[813000] Notas - Información financiera intermedia de conformidad con la NIC 34
3. Bases de preparación a)Bases de preparación
In.the.analysis.of.laminates.having.multiple.laminae,.it.is.often.necessary.to.
know. the. stress–strain. relationships. for. the. generally orthotropic lamina. in.
nonprincipal. coordinates. (or. “off-axis”. coordinates). such. as. x. and. y. in.
Figure 2.5.. Fortunately,. the. elastic. constants. in. these. so-called. “off-axis”.
stress–strain. relationships. are. related. to. the. four. independent. elastic. con-stants.in.the.principal.material.coordinates.and.the.lamina.orientation.angle..
wedge-shaped. differential. element. in. Figure. 2.12.. For. example,. the. force.
equilibrium.along.the.x.direction.is.given.by
. .
∑
Fx =σxdA−σ1dAcos2θ−σ2dAsin2θ+2τ12dAsin cosθ θ= 0. (2.28)Positive θ Negative θ
+θ
FIGURE 2.11
Sign.convention.for.lamina.orientation.
72 Principles of Composite Material Mechanics
which,. after. dividing. through. by. dA,. gives. an. equation. relating.σx. to. the.
stresses.in.the.12.system:
the. stresses. in. the. xy-coordinate. system. can. be. developed. and. written. in.
matrix.form.as
FIGURE 2.12
Differential.element.under.static.equilibrium.with.forces.in.two.coordinate.systems.
A. procedure. for. determining. the. matrix. inverse. [T]−1. is. described. in.
Appendix.A..It.can.also.be.shown.[2,3].that.the.tensor strains.transform.the.
same.way.as.the.stresses,.and.that
Substituting. Equation. 2.33. into. Equation. 2.26,. and. then. substituting. the.
resulting.equations.into.Equation.2.30,.we.find.that
where. the. components. of. the. stiffness. matrix. [Q*]. in. Equations. 2.34. are.
defined.as.Qij* =Qij.for.all.ij.except.Q66* = 2Q66.as.shown.in.Equations.2.26.
Carrying.out.the.indicated.matrix.multiplications.and.converting.back.to.
where.the.Qij.are.the.components.of.the.transformed.lamina.stiffness.matrix.
which.are.defined.as.follows:
74 Principles of Composite Material Mechanics
where. the.Sij. are. the. components. of. the. transformed. lamina. compliance.
matrix.that.are.defined.by. = S Q −1,.or.in.expanded.form
.
where.the.strain.εx.in.the.denominator.has.been.found.by.substituting.the.
stress.conditions.σx.≠.0,.σy.=.τxy.=.0.in.Equations.2.37.
By.substituting.S11.from.the.first.of.Equation.2.38.into.Equation.2.39.and. where.c.=.cos.θ.and.s.=.sin.θ.as.before..The.rest.of.Equation.2.40.for.the.off-axis.transverse.modulus.Ey,.the.off-axis.shear.modulus.Gxy,.and.the.off-axis.
major.Poisson’s.ratio.υxy.can.be.obtained.from.similar.derivations.
The.variation.of.these.properties.with.lamina.orientation.for.several.com-posites. is. shown. graphically. in. Figure. 2.14. from. Ref.. [9].. As. intuitively.
expected,.Ex.varies.from.a.maximum.at.θ.=.0°.to.a.minimum.at.θ.=.90°.for.
this.particular.material..It.is.not.necessarily.true.that.the.extreme.values.of.
such.material.properties.occur.along.the.principal.material.directions,.how- ever.[6]..What.may.not.be.intuitively.expected.is.the.sharp.drop.in.the.off-axis.modulus.Ex.as.the.angle.changes.slightly.from.0°.and.the.fact.that.over.
much.of.the.range.of.lamina.orientations.the.modulus.Ex.is.very.low..This.is.
why.transverse.reinforcement.is.needed.in.unidirectional.fiber.composites.
which.are.subjected.to.multiaxial.loading..The.maximum.value.of.the.off-axis.shear.modulus.Gxy.at.θ.=.45°.and.minimum.values.of.Gxy.at.θ.=.0°.and.
θ.=.90°.indicate.that.off-axis.reinforcement.is.also.essential.for.good.shear.
