Simulation of mechanical response during polymer crystallization around rigid inclusions and voids: homogeneous crystallization

MAllUAIJ ELSEVIER

Mechanics of Materials 21 (1995) 25-50

Simulation of mechanical response during polymer crystallization around rigid inclusions and voids: homogeneous crystallization* Ruojuan Ma, Mehrdad Negahban i Department of Engineering Mechanics and The Center for Materials Research and Analysis, 212 Bancroft Hall. University of Nebraska - Lincoln, Lincoln, NE 68588-0347, USA

Received I February 1994; revised version received 25 October 1994

Abstract The mechanical effects of homogeneous polymer crystallization around single defects are studied, showing how crystallization can develop residual stresses, and change material moduli. Defects in the form of rigid inclusions or voids are considered, either having spherical or cylindrical geometry. Problems with spherical symmetry are considered in the case of a spherical defect, ~nd plane strain problems with axial symmetry are considered in the case of a cylindrical defect. The predicted response is based on a constitutive model developed by Negahban et a l. (1993, Int. J. Eng. Sci. 31(1), 93-113). and shows that large n~,sidual stresses develop, which may result in debonding or fracture. Keywords: Polymers; Composites; Crystallization; Defects; Inclusions; Voids; Mechanics; Stress; Inhomogeneous

1. Introduction The objective o f tlais study is to estimate the influence o f crystallization on the effective mechanical response o f polymers in the presence o f defects. As shown in Fig. 1, a rigid inclusion or a void is used as the defect. The defects are selected to be in the shape of spheres or infinite cylinders. Spherically symmetric problems are studied in the case o f spherical defects, and plane-strain axially symmetric problems are studied in the case of cylindrical defects. In each case residual stresses developing due to homogeneous crystallization in the polymer are evaluated and presented for the special case of natural rubber as the crystallizing polymer. In addition to residual stresses, the stress-free moduli at each location in the polymer are evaluated by theoretically removing material, unloading the residual stresses, and perturbing. The results of these studies show that, even though it is assumed that the crystallization is homogeneous, the inhomogeneous nature of the deformation results in an inhomogeneous distribution of mechanical moduli. * This work has been partially supported by the National Science Foundation through Grant MSS-9111895. t Correspondence to." Dr Mehrdad Negahban, 309 Bancroft Hall, Department of Engineering Mechanics, Universityof Nebraska - Lincoln, Lincoln, Nebraska 68588-0347; FAX: (402) 472-8292; E-mail: [email protected] 0167-6636/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved S S D 1 0 1 6 7 -6636(94) 00072-7

26

R.-J. Ma. M. Negahban/Mechanics of Materials 21 (I 995) 25-50

Inclusion

Void

Fig. 1. Cross-section of a rigid inclusion and a void.

The study of the effects of crystallization around inclusions was motivated by the fact that many polymermatrix composites are replacing steel and other metallic materials in engineering applications, and it is, therefore, important to understand how these composite materials behave. As a first step toward this better understanding, we have considered the effects of homogeneous crystallization in the vicinity of spherical and cylindrical inclusions, representing entrapped particles and fibers, respectively. The experimental study of polymer crystallization around fibers by Incardona et al. (1993) is an indication of the growing interest in this area. As a second problem we have studied the effects of homogeneous crystallization around voids, which, in one size or another, are normal byproducts of most processing operations. This study is further justified since many common polymers undergo crystallization, and the crystallization substantially modifies the properties of these polymers. For example, there is a one hundred lbld increase in the elastic modulus of natural rubber due to crystallization, as shown by Leitncr (1955). In the problems that are considered, the required reduction in volume due to crystallization (see Gent, 1954) is the primary mechanism which induces deformation and the development of residual stresses. The constitutive theory developed by Negahban and Wineman (1992) and Negahban et al. (1993) is used to characterize the mechanical response during crystallization, and the natural inhomogeneous nature of the deformation results in the development of inhomogeneous stress fields, and inhomogeneous material properties, in spite of homogeneous crystallization. The assumption of homogeneous crystallization in the presence of large changes in strain is not particularly good, as shown by Gent (1954), G6ritz and MiJller (1973), and Stevenson (1983), who showed that stretching increases the rate of crystallization, increases the final amount of crystallinity, and affects the elastic moduli. Yet, the strain induced by crystallization around rigid inclusions seems to be small for most polymers, making the approximation an appropriate simplification. On the other hand, in some cases for the problem involving the void we see very large strains, and a large variation of the strains, suggesting that the approximation is less appropriate for some of the conditions we have selected. We are currently in the process of introducing a strain dependent model for the rate of crystallization, which should eliminate some of these shortcomings.

R.-J. Ma. M. Negahban/Mechanicsaf Materials 21 (1995)25-50

27

The details of the constitutive model are presented in Section 2. The specifics of the problems solved are described in Section 3. Section 4 contains the effect of crystallization on the development of residual stress. Section 5 contains the influence of crystallization on the distribution of mechanical moduli around the defects.

2. Kinematics and constitutive theory In this study, K0 ,viii denote the reference configuration of the body which will be selected as the initial stress-free configuration. K(s) will denote the configuration of the body at time s. The deformation gradient at time s will be denoted by F ( s ) . The volume ratio J(s) is the ratio of volume at time s to that in the reference configuration at any material point, and is defined by J(s) = d e t ( F ( s ) ) , where "det" denotes the determinant. The deformation gradient comparing the current configuration to the configuration at time s will be denoted by F.~(t) and defined as Fs(t) = F ( t ) F -j (s). The two left Cauchy strain tensors created from F ( t ) and F,.(t) will be given by B ( t ) = F ( t ) F l " ( t ) and Bs(t) = F s ( t ) F ~ ( t ) , respectively. The constitutive rrodel used is one introduced by Negahban and Wineman (1992), which gives the macroscopic value of the Cauchy stress T at current time t in the form t

T(t) = b(t)TA(t) +/a(s)TC(t,s)

ds,

(1)

t~

where b(t) is the current ratio of amorphous matter to total matter at the particular material point, a(s) is the normalized rate of mass transfer from amorphous matter to crystalline matter at time s, TA(t) is the current value of the effective stress in the amorphous portion of the material, TC(t, s) is the current value of the effective stress in the crystal created at time s, and ts is the starting time of crystallization. As shown by Negahban et al. (1993), under the assumption that each phase is incompressible and that macroscopic volume changes occur only clue to changes from one phase to the other, this model reduces to I

md(t) - 1TA m T(t) =pl + (t) + m-I m-I

[

J

TC(t,s) d J ( s ) ,

(2)

J(t)

where the integration over time has been replaced by integration over volume ratio, an indeterminate function p is introduced due to the assumption of incompressibility of each phase of the material, m = PC/RA, RA and Pc are the density of the amorphous and crystalline materials, respectively, and

b(t)---

mJ(t)-1 m-I

,

a(s) =

m m-I

dJ(s) ds

(3)

A further simplification is introduced by selecting neo-Hookean models for each of the phases of the material. Negahban et al. (19o3) have used such a model to characterize the response of natural rubber. This model uses TA(t) = q ( t ) B ( t ) ,

TC(t,s) = w ( s ) B s ( t ) ,

where

q(t)=Ql+Q2[1-J(t)l,

w(s)=Wl+W2[l-J(s)],

and Qz, Q2, wi, and W2 are constants, resulting in the final form of the constitutive equation (2) as 1

mY(t) - 1 T(t) = p! + --q(t)B(t) m-1

m + m-1

f

J J(t)

w(s)Bs(t) dJ(s).

(4)

28

R.-J. Ma, M. Negahban/Mechanics of Material.~" 21 (1995) 25-50 16 14 12

~

s

~

6 4

2

3

4

5

Fig. 2. Theoretical fit to experimental results o f M i n ( 1 9 7 6 ) on the instantaneous response ( 1 sec after loading) in uniaxial extension of natural rubber at room temperature ( 2 2 ° C ) . - - Theoretical using neo-Hookean model; * Experimental results of Min.

