VISCOELASTIC SHEAR LAG ANALYSIS OF THE DISCONTINOUS FIBER COMPOSITE

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1 THE 19TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS VISCOELASTIC SHEAR LAG ANALYSIS OF THE DISCONTINOUS FIBER COMPOSITE N. Smih 1, G. A. Medvedev 2, R. B. Pipes 1,2 * 1 Aeronauical and Asronauical Engineering, Purdue Universiy, Wes Lafayee, USA, 2 Chemical Engineering, Purdue Universiy, Wes Lafayee, USA * Corresponding auhor (bpipes@purdue.edu Keywords: Viscoelasic, Shear Lag, Disconinuous Fiber Composie, Creep, Sress Relaxaion 1 Inroducion Many models have been developed o predic homogenized maerial properies for fiberreinforced composie maerials. For dilue sysems, Eshelby s[1] elegan elasiciy soluion of he equivalen inclusion problem provides he foundaion for his work. The Mori-Tanaka[2] model provides predicions for more concenraed sysems. A heurisic model, popular for is simpliciy, is he Halpin-Tsai[3] model which predics a variey of composie properies wih a single coefficien. All of hese models are very useful for finding he overall homogenized elasic properies, bu yield lile insigh o sress-ransfer beween he fiber and he marix. The so-called shear lag model, developed by Cox[4] as a soluion for a single disconinuous fiber embedded in a marix, is also popular due o is simpliciy. While some of is assumpions limi is accuracy[5], shear lag does yield valuable insigh ino boh he homogenized maerial properies and he sress-ransfer region beween he fiber and marix. All of he above models were developed assuming elasic maerials for boh fiber and marix. Over shor ime scales and a emperaures well below he glass ransiion, his is a good approximaion. A elevaed emperaures, or long ime scales, however, creep and sress relaxaion phenomena are imporan[6]. To model he ime-dependen properies of a composie maerial, i becomes necessary o include ime-dependen maerial properies and o solve he analogous problem wih viscoelasiciy. As an iniial soluion, and o gain insigh ino he micromechanical behavior presen, he presen paper follows he elasic soluion for a disconinuous fiber suspended in an elasic marix developed by Cox[4], inroducing a viscoelasic marix maerial, while assuming he reinforcing fibers o remain elasic. 2 Viscoelasic Analysis 2.1 Time-dependence: Sress Relaxaion Consider he elasic soluion for a disconinuous fiber of lengh 2L and radius r suspended in an elasic marix and submied o an axial srain ε x as described by he radiional shear-lag model[4] shown in Fig. 1. Fig. 1. The RVE for he shear-lag formulaion used in his paper. To solve he Shear Lag Problem for a viscoelasic maerial, he correspondence principle is invoked. As shown by Lee[7] and Bio[8], equivalen viscoelasic equaions can be found for isoropic or anisoropic maerials uilizing an elasic analysis.

2 The relaionship beween he axial sress, σ f, and he shear racion a he fiber surface, τ i, is given in (1. Here he sress componens are aken as funcions of he spaial coordinae in he fiber direcion, x, and ime,. The fiber radius is r. dσ f (x, = 2τ i(x, r (1 dx By equaing he shear forces on adjacen annuli, a relaionship can be found beween he shear racion, τ i, and he shear sress a some disance, r, from he cener line of he fiber. τ( = r r τ i( (2 For a viscoelasic maerial, he shear sress can be expressed as a funcion of he insananeous Shear Modulus, G m (, and he srain rae, dγ, d as shown in (3, where γ is he shear srain. τ( = G m ( dγ d d (3 Assuming ha he shear srain, γ, is equal o du, dr where u is he axial displacemen, and differeniaing wih respec o : dγ d = d d (du dr (4 Subsiuing (4 ino (3 and subsequenly subsiuing ino (2 yields: τ i ( r r dr = G m( du d d (5 Inegraing from he edge of he fiber (r = r o he far-field locaion, R, yields: τ i ( = 1 r ln ( R G m ( d d (u R u r d (6 r Subsiuing (6 ino (1 yields: dσ f (x, = dx 2 r 2 ln ( R G m ( d d (u R u r d r (7 Differeniaion wih respec o x and subsiuion of srain-displacemen relaions: d 2 σ f (x, dx 2 = 2 r 2 ln ( R G m ( d (8 r d (ε R ε r d If a perfec bond beween he fiber and marix is assumed, ε r can be replaced wih ε f. I is also assumed ha he far-field srain, ε R is far enough from he fiber ha i is idenical o he applied srain, ε 1. d 2 σ f (x, dx 2 = 2 r 2 ln ( R G m ( d (9 r d (ε 1 ε f d Assuming he fiber o be perfecly elasic and also assuming ha he ransverse and radial sresses are negligible when compared o he axial sress, ε f can be replaced wih. d 2 σ f (x, 2 dx 2 = r 2 ln ( R r (1 G m ( ( dε 1 d 1 dσ f ( d E f d A Laplace Transform can be performed on (1 o ransform he convoluion inegral ino a simple muliplicaion. d 2 σ f (s 2 dx 2 = r 2 ln ( R G m (s r (11 (sε 1 (s + ε 1 (+ sσ f(s E f This is a second-order linear differenial equaion of a sandard form wih he soluion: σ f (s = Asinh ( n x + Bcoshh ( n x r r Where: + [E f ε 1 (s + E f s ε 1(+] n = 2sG m (s E f ln ( R r (12 (13

