thefluid pressure is transmitted directly to the solid.The cond term is the solid extra stress,and is related to deformation of the solid state.For later u we define.
The above equations are combined with the momentum balance equations,
(8)
(9) to form the governing equations.It is convenient to u a Lagrangian coordinate system that isfixed on thefiber bed and deforms with the solid.Call this coordinate.Initially at time,and for all subquent times the coordinate follows the(average)motion of thefiber bed.The time derivative
is the rate of changes offiber volume fraction atfixed in thefixed frame.But for computation we need an equation for the rate of changes atfixed,in the deforming frame,and we call this quantity.Further, we introduce the void ratio as a primary variable,which is the ratio of thefluid volume fraction to the solid volume fraction.The substitutions(21)give the consolidation governing equation in the deforming frame that we u in this study as
(10)
This equation is the same as the one propod by Gutowski(4).
Equation(10)has initial conditions of at.From the fact that the resin does notflow through the bottom into tooling,the boundary condition at the bottom is.At the top of the laminate,the resin pressure equals ,.
2.3Material Properties
In order to solve Eqn.(10),we must know and as functions of.is directly related to the solid stress,which is a function of.By studying the transver stiffness behavior of a bundle of confined fibers,Gutowski et al.(4)developed a function for thefiber extra stress as
(11)
where is called the spring constant and is the maximum possiblefiber volume fraction.Fibers which are wavy or slightly misaligned will result in a lower.The minus sign on the right-hand side of Eqn.(11) accounts for the fact that the stress on thefiber bed is compressive,and becomes more negative as increas.
In this study the modified Carman-Kozeny equation is ud for the transver permeability for afiber