6

Stretch Ratio

Fig. 2 shows that the neo-Hookean model gives a very good representation of the instantaneous elastic response of natural rubber, even though the equilibrium response may be better modeled by other forms. The neoHookean form selected for the effective stress of the crystalline portion represents the simplest selection. This is an adequate first approximation since this selection can fit existing experimental data for unconstrained crystallization and crystallization under uniaxial extension (see Negahban et al., 1993). In addition, this selection has been able to correctly predict the multi-dimensional response of natural rubber, as shown by Negahban (1994) in explaining an anomalous response observed by Kolsky and Pipkin (1980) in a torsional oscillator made of a natural rubber torsional spring which undergoes crystallization. The apparent appropriateness of the neo-Hookean form for the representation of the effective contribution of the crystals to the response of the bulk material may be a result of the reduced relation between the actual crystal properties and the effective crystal properties, which may be explained by (a) the random nature of crystallization which might result in the diminished importance of the anisotropic nature of the crystals, (b) the low degree of crystallization for natural rubber (less than 30% crystallization under the most ideal conditions), and (c) the large elasticity of the amorphous phase which may result in large variations between the macroscopic deformations and the actual detormations of the crystals, which are much more rigid. The constitutive equation given by (4) is used for all analytical results. A fit, by Negahban et al. (1993), of the material constants to experimental results tbr natural rubber has resulted in the following set of parameters: Q1 = 3.27 × 10 -I MPa, WI=5.19MPa,

Q2 = -3.19 × 10MPa,

W 2 = 1 . 1 9 × 104MPa,

m=l.10.

(5)

These specific parameters will be used in this article to generate all figures, they represent natural rubber at 0°C. QI was obtained from the instantaneous response before crystallization (Leitner, 1955). Wi and W2 were calculated by a least square fit to the experimental data on the change of elastic modulus as a function of change in density lbr unconstrained crystallization as given by Leitner (1955), shown in Fig. 3. Q2 was selected to make the stress relax to zero in uniaxial extension at about 10% crystallization, as observed by Gent (1954). m is obtained using the theoretical values of the density of the amorphous and crystalline portions (Stevenson, 1983). Fig. 3 shows both the theoretical curve obtained using the proposed values of these constants and the experimental results of Leitner (1955). The volume ratio J(t) at time t is, in general, prescribed by a functional of the history o1" temperature and deformation, normally a differential equation for the rate of change of J in terms of the current state of strain and temperature. It will be assumed that the rate of change of J is given by a function only of J. A modified Avrami relation (see Gent, 1954) is a special form of this type of equation. The original Avrami equation was introduced for crystallization in metals (Avrami, 1939, 1940, 1941 ). The modified Avrami equation is given by

29

R.-J. Ma. M. Negahban/Mechanics of Materials 21 (1995) 25-50

80

.-. '"

Table I Relation between J and percent crystallization for natural rubber

20 0

|

0

0.5

i

I 1.5 Increasein Density,%

|

2

Fig. 3. Theoretical fit to experimental results of Leitner (1955) on the elastic modulus at 0°C during unconstrainedcrystallization. -- Theoretical; • Leitner sample A; [] Leimer sample B. dJ(s) - = A[J(s) - J~], ds

J

% Crystallinity

I

0%

0.99 0.98 0.97

I 1% 22 % 33 %

2.5

(6)

where A and J,~ are constants, and represent the rate of crystallization and final degree of crystallization, respectively. This simplistic model describes uniform crystallization since one can integrate (6), independent of the equations of equilibrium, to obtain J = Jc~ -4--(l - Jcx~)e At.

(7)

Similar integrals can be evaluated for all models which assume the rate of change of J to be only a function of J, resulting in homogeneous crystallization for all such models. In this case we will present results directly in terms of J to avoid giving any specific model for the rate of crystallization. For the case of natural rubber, the relation between J and the percent of crystallization can be obtained using Eq. (3) for m = 1.1, and which is given in Table 1. For natural rubber more than 20-30% crystallization is uncommon. The maximum change in J possible in polymers is about 10-15% (i.e., the range of J is between 0.85 and 1). The larger change in J is for materials like polyethylene which undergoes 80-90% crystallization.

3. P r o b l e m d e s c r i p t i o n

Two types of problems are considered in this study, that of spherical and cylindrical defects. Fig. I shows the cross-section for typical rigid inclusion problems and for typical void problems. As can be seen in Fig. I, for each type of geollaetry we consider either a rigid inclusion in a finite domain of polymer with radial traction on the outside boundary of the polymer, or we consider a void inside the polymer with radial traction on the surface of the void and a fixed boundary on the outer surface of the polymer. We assume spherical symmetry in the case of the spherical defects, and assume plane-strain motion with axial symmetry in the case of cylindrical defects. Spherical coordinates will be used for spherical defects, where (R, O, q~) are the coordinates of particles in the reference configuration, and (r, 0, &) are the corresponding coordinates in the current configuration. Cylindrical coordinates will be used when studying cylindrical defects, in this case (R, O, Z) denote the coordinates in the reference configuration and (r, 0, z ) denote the corresponding coordinates in the current configuration. In both cases of spherical arid cylindrical defects, a0 will denote the initial radius of the defect and b0 will denote the initial radius of the outer boundary of the polymer, as shown in Fig. I.

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995) 25-50

30 0.1

0.1 o~

Hoop Strain • Radial Strain Inclusion

,,"

J "0.975

/~

Hoop Strain Radial Strain

J - 0.95

Void

_

0

0

........

#

.

.

.

.

.

.

of • | I, tl It;

-0.1

J

t ;

I

,

,

,

2

3

4

1

Normalized

5

Radius

Fig. 4. Radial and hoop strains for a spherical defect and for several values of volume ratio as a function of radius normalized to the radius of the fixed boundary (i.e., R/an for the inclusion and R/bc~ for the void). The inclusion is represented by the pan of the figure with normalized radius larger than 1 and the void is represented by the portion with normalized radius smaller than 1. I radial strain; - - hoop strain.

0

~ - 0.925 J

,/"f

"""

-0.1

0

/

1

0.9

,

,

,

2

3

4

Normalized

5

Radius

Fig. 5. Radial and hoop strains for a cylindrical defect and for several values of volume ratio as a function of radius normalized to the radius of the fixed boundary (i.e., R/ao for the inclusion and R/bo for the void). The inclusion is represented by the part of the ligure with normalized radius larger than I and the void is represented by the portion with normalized radius smaller than 1. -radial strain; - - hoop strain.

The boundary conditions for the problems representing a rigid inclusion are selected to show the effects of a single spherical particle or cylindrical fiber on the otherwise homogeneous crystallization in the polymer. The displacement at the surface of the inclusion is assumed to be zero, and a traction is assumed to be at the outer boundary. Since, for the problems under consideration, the strain field is totally described by the value of the volume ratio J, one can select the ratio of the radius of the outer boundary of the polymer to the radius of the rigid inclusion, bo/ao, in a way so as to simulate a small inclusion in a large polymer body. For spherical inclusions Fig. 4 shows the hoop and radial strains for several values of J, spanning the typical range for polymer, as a function of normalized radial location, radius R is normalized by the radius of the fixed boundary (a0 for the inclusion and b0 for the void). Values of normalized radius above 1 are associated with the inclusion problem, where the polymer is at locations outside the fixed boundary, and normalized radius values smaller than 1 are associated with the void problem, where the polymer is inside the fixed boundary. Fig. 5 is a similar plot for cylindrical defects. It can be seen from Fig. 4 that the strain field around a spherical rigid inclusion becomes homogeneous as one reaches values of R/ao of about five, and the values of radial and hoop strain converge, indicating the diminishing effect of the inclusion. Fig. 5 shows a similar effect for the cylindrical rigid inclusions. In general, for the inclusion problem we have selected bo/ao = 10, allowing the outer boundary to be outside the domain of influence of the inclusion. An additional point to note in Fig. 4 is that the strains are, in general, very small for all polymers. In the case of natural rubber J = 0.975 is representative of the value of J at the end of crystallization, indicating that one will have a maximum of 2-3% strain, which in natural rubber is too small to have a pronounced effect of strain-induced crystallization. This further justifies the assumption that homogeneous crystallization is a good approximation for the case of a rigid inclusion in natural rubber. For other polymers the minimum value for J, after the completion of crystallization, may be as low as J = 0.9. For J = 0.9 the maximum strain for the inclusion problem can be up to 10% strain, which is still relatively small. This justifies, to some extent, our assumption that effects of strain-induced crystallization may, in general, be neglected for this problem. In the case of a void, as can be seen from Fig. 1, it is assumed that the polymer is constrained from motion at the outer boundary, R = b0. The purpose of this is to simulate the growth of a void in a constrained polymer. The required volume change associated with crystallization is the mechanism which induces the growth of the

