Analysis of Lamina Hygrothermal Behavior

“Hygrothermal” – combined effects of temperature and moisture

Effects of changes in hygrothermal environment on mechanical behavior of

polymer composites:

1. Hygrothermal deformations change the stress and strain distributions.

2. Matrix – dominated properties under transverse, off-axis and shear loading are altered, particularly in the glass transition region.

Variation of stiffness with temperature for a typical polymer showing the glass transition temperature Tg and the effects of

absorbed moisture on Tg

wet

Dry

Glassy Region

Increasing Moisture Content

Rubbery Region

Temperature

Sti

ffne

ss

Tgw TgoTypical operating temperature below Tg

for structural polymers

Glass Transition Region

Operating temperature ranges for different types of polymers

Stiffness

TemperatureTg

Glassystructuralpolymers

Rubberypolymers(elastomers)

Arrow indicates hot gasses leaking from O-ring in joint on solid rocket booster

Practical example of the importance of the glass transition temperature, Tg, of polymers

1986 Space Shuttle Challenger explosion

Space shuttle system

Joints in solidrocket boostertank

Joint in Challenger solid rocket booster showing the problem O-rings

From http://en.wikipedia.org/wiki/O-ring#Challenger_disaster

The failure of an O-ring seal was determined to be the cause of the Space Shuttle Challenger disaster on January 28, 1986. A contributing factor was cold weather prior to the launch. This was famously demonstrated on television by Caltech physics professor Richard Feynman, when he placed a small O-ring into ice-cold water, and subsequently showed its loss of pliability before an investigative committee. O-rings are now examined under high-power video microscopes for defects. The material of the failed O-ring was FKM (a fluoroelastomer) which was specified by the shuttle motor contractor, Morton-Thiokol. FKM is not a good material for cold temperature applications. When an O-ring is cooled below its Tg (glass transition temperature), it loses its elasticity and becomes brittle. More importantly, when an O-ring is cooled near, but not beyond, its Tg, the cold O-ring, once compressed, will take longer than normal to return to its original shape. O-rings (and all other seals) work by creating positive pressure against a surface thereby preventing leaks. On the night before the launch, exceedingly low air temperatures were recorded. On account of this, NASA technicians performed an inspection. The ambient temperature was within launch parameters, and the launch sequence was allowed to proceed. However, the temperature of the rubber O-rings remained significantly lower than that of the surrounding air. During his investigation of the launch footage, Dr. Feynman observed a small out-gassing event from the Solid Rocket Booster (SRB) at the joint between two segments in the moments immediately preceding the explosion. This was blamed on a failed O-ring seal. The escaping high temperature gas impinged upon the external tank, and the entire vehicle was destroyed as a result.

Challenger O-ring stiffness vs. temperature

O-ringStiffness

TemperatureTg

Desired O-ring temperature is above Tg so that material is pliable enough to fill gaps and form seal proper seal against metal rocket motor case asit expanded under pressure

Actual O-ring temperature at launch was below Tg and material was too stiff to fill gaps and form proper seal against metal motor case as it expanded under pressure

Dr. Feynman's famous C-clamp experimentduring meeting of the Presidential Commission

investigating the Challenger disaster

Conclusions

• When designing with polymers, knowledge of the glass transition temperature , Tg, and the operating temperature range are essential

• Structural polymers should operate below Tg, and rubbery polymers (elastomers) should operate above Tg

• The operating temperature should not be in or near the glass transition region, as stiffness and strength are very sensitive to temperature in this region

“Free Volume” concept used to interpret glass transition

Total sample volume fo VVV

where Vo = volume occupied by molecules

Vf = free volume due to space between molecules

Temperature Tg

Vol

ume

Vf

V

Vo

Glassy region

Rubbery region

“Free Volume” concept used to interpret glass transition

Material Supplier

Saturation

MoistureContent, Mm

(Weight %)

Tgo (Dry)

[°F (°C)]

Tgw (Wet)

[°F (°C)]

Maximum

Service

Temperature

(Dry) [°F (°C)] 

Hexply® F655 bismaleimide Hexcel 4.1 550(288) 400(204) 400(204)

Hexply® 8551-7

epoxy

Hexcel 3.1 315(157) 240(116) 200(93)

