Pergamon
Int. J. Mech. Sci. Vol.37, No. 8, pp. 883 897, 1995
ElsevierScienceLtd Printed in Great Britain. 0020-7403/95 $9.50+ 0.00
0020-7403(94) E0093-X FINITE ELEMENT SIMULATION OF POWDER CONSOLIDATION IN THE FORMATION OF FIBER REINFORCED COMPOSITE MATERIALS J. XU t and R. M. McMEEK1NG ~
tConcurrent Technologies Corporation 1450 Scalp Avenue, Johnstown, PA 15904, U.S.A. and ~ Mechanical Engineering Department, University of California, Santa Barbara, CA 93106, U.S.A.
(Received 15 October 199

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Pergamon0020-7403(94) E0093-X
Int. J. Mech. Sci.
Vol. 37, No. 8, pp. 883 897, 1995ElsevierScienceLtdPrinted in Great Britain.0020-7403/95 $9.50 + 0.00
FINITE ELEMENT SIMULATION OF POWDER CONSOLIDATIONIN THE FORMATION OF FIBER REINFORCEDCOMPOSITE MATERIALS
J.
XU t and
R. M.
McMEEK1NG ~
tConcurrent Technologies Corporation 1450 Scalp Avenue, Johnstown, PA 15904, U.S.A. and ~ MechanicalEngineering Department, University of California, Santa Barbara, CA 93106, U.S.A.
(Received
15
October
1993;
and in revised form
30
October
1994)Abstract--Finite element codes are developed to analyze powder consolidation due to time-inde-pendent plasticity and time-dependent plasticity (power law creep and diffusional creep). Fortime-dependent plasticity, a new algorithm is proposed to obtain a prediction of the plastic strainincrement when solving for the nodal displacement increment. This eliminates the use of an
ad hoc
Newton loop. A set of newly developed micromechanical models are adopted as the constitutivelaws for power deformation,where the influence of deviatoric stresses on densification is taken intoaccount. The densification maps are constructed for the powder consolidation in the formation offiber reinforced composite materials. The influence of fiber volume fraction, powder particle size,temperature and fiber arrangement is investigated,1. INTRODUCTION
Owing to its potential in manufacturing high performance components, hot isostaticpressing (HIP) has been motivating theoretical and experimental research. The densifica-tion of a powder compact by HIP is dependent on many factors such as applied pressure,temperature, powder particle size and the properties of the material composing the powder.A theoretical foundation has been established for the mechanisms contributing to powderdensification and densification maps (mechanism diagrams) have been constructed I-1-5].In recent years, the fact that deviatoric stresses can influence powder consolidation hasmade it the subject of a number of micromechanical analyses [6-15]. The results of theseanalyses include the deviatoric stresses in the constitutive laws for the deformation ofa powder compact. With these micromechanical models, one can evaluate powder densifi-cation under nonuniform as well as nonhydrostatic stress states. Some of the above-mentioned models have been implemented in finite element programs to analyze problemssuch as shape change, the can effect and closed-die compaction 1-16, 17].During densification, the powder compact experiences large geometric changes and theconstitutive laws are nonlinear. The implementation of the above-mentioned constitutivelaws in processing requires the use of a nonlinear finite element analysis. The objective ofthis work is to develop a nonlinear finite element code to evaluate powder densification dueto time-independent and time-dependent plasticity. For the latter, a new algorithm isproposed to provide the first estimate of the nodal displacement increment and a guess forthe plastic strain increment simultaneously. When the nonlinear problem is solved numer-ically, the accuracy and stability of the incremental solution is an important issue. Indeed,this has long represented an interesting challenge in nonlinear finite element analysis.Several methods have been suggested for the integration of the elastoplastic constitutiveequations of time-independent and time-dependent plasticity 1-18-30]. The new algorithmformulated for the current paper exhibits excellent qualities in regard to accuracy andstability.Densification maps are constructed for powder consolidation during the formation offiber reinforced composite materials. Three mechanics, time-independent plasticity, powerlaw creep and diffusional creep, are considered. A set of newly developed micromechanicalmodels are adopted as the constitutive laws for powder deformation, where the influence of
883
884 J. Xu and R. M. McMeekingdeviatoric stresses on the densification is taken into account. The transition of the constitut-ive relations from the initial particle contact controlled stage to the final void controlleddensification stage is smoothed by a transitional interpolation stage between them. Theinfluence of fiber volume fraction, powder particle size and temperature is investigated.2. MECHANISM-BASED MICROMECHANICAL MODELSThe micromechanical models adopted in this paper are as follows. In particle contactcontrolled consolidation, known as Stage I, for time-independent plasticity, the model isdue to Fleck
et al.