R.-J. Ma. M. Negahban/Mechanicsof Materials 21 (1995)25-50

31

void under these conditions. The part of Fig. 4 associated with values of normalized radial location less than I shows the radial and hoop strains for several values of J, spanning the typical range of polymer response, tbr spherical voids. As can be seen from this figure, the strains are much larger for the case of the void. In the case of natural rubber, which has a typical value of J = 0.975 after the completion of crystallization, it can be seen that the strains are within 10% for ao/bo between about 0.6 and I. For natural rubber, strains larger than 10% will possibly have a noticeable effect on the rate of crystallization, making the assumption of homogeneous crystallization not very appropriate. The range of ao/bo between 0.6 and 1 is more representative of very large voids, approaching thick and then thin films. For some other polymers, where one can get J = 0.9, or slightly smaller, Fig. 4 show,,; that one will even have strains larger than 10% for thin film layers coating the inside of a rigid container. In general, the assumption of no effects of strain-induced crystallization is less appropriate for the case of the ,,oid if either the void radius is small relative to the fixed boundary radius, or there is a large degree of crystallization resulting in small final values of J. Yet, in all cases, homogeneous crystallization can be considered appropriate for the initial stages of crystallization where J is close to I. A second point of concern for the void problem may be that the zero-displacement boundary condition at the outer surface of the polymer is possibly lioo rigid of a constraint; it might be more appropriate to relax this condition and replace it by one of a displazcment related radial stress. We chose not to do this since we do not have a good model tbr the relation between this displacement and the radial stress. As can be seen from Fig. 5, similar statements hold tor cylindrical voids under plane-strain conditions. In presenting results for the inclusion we will normalize the value of radial locations to the value of the radius of the inclusion (i.e., R/ao). In the case of the void, we will normalize the value of radial locations to the value of the radius at the [ixed outer boundary (i.e., R/bo). This normalization leaves the strain tields independent of the radius of the free boundary. Effects of temper~tture are neglected, in general, but may have a substantial influence on the response of the material if there is large variation in the rate of crystallization at different locations of the polymer, or if the flow of heat is induced through external sources. Of particular importance might be the case where a tiber of much higher thermal conduction acts as a heat sink, developing a natural thermal gradient, and inducing inhomogeneous crystallization.

4. Distribution o f stress around a defect

To simplify the problem, it is assumed that the body force in the polymer can be ignored, and that the motion of the body during crystallization is quasi-static. Since only the boundary conditions distinguish between rigid inclusions and voids, the solution process for both rigid inclusions and voids will be studied simultaneously, first for spherical detects and then for cylindrical defects.

4. I. A spherical defect As stated above, il is assumed that there is spherical symmetry in the case of a spherical defect so that one can write the motion of the polymer body in spherical coordinates as

r(s) = r ( R , s ) ,

O(s) = 0 ,

~b(s) =q~,

(8)

where ( r ( s ) , O ( s ) , 4 . ( s ) ) are coordinates at time s of a particle at location (R,O,q~) in the reference configuration. The deformation gradients F ( s ) and F , ( t ) are given by

32

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995) 25-50

,~r(R,t)

F(s)

Or( R, s )

0

0

OR

r(R,s)

0

0

R

r(R,s)

0

0

~

=

,

Fs(t) =

a~(t¢ s)

0

0

oR 0

r(R,t)

0

r(R,s)

r(R,t)

0

r(R,s)

0

(9) for any time s C (t~, t]. The volume ratio J ( s ) = d e t ( F ( s ) ) is given by Or(R,s) r2(R,s) J(s) = - OR R2

(lO)

Since we have assumed homogeneous crystallization, so that the change in J is prescribed independent of the equations of equilibrium, one can integrate this equation assuming J to be constant. From this process one obtains

(ll)

r3(R,s) = J(s)R 3 + Dl(s),

where imposing the zero-displacement condition on the surface of the spherical inclusion requires that Dl (s) = a3[ 1 - J ( s ) ], or imposing the zero-displacement boundary condition on the outer boundary of the polymer for the case of a void requires that D l ( s ) = b03[l - J ( s ) ] . The left Cauchy strain tensors B ( t ) and B.~(t) are given by ( ar( R , t ) ) 2

\

(ar(R,~2

0

r2( R, t)

0

B(t) =

o R2

0

0

0 r2( R , t ) R~

'

B~(t) . =

0

0

r2( R, t) r2(R,s )

0

0

0 0 , r2( R , t ) r 2 ( R, s-------~ (12)

for s C (ts, t]. Substitution of B ( t ) and B s ( t ) into the constitutive equations, Eq. (4), gives the components of stress around the spherical defect as I

T~r=p +

mJ(t) - 1 m-

I

q(t)

Or(R,t) )2 aR

+

m rn- I

w(s) J(t)

( (

,~R " d J ( s ) , ate ,

Too = T~b~b I

mJ(t) - 1 =p + q(t) m-1

r ( R , t ) .2 m f '-R~) + rn- 1

r2(R,t) w ( s ) ~ dJ(s),

J(t) L . = L 4 , = To~, = O,

(13)

where r is given in Eq. ( 1 1 ). In spherical coordinates the equilibrium equations are expressed by OTrr --+ Or

2Lr - Teo - T~,/, =0, r

(14)

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995) 25-50

1 aToo - = r sin ~b O0 -

-

o,

33

(15)

10T~o = 0. r O&

(16)

Substitution of (13) into (15) and (16) requires that the indeterminate function p be only a function of radial location r and time t, or, in view of Eq. (11), a function of R and t. Let ff denote the function for p given in terms of R and t, so that p = ff(R, t).

(17)

Substituting the constitutive relations, Eq. (13), into Eq. (14) and reorganizing gives 0ff(R, t) = 7-((J(t), R, t), OR

(18)

where

7"l(J(t),R,t) =

mJ(t) - 1 [2j3(t)R 6 2 J ( t ) 4R3jl(__t) 1 m - 1 q(t) [,rT---~-~, ~ + r(R,t------~ r4(R,t) J I

m f [2R2j3(t)r4(R,_Ls) 2R2j(t) - 4R2J2(t)r(R's)] d J ( s ) , + w(s) + m- 1 [ J2(s)r7(R,t) r(R,t)r2(R,s) J(s)r4(R,t) J J(t) (19) and r(R,s) = ~ , R

3 + Dl(s). Integration of Eq. (18) gives R

ff(R,t) = ff(a0, t)

+fT-t(J(t),R*,t) dR*,

(20)

ao

where a0 is the initizl radius of the defect. Substitution of the: expression for p into the constitutive equations, Eq. (13), results in the distribution of stress around a spherical defect in a crystallizing polymer being given by R

L , ( R , t ) =ff(ao, t)

+f T-t(J(t),R*,t)dR*+ mJ(t) m-

-1 lq(t)j2(t)r4(R,t) R4

ao 1

m f J2(t)r4(R's) d J ( s ) +-~_-'-~ w(s) j2(s)r4(R,t ) , J(t)

(21)

34

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995) 25-50

7"o~j ( R, t) = Te~ ( R, t ) R

=fi(ao, t)

+ .f T-t(J(tl'R*'t) dR* +

mJ(t) - 1 m

-

r ( R , t ) ,2

q(t)(

----R-~

1

ao

I

+ m -m I .f w t, S ), rr2(R,t) 2 ( R , s ) dJ(s),

(22)

J(t)

and T,, = Tr~ = To~ = O. We will now look separately at the cases of a rigid spherical inclusion and a void. A radial traction Po is assumed to be applied on the outer boundary of the polymer in the case of a rigid spherical inclusion. This condition is imposed by requiring Trr(bo, t) = P0, resulting in hC)

7-[(J(t),R*,t) dR* +

p ( a 0 , t) = p0 -

m-

I

q(t)Je(t)

-

r4(bo, t)

uo I

+

m f m- 1

J2(t) r4(b°'s) dJ(s)] w(s) j2(s) r4(bo, t)

(23)

J(t)

Substitution of (23) into (21) and (22) gives R

Tr,.(R,t) =P0 +

fT-t(J(t),R*,t)

dR* +

mJ(t) - 1 ( R4 m- 1 q(t)j2(t) r4(-R,,t)

ha)

r4 (bo, t)