Hexply® 8552 epoxy Hexcel — 392(200) 309(154) 250(121)

Hexply® 954-3A cyanate Hexcel — 400(204) — —

CyCom® 2237 polyimide Cytec 4.4 640(338) 509(265) 550(208)

CyCom® 934

epoxy

Cytec — 381(194) 320(160) 350(177)

Avimid® R polyimide Cytec 2.8 581(305) 487(253) 550(288)

Derakane®

411-350 vinylester

Ashland 1.5 250(120) — 220(105)

Ultem® 2300

polyetherimide

Sabic 0.9 419(215) — 340(171)

Victrex® 150G

polyetherether-ketone

Cetex® polyphenylene

sulfide

Victrex plc

 

Tencate

0.5

 

 

0.02

289(143)

 

 

194(90)

—

 

 

—

 

356(180)

 

 

212(100)

Hygrothermal Properties for Various Polymer Matrix Materials

Stress–strain curves for 3501–5 epoxy resin at different temperatures and moisture contents. (From Browning, C.E., Husman, G.E., and Whitney, J.M. 1977. Composite Materials: Testing and Design: Fourth Conference, ASTM STP 617, pp. 481–496. American Society for Testing and Materials, Philadelphia, PA. Copyright, ASTM. Reprinted with permission.)

Stress–strain curves for AS/3501–5 graphite/epoxy composite under transverse loading at different temperatures and moisture contents. (From Browning, C.E., Husman, G.E., and Whitney, J.M. 1977. Composite Materials: Testing and Design: Fourth Conference, ASTM STP 617, pp. 481–496. American Society for Testing and Materials, Philadelphia, PA. Copyright, ASTM. Reprinted with permission.)

Percent weight gain due to moisture pickup vs. soaking time for several E-glass/polyester sheet-molding compounds. Materials described in table 5.2. (From Gibson, R.F., Yau, A., Mende, E.W., and Osborn, W.E. 1982. Journal of Reinforced Plastics and Composites, 1(3), 225–241. Reprinted by permission of Technomic Publishing Co.)

Variation of flexural modulus of several E-glass/polyester sheet-molding compounds with soaking time in distilled water at 21 to 24°C. (From Gibson, R.F., Yau, A., Mende, E.W., and Osborn, W.E. 1982. Journal of Reinforced Plastics and Composites, 1(3), 225–241. Reprinted by permission of Technomic Publishing Co.)

Description of Composite Materials for Figures 5.4 and 5.5 from Gibson, et al [1982]

MaterialWeight of Percentage of constituents

Chopped E-glass Fibers

Continuous E-glass Fibers

Polyester Resin, Fillers, etc.

PPG SMC-R251 25 0 75

PPG SMC-R65 65 0 35

PPG XMC-3 2550 ( x-

pattern)25

OCF SMC-R252 25 0 75

OCF C20/R30 30 20 (Aligned) 50

1Manufactured by PPG Industries, Fiber Glass Division, Pittsburgh, PA 15222.

2Manufactured by Owens-Corning Fiberglas Corporation, Toledo, OH 43659.

,5.7

Schematic representation of temperature and moisture distributions through the thickness of a plate which is exposed to an environment of temperature Ta and

moisture concentration Ca on both sides.

Plate

h

z

z

Thickness

Ta

Ca

T

C

Ambient Temperature Ta and Ambient Moisture Concentration Ca

Distribution of Temperature One dimensional case

• Fourier Heat Conduction Equation:

where = density of material C = specific heat of material

Kz = thermal conductivity of material along z direction

T = temperature t = time

z

TK

zt

TC z

(5.1)

• Fick’s Second Law of Diffusion:

where c = moisture concentration

Dz = mass diffusivity along z direction

Equations solved subject to initial and boundary conditions

Distribution of Moisture One dimensional case

z

cD

zt

cz

0,0

thzcc

TT

i

i

0,,0

thzzcc

TT

a

a

(5.2)

• For typical composites, the values of C, Kz, Dz and are such that T approaches equilibrium ~ 106 times faster than C, so temperature can usually be assumed equal to ambient temperature Ta. Moisture concentration requires further analysis,

