Concluding Remarks

Chapter 2 Nonlinearly Viscoelastic Response of Polycarbonate under Pure Shear

2.7 Concluding Remarks

The study of the time dependent constitutive behavior of polycarbonate in the nonlinear range poses several serious problems, not the least of which is the proper prescription of the boundary loads so as to assure a firm knowledge of the stress state in

the reglOn where the deformations are measured. Such determinations invariably are burdened with an iterative process that cycles between experiment and analysis subject to the expectation of eventual convergence. The determination of the linearly thermo- viscoelastic shear response is systematically free of difficulties and determines this material with considerable precision to be a thermo-rheologically simple material, except that the time-temperature shift function experiences an apparent break at the ,B-transition of70

Dc.

In contrast, passing to the investigation of nonlinear material behavior, this study draws on an initial measurement-analysis iteration to estimate the nonlinear response under pure shear stresses. Starting with simple, and then linearized, analysis the experimentally determined yield-like behavior of the material has been examined as a function of time and temperature. Creep data in the nonlinearly viscoelastic range was acquired by means of Arcan specimens and digital image correlation. It is demonstrated that nonlinearity starts to enter the constitutive law at about 1 % strain for all temperatures. Hence any engineering design involving shear strains of 1 % and beyond needs the consideration of nonlinear viscoelasticity. A consequential re-evaluation of the imposed stress states to address differences between a linearized and nonlinear stress analysis of the test configuration is accomplished based on a quasi-plastic analysis that parallels the quasi-elastic analysis for linearly viscoelastic materials exhibiting logarithmically slowly varying creep or relaxation behavior. Corrections of the stresses associated with that iterative evaluation are on the order of 10% and reasonably close to the expected values to provide confidence in the evaluation of the measured data.

Although polymers, including polycarbonate, are often treated as plastically deforming solidso, there exists no well defined stress or strain similar to the yield-stress or strain for metals at which the material undergoes permanent set. While a seemingly permanent deformation set can be induced in glassy polymers, many if not most of such situations can be reversed through suitable degrees of heat addition, i.e., no measurable set remains. Rather, the stress at which large amounts of strain is accumulated -in comparison to linear or small strain behavior- is apparently a continuously changing function of stress, time and temperature. While the time required to achieve large strains may be impractically large at some stress levels on the order of decades of years, temperature accelerates this process, though not in an as yet closely or quantitatively predictable manner. As a means of interpreting the data acquired for design purposes, it appears thus useful to deal with this "yield-like" process as a function of temperature, under which conditions the large flow regime requires decreasing stress levels as the temperature increases. That there is a connection between this flow stress, the temperature and the time to achieve a given strain is supported by the observation that the analysis of the pertinent data seems to reflect a special transition (p-transition) temperature without introducing any particular reference to it.

References

2.1 Lu, H. and Knauss, W.O., "The Role of Dilatation in the Nonlinearly Viscoelastic Behavior of PMMA under Multiaxial Stress States," Mechanics of Time-Dependent Materials, 2, 307-334 (1999)

o It is an unfortunate twist of fate that "plastics" -short for thermo-plastics- derive their name from their thermal softening at elevated temperatures, and not from their rate-independent plastic flow characteristics as metals do.

2.2 McKinney, J.E. and Belcher, H.V., "Mechanical Properties of Toughened Epoxies," J. Res. Nat. Bur. Stand. A. Phys. Chem., 67 A, 43 (1963)

2.3 Deng, T.H. and Knauss, W.G., "The Temperature and Frequency Dependence of the Bulk Compliance of Poly (Vinyl Acetate). A Re-Examination,"

Mechanics of Time-Dependent Materials, 1,33-49 (1997)

2.4 Sane, S. and Knauss, W.G., "The Time-Dependent Bulk Response of Poly (Methyl Methacrylate) (PMMA)," in preparation

2.5 Bland, D.R, The Theory of Linear Viscoelasticity, Pergamon Press, New York (1960)

2.6 Gross, B., Mathematical Structure of the Theories of Viscoelasticity, Hermann, Paris (1968)

2.7 Christensen, RM., Theory of Viscoelasticity: An Introduction, Academic Press, New York (1971)

2.8 Flugge, W., Viscoelasticity, Springer-Verlag, New York (1975)

2.9 Ferry, J.D., Viscoelastic Properties of Polymers, 3^rd edition, John Wiley &

Sons, New York (1980)