I

+m -m 1 ./

"ra(R's) w(s) .--:~7-7-.__J2(t) JL(s) ( r 4 ( R , t )

r4 (bo, s ) r4(bo, t) ) dJ(s),

(24)

J(t)

To~ ( R, t ) = Tc~ ( R, t ) R

=po+ f ~(J(t),R*,t)

dR*+

mJ(t)- - -~ lq (t) { r 2 ( R , t ) m ~. R2

b4)

J2(t) r4 (b0, t-~

ho

+ m m- I /

I

,t) w(s) \( r~2(( RR--~

j2(t) r4(bo, s__)'~ j2(s ) ra(bo, t) j d J ( s ) ,

(25)

J(t)

where 7-( is given in (19), and r( R, s ) = ~/ J ( s ) R 3 + a3 ( I - J ( s ) ). For the case of a spherical void it will be assumed that a radial stress (pressure) Pi exists on the surface of the void, and the outer boundary of the polymer is fixed. The traction on the surface of the void can be imposed by the condition Trr(ao, t) = Pi, resulting in 1

,0(ao, t)

Imp_Q- 1 = Pi -

i

a4 m q(t)J2(t) r'tao, . + ~

/ J(t)

Substitution of (26) into (21) and (22) will give

j2(t) r4(ao, s) ] w(s)--j2(s ) r4(ao, t)-dJ(s) .

(26)

35

R.-J. Ma, M. Negahban/Mechanics of Materials 21 ( 1 9 9 5 ) 2 5 - 5 0

04 )

R

Trr(R,t)=pi+

7"[(J(t),R*,t) d R * ÷

m -- 1

q(t)J2(t)

r4~,,

r4(ao, t)

t)

41o 1

m f ÷m-- 1

j2(t) : r4(R's) w(s)--~--~ \ r 4 ( R , t )

r4(a°'s)~ d J ( s ) , r4(ao, t) j

(27)

J(t)

Too( R,t) = T~,~ ~R,t) R

=p, + f

R., t)

mJ(t) {r2(R, t) m-- I l q(t) \ R2

dR* +

j2(t ) a_........~ ) r4 (ao, t)

¢/0 1

+-.

mr f2 ( R '(t ) l m"~

w(s)

J2(t) r 4 ( a ° ' s ) ) d J ( s ) , J2(s) r4(ao, t) J

"

r2~,1~, s )

(28)

J(t)

where 7-[ is given in (19), and r ( R , s ) = ~ / J ( s ) R 3 -t- b~o(I - J ( s ) ) . 4.2. A cylindrical defect As stated above, cylindrical coordinates will be used to study cylindrical defects. The plane-strain motion of the polymer under axially symmetric conditions may be written as r(s) = r ( R , s ) ,

O(s) = 0 ,

z ( s ) =Z,

(29)

where ( r ( s ) , O ( s ) , z ( s ) ) are the cylindrical coordinates at time s of a particle at location ( R , O , Z ) in the reference configurat:.on. The deformation gradients F ( s ) and F,.(t) are, therefore, given by

8r(R, 0 F(s) =

aR 0

r(R,s) R

0

0

0

1

,

Fs(t) =

1

0

r(R,t) r(R,s)

0

0

0

1

(30)

'

for every s C (ts, t]. The two left Cauchy strain tensors needed to evaluate the stress are given by

0/

0

r2(R, t) R2

0

0

0

1

B(t) =

"

B~( t) = "

0

r2( R, t) r2(R,s )

0

0

0

!

'

(31)

for every s E (ts, t]. The volume ratio J(s) = d e t ( F ( s ) ) is given by J(s)

=

ar( R,s) r( R,s) 8R R

-

(32)

which integrates to give r2( R,s) = J ( s ) R 2 ÷ D2(s),

(33)

36

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995)25-50

where imposing the zero-displacement condition on the surface of the cylindrical inclusion requires that D2 (s) = ag[ I - J ( s ) ], or imposing the zero-displacement boundary condition on the outer boundary of the polymer for the case of a void requires that D2(s) = b02[1 - J(s)1. Substitution of B(t) and B , ( t ) into the constitutive equation, Eq. (4), gives the components of stress around the cylindrical defect as I

mJ(t) - I ar(R,t) )2 m T~=p+ q(t)(-+ m-1 OR m-I

( o~R,,o)2

/

w(s) ( ~ )OR2

dJ(s),

](t)

Teo =p +

mJ(t) - 1 m-

1

q(t)

"r(R't))

+

~ R

m m-

/

I

1

r2(R, t) w(s) - dJ(s), r2( R,s)

J(t)

T-. =p +

mJ(t) - I

""

m-1

q(t) +

m

/

I

m-I

w(s) d J ( s ) ,

J(t)

Tm =T~: =T0= = 0,

(34)

where r is given in Eq. (33). In cylindrical coordinates, the equilibrium equations are given by ~rr

- -

Or

+

Trr - Too -

10Too

---+ r 00

-

-

r

2Tro r

0,

(35)

= O,

(36)

07"..= = 0. 0z

(37)

The last two equations require that the indeterminate p only be a function of location r and time t, or, in view of Eq. (33), p is a function of R and t. Letting ,6 denote this function of p, such that p = p(R, t), after substitution of the constitutive relations into Eq. (35) and integration, one will obtain

R

p(R,t) = p ( a o , t) + / f(J(t),R*,t) dR',

(38)

610

where

.T(j(t),R,t)= -

{ mJ(t) - I 12J2(t)R m--- 1 q(t) [r2(R,t) 1

J3(t)R3

J(t)]

~(R,/)

R

m / [ 2j2(t)R + m - 1 w(s) [J(s)r2(R,t)

j3(t)r2(R,s)R J2(s)ra(R,t)

J ( t ) R ] d J ( s ) }. r2(R,s)J

(39)

J(t)

Substitution back into the constitutive equation yields the distribution of stress around a cylindrical defect as

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995) 25-50 R

Trr(R,t) =p(ao, t) +

.T'(J(t),R*,t) dR* +

37

mJ(t) - ! J2(t)R2 q(t) m- 1 r2tt~,t)'~ "

ao I

+ m 1 f w(s)J2(t)r2(R'{s)l ~ {R ~ dJ(s), m_2,s,r2,..,t,

(40)

J(t) R

Too(R,t) =/~(o<,,t) + f .T'(J(t),R*,t) dR*+ mJ(t) ~--_~- lq(t) r2(R,t) R2 ao

I

+ ram_ 1 f w(S)r2,s,r2(R' (R at) dJ(s),

(41)

J(t) R

Tzz(R,t) =/~(ac., t)

dR*+

mJ(t) - 1 q(t) m-I

120

I

+ i'nm_l f w(s) dJ(s),

(42)

J(t)

and Tro = Trz = Toz = O. For a rigid inclusion, a radial traction P0 will be imposed on the outer boundary of the polymer body through the condition T~(bo, t) = Po. This results in R

Tr'=P°+f 7'J(t)'R*'t)dR*+mJ(t)-lm 1 -

q(t)J2(t) (

oo

I

m

/"

JZ(t) (r2(R's)

+m- 1 j w(s)~

\r2(R,t)

R2 r2(R,,t)

r2(b0, t) )

r2(b°'s)) dJ(s), r2(bo,t)

(43)

J(t) R

roo=po+f f(J(t),R*,t) dR*+ mJ(t)m_-1 lq(t) I

m F (r2(R,t) + m - 1 j w(s) r2(R,s)

(r2(R't) \ R2

j2(t)b~ r2 (b0, t) )

j2(t) r2(bo, S)) dJ(s) JZ(s) fl(bo, t)

(44)

J(t) R

Tzz=P°+f "~(J(t)'R*'t)dR*+mJ(I)-m-1 lq(/) (1 bo I s) ) + m m- 1 j ¢" w(s) ( 1 j2(t)r2(bo, j2(s)r2(bo, t ) dJ(s). J(t)

r2(bo, t)J2(t)b2) (45)

38

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995)25-50

For a cylindrical void, a radial traction in the equations

Pi will

be imposed on the surface of the void through the relation

Trr(ao, t) = Pi, resulting

/ 5F(J(t),R*,t) R

Trr=Pi+

dR*+

m- 1

q(t)J2(t)

