Three dimensional thermoset composite curing simulations involving heat conduction, cure kinetics, and viscoelastic stress strain response
Harrison Poon, Seid Koric, M. Fouad Ahmad National Center for Supercomputing Applications University of Illinois at Urbana-Champiagn
405 N. Mathews Avenue Urbana, IL 61801, USA
ABSTRACT
1. Introduction
general background on composite curing
literature review of previous work, including recent attempts in resin flow study purpose and scope of present work
2. Constitutive model
The material used in the present work is AS4/3501-6 graphite/epoxy, modeled as transversely isotropic, with a fiber volume fraction of 0.67. The governing equations for heat transfer and chemical curing are the widely-used ones in related literature:
where is density, c is specific heat, T is temperature, t is time, k is thermal conductivity tensor, and is internal heat generation (per unit volume).
The heat generation comes from the curing reaction:
where Hr is the total heat of reaction (per unit mass), and the degree of cure is the ratio of heat generated so far to the total heat generated.
This portion of the numerical modeling is accomplished using the general purpose finite element code ABAQUS by invoking the HEAT TRANSFER procedure in conjunction with user subroutine HETVAL to specify the volumetric heat generation indicated in above equation. We use graphite/epoxy
(AS4/3501-6) in our simulations and the required material parameters are listed in Bogetti and Gillespie (1991).
Next, a temperature and cure dependant viscoelastic stress analysis is performed, making use of the thermal and cure histories solved for in the previous heat transfer analysis. Invoking the assumption of rheologically simple (both thermo and chemo) behavior, the hereditary integral relationship between
stress and strain can be written, in a general anisotropic setting, as:
where is the stress, is the total strain, is the relaxation moduli. Other symbols are defined immediately below.
For anisotropy, the reduced time can be different for different directions. Hence we write
( has a similar definition, with integration up to t'.) Note that the shift factor A is made dependant on temperature T and degree of cure .
The non-mechanical, stress-free strain is expressed as
where are the thermal expansion coefficients (so the first term is thermal strain) and are the cure analogs.
To model this constitutive behavior, our simulation adopts a novel scheme, proposed by the authors (1997), that is suitable for implementation into existing nonlinear solid mechanics finite element codes. Previously, anisotropic cure dependant viscoelastic stress analysis have been carried out by Yi and his co-workers (see Yi and Hilton (1995), Yi et al. (1996), Yi et al. (1997), for representative samples of their work) and Kim and white (1997). Their finite element algorithm is detailed in Yi and Hilton (1994) and Yi (1991). In brief, the approach starts from the constitutive relation, apply viscoelastic version of the variational principle, invokes finite element discretization, arrives at a system of integral equations in the nodal unknowns, and figures out tricks (such as Laplace transforms) of solving this system without too much memory burden.
On the other hand, our (1997) proposed an attractive alternative which is strictly an integration point constitutive update algorithm and can be readily incorporated into any quasy-static, nonlinear, implicit structural FEM code. It is amenable to adaptive time stepping and features a consistent tangent operator (Simo and Taylor, 1985) which should minimize the number on Newton iterations within each time step. We again use ABAQUS as the modeling engine, to simulate the stress development during the composite curing process. The quasi-static procedure is invoked in conjunction with a user material subroutine UMAT which implements the aforementioned novel constitutive update algorithm.
The relaxation moduli tensor is typically assigned the form of an exponential (Prony) series:
To simplify the computations, we reduce the Prony series to just one exponential term:
where are the equilibrium moduli, the magnitudes of transient decay, and the relaxation
times.
In this simplified setting the material parameters required to characterize the viscoleastic response are then: equilibrium moduli ● glassy moduli ● relaxation times ● shifting functions ●
To the author's best knowledge, experimental data for determining these parameters, for the
graphite/epoxy composite, do not exist. As an alternative, we try to concoct resonable values based on the properties of the constituent fiber (AS4 graphite), matrix (3501-6 epoxy), and a fiber volume fraction of 67%. This is discussed below.
The graphite fiber is assumed to be elastic, with transversely isotropic engineering constants taken from column 3, Table 4 of Bogetti and Gillespie (1992).
The epoxy matrix is assumed to be isotropic, with aspects of its viscoelastic behavior documented in the experimental work of Kim and White (1997). More precisely, master relaxation curves for the Young's modulus E (at 30 C), together with its time-temperature shift functions, are shown at various cure levels. Owing to the lack of information regading the relaxation of Poisson’s ratio , we take it to be constant, with a value of 0.35. (Table 4 of Bogetti and Gillespie, 1992).
The composite’s glassy moduli is then computed based on:
graphite elastic moduli
epoxy’s glassy E, constant
●
theory from micromechanics to find effective moduli, see e.g. chapter 9 in Tsai and Hahn (1980)
●
The composite’s equilibrium moduli is computed in an analogous manner.
Rough hints for the relaxation times are taken from the epoxy master relaxation curves, Finally, the
shifting functions are concocted based on the series of epoxy time temperature shift
functions.
With this set of material parameters, we carried out simulations to investigate the evolution of
temperature, degree of cure, and stresses during autoclave curing. A 3D right-angle bend is subjected to a temperature history (that of the autoclave) on the entire surface, and a pressure history on the top and bottom faces. The object is made up of four plays, each with its own fiber orientation. Parametric studies were performed to investigate the effect of autoclave profile, laminate thickness, and other
process-related parameters, on the final residual stress, which has a direct bearing on the product quality. An example numerical results is shown in Figure 1, where the temperature distribution at a moment of rapid curing is illustrated. It is hoped that such parametric studies will yield a database of useful
information, which will ultimately allow us to engage in the clever optimal design of process conditions for the manufacture of composite parts.