2.10 Leaderman, H., Elastic and Creep Properties of Filamentous Materials and Other High Polymers, the Textile Foundation, Washington, D.C. (1943)

2.11 Tobolsky, A. and Eyring, H., "Mechanical Properties of Polymeric Materials,"

J. Chem. Phys., 11, 125 (1943)

2.12 Williams, M.L., Lande!, RF. and Ferry, J.D., J. Am. Chem. Soc., 77, 3701 (1955)

2.13 Green, AE. and Rivlin, RS., "The Mechanics of Non-Linear Materials with Memory, Part I," Arch. Ration. Mech. Anal., 1, 1-21 (1957)

2.14 Bernstein, B., Kearsley, E.A and Zapas, L.J., "A Study of Stress Relaxation with Finite Strain," Trans. Soc. Rheol., 7,391-410 (1963)

2.15 Schapery, RA, "An Engineering Theory of Nonlinear Viscoelasticity with Applications," Int. J. Solids Struct., 2, 407-425 (1969)

2.16 Hasan, O.A and Boyce, M.e., "A Constitutive Model for the Nonlinear Viscoelastic Viscoplastic Behavior of Glassy Polymers," Polym. Eng. Sci., 35, 331-344 (1995)

2.17 Lustig, S.R, Shay, RM. and Caruthers, J.M., "Thermodynamic Constitutive Equations for Materials with Memory on a Material Time Scale," J. Rheol., 40, 69-106 (1996)

2.18 Knauss, W.G. and Emri, I., "Non-Linear Viscoelasticity Based on Free Volume Consideration," Comput. Struct., 13, 123-128 (1981)

2.19 Doolittle, AK., "Studies on Newtonian Flow. II. The Dependence of the Viscosity of Liquids on Free-Space," J. Appl. Phys., 22,1471-1475 (1951) 2.20 Cohen, M.H. and Turnbull, D., "Molecular Transport in Liquids and Glasses,"

J. Chem. Phys., 31, 1164 (1959)

2.21 Knauss, W.G. and Kenner, V.H., "On the Hygrothermomechanical Characterization of Polyvinyl Acetate," J. Appl. Phys., 51, 5131-5136 (1980) 2.22 Moonan, W.K. and Tschoegl, N.W., "Effects of Pressure on the Mechanical

Properties of Polymers 2. Expansivity and Compressibility Measurements,"

Macromolecules, 16,55 (1983)

2.23 Struik, L.C.E., Physical Aging in Amorphous Polymers and Other Materials, Elsevier Scientific Publishing Company, Amsterdam (1978)

2.24 Lu, H., "Nonlinear Thermo-Mechanical Behavior of Polymers under Multiaxial Loading," Ph.D. Thesis, California Institute of Technology, Pasadena (1997) 2.25 Adams, G. and Gibbs, J.H., "On the Temperature Dependence of Cooperative

Relaxation Properties in Glass-Forming Liquids," J. Chern. Phys., 43, 139-144 ( 1965)

2.26 O'Connell, P.A. and McKenna, G.B., "Large Deformation Response of Polycarbonate: Time-Temperature, Time-Aging Time and Time-Strain Superposition," Polym. Eng. Sci. 37, 1485 (1997)

2.27 Veazie, D.R. and Gates, T.S., "Compressive Creep of IM7/K3B Composite and the Effect of Physical Aging on Viscoelastic Behavior," Exp. Mech., 37, 62 (1997)

2.28 Plazek, D.J. and Ngai, K.L., "The Glass Transition," Physical Properties of Polymers Handbook (edited by Mark, J.E.), AlP Press, Woodbury, New York (1996)

2.29 Washabaugh, P.D. and Knauss, W.G., "A Reconciliation of Dynamic Crack Velocity and Rayleigh-Wave Speed in Isotropic Brittle Solids," Int. 1. Fract., 65,97-114 (1994)