FIGURE 2.13
Off-axis.loading.for.determination.of.off-axis.longitudinal.modulus.of.elasticity.Ex.
76 Principles of Composite Material Mechanics
off-axis. reinforcement. is. required. for. acceptable. shear. stiffness.. Later. in.
Chapter.7,.it.will.become.clear.that.off-axis.laminae.are.essential.in.the.design.
of.laminates.which.are.subjected.to.shear.loading.
The.shear-coupling.effect.has.been.described.previously.as.the.generation.
of. shear. strains. by. off-axis. normal. stresses. and. the. generation. of. normal.
strains.by.off-axis.shear.stresses..One.way.to.quantify.the.degree.of.shear.
coupling.is.by.defining.dimensionless.shear-coupling.ratios.[4,5].or.mutual.
influence. coefficients. [10]. or. shear-coupling. coefficients. [11].. For. example,.
when.the.state.of.stress.is.defined.as.σx.≠.0,.σy.=.τxy.=.0,.the.ratio
η ε ν ν
FIGURE 2.14
Variations.of.off-axis.engineering.constants.with.lamina.orientation.for.unidirectional.carbon/
epoxy,.boron/aluminum.and.glass/epoxy.composites..(Sun,.C.T..Mechanics of.Aircraft Structures, 1998,.John.Wiley.&.Sons,.New.York..Reproduced.with.permission.)
is.a.measure.of.the.amount.of.shear.strain.generated.in.the.xy.plane.per.unit.
normal.strain.along.the.direction.of.the.applied.normal.stress,.σx..Thus,.the.
shear-coupling.ratio.is.analogous.to.the.Poisson’s.ratio,.which.is.a.measure.of.
the.coupling.between.normal.strains..As.shown.in.Figure.2.14,.ηx,xy.strongly.
depends. on. orientation. and. has. its. maximum. value. at. some. intermediate.
angle.which.depends.on.the.material..Since.there.is.no.coupling.along.prin-cipal.material.directions,.ηx,xy.=.0.for.θ.=.0°.and.θ.=.90°..As.the.shear-coupling.
ratio.increases,.the.amount.of.shear.coupling.increases..Other.shear-coupling.
ratios. can. be. defined. for. different. states. of. stress.. For. example,. when. the.
stresses.are.τxy.≠.0,.σx.=.σy.=.0,.the.ratio
.
strain.relationship.for.the.normal.strain.εx.in.terms.of.off-axis.engineering.
constants.can.be.expressed.as:
with. similar. relationships. for.εy. and.γxy.. As. with. the. specially. orthotropic.
case.and.the.general.anisotropic.case,.the.stiffness.and.compliance.matrices.
are.still.symmetric..So,.for.example,.the.off-axis.compliances.S12 = S21,.or.in.
terms.of.off-axis.engineering.constants,.(νyx/Ey).=.(νxy/Ex).
Example 2.3
A 45° off-axis tensile test is conducted on a generally orthotropic test specimen by applying a normal stress σx as shown in Figure 2.15. The specimen has strain gages attached so as to measure the normal strains εx and εy along the x and y directions, respectively. How many engineering constants for this material can be determined from the measured parameters σx, εx, and εy?
SOLUTION
The off-axis modulus of elasticity along the x direction is determined from
Ex x x
= σε
78 Principles of Composite Material Mechanics
The off-axis major Poisson’s ratio is given by
ν ε
xy εy
x
= −
Although it may not seem obvious, the in-plane shear modulus G12 may also be found from these measurements by using the appropriate transformations of stress and strain. Using θ = 45° in Equation 2.33, the shear strain γ12 is found from
γ12 ε ε 2 2 γ
2 45 45 45 45 45 45
= −cos sin +cos sin +(cos −sin ) 2
x y xy
or
γ12 = − +εx εy
Note that the γxy term drops out of the above equation only when θ = 45°. For any other value of θ, we would need to know γxy as well as εx and εy in order to find γ12 from this transformation equation. Likewise, the shear stress τ12 is found by substituting the tensile test conditions σx ≠ 0, σy = τxy = 0 in Equation 2.31:
τ σ σ
12 45 45
= −cos sin = − 2
x x
Finally, the shear modulus G12 may now be calculated from the third line of Equation 2.24 as
G12 S66 12 12
= 1 = τ γ
Note that this equation holds even though the stresses σ1 ≠ 0 and σ2 ≠ 0 in the first two lines of Equation 2.24.
y 1
45° x
2 σx
Strain gage for measuring εx Strain gage for
measuring εy
45° off-axis tensile test specimen
FIGURE 2.15
Strain-gaged.specimen.for.a.45°.off-axis.tensile.test.