(,'r2(R,t)

a__0' r2 (ao, t) J

aO I

+ m f .j2(t) {r2(R,s) m-1 w(s)--fi-~ \r~(R,- ~

r2(ao, s)) dJ(s ) r2(ao, t)

(46) ,

J(t) R

Too=Pi+ f .~(J(t),R*,t)

dR*+

m J ( t ) - I q(t) {r2(R,t) m- I ~ -~

J2(t)a~) r2( ao, t )

a0 I

-4m- i

w(s)

\r2(R,s)

(t) r2(ao, s) J2(s) r2(ao, t)

dJ(s)

(47)

J(t) R

Tzz=pi+ f .T'(J(t),R*,t)

dR*+

m J ( t ) - I q(t) ( j12 ( t ) a ~ ) m- 1

r2(a0, t)

a0

mf w(s) ( j12-( t J2(s)r2(ao, )r2(ao, s))dJ(s). t)

+m - 1

(48)

J(t)

4.3. Simulation of stress response It can be seen from the final expressions for the values of stress that the components of stress are linear functions of the traction on the boundary. The traction term only shifts the values of stress, independent of the influence of crystallization. To avoid duplication, we will only present results for zero traction on the boundary. As stated earlier, the results will be presented for material parameters associated with the uniform crystallization of natural rubber, and that the radial coordinate is normalized to the radius of the fixed boundary (i.e., R/ao for the inclusion and R/bo for the void). Fig. 6 shows the distribution of radial and hoop stresses around a rigid spherical inclusion, for several values of bo/ao, and for a value of J = 0.98, typical at the end of crystallization in natural rubber. As can be seen, with crystallization a compressive radial stress develops around the inclusion, and diminishes to zero to satisfy the zero-traction boundary condition on the outer surface of the polymer. Also, a tensile hoop (circumferential) stress develops around the rigid inclusion, which diminishes as one moves away from the inclusion. This shows that the inclusion is acting as a source of stress concentration during the crystallization process. It can also be seen that the stress distribution becomes insensitive to the location of the outer boundary for values of bo/ao greater than 5. In the case of natural rubber, the ultimate stress is in the order of 15 MPa. Since our results indicate stresses much smaller than 15 MPa, it may be concluded that the crystallization process in natural rubber may, in itself, not induce failure, yet may contribute to the acceleration of failure under superimposed loads. Fig. 7 shows the distribution of radial, hoop, and axial stresses around a rigid cylindrical inclusion, for several values of bo/ao, and for a value of J = 0.98, typical of the end of crystallization in natural rubber. As can be seen, the response around the cylindrical inclusion is similar to that of the spherical inclusion, with slightly smaller radial stresses, and slightly larger hoop stresses. The axial stress is fairly constant. In addition, the stresses are still slightly sensitive to the value of bo/ao even after bo/ao = 5. This can also be observed in the

R.-J. Ma. M. Negahban/Mechanics of Materials 21 (1995) 25-50 0.2

39

0.3 bo

J = 0.98

I ,**=2 ~ ~,~ $

0.1

b ~-----5,10

0

J- 098

|" " ~ , . ~ " " ~ - " "~k" ' ' "

0.2

V

......

-

-0.I

0

-0.2

-0.1

-0.3

.....

-0.2 0

1

2

3

4

5

6

7

8

9

10

0

'

'

'

'

'

1

2

3

4

5

Normalized Radius, R/ao

6

7

8

9

10

Normalized Radius, R/ao

Fig. 6. Radial and hoop stresses in natural rubber around a spherical rigid inclusion as a function of normalized radius, R/ao, for several values of the ratio of the radius of the outer boundary to the radius of the inclusion, bo/ao = 2,5, 10, and for a volume ratio of J = 0.98. - - radial stress; - - hoop stress.

Fig. 7. Radial, hoop, and axial stresses in natural rubber around a cylindrical rigid inclusion as a function of normalized radius, R/ao, for several values of the ratio of the radius of the outer boundary to the radius of the inclusion, bo/ao = 2, 5, 10, and for volume ratio J = 0.98. - - radial stress; - - hoop stress; - - axial stress. 0.7

t

I = 0.97

0.3

~

~J = 0 97

J-098

0.35

~_.~.~.~.. -- _ _

j = o.9s~

u

0

-0.3 l- 0

J-098

~

-0.35

-0.6

I J = 0.97

r = 097

-0.9 0

1

" 2

' 3

' 4

' 5

' 6

' 7

-0.7 8

9

10

0

Normalized Radius, R/ao Fig. 8. Radial and hoop stresses in natural rubber around a spherical rigid inclusion as a thnction of normalized radius, R/ao, for a value of the ratio of the radius of the outer boundary to the radius of the inclusion bo/ao = 10, and for several volume ratios, J = 0.97, 0.98, 0.99. - - radial stress; - - hoop stress.

. . . . . 1 2 3 4

. . . . 5 6 7

8

9

I0

Normalized Radius, R/no Fig. 9. Radial and hoop stresses in natural rubber around a cylindrical rigid inclusion as a function of normalized radius, R/tvoo for a value of the ratio of the radius of the outer boundary to the radius of the inclusion bo/ao = 10, and for several volume ratios, J = 0.97,0.98, 0.99. - - radial stress; - - hoop stress.

s t r a i n field ( s e e F i g . 5 ) w h i c h s h o w s t h a t t h e r a d i a l a n d h o o p s t r a i n s d o n o t f u l l y c o n v e r g e e v e n f o r n o r m a l i z e d r a d i u s v a l u e s o f 5. Figs. 8 and 9 show the distribution of radial and hoop stresses around rigid spherical and cylindrical inclusions, respectively, for

bo/ao = 10, a n d f o r s e v e r a l v a l u e s o f v o l u m e r a t i o , J = 0 . 9 7 , 0 . 9 8 , 0 . 9 9 , s h o w i n g t h e g r a d u a l

i n c r e a s e in r e s i d u a l s t r e s s as c r y s t a l l i z a t i o n p r o g r e s s e s . F i g . 10 s h o w s h o w t h e r a d i u s o f s p h e r i c a l a n d c y l i n d r i c a l v o i d s i n c r e a s e s w i t h t h e d e c r e a s e o f v o l u m e r a t i o , o v e r a r a n g e t y p i c a l f o r all p o l y m e r s , a n d f o r s e v e r a l i n i t i a l v o i d s i z e s ,

ao/bo = 0 . 0 5 , O. 1 , 0 . 2 , 0 . 5 . A s c a n b e

s e e n , s p h e r i c a l v o i d s g r o w f a s t e r t h a n c y l i n d r i c a l v o i d s . F i g s . 11 a n d 12 s h o w t h e d i s t r i b u t i o n o f r a d i a l a n d h o o p stresses around a spherical void and a cylindrical void, respectively, for several values of the initial void size,

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995)25-50

40 1 ~°0"9

25

,~ 0.8 ~0.7

20

~

15

[/,,~ ~=o98

0.5

0.4 _ 0.3

10

0.2 0.1

5

i

0

.......

0 0

0.02

0.04

0.06

0.08

0.1

0

0.2

Decrease in V o l u m e Ratio, (1 - J)

0.4

0.6

0.8

1

N o r m a l i z e d Radius, R/b,,

Fig. 10. Radius of spherical and cylindrical voids as a function of decrease in volume ratio, ( i - J), for several values of initial void size, ao/bo = 0.05,0. I, 0.2, 0.5. - - spherical void; - - cylindrical void.

Fig. 11. Radial and hoop stresses in natural rubber around a spherical void as a function of normalized radius, R/bo, for volume ratio J = 0.98, and for several values of initial void size, ao/bo = 0.05,0.1,0.2,0.5. - - radial stress; - - hoop stress. 9

25

~

20

j-0.gs

~,

8

7

\

~ ....~

6

"~

5

J = 0.97

~4 10

.... ---7 ."T ~ ' . - I. w-tP-p . , , , - . . . .

....... ,

-

3

---'I

.

2 1

0

// 0

~ - " 0.2

'--"~' "~'~" 0.4

0.6

-

0

0.8

1

N o r m a l i z e d Radius, R/bo

Fig. 12. Radial and hoop stresses in natural rubber around a cylindrical void as a function of normalized radius, R/bo, for volume ratio J = 0.98, and for several values of initial void size, ao/bo = 0.05,0.1,0.2,0.5. - - radial stress; - - hoop stress; - - - axial stress.