2.30 Saunders, KJ., Organic Polymer Chemistry: An Introduction to the Organic Chemistry of Adhesives, Fibers, Paints, Plastics, and Rubbers, Chapman &

Hall, London (1988)

2.31 Lee, S. and Knauss, W.G., "A Note on the Determination of Relaxational Creep Data from Ramp Tests," Mechanics of Time-Dependent Materials, 4, 1-7 (2000)

2.32 Kenner, V.H., Knauss, W.G. and Chai, H., "A Simple Creep Torsiometer and Its Use in the Thermorheological Characterization of the Structural Adhesive,"

Exp. Mech., 22, 75-82 (1982)

2.33 Schwarzl, F.R. and Zahradnik, F., "The Time Temperature Superposition of the Glass-Rubber Transition of Amorphous Polymers and the Free Volume,"

Rheol. Acta, 19, 137-152 (1980)

2.34 Fried, J .R., "Sub-T g Transitions," Physical Properties of Polymers Handbook (edited by Mark, J.E.), AlP Press, Woodbury, New York (1996)

2.35 Arcan, M., Hashin, Z. and Voloshin, A., "A Method to Produce Uniform Plane- Stress States with Applications to Fiber-reinforced Materials," Exp. Mech., 18,141-146 (1978)

2.36 Sutton, M.A., Wolters, W.J., Peters, W.H., Ranson, W.F. and McNeil, S.R.,

"Determination of Displacements U sing an Improved Digital Image Correlation Method," Image Vis. Comput., 1, 133-139 (1983)

2.37 Vendroux, G. and Knauss, W.G., "Sub micron Deformation Field Measurements: Part 2. Improved Digital Image Correlation," Exp. Mech. 38, 86-92 (1998)

2.38 Huang, Y., "Scanning Tunneling Microscopy and Digital Image Correlation in Nanomechanics Investigations," Ph.D. Thesis, California Institute of Technology, Pasadena (2001)

2.39 Gonzalez, J. and Knauss, W.G., "Strain Inhomogeneity and Discontinuous Crack Growth in a Particulate Composite," J. Mech. Phys. Solids, 46, 1981- 1995 (1998)

3 ,---,---.---~-c:-) -no-n---;~~Id~d pl~te, trace 1 I

x non-remolded plate, trace 2 i

E 5

2.5

1: 1.5 . iii Ol I

0.5

+

non-remolded plate, trace 3

• remolded plate, trace 1

• remol~ed plate, trace 2

-20 -10 o

Length (mm)

20 30

Fig. 2-1: Profile of non-remolded and remolded polycarbonate sheets (both after annealing). Shapes are very close to cylindrical; traces all follow parallel lines.

-3.25

x -3.3 u Q)

c

.~

C. E -3.35 u o

0..

Q)

U Q) -3.4

Ol o

--' -3.45

U specimen in direction 1 x specimen in direction 2

x

x

I

.

_3.5L--~--~--~---L--~---~-~

0.5 1.5 2 2.5 3 3.5 4 4.5

Log (Time/Sec.)

Fig. 2-2: Creep at 22°C under 38.6 MPa tensile stress for dogbone specimens cut along and orthogonal to traces in Fig. 2-1.

0.025

0.02

.~ 0.015

Vi ~

ro ClJ .c

(f) 0.01

0.005

() specimen 1, first test

x specimen 1, second test after annealing and aged for 2 days

+

specimen 1, third test after annealing and aged for 15 days

n specimen 1, fourth test after annealing and aged for 60 days

• specimen 2, first test ... specimen 3, first test

~ specimen 4, first test

O~~==~======~==================~~

0.5 1.5 2 2.5 3 3.5

Log (Time/Sec.)

4 4.5 5 5.5

Fig. 2-3: Experimental precision of the digital image correlation. Repeatability of strain measurements using 1) reconditioned specimen 1

2) different, fresh specimens 2, 3 and 4.

All under 19.4 MPa shear stress at 80°C. Error bar corresponds to a strain range of 0.2%.