The.effects.of.lamina.orientation.on.stiffness.are.difficult.to.assess.from.a.
quick.inspection.of.stiffness.transformation.equations.such.as.Equations.2.36.
and. 2.40.. In. addition,. the. eventual. incorporation. of. lamina. stiffnesses. into.
laminate.analysis.requires.integration.of.lamina.stiffnesses.over.the.laminate.
thickness,.and.integration.of.such.complicated.equations.is.also.difficult..In.
view.of.these.difficulties,.a.more.convenient.“invariant”.form.of.the.lamina.
stiffness. transformation. equations. has. been. proposed. by. Tsai. and. Pagano.
[12].. By. using. trigonometric. identities. to. convert. from. power. functions. to.
cos cos
cos
cos cos
θ θ
sin sin
sin sin
( ) c
of.the.Qij,.are.invariant.to.rotations.in.the.plane.of.the.lamina..Note.that.there.
are.four.independent.invariants,.just.as.there.are.four.independent.elastic.
80 Principles of Composite Material Mechanics
cos cos
cos
cos cos
θ θ
sin sin
sin sin
( ) c
normal.stress.component.σx.can.be.described.by.the.invariant.formulation
. σx.=.I1.+.I2.cos.2θp. (2.48)
In.this.case.the.invariants.are.I1,.which.defines.the.position.of.the.center.of.
the.circle,.and.I2,.which.is.the.radius.of.the.circle..Note.the.similarity.between.
Equation.2.48.and.the.first.of.Equation.2.44.in.that.the.invariant.formulation.
typically.consists.of.a.constant.term.and.a.term.or.terms.that.vary.with.ori-entation..Similarly,.the.invariant.forms.of.the.stiffness.transformations.can.
also.be.interpreted.graphically.using.Mohr’s.circles..For.example,.Tsai.and.
Hahn.[11].have.shown.that.the.stiffness.transformation.equation
. Q11 =U1+U2cos2θ+U3cos4θ. (2.49)
can. be. represented. graphically. by. using. two. Mohr’s. circles,. as. shown. in.
Figure 2.17..The.distance.between.points.on.each.of.the.two.circles.represents.
τ σx
σ 2θp
τxy
I2 I1
FIGURE 2.16
Mohr’s.circle.for.stress.transformation.
Q11 2θ
4θ
U2 U4
U1 U3
Q11
FIGURE 2.17
Mohr’s.circles.for.stiffness.transformation..(From.Tsai,.S.W..and.Hahn,.H.T..1980..Introduction to Composite Materials.. Technomic. Publishing. Co.,. Lancaster,. PA.. Reprinted. by. permission. of.
Technomic.Publishing.Co.)
82 Principles of Composite Material Mechanics
the.total.stiffness.Q11,.whereas.the.distance.between.the.centers.of.the.two.
circles.is.given.by.U1..The.radius.and.angle.associated.with.one.circle.are.U2. and.2θ,.respectively,.and.the.radius.and.angle.associated.with.the.other.circle.
are. U3. and. 4θ,. respectively.. Thus,. the. distance. between. the. centers. of. the.
circles.is.a.measure.of.the.isotropic.component.of.stiffness,.whereas.the.radii.
of.the.circles.indicate.the.strength.of.the.orthotropic.component..If.U2.=.U3.=.0,.
the.circles.collapse.to.points.and.the.material.is.isotropic.
Invariants.will.prove.to.be.very.useful.later.in.the.analysis.of.randomly.
oriented.short.fiber.composites.and.laminated.plates..For.additional.applica-tions.of.invariants.in.composite.analysis,.the.reader.is.referred.to.books.by.
Halpin.[5].and.Tsai.and.Hahn.[11].
