Investigation of the rheological properties of short glass fiber-filled polypropylene in extensional flow

Abstract

The behavior of short glass fiber–polypropylene suspensions in extensional flow was investigated using three different commercial instruments: the SER wind-up drums geometry (Extensional Rheology System) with a strain-controlled rotational rheometer, a Meissner-type rheometer (RME), and the Rheotens. Results from uniaxial tensile testing have been compared with data previously obtained using a planar slit die with a hyperbolic entrance. The effect of three initial fiber orientations was investigated: planar random, fully aligned in the stretching flow direction and perpendicular to it. The elongational viscosity increased with fiber content and was larger for fibers initially oriented in the stretching direction. The behavior at low elongational rates showed differences among the various experimental setups, which are partly explained by preshearing history and nonhomogenous strain rates. However, at moderate and high rates, the results are comparable, and the behavior is strain thinning. Finally, a new constitutive equation for fibers suspended into a fluid obeying the Carreau model is used to predict the elongational viscosity, and the predictions are in good agreement with the experimental data.

Appendices

Appendix 1

From the Dinh and Armstrong expression (Dinh and Armstrong 1984), the normal stress components are defined as:

$$ \sigma _{11} =2\eta _0 \dot {\varepsilon }+\eta _0 \phi \mu _2 \big( {2a_{1111} -a_{1122} -a_{1133} } \big)\dot {\varepsilon }, $$

(17)

$$ \sigma _{22} =-\eta _0 \dot {\varepsilon }-\eta _0 \phi \mu _2 \big( {-2a_{1122} +a_{2222} +a_{2233} } \big)\dot {\varepsilon }, $$

(18)

$$ \sigma _{33} = -\eta _0 \dot {\varepsilon }-\eta _0 \phi \mu _2 \big( {-2a_{1133} +a_{2233} +a_{3333} } \big)\dot {\varepsilon }. $$

(19)

According to the definition of Eq. 13, the transient elongational viscosity is written as:

$$ \begin{array}{lll} &&{\kern-6pt} \eta _{\rm E}^+ =3\eta _0 +\eta _0 \phi \mu _2 \left(\vphantom{+\frac{1}{2}a_{2222}+\frac{1}{2}a_{3333}}2a_{1111} -2a_{1122} -2a_{1133} +a_{2233}\right.\\ &&{\kern76pt} \left. +\,\frac{1}{2}a_{2222}+\frac{1}{2}a_{3333} \right). \end{array} $$

(20)

Thanks to the equation of change for ψ, the fourth-order orientation tensor components are determined easily and the transient extensional viscosity is expressed without the need of a closure approximation for the fourth-order tensor.

Appendix 2

Equation 11 is used to express the steady extensional flow in the case where fibers are perfectly aligned in the principal stretching direction (x ₁). Therefore, the orientation distribution function is simply a Dirac function:

$$ \psi =\delta \left( {{\bf p}-{\bf x}_1 } \right). $$

(21)

Then the normal stress components are given by:

$$ \begin{array}{lll} \sigma _{11} &=&2\dot {\varepsilon }\eta _0 \left[ {1+\left( {\lambda _m \sqrt 3 \dot {\varepsilon }} \right)^2} \right]^{\frac{m-1}{2}}\\&& +\,2\dot {\varepsilon }\eta _0 \phi \mu _2 \left[ {1+\left( {\lambda _f 2\dot {\varepsilon }} \right)^2} \right]^{\frac{m-1}{2}}, \end{array} $$

(22)

$$ \sigma _{22} =\sigma _{33} =-\dot {\varepsilon }\eta _0 \left[ {1+\left( {\lambda _m \sqrt 3 \dot {\varepsilon }} \right)^2} \right]^{\frac{m-1}{2}}. $$

(23)

Consequently, the steady elongational viscosity is written as:

$$ \begin{array}{lll} \eta _{\rm E} &=&3\eta _0 \left[ {1+\left( {\lambda _m \sqrt 3 \dot {\varepsilon }} \right)^2} \right]^{\frac{m-1}{2}}\\&&+\,2\eta _0 \phi \mu _2 \left[ {1+\left( {\lambda _f 2\dot {\varepsilon }} \right)^2} \right]^{\frac{m-1}{2}}, \end{array} $$

(24)

where λ _m , m, μ ₂, and λ _f are used as fitting parameters.

Appendix 3

In order to achieve model predictions for the transient elongational viscosity, initial orientation functions have to be defined to take into account the three initial fiber orientations of Fig. 1. Following Advani and Tucker (1987), highly oriented states (i.e., parallel and perpendicular to the stretching direction) are obtained by a nonconventional distribution function having the form:

$$ \psi \left( {\varphi ,\theta } \right)=K_\bot \sin ^p\theta \;\sin ^q\varphi \mbox{,}\quad \mbox{for PERP,} $$

(25)

$$ \psi \left( {\varphi ,\theta } \right)=K_= \sin ^p\theta \;\cos ^q\varphi \mbox{,}\quad \mbox{for PARA,} $$

(26)

where $K_{\bot }$ and K _{_} are constants that satisfy the normalization condition. Both p and q are taken equal to 60 to give aligned fiber orientation. The planar random orientation state (not to be confused with a biplanar orientation obtained by adding Eqs. 25 and 26 with respect to the normalization condition) is reached with the following distribution function:

$$ \psi \left( {\varphi ,\theta } \right)=K_\times \delta \left( {\theta -\frac{\pi }{2}} \right),\quad \mbox{for PRO,} $$

(27)

where δ is the Dirac function and K _x is introduced to respect the normalization condition.