I , 19.7

fillet ra dillS of 1 .6 for tra ns ition from s qua re to cylinder geometry" -',_

/ /

/ /

35.6

round-to-s qua re steel adaptors to accomodate tors iometer grips '''"

19.7

Fig. 2-4: Geometry of the solid cylindrical specimen (dimensions in mm).

-2.8~i ~'---'--'---'---'.

-2.82

m c..

-2.84

:z: -2.86 x

Cll ~ -2.88

t

OJ ^-2.9(1

^~

~

^-2.92

1! '

i-

^2.94

^r

-2.96-

1 '

-2.98,

J

_0.5

>1\8~~~~~ ^I

'lls(ll>e>f<>

'i1~~~~§§~~g ~e>

~ ~ ~~ ~~~~~~l~~~~~§~S~8

1.5 2 2.5 3 3.5

Log (Time/Sec.)

4 4.5 5 5.5

Fig. 2-5: Multiple measurements of torsional creep of the same specimen (annealed and physically aged after each test) under 6.88 MPa maximum shear stress at 80°C (Error bar corresponds to a range of 0.0001 for maximum shear strains).

0ic========i----~---~---~---~~

-0.5

-2.5

_3L---~---L---~---~---~--~

o 2 3 4 5

Log (Time/Sec.)

Fig. 2-6: Creep curves at different temperatures shifted for clarity by constants A along the ordinate; A increases by 0.05 with temperature starting from 0 at 0 DC (A

=

0.55 at 140 DC). Precision of strain measurement

=

0.0001 (smaller than the size of the symbols).

-1.6

-1.8

ro

~ -2 x u Q)

.~ -2.2

"is..

o E

c:

_OJ ^-24

.r: Q) (j)

';;-2.6

...J o -2.8

-3~--~~~---~---~---~---~

-10 -5 o 5

Log (Time/Sec.)

10 15

Fig. 2-7: Master curve derived from small strain creep data in Fig. 2-6 with 22°C as the reference temperature. The isolated solid point has been derived from stress wave propagation via an ultrasonic analyzer at 5 MHz.

5 ^I

-5 ^-"~

"'0.,

o

_10~----~--__ ^~ ____ ^{- L _ _ _ _ ~ _ _ _ _ - L _ __ _ - L _ __ _ ~}

o 20 40 60 80 100 120 140

Temperature FC)

Fig. 2-8: Shift factors as a function of temperature for producing Fig. 2-7 from Fig. 2-6.

The creep curves, superposed with the precision of strain measurement (0.0001), were shifted to deduce the error bars.

x 0

-2.7 r0-o..

~ x

Q)

~-2.75 .~ c..

E a

U

ro Q)

.r::.

~

Ol a

...J

-2.85

0

X 0.5

maximum shear stress 6.88 MPa maximum shear stress 2.75 MPa

1.5 2 2.5 3 3.5

Log (Time/Sec.)

X

4 4.5 5 5.5

Fig. 2-9: Linearity of shear responses at 110 DC for two different stress levels. Error bar corresponds to an error of 0.01 % strain.

-2.3

ro-

~ -2.4 x

Q) u

.~ -2.5

c.. E a ': -2.6

co .r::. Q)

~ 01-2.7

...J a -2.8

0.5

---,---,--c---~--__,___---... - .. -

xmaxim~m she~r stress 2.75 MPa I

o

maximum shear stress 1.38 MPa I

1.5 2 2.5 3 3.5

Log (Time/Sec.)

4 4.5 5 5.5

Fig. 2-10: Linearity of shear responses at 130 DC for two different stress levels. Error bar corresponds to an error of 0.01 % strain.

,

...

Test Section

.... 10.1~

I ,

I

;R=IO.16 typo

, 15.24 ~ ,

30.48

"<:t

----C'i V)

N

,

Fig. 2-11: Geometry of the Arcan specimen (dimensions ^III mm, thickness 3.00 mm).

Hatch marks indicate areas clamped (bonded) during test.

+5.310e-02 +4.884e-02 +4.460e-02 +4.036e-02 +3.612e-02 +3.188e-02 +2.764e-02 +2.340e-02 +1. 916e-02 +1.492e-02 +1.068e-02 +6.435e-03 +2.193e-03

Y,V

x,U h

Fig. 2-12: Shear strain field in an Arcan specimen for a linearly elastic solid under simple shear of EX).

