The.most.widely.used.numerical.approach.to.micromechanics.modeling.is.
finite.element.analysis.(FEA)..The.previously.mentioned.FEA.of.Caruso.and.
Chamis.[19].and.Caruso.[23].is.one.example.of.such.an.approach..In.this.case,.
a. single-cell. (SC). finite. element. model. was. developed. from. 192. 3-D.
.isoparametric.brick.elements.(Figure.3.13)..This.SC.model.was.then.used.as a. Composite shear stiffness (G12/Gm)
2
Circular filaments in a square array
Filament volume (υf) [Filament spacing (δ/r)]
78% (0.012)
75% (0.044)
70% (0.118)
55% (0.39)
FIGURE 3.11
Normalized. composite. shear. stiffness,. G12/Gm,. vs.. shear. modulus. ratio,. Gf/Gm,. for. circular.
fibers.in.a.square.array..(From.Adams,.D..F..and.Doner,.D..R..1967..Journal of Composite Materials,.
1,.4–17..With.permission.)
nine-cell.MC.model.for.the.calculations..Boundary.and.load.conditions.were.
[21,22],. the. quarter. domain. model. of. a. representative. volume. can. also.
be analyzed.by.using.finite.elements..Typical.2-D.and.3-D.quarter.domain.
finite.element.models.are.shown.in.Figure.3.14.from.[24]..For.example,.in.one.
study,.3-D.finite.element.quarter.domain.models.similar.to.the.one.in.Figure.
3.14.were.subjected.to.transverse.normal.loading.as.in.Figure.3.15.[24,25],.and.
the. effect. of. model. aspect. ratio. L/(D/2). on. the. transverse. modulus. was.
.determined,.as.shown.in.Figure.3.16..In.this.model,.the.fiber.axis.is.assumed. Normalized composite transverse stiffness (E2/Em)
2
Circular filaments in a square array
78% (0.012)
Constituent stiffness ratio (Ef/Em)
40 60 100 200 400 600 1000 σx
Filament volume (υf) [Filament spacing (δ/r)]
FIGURE 3.12
Normalized.composite.transverse.stiffness,.E2/Em,.versus.modulus.ratio,.Ef/Em,.for.circular.
fibers.in.a.square.array..(From.Adams,.D..F..and.Doner,.D..R..1967..Journal of Composite Materials,.
1,.152–164..With.permission.)
118 Principles of Composite Material Mechanics
the.plane.y.=.0,.displacements.perpendicular.to.that.plane.are.also.prevented..
The. transverse. modulus. was. calculated. by. imposing. a. uniform. displace-ment.Ux.along.the.plane.x.=.D/2.and.then.using.the.calculated.stresses.from.
the.finite.element.model.to.evaluate.Equation.3.62:
where.σx =.average.stress.acting.along.plane.x.=.D/2.in.Figure.3.15,
.
εx Ux
D x D
= = =
( / ) /
2 average strain along plane 2
3, Z, w
FIGURE 3.13
3-D.finite.element.models.of.RVEs..(a).The.single-cell.model.used.for.the.SC.calculation,.(b).the.
multicell.model.which.is.used.for.the.MC.calculation,.and.(c).the.multicell.model.which.only.
the.center.cell.is.used.for.the.CCMC.calculation..(From.Caruso,.J..J..and.Chamis,.C..C..1986..
Journal of Composites Technology and Research,.8(3),.77–83..Copyright.ASTM..With.permission.)
where
D/2.=.dimension.defined.in.Figure.3.15 Ux.=.imposed.displacement.along.x.=.D/2 L.=.length.of.model.along.z,.the.fiber.direction V.=.volume
FIGURE 3.14
Examples. of. 2-D. and. 3-D. finite. element. quarter. domain. micromechanics. models.. (From.
Finegan,.I..C..and.Gibson,.R..F..1997..In.Farabee,.T..M..ed..Proceedings of ASME Noise Control and Acoustics Division..NCA-Vol..24,.pp..127–138..With.permission.)
D/2 Dc/2 (Di/2)
Df/2
L Matrix
Coating (interphase)
Fiber
y x z
Transverse loading
FIGURE 3.15
Quarter.domain.of.RVE.under.transverse.normal.loading..(From.Finegan,.I..C..and.Gibson,.