0.5

0.6

. . l J= °99

•

m

|

0.7

0.8

0.9

I

N o r m a l i z e d Radius, R / ~

Fig. 13. Radial, hoop, and axial stresses in natural rubber around a spherical void as a function of normalized radius, R/bo, for several volume ratios, J = 0.97,0.98,0.99, and for initial void size ao/bo = 0.5. - - radial stress; - - hoop stress.

ao/bo = 0 . 0 5 , 0 . 1 , 0 . 2 , 0.5, a n d for a v o l u m e ratio o f J = 0.98. A s can b e seen, t h e radial stress starts at zero o n t h e v o i d surface a n d i n c r e a s e s to a finite tensile v a l u e at t h e fixed o u t e r b o u n d a r y , a n d t h a t t h e radial stress at the fixed b o u n d a r y can b e s u b s t a n t i a l , i n d i c a t i n g the p o s s i b i l i t y o f d e b o n d i n g at this surface. A l s o , a t e n s i l e h o o p stress d e v e l o p s d u e to c r y s t a l l i z a t i o n , w h i c h has a m a x i m u m , n o r m a l l y , at the void surface a n d d e c r e a s e s to a final, n o n - z e r o , v a l u e at t h e o u t e r b o u n d a r y . A g a i n , t h e h o o p stress is large relative to the 15 M P a u l t i m a t e stress for natural rubber, i n d i c a t i n g t h e p o s s i b i l i t y o f the d e v e l o p m e n t o f radial c r a c k s o n t h e v o i d surface. In the case o f a s p h e r i c a l v o i d ( s e e Fig. 11) a n d for t h e s m a l l e s t v o i d size, a o / b 0 -- 0.05, it is o b s e r v e d t h a t t h e m a x i m u m h o o p stress h a s m o v e d away f r o m t h e v o i d surface, i n d i c a t i n g t h e p o s s i b i l i t y o f d e f e c t g r o w t h b y n e w v o i d n u c l e a t i o n . It m u s t b e n o t e d t h a t for s u c h small v o i d sizes o n e o b t a i n s very large s t r a i n s a r o u n d the void, w h i c h s h o u l d m a k e t h e a s s u m p t i o n o f h o m o g e n e o u s c r y s t a l l i z a t i o n i n a p p r o p r i a t e for this case. T h e effect

R.-J. Ma, M. Negahban/Mechanicsof Materials21 (1995)25-50

41

5 4.5 '., 3.5

"" .-..

...........

7.7.':

:-: .--..--.-.

,,r 2.5

J -

0.97

l.sl 0.5 0 0.5

0.6 0.7 0.8 0.9 Normalized Radius, R/bo

1

Fig. 14. Radial, hoop, and axial stresses in natural rubber around a cylindrical void as a function of normalized radius, R/bo, for several volume ratios, J = 0.97,0.98,0.99, and for initial void size ao/bo = 0.5. - - radial stress; - - hoop stress; - - - axial stress.

on the radial and hoop stresses of changing the degree of crystallization, by reducing the volume ratio J, for spherical and cylind:ical voids is shown in Figs. 13 and 14, respectively, for a value o f ao/bo = 0.5. The hydrostatic pressure (not shown) is fairly uniform for the case of inclusions and most voids, showing an inhomogeneous zone around the smaller voids.

5. Distribution of mechanical m o d u l i To evaluate the distribution of mechanical moduli in the polymer around the defect, a three-step process is adopted. At each radial point, first the material is theoretically unloaded to a residual stress-free shape; next a stress characterizing that which would be imposed to evaluate the modulus is imposed on the material to obtain the function relating the stress to the stretch ratio under these conditions; and finally the slope o f this function is evaluated at the values o f stretch ratio associated with the residual stress-free shape, to obtain the desired modulus. It is assumed that there is no crystallization in the unloading and reloading process; therefore, we assume elastic response for all stages o f this three-step process.

5.1. Distribution of mechanical moduli around a spherical defect As stated above, at each material point we first elastically unload the sample, assuming J to be constant (i.e., no further crystallization). Once we set the stresses equal to zero in the constitutive equations, the residual stress-free shape at each material point in the polymer for a spherical defect is obtained in terms o f stretch ratios from the initi~.l reference configuration as

A0 = ( K 2 J ( t ) x 1/3

(49)

and

ao= ao,= \ Kl I2(t) /~ I/6 , where

(50)

42

R.-J. Ma, M. Negahban/Mechanics o f Materials 21 (1995) 2 5 - 5 0 I

mJ(t) - 1 KIq(t) + mm_ 1 f m-I

w(s)--[~21

J(t)

\

1

K2-

mJ(t) - 1 q(t) + / m-1

R2

w(s)~

(5~)

dJ(s)

aR

]

dJ(s).

(52)

J(t)

For a rigid spherical inclusion, the radius at any time s during the process of crystallization is given by

r(R,s) = ~/J(s)R 3 +a3(

1 - J(s)),

(53)

therefore resulting in I

mJ(t) -11 m 1/ Klq(t) + mm-

(R3j(s) +a3(l'" "~--

w(s)

j2(S)K4

J(s))4/3 d J ( s ) ,

(54)

J(t) I

f

K2- mJ(t) m - -I lq(t) +

w(s)(j(s)R3+a~(l

j(s)))2/3

(55)

dJ(s)

J(t)

For a spherical void, the radius at any time s during the process of crystallization is given by

r(R,s) = ~/J(s)R 3 + b3(l - J(s)),

(56)

therefore resulting in

mJ(t) -11 m Kiq(t) + mm-

If

1

(R3J(s) +b3(I...,-.a--J(s))4/3 dJ(s), a2ts)tc"

w(s)

(57)

J(t)

f

K~ mJ(t) - l.q(t) + -= m - 1

1

w(s)

R2

(j(s)R3 +b3(l _ J(s)))2/3 dJ(s).

(58)

J(t)

5. I. 1. Change in elastic moduli To measure the elastic modulus, a stress is imposed along the desired direction, keeping all other surfaces stress free. For the measurement of elastic modulus along the radial direction, the stress is, therefore, given by

0 0/ T(t) =

0

0

0

0

0

0

.

(59)

The deformation gradient and the left Cauchy strain tensor are given by

F(t) =

0

Ao(t)

0

0

0

,t~(t)

,

B(t) =

0

,~20(t)

0

0

0

A~(t)

,

(60)

and the deformation gradient and the left Cauchy strain tensor comparing the current configuration to the configuration at time s during crystallization are given by

( F,(t) =

43

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995)25-50

o

o

ate 0

I~o(t)R r(R,s)

0

0

0

h~(t)

B~(t) = •

'

o

o

0

a~(t) R2 r2(R,s )

0

0

0

,~(t)

"

(61) The current volume ratio is given by J ( t ) = ,~r(t),~o(t),~(t), resulting in the relation J ~

(62)

ao(t) = a,~(t) =V A,(t)'

due to the symmetry of the deformation. Substitution of the above relations into the constitutive equations, Eq. (4), results in the following two independent equations 1

mJ( t) - 1 Trr(t)=p+ .... i q ( t ) , ~ ( t ) + m -m I f J(t)

w(s) ,ar~R.s~\AZ(t~) ,....,.._,,2 d J ( s ) , k

(63)

P,R ]

I

rnJ(t) - 1 0=p + --q(t) r,~ - 1

J(t) / + ~

w(s)

R2 J(t) dJ(s). r2(R,s) ,~r(t)

(64)

J(t)

Elimination of p from these equations yields Trr(t)

J(t)

=

Kih.2r(t ) - Kear(t------S

,

(65)

where Ki and K2 are given in (51) and (52). This equation would represent the one-dimensional stress-stretch ratio relation of the l:~lymer along the radial direction if the polymer would undergo no further crystallization. The incremental elastic modulus Er along the radial direction will be defined as

Er -

dTrr(t) _ 2KIAr(t) + v J ( t ) • d,,~r (t~ r,2 ArT(t)

(66)

At the stress-free configuration, the incremental elastic modulus along the radial direction, E~, can be obtained by replacing At(t) by ,~0 from (49), and is given by E~r = 31K~K2J(~)] t/3.

(67)

To obtain the hoo F (circumferential) elastic modulus, a stress will be imposed of the form (0 T(t) =

0 0

0 T0o(t) 0

0) 0 0

.