=

^t::..u

=

0.05 parallel to the x-axis.

. 2h

+6.805e-02 +6.250e-02 +5.695e-02 +5.140e-02 +4.583e-02 +4.028e-02 +3.472e-02 +2.916e-02 +2.361e-02 +1. 805e-02 +1.250e-02 +6.935e-03 +1. 382e-03

Y,V

x,u h

Fig. 2-13: Shear strain field in an Arcan specImen for an ideally plastic solid under . 1 h f - /),.U

sImp e sear 0 C

= - =

0.05 parallel to the x-axis.

'Y 2h

0.08

0.07

0.06

OOOOO()()O()()OOOOO

.£: 0.05 xxxxxxxxxxxxx

x x

(j) 1::

.... 0.04 co Q)

.r: Vl

0.03

0.02 x x

0

o

0.01, r - - - ,

II x quasi-elastic

I

o~II=O=·==q=u=a~si=-=PII=a=st=ic=-~ _ _ _ ^{L -_ _ _} L -_ _ ~ ^{_ _ _} ~

-15 -10 -5 0 5 10 15

x-Coordinate along the Central Line (y=O) of the Arcan Specimen (mm)

Fig. 2-14: Shear stress distribution along the central line (y

=

0) for quasi-elastic and quasi-plastic responses in Fig. 2-12 and 13.

o o

"'&,

"

o

t

Fig. 2-15: Test fixture for the Arcan specimen.

A-A

y, V

2.36 2.35 2.51 2.27 2.07 1.89 1.98 ! 2.07 2.11 2.14 2.17 ^! I 1.88 ,

~ . ^---

1.85 2.35 2.46 2.45 2.37 2.32 2.35 • i2.36 2.43 2.46 2.33 1.91

h

: X,

I

I 1.80 2.35 2.51 2.54 2.45 2.48 2.51 ^I I ~.50 2.63 2.72 2.61 2.09

I

^{_ . - - - - - r-} _________ L: ^{I •} -_ .. _--- ( - -

I, I

/

^{1.69 2.10} 2.32 2.38 2.36 2.31 2.39 :2.51 2.64 2.76 2.61 2.24 ^\

--- ^{_______ ~.;. _.t} \

1.9412.05

\

1.65 1.86 2.06 ! ^{I 2.07 1.99 2.14 2.38 2.48 2.54 1.82}

\

/ ^\

Fig. 2-16: Experimental shear strain field (%) for a gross strain of £XI'

= ~u ⁼

2.4% (each . 2h

square with 40 pixels on a side corresponds to 1.41 mm). The strains averaged over the area within the dashed lines (100x100 pixels) are defined as the strains in the measurements .

. S m

~ ro OJ

.r: (f)

<J)

<J)

19

e

~ 0.8~

« I

x

C)

c -'

•

....

•

....

•

0 o.

• •

1

⁰

....

•

^(")

OJ 0

-50.6 L

01 c .2 m

c .~ 0.4- (j) ....

m OJ

.r:

(f) 0.2

'0 OJ 01

~ OJ

>

«

• experimental at 2.4% gross shear strain (80 DC 20.2 MPa)

• experimental at 3.8% gross shear strain (100 DC 17.5 MPa) .... experimental at 4.3% gross shear strain (35 DC 31.88 MPa)

C; numerical (quasi-plasticity) at 5% gross shear strain

o

y-Coordinate on the Arcan SpeCimen (mm)

o

5

Fig. 2-17: Homogeneity of the experimental and numerical strain field for the Arcan specimen. The ordinate represents the ratio of the shear strain averaged over the specimen parallel to the x-axis and the gross shear strain for various coordinates.

0.1 0.09 0.08 0.07

.~ 0.06

Vi ~

~ 0.05 ro Q)

.<:

(/) 0.04 0.03 0.02

(J

X

<]

+ ::>

13.4 MPa 23.3 MPa 27.6 MPa 31.9 MPa 35.1 MPa

0.01

o

0 0 C) 0 ()

c e o

0 :)

r)

^{C) C)}

n

0 0 () () CD

0 0.5 1.5 2 2.5 3 3.5 4

Log (Time/Sec.)