Example 2.4
A filament-wound cylindrical pressure vessel (Figure 2.18) of mean diameter d = 1 m and wall thickness t = 20 mm is subjected to an internal pressure, p. The filament-winding angle θ = 53.1° from the longitudinal axis of the pressure vessel, and the glass/epoxy material has the following properties: E1 = 40 GPa = 40(103) MPa, E2 = 10 GPa, G12 = 3.5 GPa, and ν12 =0.25. By the use of a strain gage, the normal strain along the fiber direction is determined to be ε1 =0.001. Determine the inter-nal pressure in the vessel.
SOLUTION
From mechanics of materials, the stresses in a thin-walled cylindrical pressure vessel are given by
σ
These equations are based on static equilibrium and geometry only. Thus, they apply to a vessel made of any material. Since the given strain is along the fiber
y
x θ
1
2
FIGURE 2.18
Filament-wound.vessel.
direction, we must transform the above stresses to the 12 axes. Recall that in the
“netting analysis” in Problems 5 and 6 of Chapter 1, only the fiber longitudinal normal stress was considered. This was because the matrix was ignored, and the fibers alone cannot support transverse or shear stresses. In the current problem, however, the transverse normal stress, σ2, and the shear stress, τ12, are also consid-ered because the fiber and matrix are now assumed to act as a composite. From Equations 2.31, the stresses along the 12 axes are
σ1 σ 2θ σ 2θ τ θ θ
cos sin sin cos
( . )( . ) ( )( . ) 00 20 5
sin cos sin cos
( . )( . )
where the pressure p is in MPa. From the first of Equation 2.24, the normal strain ε1 is
. ε
and the resulting pressure is p = 2.46 MPa.
Example 2.5
A tensile test specimen is cut out along the x direction of the pressure vessel described in Example 2.4. What effective modulus of elasticity would you expect to get during a test of this specimen?
SOLUTION
The off-axis modulus of elasticity, Ex, associated with the x direction is given by the first of equations (2.40) with θ = 53.1°.
E E c E G c s E s
Example 2.6
A lamina consisting of continuous fibers randomly oriented in the plane of the lamina is said to be “planar isotropic,” and the elastic properties in the plane are
84 Principles of Composite Material Mechanics
isotropic in nature. Find expressions for the lamina stiffnesses for a planar isotropic lamina.
SOLUTION
Since the fibers are assumed to be randomly oriented in the plane, the “planar isotropic stiffnesses” can be found by averaging the transformed lamina stiffnesses as follows:
Q Q
where the superscript (~) denotes an averaged property.
It is convenient to use the invariant forms of the transformed lamina stiffnesses because they are easily integrated. For example, if we substitute the first of Equation 2.44 in the above equation, we get
Q Q U U U lamina stiffness, and that the orthotropic parts drop out in the averaging process.
Similarly, the other averaged stiffnesses can be found in terms of the invariants.
The derivations of the remaining expressions are left as an exercise.
PROBLEMS
. 1.. A. representative. section. from. a. composite. lamina. is. shown. in.
Figure. 2.19. along. with. the. transverse. stress. and. strain. distribu-
. 2.. Derive.the.first.of.Equation.2.40.for.the.off-axis.modulus,.Ex. . 3.. Derive. the. third. of. Equation. 2.40. for. the. off-axis. shear.
modulus, Gxy.
. 4.. Using.the.result.from.Problem.3:
. a.. Find.the.value.of.the.angle.θ.(other.than.0°.or.90°).where.the.
curve. of. Gxy. vs..θ. has. a. possible. maximum,. minimum,. or.
inflection.point.
. b.. For.the.value.of.θ.found.in.part.(a),.find.the.bounds.on.G12.which.
must.be.satisfied.if.Gxy.is.to.have.a.maximum.or.minimum.
. c.. Qualitatively.sketch.the.variation.of.Gxy.vs..θ.for.the.different.
cases.and.identify.each.curve.by.the.corresponding.bounds.on.
G12,.which.give.that.curve.
. d.. Using.the.bounds.on.G12.from.part.(b),.find.which.conditions.
apply.for.E-glass/epoxy.composites..The.bounds.on.G12.in.part.
(b).should.be.expressed.in.terms.of.E1,.E2,.and.ν12.
. 5.. Describe.a.series.of.tensile.tests.that.could.be.used.to.measure.the.
four.independent.engineering.constants.for.an.orthotropic.lamina.
without.using.a.pure.shear.test..Give.the.necessary.equations.for.
the.data.reduction.