However,if diffusivity is constant, Fick’s second law is

Distribution of Temperature and Moisture – one dimensional case

2

2

z

cD

t

cz

(5.3)

Solution

02

22)12(exp

)12(sin

12

141

j

z

im

i

h

tDj

h

zj

jcc

cc

where cm = moisture concentration at surface Equation (5.4) gives local concentration c(z,t), but we normally measure total amount of moisture averaged over sample. Average concentration is

h

im ccdztzch

c0

)(),(1

i

j

z cj

htDj

02

222

2 12

)12(exp81

(5.4)

(5.5)

Weight percent moisture M, is what we actually measure, and since c is linearly related to M, we can write,

0

2

2

22

2 )12(

)12(exp

81

j

z

im

i

i

tDh

j

MM

MMG

(5.6)

Where M = average weight percent of moisture at time t

Mi = initial average weight percent moisture

Mm = average weight percent moisture at fully saturated equilibrium condition Series converges rapidly.

Predicted moisture profiles through the thickness of a graphite/epoxy plate after drying out for various periods of time. From Browning, C.E., Husman, G.E., and Whitney, J.M. 1977. Composite Materials: Testing and Design: Fourth Conference, ASTM STP 617, pp. 481–496. American Society for Testing and Materials, Philadelphia, PA. Copyright, ASTM. With permission.)

Comparison of predicted (eq. [5.6]) and measured moisture absorption and desorption of T300/1034 graphite/epoxy composites. Open symbols represent measured absorption and dark symbols represent measured desorption. (From Shen, C.H. and Springer, G.S. 1976. Journal of Composite Materials, 10, 2–20. Reprinted by permission of Technomic Publishing Co.)

Hygrothermal Degradation of Strength or Stiffness

Chamis Equation (empirical)

21

ogo

gw

om TT

TT

P

PF (5.7)

Where Fm = matrix mechanical property retention ratio P = degraded property of matrix Po = reference undegraded property of matrix

T = temperature at which P is predicted Tgo = glass transition temperature for reference condition (dry) Tgw = glass transition temperature for wet matrix material at moisture content corresponding to property P To = test temperature at which Po was measured

Empirical equation for Tgw:

where Mr = percent moisture in matrix by weight

Procedure: Use equations (5.8) and (5.7) to find degraded matrix property, than use degraded property in appropriate micromechanics equation to predict degraded composite property.

gorrgw TMMT 0.110.0005.0 2 (5.8)

Example: longitudinal modulus

mmomff vEFvEE 11(5.9)

Where Emo = reference value of matrix modulus in dry condition

Caution: these are empirical equations

Variation of glass transition temperature with equilibrium moisture content for several epoxy resins. (From DeIasi, R. and Whiteside, J.B. 1987. In Vinson, J.R. ed., Advanced Composite Materials — Environmental Effects, ASTM STP 658, pp. 2–20. American Society for Testing and Materials, Philadelphia, PA. Copyright, ASTM. Reprinted with permission.)

Comparison of predicted (eq. [5.7]) and measured strengths of several hygrothermally degraded composites. (From Chamis, C.C. and Sinclair, J.H. 1982. In Daniel, I.M. ed., Composite Materials: Testing and Design (Sixth Conference), ASTM STP 787, pp. 498–512. American Society for Testing and Materials, Philadelphia, PA. Copyright, ASTM. With permission.)

Effect of temperature on rate of moisture absorption in AS/3501–5 graphite/epoxy composite. (From DeIasi, R. and Whiteside, J.B. 1987. In Vinson, J.R. ed., Advanced Composite Materials — Environmental Effects, ASTM STP 658, pp. 2–20. American Society for Testing and Materials, Philadelphia, PA. Copyright, ASTM. Reprinted with permission.)

Diffusion is a thermally activated process.

Arrhenius Relationship for diffusivity:

Where Do = material constant

Ea = activation energy

R = universal gas constant

T = absolute temperature

log D Vs should be a straight line – see Fig (5.11)

RTEDD ao exp (5.10)

T

1

Variation of transverse diffusivity with temperature for AS/3501-5 graphite/epoxy composite. (From Loos, A.C. and Springer, G.S. 1981. In Springer, G.S. ed., Environmental Effects on Composite Materials, pp. 34–50. Technomic Publishing Co., Lancaster, PA. Reprinted by permission of Technomic Publishing Co.)