Fig. 2-18: Creep strains for indicated stress levels at 22°C.

0.1 r---c- -- - - - . l _

(J 13.3 MPa 0.09

0.08 0.07 .~ 0.06

Vi ~

~ 0.05 ro Q)

.<:

(/) 0.04 0.03 0.02 0.01

X 19.7 MPa

+

^{23.4 MPa}

<

^{27.6 MPa}

: >

^{31.9 MPa}

4.5 5 5.5

o~---~-~-~---~--~--~-~-~

0.5 1.5 2 2.5 3 3.5

Log (Time/Sec.) 4

Fig. 2-19: Creep strains for indicated stress levels at 35°C.

4.5 5 5.5

0.1 0.09 0.08 0.07

~ 0.06r I

~ 0.05i

~ ^I

(f) 0.04 0.03 0.02 -

() X

j

<J

[>

13.4 MPa

19.7 MPa

r''''

23.2 MPa 27.7 MPa 31.9 MPa

/J?.-> A1

t f -t>,-f> <l~~

[7 .>-;:> ~ ^{/ /1} L-L~cL4<;-<~l<.j-<t<t'<

<J-~ ~::-<"r"8--E.::J -<,r"~ .~

+-l...o....-.-.+-T-+ ~

=+ ++ +4--+-t-,

+1

X~--?( X X X~X--cX--X--X. >E-*-X---1<-X---X-'XcX 0.011 GB 0 CJ 0 0 0 0 () G 0 U 0 :::'}8-B e-o-e-C-f)

o ~' __ .. _L---L... L---~

0.5 1 1.5 2 2,5 3 3.5 4

Log (Time/Sec.)

Fig. 2-20: Creep strains for indicated stress levels at 50°C.

0.1 ~:--_----"----,

CJ 13.4MPa!

0.09 0.08 0.07 .0: 0.06

iii r::

~ 0.05 m Cl) ..c (f) 0.04

0.03 0.02 0.01

X 19.4 MPa

+

^{23.2 MPa}

<

^{27.6 MPa}

1.5 2 2,5 3 3.5

Log (Time/Sec.) 4

Fig. 2-21: Creep strains for indicated stress levels at 80°C.

4.5 5 5.5

4.5 5 5.5

0.1 r-;:::---:-::--::-:-:-::-,

o

^{10.3 MPa}

0.09 0.08 0.07

<:0.06

.~

Vi ~ 0.05 m Cl

(/) 0.04 .c

0.03 0.02 0.01

x 13.3 MPa

+

^{17.5 MPa}

<] 19.5 MPa i

C> 23.3 MPa .

O~---~--~L---~--~---~--~--~--~

0.5 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Log (Time/Sec.)

Fig. 2-22: Creep strains for indicated stress levels at 100

Dc.

0.1 ,----:~_:_:---:-:-="-,

o

6.6 MPa 0.09 x 9.8 MPa

+ ^13.4MPa

0.08 0.07 .S: 0.06 Vi r:

~ 0.05

m Cl

.c (/) 0.04

0.03 0.02 0.01

OL---~--~----~--~--______ ^~ __ ^~ __ ^~ __ ^~

0.5 1.5 2 2.5 3 3.5

Log (Time/Sec.) 4

Fig. 2-23: Creep strains for indicated stress levels at 120 DC.

4.5 5 5.5

ro

40

35-

30-

i

~ 25

~ 20~

1

(/) I

1! ~ 15-

(/)

10

o o

_ - _ e

-

16xl0³ S 10 s

_ _ --L _ _ _ _ ~~_

0.02 0.04 0.06 0.08 0.1

Shear Strain

Fig. 2-24: Isochronal data at 22

0c.

^{~ and J} are creep strain increments at the same stress level for the linear and nonlinear viscoelastic responses, respectively. The ratio of ~ and J is a measure of nonlinearity of the viscoelastic behavior. The encircled marker at 6.9 MPa (actually 21 individual points) is derived from torsional measurements.