. 6.. A.balanced.orthotropic,.or.square.symmetric.lamina,.is.made.up.
of.0°.and.90°.fibers.woven.into.a.fabric.and.bonded.together,.as.
shown.in.Figure.2.9.
. a.. Describe.the.stress–strain.relationships.for.such.a.lamina.in.
terms.of.the.appropriate.engineering.constants.
. b.. For.a.typical.glass/epoxy.composite.lamina.of.this.type,.sketch.
the.expected.variations.of.all.the.engineering.constants.for.the.
lamina.from.0°.to.90°..Numerical.values.are.not.required.
. 7.. An. element. of. a. balanced. orthotropic. carbon/epoxy. lamina. is.
under.the.state.of.stress.shown.in.Figure.2.20..If.the.properties.of.
the.woven.carbon.fabric/epoxy.material.are.E1.=.70.GPa,.ν12.=.0.25,.
G12.=.5.GPa,.determine.all.the.strains.along.the.fiber.directions.
. 8.. Derive.Equation.2.27.
. 9.. Express.the.stress–strain.relationships.in.Equation.2.37.in.terms.
of.off-axis.engineering.constants.such.as.the.moduli.of.elasticity,.
shear.modulus,.Poisson’s.ratios,.and.shear-coupling.ratios.
. 10.. Derive.the.first.two.equations.of.Equations.2.44.
. 11.. Find.all.components.of.the.stiffness.and.compliance.matrices.for.a.
specially.orthotropic.lamina.made.of.AS/3501.carbon/epoxy.
. 12.. Using.the.results.of.Problem.11,.determine.the.invariants.Ui.and.Vi. for.the.AS/3501.lamina,.where.i.=.1,.2,.3,.4.
. 13.. Using.the.results.of.Problem.11.or.Problem.12,.compare.the.trans-formed. lamina. stiffnesses. for. AS/3501. carbon/epoxy. plies. ori-ented.at.+45°.and.45°.
10
0.1 mm
0.1 mm
Stress, σ2 (MPa) Strain, ε2 0.0015 0.001
1 mm 20
σ2 σ2
x3 x3
FIGURE 2.19
Transverse.stress.and.strain.distribution.over.a.section.of.lamina.
86 Principles of Composite Material Mechanics
. 14.. Show.how.the.Mohr’s.circles.in.Figure.2.17.can.be.used.to.interpret.
the.transformed.lamina.stiffness.Q12.
. 15.. Using.the.approach.described.in.Example.2.5,.derive.the.expres-sions.for.all.the.averaged.stiffnesses.for.the.planar.isotropic.lamina.
in.terms.of.invariants..Use.these.results.to.find.the.corresponding.
averaged.engineering.constants.(modulus.of.elasticity,.shear.mod-ulus,.and.Poisson’s.ratio).in.terms.of.invariants.
. 16.. For. a. specially. orthotropic,. transversely. isotropic. material. the.
“plane.strain.bulk.modulus,”.K23,.is.an.engineering.constant.that.
is.defined.by.the.stress.conditions.σ2.=.σ3 .=.σ.and.the.strain.condi-tions.ε1.=.0,.ε2.=.ε3.=.ε..Show.that.these.conditions.lead.to.the.stress–
strain.relationship.σ.=.2K23.ε,.and.find.the.relationship.among.K23,.
E1,.E2,.G23,.and.ν12.
. 17.. Describe.the.measurements.that.would.have.to.be.taken.and.the.
equations.that.would.have.to.be.used.to.determine.G23,.ν32,.and.E2. for.a.specially.orthotropic,.transversely.isotropic.material.from.a.
single.tensile.test.
. 18.. A.45°.off-axis.tensile.test.specimen.has.three.strain.gages.attached..
Two.of.the.gages.are.mounted.as.shown.in.Figure.2.15.so.as.to.mea-sure.the.normal.strains.εx.and.εy,.and.a.third.gage.is.mounted.at.
θ.=.45°.so.as.to.measure.the.normal.strain.ε1..If.the.applied.stress.