Effects of applied stress on moisture diffusion in polymers and polymer

composites

Diffusivity D increases under tensile stress and decreases under compressive stress

So, preferred diffusion path is through a tensile stress-field.

Stress-Strain Relationships Including Hygrothermal Effects

Thermal strains – isotropic material

0

TTi

if i = 1, 2, 3

if i = 4, 5, 6(5.11)

Where i = 1, 2, 3, 4, 5, 6 (contracted notation) T = temperature change = T – To

T = final temperature To = initial temperature where i

T = 0 for all i = coefficient of thermal expansion (CTE)

• Assumption of linearity and constant is valid over sufficiently narrow T

• No shear distortion – uniform expansion or contraction

Stress-Strain Relationships Including Hygrothermal Effects

Thermal expansion vs. temperature for 3501-6 epoxy resin. (From Cairns, D.S. and Adams, D.F. 1984. In Springer, G.S. ed., Environmental Effects on Composite Materials, vol. 2, pp. 300–316. Technomic Publishing Co., Lancaster, PA. Reprinted by permission of Technomic Publishing Co.)

Hygroscopic Strains for Isotropic Material

0

cMi

if i = 1, 2, 3

if i = 4, 5, 6(5.12)

Where i = 1, 2, 3, 4, 5, 6 (contracted notation) c = moisture concentration

= mass of moisture in unit volume/ mass of dry material in unit volume = coefficient of hygroscopic expansion (CHE)

Hygroscopic Strains forIsotropic Material

• Reference condition c = 0, iM = 0

• Linearity and constant valid if range of c is not too wide

Total hygrothermal strains – isotropic case

Mi

Ti

Hi (5.13)

Hygroscopic expansion vs. moisture content for two epoxy resins. (From Delasi, R. and Whiteside, J.B. 1987. In Vinson, J.R. ed., Advanced Composite Materials — Environmental Effects, ASTM STP 658, pp. 2–20. American Society for Testing and Materials, Philadelphia, PA. Copyright, ASTM. Reprinted with permission.)

• Composite lamina – orthotropic in both mechanical and hygrothermal properties due to difference in fiber and matrix properties (see tables 3.1 and 3.2)

• Subscript needed on and

Total hygrothermal strains for specially orthotropic lamina:

If transversely isotropic, 2 = 3, 2 = 3

0

cT iiHi

if i = 1, 2, 3

if i = 4, 5, 6(5.14)

Variation of measured longitudinal and transverse thermal strains for unidirectional Kevlar 49/epoxy and S-glass/epoxy with temperature. (From Adams, D.F., Carlsson, L.A., and Pipes, R.B., 2003. Experimental Characterization of Advanced Composite Materials. CRC Press, Boca Raton, FL. With permission.)

Typical Thermal and Hygroscopic Expansion Properties. From Graves, S.R. and Adams, D.F. 1981. Journal of Composite

Materials, 15, 211–224. With permission..

Material

Thermal Expansion coefficients

(10-6 m/m)/C

Hygroscopic Expansion Coefficients

(m/m)

1 2 1 2

AS graphite/epoxy

0.88 31.0 0.09 0.30

E-glass/epoxy 6.3 20.0 0.014 0.29

AF-126-2 adhesive

29.0 29.0 0.20 0.20

1020 steel 12.0 12.0 --- ---

Total strains (mechanical + hygrothermal) for specially orthotropic lamina

cT

S

SS

SS

0000

0

0

2

1

2

1

12

2

1

66

2221

1211

12

2

1

(5.15)

or in matrix notation,

cTS (5.16)

so, stresses are

cTS 1 (5.17)

No shear coupling in mechanical or hygrothermal behavior for specially orthotropic case ( not so for generally orthotropic case as shown later.)