4 0 : - - - ---,----~ ----T~----~

35

30-

ro 10 s

0... 25

6

<J)

<J)

~ 20

tn 1! Oi 15

(/)

:-.-= ••.• ~,~c~ .• ~.--- --...~-_--<--.,----..(}-- - - --0-

•

16xl0³ S

0.02 0.04 0.06 0.08

Shear Strain

0.1

Fig. 2-25: Raw and adjusted isochronal data at 22°C fitted with exponential curves. The adjustment is based on C in Fig. 2-37.

40

35

30-

m a.. 25

6

'"

'"

E: 20 Vi l! ~ 15

Vl

10

5-

0 0 10 s

0.02

;'-;'~:;;::o;~-:!~-~--- . • - - -

X}~ -_-cr_:.. -C)-:.. _ -5) -=- -::... ^{-~ ~~} -- -- - - - -()-

..;~,...~'"'"-.~oo.~ = . - - 16x103 S

0.04 0.06

Shear Strain

() raw data

• adjusted

0.08 0.1

Fig. 2-26: Raw and adjusted isochronal data at 35 DC fitted with exponential curves.

40

35 ~

30-

5

o o 0.02

'/~;; ~- -.~""

4- ~

~Ox103 ^S

0.04 0.06

Shear Strain

631 s

0.08 01

Fig. 2-27: Raw and adjusted isochronal data at 50 DC fitted with exponential curves.

40

35 -

30 (iJ CL 25

6

'"

'"

~ 20

Vi

1! ~ 15-

(j)

0,02

/~~~~~~~~~~~-~~-.-

-: ^~ "- -=--,oro __ = ^~ - - - e- 158 s

0,04 0,06 0,08 0,1

Shear Strain

Fig. 2-28: Raw and adjusted isochronal data at 80°C fitted with exponential curves.

30

25

(iJ20- CL

6 '"

'"

~15-10s

Vi

~

Q)

.r::

(j) 10

5

° _°

^0,02

251 s

.~

80x10³ S

0,04 0,06

Shear Strain

o raw data I!

adjusted ; i I

0,08 0,1

Fig. 2-29: Raw and adjusted isochronal data at 100°C fitted with exponential curves.

16

14-

",,[ll0S

c...l0

6

~ 8~

(j) ,

Co I .r. Q) (j)

° °

^{0.02 0.04} Shear Strain ^0.06

8 raw data

• adjusted

0.08

^0.1

Fig. 2-30: Raw and adjusted isochronal data at 120°C fitted with exponential curves.

-1.8 --

co

^-2

c...

:2:

x

Q)

~ -2.2 .0:2 Ci E u o -2.4

0..

Q)

e:

u

~ -2.6

.r.

~ OJ o

..J -2.8

2

-~---T

.. <I

<

4 6 8 Log (Time/Sec.)

10 12

Fig. 2-31: Compliance data at different temperatures and stress levels shifted along the log(time) axis according to a stress-shift factor au. For clarity of presentation the "master- curves" have been offset relative to the ordinate by a constant A as indicated.

-1.8r - - - , - - - - . - - - . - - - , - - - , - - - -,

ro

-2

Cl...

~ X

C]) ~ -22

.~

Ci E u -2.4 o

0..

C])

~ U

~ -2.6

~

~

CJl o

--l -2.8

23.4 MPa A=O.l

~

_3L---~---~----L---~---~--~

o 2 4 6 8 Log (Time/Sec.)

10 12

Fig. 2-32: Compliance data at different temperatures and stress levels shifted along the log (time) axis according to a temperature-shift factor ^aT' For clarity of presentation the

"master-curves" have been offset relative to the ordinate by a constant A as indicated.

+6.473e+07 +5.956e+07 +5.438e+07 +4.921e+07 +4.404e+07 +3.887e+07 +3.370e+07 +2.853e+07 +2.336e+07 +1.81ge+07 +1.302e+07 +7.845e+06 +2.674e+06

Y,V

x,u h

Fig. 2-33: Shear stress field in an Arcan specimen for a quasi-elastic solid under simple shear of EX).