(68)

Substitution of this stress and the above strain tensors into the constitutive equations will give the following three equations

44

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995)25-50

m

0=p +

-1

m q(t) + m - 1

i

! w(s) (arCR,s~]2 dJ(s)

J(t)

Too=P +

m

-

1

i q(t) +

:

k

w(s)~

R2

dR

dJ(s)

J(t) m

-

O=p+

i

1

]

Rz w(s)~dJ(s)

q(t)+

a~(t),

(69)

]

A~(t),

(70)

A~(t).

(711

J(t)

From these equations, after the elimination of p and using stretch ratio relation along the hoop direction as

J(t) = Ar(t)Ao(t)h~(t),

J(t) Too= -~/KIK2 A-~~ + K2A2o(t)

The incremental elastic modulus

dToo(t__.____)_ Kv/-~IK2J(t), Eo- dAo(t) a~tt)

one obtains the stress-

(72)

Eo along the hoop direction is given by +

2K2ho(t).

(73)

At the stress-free configuration, the incremental elastic modulus along the hoop direction, E~0, can be obtained by replacing a0(t) by A° from Eq. (50), resulting in the relation

E~° = 3[ KIKSj2( t) ] I/6.

(74)

5.1.2. Change in shear modulus As before, first a sample is theoretically taken from each point in the polymer and unloaded, assuming no further crystallization by imposing J to be constant during this process. Then a simple shear deformation is imposed on this stress-free shape by requiring T4,~, = 0 and At(t) = A°, and a deformation gradient of the form

F(t)

=

a

yAo(t)

0 0

Ao(t) 0

0 ) 0 A~(t)

(75)

.

The deformation gradient comparing the current configuration to the configuration at any time s during the process of crystallization is given by

I ~r~A°r R,s)

yAo(t)R r( R, s)

0

0

Rao(t) r(R,s)

0

0

0

RAm(t) r(R,s)

OR

Fs(t) =

These result in the expressions

I .

(76)

( 22 B(t) =

)

R.-J. Ma. M. Negahban/Mechanics of Materials 21 (1995)25-50

A°2 q- y Ao(t)

ya2(t)

0

T(,,) A2(t)

A2(t)

0

0

0

A~(t)

,

45

(77)

and

a°2 (Or(R,.~Q)2 --

y2Azo(t)R2 yAzo(t)R2 + fi(R,s) r2(R,s)

0

OR

TA2(tlR2 r2( R,s)

B.,.(t) =

R2A~(t) r2( R,s)

0

(78)

R2A~(t) 0

r2( R,s)

The stress field at this point is assumed to be of the form

( Trr(t) T~o(t) T(t) = Tro(t) Too(t) 0 0

0) 0 . 0

Substituting into the constitutive equations, Eq. (4), yields the expression for shear stress of the shear strain y, given by Tin(t) = K2TA2(t),

(79)

Tro(t) as a function (80)

where K2 is given in ( 5 2 ) . The incremental shear modulus will be defined by

G = dTro(t)dT .%,,= K2A02 = [KI K2j2(t)] I/3

(81)

where Kt and K2 are given in (51) and (52), respectively.

5.2. Distribution of mechanical moduli around a cylindrical defect For the case of a cylindrical defect, the residual stress-free shape after elastically unloading the stress at each point in the polymer is given by

aOr= (K2K3J2(t) ) t/6

(82)

K~ 1/6

boa= [{ KI K3J2 ( t ) \

(83)

K~

( KIK2J2(t ) .~ 1/6 \

K~

]

'

(84)

46

R.-Z Ma, M. Negahban/Mechanics t ( Materials 21 (1995) 25-50

where K~, K2 are given in (51) and (52),and I

K ~ - mJ(t) - 1 •

m

-

1

+ / q(t)

w(s) dJ(s).

(85)

J(t)

For a rigid inclusion, the radius at time s during the process of crystallization is given by

r(R, s) = ~//J(s)R 2 + a2( 1 - J(s) ).

86)

For the void, the radius at time s during the process of crystallization is given by

r( R, s)

=

~/J(s) R 2 + b~( 1 - J(s) ).

87)

Using these expressions one can evaluate Kl and K2 from Eqs. (51) and (52), respectively, for the cylindrical inclusion and the cylindrical void. Following the same procedure described above to evaluate the mechanical moduli for spherical defects, one can obtain the mechanical moduli for cylindrical defects as follows.

5.2. I. Change in elastic moduli The radial incremental elastic modulus is given by dTrr(t) ~ J(t) E r - d~r('~ - x/K2K3 A2-~ + 2KIAr(t),

(88)

and at the stress-free configuration the incremental elastic modulus along the radial direction, E~, is given by replacing At(t) by ,~o from (82) to obtain

E•r

3[ K4K2K3fi(t) ] 1/6.

(89)

The hoop incremental elastic modulus is given by

dToo(t) E 0 -

-

-

~ J(t) - ~/KiK3hz-~ + 2Kffto(t),

(90)

and at the stress-free configuration has the value

3[ KIK~K3J2( t) ] I/6.

(91)

The axial incremental elastic modulus is given by E~,

-

J(t) d~z(t)daz(t) - ~/KjKz A2-~ + 2K3az(t),

-

(92)

and at the stress-free configuration has the value

~.= 3[ KIK2K4jZ(t) ]U6.

(93)

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (I 995) 25-50

47

5.2.2. Change in shear modulus As before, a simple shear deformation is imposed on the residual stress-free configuration of the material by setting 7":- = 0 , At(t) = A°, and using the deformation gradient ( At°

F(t) =

0 0

y'~"(t)

0

Ao(t) 0

0 ,~:(t)

)

.

(94)

The de/brmation gradient comparing the current configuration to the configuration at time s during the process of crystallization is ~iven by

I a~(e,.,.) A'? F., (t) =

r( R, s) yAo(t)R

0

I

0

Rao(t) r(R, s)

0

0

0

Az(t)

"

(95)

These result in the expressions

B(t) =

h 2 + y2A~(t) y(t)A2o(t) 0

Taro(t)

0

h2o(t) 0

0 A:

(96)

and

.~,Or2

T2A2(t)R2 (~r(R.O~2 + r2(R,s)

TA2(t)R 2

TA2( t ) R2 r2(R, s)

R2A20(t ) r2( R, s)

0

0

0

A2z(t)

3R

B,.(t)

=

I

r2(R,s) (97)

The stress is assuraed to take the form

( T~r(t'j Tro(t) T(t) = Tro(t} Too(t) 0 0

0I 0 . 0

(98)

Substitution of the above relations into the constitutive equations, Eq. (4), yields the expression for shear stress

Tro(t) as a function of shear strain Y, and is given by T~0(t) = K27A~(t),

(99)

where K2 is given in (52). The incremental shear modulus is then given by

G = dTro(t) dy a~,:= K2AO2 = [KIK2K3j2(t)]I/3

(l()o)

R.-J. Ma, M. Negahban/Mechanics of Materials 21 (1995) 25-50

48

200

200 J

-

J : 0.97

0.97

180

180

,g 160

~=~160

•~

140

•~

120

.,~ ~ 120

u

g:l

140

100

100 J =

80

I~'

•

m

1

2

3

.

4

.

5

.

6

.

098

J = 0,98

80

.

7

8

9

J'-

i

2

10

n

l

n

4

6

8

10

Normalized Radius, R/ao

Normalized Radius, R/ao Fig. 15. Radial and hoop elastic moduli in natural rubber around a rigid spherical inclusion as a function of normalized radius, R/an, for two volume ratios, J = 0.97,0.98, and for bo/ao = I0. - radial elastic modulus; - - hoop elastic modulus. 600

Fig. 16. Radial and hoop elastic moduli in natural rubber around a rigid cylindrical inclusion as a function of normalized radius, R/ao, for two volume ratios, J = 0.97,0.98, and for bo/ao = 10. - - radial elastic modulus; - - hoop elastic modulus. 250

J =

I e~

0.98

J - 098

200

I I I

400

,_~ 150

c

._~

I~ 100

% \

200

\

I1 I

50

0

I

I

|

I

0.2

0.4

0.6

0.8

1

N o r m a l i z e d R a d i u s , R/bo Fig. 17. Radial and hoop elastic moduli in natural rubber around a spherical void as a function of normalized radius, R/bo, for volume ratio J = 0.98, and for several values of initial void size, ao/ho = 0.1,0.2,0.5. - - radial elastic modulus; - - hoop elastic modulus.