3 Applying he boundary condiion of zero fiber sress a he fiber erminaions (σ f = a x=±l yields: σ f (s = [E f ε 1 (s + E f s ε 1(+] {1 cosh ( n x sech ( n L (14 } r r Since deerminaion of he global average sress wihin he body, σ x, is essenial o deermine he overall composie response, i is necessary o deermine he average fiber sress over is lengh, 2L, σ f. σ f (s = 1 2L σ f(sdx L L (15 σ (s f = [E f ε 1 (s + E f s ε (16 1(+] {1 anh (n L/r n } L/r If he global sress is assumed o be found by a simple rule of mixures relaionship ( σ 1 = ν f σ f + (1 ν f σ, m hen σ 1 can be expressed as: σ 1 = ν f [E f ε 1 (s + E f s ε 1(+] {1 anh (n L/r (17 n } + (1 ν L/r f σ m Since here is no closed-form soluion for an inverse Laplace ransform of an equaion of his form, i mus be inegraed numerically. A simple and compuaionally efficien algorihm was developed by Trefehen e al.[9] and adaped for use in his problem. 2.2 Time-dependence: Creep A similar formulaion can be developed by relaing he shear srain o he insananeous creep compliance, J m (, and he shear sress rae. γ( = J m ( d d [τ(]d (18 Following he same approach as oulined in (1 - (17, he fiber srain is deermined as follows: ε 1 σ f = J E m ( f d d [r 2 ln (R/r d 2 (19 σ f ( 2 dx 2 ] d Again, performing a Laplace Transform yields a second-order ordinary differenial equaion: ε 1 (s σ f(s = E f J m (s r 2 ln(r/r s d 2 (2 σ f (s 2 dx 2 Wih he soluion: σ f (s = E f ε 1 {1 cosh ( ncreep x sech ( ncreep L (21 } r r This is similar o he sress relaxaion soluion, wih he excepion ha n creep is given by: n creep 2 = E f J m (sln(r/r s (22 As before he average fiber sress is found by inegraion: anh ( ncreep L r σ (s f = E f ε 1 (s {1 n creep } (23 L r In he creep formulaion, however, he inpu global sress is relaed o an oupu srain, hence: σ 1 = ν f σ f + (1 ν f σ m (24 Subsiuing Hooke's Law for a viscoelasic maerial in creep yields: σ 1 ( = 1 dε ν f σ f + (1 ν f D m ( d d (25 And performing a Laplace ransform o handle he convoluion inegral: σ 1 (s = ν f σ f + (1 ν f ε 1(ss D m (s Which can be expressed in erms of ε 1 : σ 1 (1 = ν f E f ε 1 (s [1 anh(ncreep L/r ] + (1 ν f ε 1(ss (26 (27 n creep L/r D m (s Solving for ε 1 and including he ime-zero-value of σ 1 yields:

4 ε 1 (s = σ 1 (s + 1 s σ 1(+ ν f E f [1 anh(ncreep L/r n creep L/r ] + (1 ν fs D m (28 As before for he ime-domain soluion, he inegrals in (36 and (37 will generally be evaluaed numerically. 2.4 Frequency Domain Calculaion Example 2.3 Frequency Domain I is desirable o develop a frequency domain soluion. The following derivaion follows a similar derivaion used by Sims and Halpin[6] o equae separae experimens for composies. Recall Hooke s law for creep (shown here in ension: ε( = S( σ d (29 Where S( is he creep compliance. For a sinusoidal inpu sress, σ, of frequency ω: σ( = E e iω (3 σ = E iωe iω (31 Subsiuing (31 ino (29: ε( = E iω Change variables: u = S( e iω d ε( = E iω S(ue iω( u ( du ε( = E iωe iω S(ue iωu du (32 (33 (34 Dividing by σ( and convering o rigonomeric form: ε( = iω S(u[cos(ωu isin(ωu]du (35 σ( The sorage, J 1, and loss, J 2, compliances can now be defined as: J 1 = ω J 2 = ω S(usin(ωudu S(ucos(ωudu (36 (37 Where he complex compliance, J ( ε(ω = J σ(ω: J = J 1 ij 2 (38 To simplify he calculaion of he improper inegral in Equaions (36 and (37, a curve can be fi o he ime-domain creep daa of he form: S( = a b (39 Subsiuing (39 for S( in (36 and (37 gives he following: J 1 = ω a b sin(ωudu (4 J 2 = ω a b cos(ωudu (41 These can be evaluaed explicily using he gamma funcion, which was demonsraed by Lebedev and Halpin[1]. e cu u d du = Γ(d + 1 c d+1 (42 Where Γ(d + 1 represens he gamma funcion. To uilize his relaionship, le c = iω: e cu = cos(ωu i sin (ωu (43 Also: 1 c d+1 = c d 1 = (iω d 1 = (ωe iπ d 1 2 = i(ω d 1 (cos ( dπ 2 isin ( dπ 2 Finally, muliplying (38 by i: ij 1 + J 2 = iω a b sin(ωudu + ω a b cos(ωudu = aω b e iω b d (44 (45 (46 This is now in he form presened in (42 wih c = iω, u = and d = b herefore, using (44 and (42:

5 ij 1 + J 2 = aω b ( i cos ( bπ 2 sin ( bπ 2 Γ(1 + b (47 The real and imaginary pars can hen be separaed o find J 1 and J 2 independenly. To perform his procedure in reverse (for example, o obain he creep compliance curve of a maerial only using dynamic experimens, a curve of he form J 1 (ω = cω b would need o be fi o he experimenal daa. This curve fi will auomaically provide he b used for he ime-domain compliance S( = a b, since he only power of ω found in (47 is b. I is hen simple algebra o find a: J 1 a = ω b cos ( bπ 2 Γ(1 + b (48 To idenify he proper range of frequencies over which o perform an experimen, i is helpful o develop a relaion such ha S( = J 1 (ω. If he compliance, S(, is again assumed o have a power law form: a b = (aω b cos ( bπ 2 Γ(1 + b (49 The angular frequency can be solved for explicily: 1 (cos ( bπ 2 Γ(1 + b b (5 ω = 3 Resuls 3.1 Boundary Layer Elasic fiber-marix shear sress ransfer is a boundary layer phenomenon. If he inerfacial shear sress, τ i from he Shear-Lag Model is shown as a funcion of axial disance, x, normalized by he fiber diameer, D for differen fiber aspec raios (2L/D, i can be seen ha he sress is idenical regardless of aspec raio. This effec can be referred o as a boundary layer, wih all sress ransfer occurring wihin a cerain disance from he fiber end. The normalized shear sress as a funcion of aspec raio is shown in Fig. 2 for a maerial wih a fiber volume fracion, v f, of.3, a marix Poisson s Raio, ν m, of.3 and raio of fiber o marix siffness, E f E m, of 2. Fig. 2. The boundary layer for inerfacial shear sress in he elasic shear-lag model. The boundary layer can be solved for explicily if a specific, non-zero value is aken as a hreshold zero-value. In he presen case, he value of.1 was used, which (for he same maerial properies yields a consan boundary layer of 5.5 fiber diameers for all fiber lenghs greaer han wo imes he boundary layer. For fibers less han wo imes he boundary layer, here is an insufficien lengh o develop he full sress profile. For he viscoelasic Shear-Lag Model, since numerical inegraion is necessary o obain a soluion, several addiional assumpions concerning he maerial properies mus be made. Consider an isoropic viscoelasic marix maerial, whose sress relaxaion modulus is given as a simple exponenial decay funcion, E( = E exp ( 1 τ. The shear sress for several ime seps of he soluion is ploed wih respec o he normalized disance from he end of he fiber, as before, for a fiber wih an aspec raio of 1 is shown in Fig. 3.