=

^/).u

=

0.05 parallel to the x-axis.

. 2h

x x x x x x x x x x x x x 9 ° 0 0 0 0 0 0 0 0 0 0 0 ° 9

1- 0 0

on ~ 0.8- ill Cii c

E 0.6- 2 a Cii _on

<lJ

i7i ^0.4-

0.2-

x

o

x

x quasi-elastic

o quas,i-plastic

o '

x

o

x

-15 -10 -5 0 5 10 15

x-Coordinate along the Central Line (y=O) of the Arcan Specimen (mm)

Fig. 2-34: Shear stress distribution along the central line (y = 0) for quasi-elastic and quasi-plastic responses in Fig. 2-33 and 37.

)1' Y

\ \ \ \ \ \ \

^\

I 1 1 / / 1 1 1

\ \ \ \ \ \ \ / I I I 1

^j

\ \ \ \

^{I I I /}

r0 ^vv

^ILL--'

~

V / L

'"

V / ^{. / r}

t:0 ^v v;:

^x

/ / I I

\ \ \

\

/ / / / / I \ \ \ \ \ \

111111/

I

\ \ \ \ \ \ \ \

Fig. 2-35: The finite element mesh for the numerical analysis of the Arcan specimen (three layers).

40

35

30

co Cl.. 25

6

'"

'"

~ 20 lii

~ ro 15

C/)

10

5

+ + +++++

10

oL---~---~----~----==~======~

o 0.02 0.04 0.06 0.08 0.1

Shear Strain

Fig. 2-36: Experimental isochronal data corresponding to 16x10³ seconds (4.4 hours) of creep at 22 °C and the numerical simulation results of quasi-plasticity for pure shear response. In the numerical simulations, the shear strain/stress is defined as the averaged value of shear strains/stresses in the central 3x3 elements (the shaded area in Fig. 2-35).

+3.424e+07 +3.162e+07 +2.900e+07 +2.638e+07 +2.375e+07 +2.113e+07 +1.8S1e+07 +1.S88e+07 +1.326e+07 +1.064e+07 +8.014e+06 +S.391e+06 +2.768e+06

y,v

x,U

h

Fig. 2-37: Shear stress field in an Arcan specimen for a quasi-plastic solid under simple shear of E'J =

~~

⁼ 0.05 parallel to the x-axis.

1,04 1,03

1,02 creep times of () 10 seconds

x 100 seconds 1,01 \ 1000 seconds

~<J 16xl000 seconds

1 ^~ L - - _ - - " - - - _ - - - L _ _ _ _ _ L - _________ , __ • ___ , __ , __

0.Q1 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1 Shear Strain

Fig. 2-38: Coefficient

C

_y at 22°C. C_xy is the ratio of the shear stress over the central area and the nominal shear stress so that it is a measure of the stress field inhomogeneity.

107 1,06

1,04 1,03 1,02

o

^{raw data i}

+

^adjusted!

0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,1 Shear Strain

Fig. 2-39: Coefficient Cxy based on raw data (dashed curves in Fig. 2-25) and on adjusted data (solid curves in Fig. 2-25) at 16000 seconds (4.4 hours) creep time and 22°C.

40

---

35 - - - -22 ^CY

e - ^-!

l

30

- -35 DC I

~;;;.-';;;.--~~-~~~- - - - 50

DC - l X"£~dI~~~-=*---- ~

^{- - . -}

80 -DC 1

- - - -- - - I

_{100 DC ~}

!

10

0.02 0.04 0.06 0.08 0.1

Shear Strain

Fig. 2-40: (Adjusted) isochronal curves and extrapolations to obtain the stresses producing 10% shear strain after 22 hours creep at different temperatures.

~ ,

'"

^:

0... 25r

~ I

<f)

<f)

~ 20 if)

1! ro 15

(/)

10

O~---~---L----~---~---~~

20 40 60 80 100 120

Temperature ('C)

Fig. 2-41: Stress producing 10% shear strain in one day as a function of temperature (from Fig. 2-40).

Chapter 3 The Role of Volumetric Strain in Nonlinearly