Mesoscopic Localized Deformations in Rubber-Toughened Blends

Chapter 3

Mesoscopic Localized Deformations in Rubber-Toughened Blends 1

K. G. W. Pijnenburg and E. Van der Giessen

Downloaded by PURDUE UNIVERSITY on August 7, 2013 | http://pubs.acs.org Publication Date: August 8, 2000 | doi: 10.1021/bk-2000-0759.ch003

Delft University of Technology, Koiter Institute, Mekelweg 2, 2628 CD Delft, The Netherlands

This paper is concerned with a numerical investigation of the deformation patterns in an amorphous polymer-rubber blend at a mesoscopic scale, i.e. in between the rubber particles. The computations employ a material model for shear yielding that incorporates rate-dependent yield, strain softening and anisotropic hardening at large plastic deformations. Periodic cell calculations are reported that demonstrate that shear yielding in a cavitated blend proceeds by the initiation and propagation of mesoscopic shear bands. The implications of these localized deformations on the competition with crazing are also discussed.

It is a well established fact that the fracture toughness of polymers can be greatly enhanced by adding a dispersion of rubber particles. The toughening is commonly assumed to involve a number of mechanisms: crazing, cavitation and shear yielding (7, 2). Cavitation of the rubber particles relieves the stress triaxiality in the matrix polymer. This suppresses the likelihood of matrix crazing and promotes plastic deformation in the matrix by shear yielding. The toughening effect is generally enhanced when a region of large plastic deformation spreads out over a large volume in the material. Toughening in blends thus involves a range of length scales. The 'macroscopic scale' is the scale at which plastic deformation takes place in the neighborhood of a propagating crack in a blend. The relevant 'microscopic' scale here is the molecular scale at which shear yielding takes place. The intermediate 'mesoscopic' scale is the size scale at which the individual rubber particles can be distinguished. This is a crucial scale for our understanding of toughening in blends since it is at this size scale that rubber cavitation, crazing and shear yielding compete with each other

36

© 2000 American Chemical Society

In Toughening of Plastics; Pearson, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2000.


37

Figure 1. Deformation zones in ABS (a) at a distance of ΙΟΟμπι from, and (b) near the fracture surface. From (6), with kind permission from Kluwer Academ Publishers. and determine which one(s) of them dominate(s). Evidence of massive plastic deformation at this scale during fracture is given in the TEM micrographs in Figure 1, taken near the fracture surface in notched samples of ABS (with polybutadiene rubber particles). After cavitation, the initially spherical particles in Figure la are seen to develop rather bulgy shapes. Similar shapes have been observed in (J, 4, 5) in various other materials and are expected to be relevant also at some distance ahead of a crack tip. The shape of the particles near the fracture surface (Figure lb) is quite different, however, indicating that the mesoscopic deformation processes are significantly different in this regime. The objective of this paper is to summarize recent insight about how plastic deformation occurs in a blend at this mesoscopic scale. This insight is obtained by numerical simulation using a realistic material model for shear yielding in the amorphous glassy matrix of the blend. We shall show how plastic deformation pro ceeds by the propagation of shear bands in between the rubber particles. Finally, we briefly investigate the consequences of the observed localized plastic deformation on the competition with crazing in the matrix.


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Material Model We model the polymer blend as a two-dimensional material (i.e. a material with infinitely long cylindrical particles under plane strain conditions perpendicular to the long axis). To reduce the computational effort further, we take the particles to be arranged in a regular square array. Therefore we can restrict our attention to a unit cell as shown in Figure 2. The relevant parameters are the particle radius a and half of the particle spacing, b. The ratio of a and 5 is related to the volume (or area) fraction of particles, / , in the following way: / = (π/4)(α/6) . It is commonly accepted that toughening in amorphous blends must involve the cavitation of the rubber particles or debonding of the particles from the matrix. Since the shear modulus of the rubber commonly used in blends is much lower than the yield stress in the matrix, the presence of already cavitated particles affects the subsequent deformations in the matrix only slightly. A detailed study of this in (7) has revealed that a cavitated rubber particle is mechanically equivalent to a void up to relatively large ratios between rubber modulus and matrix yield strength. Therefore, we can replace the rubber particles by traction-free voids. The applied loading consists of macroscopic strain rates En and E22 in &e χι and X2 direction, respectively, as well as a shear rate E\2. Therefore the macro scopic velocity gradient Lij reads