[7]; for power law creep, Kuhn and McMeeking [9]; for diffusional creep,McMeeking and Kuhn [13]. In void controlled consolidation, known as Stage II, fortime-independent plasticity, the model is by Gurson [31]; for power law creep, Sofronis andMcMeeking [8]; for diffusional creep, Riedel [14]. According to the micromechanicalmodels, the deformation rate of the porous powder compact caused by different mechan-isms is governed by corresponding potentials. For time-independent plasticity, the macro-scopic strain rate ~ is determined by= 2 ~, (1)where (I) is the plastic potential, o is the macroscopic stress tensor and 2 is the plasticmultiplier [32]. For power law creep and diffusional creep,8LF
= O~ (2)
where W is the creep potential [27]. Here (I) and W are functions of the macroscopic stresstensor a, temperature 0, the relative density of the powder compact D, and materialparameters, i.e.= ~(~, 0, D, material parameters) (3)andqJ = W(g, 0, D, material parameters). (4)The analytical expressions for all the plastic and creep potentials are listed in the Appendix.The derivations of the micromechanical models were performed with geometries charac-terizing different specific densification stages, either I or II. Since the topology of porosityexperiences a gradual but significant change from Stage I to Stage II, the model for theinitial stage of particle contact controlled deformation must be dispensed with completely infavor of the one for the final stage of void controlled consolidation above some transitionlevel of porosity. One way to simulate this shift of constitutive relations is to envisiona transitional stage (D1 < D < D2), where the plastic and creep potentials are obtained byinterpolation [7]. For example, let ~c be the plastic potential for Stage I due to Fleck
et al.
[7] and O~ that due to Gurson [31] for Stage II. In the transition range D1 < D < D2, theplastic potential isD2 -- D D - Djã = Oc + O~. (5)D2 - D1 D2 - D1The choice of D 1 and D2 should best reflect the transition and depends on the stress stateand the deformation mechanism. For example, Ashby [4] suggests that D, = 0.8, andD2 -- 0.9 for hot isostatic pressing. In this analysis, we choose D1 = 0.75 and DE = 0.9.3. FORMULATION OF THE BOUNDARY-VALUE PROBLEMThe formation of fiber reinforced composite materials through powder processing ismodeled by analyzing the densification of the powder compact around long rigid cylindricalinclusions. Due to the rigidity of the fiber, plane strain is assumed for the deformation of thepowder compact. The finite deformation of the powder compact is allowed for through the
Finite element simulation of powder consolidation 885
updated Lagrangian formulation [33] governed by the virtual work expression
fv[~: rM +~6: 6(LXL- 2MM)]dV= s'rrvdS,
(6)where V is the current volume of the plain strain problem, S is the boundary of V, 6 is theCauchy stress, ~ denotes the Jaumann rate of Kirchhoff stress, L is the velocity gradient, i.e.~v
L ~x' (7)
M is the deformation rate which is the symmetric part of L, T is the surface traction onS and 6 denotes an arbitrary virtual variation, tr and ~ are related by
= Jo,
(8)
where J is the ratio of volume in the current state to that in the reference state.A representative square unit cell with symmetry and periodicity conditions on theperimeter is used in the analysis as shown in Fig. 1. A comparative study is also performedwith a hexagonal unit cell (Fig. 2) to test the appropriateness of the choice of unit cell andthe influence of fiber arrangement on the densification. In the process of densification, theperimeters (four sides for the square unit cell and six sides for the hexagonal unit cell) areconstrained to remain straight. Because of symmetry, the perimeters are shear stress free.A perfect bonding between the powder compact and the inclusion is assumed. The unit cellsare equally loaded in the x and y directions. Using the symmetry of geometry and loadingcondition, it is sufficient to deal with one-eighth of the square unit cell and one-twelfth of thehexagonal unit cell. However, the finite element code for the square unit cell is developed forone-fourth of the unit cell, leaving leeway for future investigation of densification underunequal loading. In this work, the volume fractions of the consolidated composite materialare from 10% to 50%.
P
---0, @,-.,-,
I I
-,-@, @,.--,
t t t
powder compactfiber
mFig. 1. The unit cell for fibers packed in a square array and a typical mesh used in the finite
element
analysis.
886 J. Xu and R. M.
McMeeking
powder compact
_.
t t t
II
iI
Fig. 2.
The unit cell for fibers packed in hexagonal arrayelement analysis.
1
and a typical mesh used in the finite
In the finite element analysis, eight-noded second-order isoparametric elements with fourstations for the integration are used [34]. The number of elements and nodes used in theanalysis for different unit cells and volume fractions are listed in Table 1.In the construction of densification maps, the contribution of time-independent plasticityis evaluated first. To do this, displacement boundary conditions are imposed at theperimeters of the unit cells. By use of the resulting stresses in the powder compact, thenormal surface traction on the perimeter of the unit cell can be obtained. The average valueof the normal surface traction is taken to be the applied pressure, which, together with therelative density of the powder compact, is the main output of the calculation. After theapplied pressure has reached a certain value, power law creep and diffusional creep start tocontribute to the densification. The total strain rate is the sum of the creep rates due to
Table 1. Numbers of elements and degrees of freedom in the finiteelement analysisVolume Number offraction Number of degrees of
(%)
elements freedomSquare unit cell
64 450256 166630 72 50650 72 506
Hexagonal unit cell
10 50 36250 36 266

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