If material is unrestrained during hygrothermal exposure, { } = 0 and the strains are

cT (5.18)

If material is completely restrained during hygrothermal exposure, {} = 0 and the stresses are found from cTS 0 (5.19)

or cTS 1

cTQ since 1 SQ

Total Strains (mechanical + hygrothermal) for generally orthotropic lamina

cT

SSS

SSS

SSS

xy

y

x

xy

y

x

xy

y

x

xy

y

x

662616

262212

161211

(5.21)

(shear coupling present)CTEs and CHEs must transform like tensor strains, not engineering strains.

0

][

2/2

11

T

xy

y

x

(5.22)

And similarly for the CHEs

Variation of lamina thermal expansion coefficients with lamina orientation for a lamina having 2 > 1> 0.

Micromechanics Models for Hygrothermal Properties

Example: Longitudinal CTE, 1

Average composite longitudinal strain is

TE

11

11 (5.23)

stress TE 1111 (5.24)

Similarly, for fiber TE ffff 1111

and for matrix TE mmmm 1111

Recall Eq. (3.23), “rule of mixtures” for longitudinal stress

mmff vv 111 or

mmmmffff vTEvTETE 111111111 (5.25)

Recall Eq. (3.26)

111 mf (3.25)

And Eq. (3.27)

mmff vEvEE 11(3.26)

mmff

mmmfff

vEvE

vEvE

11

11111

For isotropic constituents:

mmff

mmmfff

vEvE

vEvE

1

(5.26)

(5.27)

Combining Eqs. 5.25, 3.26 and 3.27, and solving for 1:

1

11111 E

vEvE mmmfff

(5.26)

Variation of predicted longitudinal and transverse coefficients of thermal expansion with fiber-volume fraction for typical unidirectional graphite/epoxy composite. (From Rosen, B.W. 1987. In Reinhart, T.J. ed., Engineered Materials Handbook, vol. 1, Composites, Sec. 4. ASM International, Materials Park, OH. Reprinted by permission of ASM International.)

Micromechanics Models for Hygrothermal Properties

Example: Transverse CTE, 2

Average composite transverse strain is

22 2

2

cc T

E

(5.28)

Substituting equations similar to equation (5.28) for composite, fiber and matrix, respectively, in the transverse geometric compatibility condition given by equation (3.37), the result is

22 22 2 2

2 2 2

fc mf f m m

f m

T T v T vE E E

(5.29)

Combining equation (5.29) with equation (3.39) and assuming that the stresses in the composite, fiber and matrix are all equal we get another rule of mixtures type equation

2 2 2f f m mv v (5.30)

For the case of isotropic fiber and matrix, this reduces to

2 f f m mv v (5.31)

However, due to the assumption of equal stresses incomposite, fiber and matrix, Equations (5.30) and (5.31)are only rough estimates

Finite element analysis (FEA) micromechanical models for CTEs

Typical FEA unit cell for prediction of composite longitudinal and transverse CTEs. (From Karadeniz, Z., and Kumlutas, D. 2007. Composite Structures, 78, 1-10. Reprinted with permission of Elsevier.

FEA unit cells similar to this are subjected to a temperature change, , then the resulting FEA-calculated thermal displacements are used to determine the CTEs

T

Examples: FEA-calculated longitudinal CTE

10

1XX

l

l T

20 0

1 1Z YZ Y

l l

l T l T

FEA-calculated transverse CTE

(5.33)

(5.34)

Comparison of analytical and FEA numerical predictions of longitudinal and transverse CTEs for SiO2/epoxy composites of various SiO2 volume fractions.From Gibson and Muller [27].

Longitudinal CHE,

mmff

mmmfff

vEvE

vEvE

11

11111

(5.35)

Longitudinal transport Properties Thermal Conductivity:

mmff vKvKK 1(5.39)

Diffusivity:mmff vDvDD 1

(5.40)

Transverse Transport Properties K2, D2, etc. can be found by using Halpin – Tsai eqs.

Hygrothermal Degradation of , , K and c

Effect of temperature on these properties is opposite from effect on mechanical properties. Chamis empirical equation is,

Where

Fh = Matrix hygrothermal property retention ratio R = Matrix hygrothermal property after hygrothermal degradation Ro = Reference matrix hygrothermal property before

degradation

Degraded matrix property is used in micromechanics equations to estimate degraded composite property.

21

TT

TT

R

RF

gw

ogo

oh

(5.41)