σx.=.100.MPa.and.the.measured.strains.are.εx.=.0.00647,.εy.=.−0.00324.
and.ε1.=.0.00809,.determine.the.off-axis.modulus.of.elasticity.Ex,.the.
off-axis.major.Poisson’s.ratio.νxy.and.the.shear.coupling.ratio.ηx,xy. . 19.. A. off-axis. tensile. test. (Figure. 2.13). of. a. unidirectional. AS/3501.
carbon/epoxy.specimen.is.conducted.with.θ.=.45°.and.the.applied.
stress. is. found. to. be.σx.=.15.44. MPa.. Determine. the. resulting.
strain.εx.
100 MPa 50 MPa
50 MPa
2
1 30°
y
x FIGURE 2.20
Stresses.acting.on.an.element.of.balanced.orthotropic.lamina.
. 20.. An. element. of. an. orthotropic. lamina. is. subjected. to. an. off-axis.
shear. stress.τxy. at. an. angle.θ. as. shown. in. Figure. 2.21.. (a). for. an.
angle.θ.=.45°,.determine.the.value.of.the.applied.shear.stress.τxy. that. would. generate. the. following. stresses. along. the. 1,2. axes:.
σ1.=.1000.MPa,.σ2.=.−1000.MPa,.τ12 .=.0..(b).Assuming.that.the.lam-ina.described.in.part.(a).is.made.from.T300/934.carbon/epoxy.and.
that.the.stresses.are.also.given.in.part.(a),.determine.the.strains.
along.the.1,2.axes.
. 21.. Use.invariants.to.find.the.optimum.lamina.orientation.for.maxi-mizing.the.shear.stiffness.Q66 ,.then.find.the.corresponding.maxi-mum.shear.stiffness.in.terms.of.invariants.
References
. 1.. Christensen,.R..M..1979..Mechanics of Composite Materials..John.Wiley.&.Sons,.
New.York,.NY.
. 2.. Crandall,. S.. H.,. Dahl,. N.. C.,. and. Lardner,. T.. J.. 1978.. An Introduction to the Mechanics of Solids,.2nd.ed..with.SI.units..McGraw-Hill,.Inc.,.New.York,.NY.
. 3.. Sokolnikoff,.I..S..1956..Mathematical Theory of Elasticity..McGraw-Hill,.Inc.,.New.
York,.NY.
. 4.. Ashton,.J..E.,.Halpin,.J..C.,.and.Petit,.P..H..1969..Primer on Composite Materials:
Analysis..Technomic.Publishing.Co.,.Lancaster,.PA.
. 5.. Halpin,. J..C..1984..Primer on Composite Materials: Analysis,.rev.. ed.. Technomic.
Publishing.Co.,.Lancaster,.PA.
. 6.. Jones,.R..M..1999..Mechanics of Composite Materials,..2nd.ed..Taylor.and.Francis,.
Inc.,.Philadelphia,.PA.
. 7.. Vinson,.J..R..and.Sierakowski,.R..L..1986..The Behavior of Structures Composed of Composite Materials..Martinus.Nijhoff.Publishers,.Dordrecht,.The.Netherlands.
. 8.. Adams,.D..F.,.Carlsson,.L..A,.and.Pipes,.R..B..2003..Experimental Characterization of Advanced Composite Materials,.3rd.ed.,.CRC.Press,.Boca.Raton,.FL.
. 9.. Sun,.C..T..1998..Mechanics of Aircraft Structures..John.Wiley.&.Sons,.New.York,.NY.
1
τxy (MPa)
τxy (MPa)
x 2
θ y
FIGURE 2.21
Description.of.off-axis.lamina.and.stresses.for.Problem.20.
88 Principles of Composite Material Mechanics
. 10.. Lekhnitski,.S..G..1981..Theory of Elasticity of an Anisotropic Body..Mir.Publishing.
Co.,.Moscow,.USSR.
. 11.. Tsai,.S..W..and.Hahn,.H..T..1980..Introduction to Composite Materials..Technomic.
Publishing.Co.,.Lancaster,.PA.
. 12.. Tsai,.S..W..and.Pagano,.N..J..1968..Invariant.properties.of.composite.materials,.
in.Tsai,.S..W.,.Halpin,.J..C.,.and.Pagano.N..J..eds.,.Composite Materials Workshop,.
pp..233–253..Technomic.Publishing.Co.,.Lancaster,.PA.
89