0

I

I

i

I

0.2

0.4

0.6

0.8

I

Normalized Radius, R/~ Fig. 18. Radial and hoop elastic moduli in natural rubber around a cylindrical void as a function of normalized radius, R/bo, for volume ratio J = 0.98, and for several values of initial void size, ao/bo = 0.1,0.2,0.5. - - radial elastic modulus; - - hoop elastic modulus.

where KI, K2 and K3 are given in (51), (52), and (85).

5.3. Simulation of the distribution of mechanical moduli As before, we will present results for material parameters associated with natural rubber. Figs. 15 and 16 show the distribution of radial and hoop moduli for, respectively, spherical and cylindrical rigid inclusions, and for two values of volume ratio, J = 0.97,0.98. As can be seen, in both cases there is a small region of inhomogeneous distribution of elastic modulus right around the inclusion, rapidly becoming homogeneous as one moves away from the inclusion. Figs. 17 and 18 show the radial and hoop moduli for, respectively, spherical and cylindrical voids at a volume

R.-1 Ma, M. Negahban/Mechanics of Materials 21 (1995) 25-50 250

49

200 J -- 0.9/

180 m 200

~

°

J - 0.97

m

160 140

•~

120

150

i00 J - 0,9g

100

80

J = 098

60 50

40 J~099

0

0.5

i

i

i

i

0.6

0.7

0.8

0.9

J - 0.99

20 0

i

0.5

Normalized Radius, R/bo Fig. 19. Radial and hoop el&qtic moduli in natural rubber around a spherical void as a function of normalized radius, R/bo, for several volume ratios, J :-- 0.97,0.98,0.99, and for initial void size ao/bo = 0.5. - - ra:lial elastic modulus; - - hoop elastic modulus.

i

i

i

0.6 0.7 0.8 0.9 Normalized Radius,R/bo

Fig. 20. Radial and hoop elastic moduli in natural tubber around a cylindrical void as a function of normalized radius, R/bo, for several volume ratios, J = 0.97, 0.98, 0.99, and for initial void size ao/bo = 0.5. - - radial elastic modulus; - - hoop elastic modulus. 65

1800

~

1600

"0

=

02

1400

i

~

/

o.n

1200 lO~

J - 0,97

60

/

0.05

55

~" 50

/ / 0.5 /

"~ 45

~ 40

600 400

~

N 35

/

30

200 0

~

0

0.02

25

a

0.04

0.06

0.08

O.1

Decreg~e in Volume Ratio, (1 - J) Fig. 21. Radial and hoop elastic moduli in natural rubber at the surface of a cylindrical void as a function of decrease in volume ratio, (I - J), for several values of initial void size, ao/bo = 0.05, 0.1,0.2, 0.5 - - radial elastic modulus; - - hoop elastic modulus.

J - 0.98 i

i

i

i

0.2

0.4

0.6

0.8

l

Normalized Radius, Fig. 22. Shear modulus natural rubber around spherical and cylindrical voids -qs a function of normalized radius, R/bo, for two values of volume ratios, J = 0.97, 0.98, and for an initial void size ao/bo = 0. I. - - spherical void; - - cylindrical void.

ratio o f J = 0.98, and for several void sizes, ao/bo = 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 5 . A s c a n b e seen in Fig. 17, in t h e case o f a s p h e r i c a l v o i d the d i s t r i b u t i o n o f m e c h a n i c a l m o d u l i are i n d e p e n d e n t o f void size, a n d s h o w a g r a d u a l d e c r e a s e o f t h e h o o p m o d u l u s a n d i n c r e a s e o f t h e radial m o d u l u s as o n e m o v e s f r o m the surface o f t h e void t o w a r d s the fixed b o u n d a r y . I~a the case o f a c y l i n d r i c a l void, as can b e seen in Fig. 18, there is a p r o n o u n c e d effect o f v o i d size, a n d also a p r o n o u n c e d i n h o m o g e n e o u s z o n e for the elastic m o d u l i , r i g h t a r o u n d the void. Figs. 19 a n d 2 0 s h o w t h e effect o f the d e g r e e o f c r y s t a l l i z a t i o n o n the radial a n d h o o p elastic m o d u l i for s p h e r i c a l a n d c y l i n d r i c a l voids, respectively. A s can be seen, c r y s t a l l i z a t i o n increases the i n h o m o g e n e o u s zone. Fig. 21 s h o w s the elastic m o d u l i at the v o i d surface as a f u n c t i o n o f d e g r e e o f v o l u m e r e d u c t i o n ( c r y s t a l l i z a t i o n ) , for several d i f f e r e n t void sizes. A s c a n b e seen, for any given v o l u m e ratio, the radial m o d u l u s d e c r e a s e s w i t h i n c r e a s i n g void size a n d the h o o p m o d u l u s i n c r e a s e s w i t h i n c r e a s i n g void size. T h e axial m o d u l u s ( n o t s h o w n ) s h o w s a

50

R.-,I. Ma. M. Negahban/Mechanics of Materials 21 (1995) 25-50

small inhomogeneous zone around the void, showing abrupt changes, but o f the size of only one MPa. Fig. 22 shows the shear modulus for spherical and cylindrical voids for two different volume ratios. Again, a region of inhomogencous response is seen right around the void, in which the shear modulus of the spherical void is slightly larger than that o f the cylindrical void.

6. Concluding remarks Using a model to characterize the mechanical response during crystallization in polymers, we have evaluated, based on the assumption of homogeneous crystallization, the stress distribution and mechanical moduli distribution around rigid inclusions and voids, of spherical and cylindrical shapes. The results o f studying the strain field indicate that effects of strain on the rate o f crystallization can be ignored, for the most part, for the case of rigid inclusions of either spherical or cylindrical shapes. This was shown not to be the case for voids, where the variation in strain may be very large even for initially very large voids. Stresses o f about the same magnitude were predicted for spherical and cylindrical inclusions and for spherical and cylindrical voids, with the stress developed around the void being much larger than stresses developed around the inclusions. Thermal effects and effects o f strain-induced crystallization were ignored for this study, yet they can be very important, and must be addressed to provide a full picture o f the effects of crystallization.

References Avrami, M. (1939), Kinetics of phase change. 1. General theory, J. Chem. Phys. 7, 1103-1112. Avrami, M. (1940), Kinetics of phase change. II. Transformation-time relations for random distribution of nuclei, J. Chem. Phys. 8, 212-224. Avrami, M. ( 1941 ), Kinetics of phase change. 111.Granulation, phase change, and microstructure, J. Chem. Phys. 9, 177-184. Gent, A.N. (1954), Crystallization and the relaxation of stress in stretched natural rubber vulcanizates, Trans. Faraday Soc. 50, 521. GOritz, Von D. and EH. Miiller (1973), Zustandstindemngen von polymeren Netzwerken bei Orientierung. II. Analyse der Dehnungskristallisationauf der Grundlage kalorimetrischer Untersuchungen, Kolloid-Z u. Z Polymere 251,892-900. Incardona, S.D., R. Di Maggio, L. Fambri, and C. Mifliaresi (1993), Crystallization in J-I polymer/carbon-fiber composites: bulk and interface processes, J. Mater. Sci. 28. 4983-4987. Kolsky, H. and A.C. Pipkin (1980), The viscoelastic response of highly stretched rubber specimens to superposed torsional deformation, Ingenieur-Archiv 49. 337-345. Leitner, M. (1955), Young's modulus of crystalline, unstretched rubber, Trans. Faraday Soc.. 51, 1015. Min, B.K. (1976), Dynamic behavior of some solids and liquids, Doctoral dissertation, Brown University, Providence, RI. Negahban, M. and A.S. Wineman (1992), Modelling the mechanical response of a material undergoing continuous isothermal crystallization, hlt. J. Eng. Sci. 30(7), 953-962. Negahban, M., A.S. Wineman and R.J. Ma (1993), Simulation of mechanical response in polymer crystallization, Int. J. Eng. Sci. 31(1), 93-113. Negahban, M. (1994), Theoretical simulation of an anomalous response in a torsional oscillator, J. Appl. Mech. 61, 124-130. Stevenson, A. (1983), The influence of low-temperature crystallization on the tensile elastic modulus of natural rubber. J. Polymer Sci: Polymer Phys. Ed. 21,553-572.