6 Fig. 3. The effec of a viscoelasic marix maerial on he boundary layer for inerfacial shear sress. I is ineresing o see ha while he peak sress decreases wih ime, as expeced in he sressrelaxaion model, shear sress nearer o he cener of he fiber acually increases wih ime, creaing a ime-sensiive boundary layer ha increases wih ime. 3.2 Sress Relaxaion I is ineresing o now examine he viscoelasic shear lag model, varying as many parameers as possible, o demonsrae relaionships beween he inpu variables and he properies of he composie. Fig. 4 shows normalized sress relaxaion curves for a variaion in aspec raio. The modulus is normalized o more easily compare he differen degradaion beween he differen raios (as opposed o he more obvious difference in acual modulus. Fig. 4. Normalized sress relaxaion for various aspec raios. Noice ha for aspec raios of 5 or less, he modulus degrades almos as quickly as he nea marix. As he fiber lengh is increased, he composie sress relaxaion modulus reducion wih ime is grealy rearded, even for only 33 % volume fracion of fibers. I is also ineresing o examine he influence of fiber volume fracion. Fig. 5 shows a similar plo, bu he fiber aspec raio is held consan a 2, while fiber volume fracion is varied from o 6%. I is significan ha, in his case, he degradaion of he modulus of a composie shows very lile dependence on he volume fracion, so when modulus degradaion is he primary concern, a small percenage of very long fibers could suffice.

7 Fig. 5. Normalized sress relaxaion for various fiber volume fracions 3.3 Creep As should be expeced, he normalized creep resuls for variaion in aspec raio, shown in Fig. 6, appear similar o he sress-relaxaion resuls (alhough invered. Fig. 6. The effec of varying aspec raio on creep compliance. Noe ha he creep compliance is normalized as a percenage of he maximum modulus, o more easily compare beween differen aspec raios. 4 Conclusion The simpliciy of he Shear-Lag model lends iself well o adapaion for a viscoelasic soluion. Valuable insigh is gained boh a he microscopic and macroscopic levels by a close examinaion of he inerfacial shear sress and he far-field sress and srain. From he inerfacial shear sress, he local effecs of a viscoelasic marix maerial coninuously increase he boundary layer lengh wherein sress ransfer occurs, compared o he finie boundary layer observed in an elasic marix. Far field sress and srain show he global effecs of a viscoelasic marix maerial, indicaing ha aspec raio will have a much sronger effec on he creep and sress relaxaion of a fiber-reinforced composie han fiber volume fracion. Volume fracion is sill a driving force in deermining he magniude of he modulus, as in he elasic soluion, bu he modulus degradaion and increase in compliance show a much sronger dependence on aspec raio han volume fracion. These resuls all apply o a single fiber embedded in a viscoelasic marix, aligned wih he direcion of applied srain. Furher work is necessary o examine he viscoelasic properies when fibers are oriened away from he axis of applied srain and how o combine hese resuls for a disribuion of fibers, as would be seen in a shor-fiber composie. Physical experimens of creep versus aspec raio and volume fracion are also imporan o verify he predicions. These will be addressed in fuure publicaions. Fig. 1. The RVE for he shear-lag formulaion used in his paper. Fig. 2. The boundary layer for inerfacial shear sress in he elasic shear-lag model. Fig. 3. The effec of a viscoelasic marix maerial on he boundary layer for inerfacial shear sress. Fig. 4. Normalized sress relaxaion for various aspec raios. Fig. 5. Normalized sress relaxaion for various fiber volume fracions Fig. 6. The effec of varying aspec raio on creep compliance. Noe ha he creep compliance is normalized as a percenage

8 of he maximum modulus, o more easily compare beween differen aspec raios. References [1] J. D. Eshelby. "The Deerminaion of he Elasic Field of an Ellipsoidal Inclusion, and Relaed Problems." Proceedings of he Royal Sociey A 241, (1957. [2] T Mori & K. Tanaka. "Average sress in marix and average elasic energy of maerials wih misfiing inclusions." Aca Meallurgica 21, (1973. [3] J.C. Halpin. Effecs of Environmenal Facors on Composie Maerials. AFML-TR [4] Cox, H. L. "The elasiciy and srengh of paper and oher fibrous maerials." Briish Journal of Applied Physics 3, (1952. [5] John A. Nairn. "On he use of shear-lag mehods for analysis of sress ransfer in unidirecional composies." Mechanics of Maerials 26, 63 8 (1997. [6] D. F. Sims and J. C. Halpin. "Mehods for deermining he elasic and viscoelasic response of composie maerials." Composie Maerials: Tesing and Design [Third Conference] (1974. [7] Lee, E. H. "Sress Analysis in Viscoelasic Maerials." Journal of Applied Physics 27, (1956. [8] Bio, M. A. & Company, S. D. "Linear hermodynamics and he mechanics of solids." in Proceedings of he Third U. S. Naional Congress of Applied Mechanics, American Sociey of Mechanical Engineers 1 18 (1958. [9] Trefehen, L. N., Weideman, J. a. C. & Schmelzer, T. "Talbo quadraures and raional approximaions." Bi Numer Mah 46, (26. [1] Halpin, J.C. & Lebedev, Nicolais. L Ingegnere Chimico Ialiano 7, 173 (1971.