2

These macroscopic loading conditions are applied through periodic boundary con ditions for our unit cell. Details of this procedure may be found in (8). The corresponding macroscopic stress components are obtained by averaging of the stresses at the mesoscale, σ^, as Σι = ^ J

2b

L

= ^ J

σ η (±6, x )dx ,

(2a)

a 2(xi,±b)dxi

(2b)

2

2

2

,

^i {x\,±b)dxi. 2

(2c)

We can write ±6 because, by virtue of equilibrium, it does not matter whether the integration is performed over the upper or lower boundary (left/right in the case of Σι). The material model taken for the polymer matrix material is a viscoplastic one, based on the following relation between the local (mesoscopic) shear stress r and the resulting shear rate 7 , as originally derived by Argon (9): P

7

P

= 70

exp

(3)

Here, 70 is an pre-exponential factor, A is a material parameter related to the acti vation volume, Τ is the absolute temperature and SQ is the athermal shear strength.


39

To include the effect of pressure and strain softening, we use s 4- ap instead of so, where ρ is the pressure and α is a pressure dependence coefficient. Softening upon yield is incorporated by letting s evolve with plastic straining from the initial value so to a steady-state value s , via ss

p

i = h(l - s/s )7 .

(4)

ss


The rate of softening is governed by the material parameter h. The one-dimensionalflowrule (equation 3) is adopted in a fully three-dimensional constitutive model by letting the driving shear stress τ be determined from T

=

'

=

3 ~ J '

Gi

bi

iJ ~ 3 ~ \°kkàij

â

BI

,

(5)

where bij is the back stress (i, j G 1,2,3 and Sij is the Kronecker delta). See reference (10) for further details. The back stress accounts for the progressive strain

Figure 2. Sketch of the model, showing the unit cell containing a single parti cle/void and thefiniteelement mesh.


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hardening in amorphous polymers at continued plastic deformation. Physically, it is an internal stress associated with the stretching of the entanglement network upon continued plastic deformation. By exploiting the analogy with rubber elasticity, the back stress is modeled using non-Gaussian network theory. With reference to (7) for details it suffices to note here that the resulting constitutive description involves two material constants: the average number of links between entanglements, N, and the rubber hardening modulus C . For reference purposes, the stress-strain curve of a homogeneous (i.e. unvoided) piece of matrix material under macroscopic simple shear (En = Ε22 = 0) is shown in Figure 3. Here, as well as in the sequel, we have used the effective shear strain defined by Downloaded by PURDUE UNIVERSITY on August 7, 2013 | http://pubs.acs.org Publication Date: August 8, 2000 | doi: 10.1021/bk-2000-0759.ch003

R

l

Γ = ^El El + -El 1+

2

(6)

2

as the scalar measure for the macroscopic strains. The material parameters used in the calculations are taken to be representative of SAN and are: E/SQ — 12.6, ν = 0.38, s = 120 MPa, 70 = 1-06 χ MPs' , s /so = 0.79, As /T = 52.2, h I s = 12.6, a = 0.25, Ν = 12.0 and C /s = 0.033 Clearly visible in Figure 3 is the drop in stress after yield which is typical for amorphous glassy polymers. This softening will cause highly localized deformations in the model blend material, as will be seen in the next Section. 1

0

ss

0

R

0

0

0.3 -

0.2

i

0.1 -

0.0

t—•—'—«—

Mesoscopic Localized Deformations in Rubber-Toughened Blends

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