R. F..1998..Journal of Vibration and Acoustics,.120(2),.623–627..With.permission.)
120 Principles of Composite Material Mechanics
It.is.seen.from.Figure.3.16.that.the.transverse.modulus.varies.from.a.mini-mum.value.for.low-model.aspect.ratios.to.a.maximum.for.high-model.aspect.
ratios.[24,25]..It.was.also.shown.that.the.low-model.aspect.ratio.results.from.
3-D.models.coincided.with.the.results.obtained.by.using.2-D.plane.stress.
elements.(i.e.,.with.longitudinal.stress.σz.=.0),.while.the.high-model.aspect.
ratio.results.from.3-D.models.coincided.with.the.results.obtained.by.using.
2-D.plane.strain.elements.(i.e.,.with.longitudinal.strain.εz .=.0)..For.a.unidirec-tional.composite.having.continuous.fibers.oriented.along.the.z.direction,.the.
plane.strain.condition.is.more.realistic.than.the.plane.stress.condition..The.
importance.of.this.observation.is.that.2-D.plane.strain.elements.can.be.used.
for.these.types.of.models.instead.of.3-D.elements,.and.this.leads.to.signifi- cant.reductions.in.the.number.of.elements.and.the.corresponding.computa-tion.time.
Proper.treatment.of.boundary.conditions.on.the.RVEs.of.micromechani-cal. FEA. models. is. particularly. important. [26].. Since. the. composite. is.
assumed. to. consist. of. large. numbers. of. identical. and. adjacent. RVEs. as.
repeating. elements,. the. deformations. on. the. boundaries. of. a. given. RVE.
must.be.geometrically.compatible.with.those.of.the.adjacent.RVEs..Figure.
3.17a. shows. an. example. of. a. boundary. deformation. pattern. that. is. not.
.geometrically. compatible. with. that. of. the. adjacent. identical. RVEs. (i.e.,.
7.00.0001 0.01 0.1
Model aspect ratio (L/(D/2)) Plane stress
Plane strain
1 10 100
7.5 8.0 8.5 9.0 9.5 10.0
Transverse modulus (GPa)
FIGURE 3.16
Variation.of.transverse.modulus.with.model.aspect.ratio.for.graphite/epoxy.composite.from.
3-D.finite.element.models..(From.Finegan,.I..C..and.Gibson,.R..F..1998..Journal of Vibration and Acoustics,.120(2),.623–627..With.permission.)
.symmetry.arguments.tell.us.that.the.deformed.RVEs.will.not.“fit.together”.
to. form. a. continuous. composite. without. gaps).. By. contrast,. Figure. 3.17b.
shows. a. boundary. deformation. pattern. that. is. geometrically. compatible.
with. that. of. its. neighboring. RVEs. (i.e.,. the. deformed. RVEs. will. all. “fit.
together”.to.form.a.continuous.composite.without.gaps)..These.are.often.
referred.to.as.“periodic.boundary.conditions,”.since.they.are.repeated.over.
and.over.again.from.one.RVE.to.another.as.we.move.through.the.compos-ite..For.normal.loading.as.in.Figure.3.15,.geometric.compatibility.is.assured.
if. each. boundary. surface. is. constrained. to. deform. parallel. to. itself. as. in.
Figure.3.17b..Such.constraints.on.displacements.of.a.plane.can.be.accom-plished. in. FEA. by. using. the. so-called. multipoint. constraint. (MPC). ele-ments. which. are. available. in. many. FEA. codes.. MPC. eleele-ments. allow. the.
user.to.specify.a.“master.node”.in.a.plane.such.that.all.nodes.in.that.plane.
will.have.the.same.displacement.under.load..For.shear.loading.as.in.Figure.
3.10,.geometric.compatibility.is.assured.if.the.deformed.shape.is.a.paral- lelogram,.although,.as.pointed.out.by.Sun.and.Vaidya.[26],.it.is.not.neces-sary.for the.edges.of.the.parallelogram.to.remain.straight.as.they.rotate.in.
shear.distortion..Note.that,.when.the.displacement.is.imposed,.as.in.the.
example.described.above.and.in.[25],.geometric.compatibility.is.assured.by.
the. imposed. planar. displacement.. When. the. applied. stress. is. imposed,.
however,.displacement.constraints.must.be.specified.to.assure.geometric.
compatibility.