Effective viscoelastic behavior of short fibers composites using virtual DMA experiments

Abstract

May it be for environmental or economic reasons, mass reduction has become one of the main goals in mechanics. The short fiber thermoplastics composite is an interesting possibility since it presents a good compromise between a relatively easy process and mechanical properties. The aim of this work is to estimate and model the viscoelastic behavior at small strain of PC Lexan/Glass fiber composites. To meet this goal, a full field homogenization method based on solving the boundary problem through FFT is used. Virtual DMA experiments are used to build the master curve of the composite. They are later used to identify a macroscopic model for transverse isotropic short fiber composites. Finally, a meta-model is built to estimate the behavior of the composite at any given fiber volume ratio.

Appendices

Appendix A: Hashin–Shrikman lower bound

The aim of this appendix is to derive the Hashin–Shtrikman lower bound for the storage and the loss moduli of the composites described in Sects. 2 and 3. These composites are made of an isotropic and incompressible Maxwellian matrix reinforced by short fibers parallel to the $\mathbf{n}$ direction. The constitutive behavior of the matrix is given by the following differential equation:

$$ \frac{\mathbf{{s}}}{2 \eta }+\frac{\dot{\mathbf{{s}}}}{2 \mu }= \dot{\mathbf{{e}}}, $$

(25)

with $\mathbf {s}$ the deviatoric part of the stress tensor, $\mathbf {e}$ the deviatoric part of the strain tensor, $\mu$ the shear modulus and $\eta$ the viscosity. Equation (25) can be transform by using the Laplace–Carson transform (14) which gives the constitutive behavior in the Laplace domain:

$$ \hat{\mathbf{{s}}}(p)=\mathbf{{L}}_{\mathit{ve}}(p):\hat{\mathbf{{e}}}(p), \quad \text{with } \mathbf{{L}}_{\mathit{ve}}(p)=2\mu _{\mathit{ve}}(p) \mathbf{{K}}, \ \mbox{and} \ \mu _{\mathit{ve}}(p)=\frac{p \mu }{p+ \mu /\eta }. $$

(26)

In this expression, the relation between the strain and the stress is linear (with modulus $\mu _{\mathit{ve}}(p)$) and one can derive the Hashin–Shtrikman lower bound for the macroscopic modulus tensor in the Laplace domain as in Ricaud and Masson (2009). In the case of a matrix, denoted by superscript 2, its modulus tensor and concentration are, respectively, $\mathbf{{L}}_{\mathit{ve}}^{(2)}(p)$ and $c^{(2)}$, reinforced by identical aligned ellipsoidal and stiffer fibers ($\mathbf{{L}}_{\mathit{ve}}^{(1)}(p)$ and $c^{(1)}$); Ponte Castañeda and Willis (1995) give the following expression for the Hashin–Shtrikman lower bound:

$$ \tilde{\mathbf{{L}}}_{\mathit{ve}}^{(\mathit{HS})}(p)=\mathbf{{L}}_{\mathit{ve}}^{(2)}(p)+c^{(1)} \bigl[ \bigl( \mathbf{{L}}_{\mathit{ve}}^{(1)}(p)-\mathbf{{L}}_{\mathit{ve}}^{(2)} \bigr) ^{-1} +c ^{(2)} {\mathbf{{P}}} \bigr] ^{-1}, $$

(27)

in which the microstructural tensor $\mathbf{{P}}$ depends on the shape of the ellipsoidal inclusion and the matrix modulus tensor. When the matrix is isotropic the $\mathbf{{P}}$ tensor is transversely isotropic and if the matrix is also incompressible it can be written

$$ {\mathbf{{P}}}=\frac{3}{2}\alpha _{P} {\mathbf{{K}}}_{E}+ \delta _{P} {\mathbf{{K}}}_{T}+\gamma _{P}{ \mathbf{{K}}}_{L}, $$

(28)

with tensors $\mathbf{{K}}_{E}$, $\mathbf{{K}}_{T}$ and $\mathbf{{K}} _{L}$ already defined in Sect. 2 and scalars $\alpha _{P}$, $\delta _{P}$ and $\gamma _{P}$ defined by

$$\begin{aligned} & h(x)=\frac{x ( x \sqrt{x^{2}-1}-\cosh ^{-1}(x) ) }{ ( x^{2}-1 ) ^{3/2} }\quad \mbox{and} \quad x>1, \end{aligned}$$

(29)

$$\begin{aligned} & \alpha _{P}=\frac{-2 x^{2} h(x)-h(x)+2 x^{2}}{2 \mu ( x^{2}-1 ) }, \end{aligned}$$

(30)

$$\begin{aligned} & \delta _{P}=\frac{2 x^{2}-3 h(x)}{8 \mu ( x^{2}-1 ) }, \end{aligned}$$

(31)

$$\begin{aligned} & \gamma _{P}=\frac{ ( x^{2}+1 ) (3 h(x)-2)}{4 \mu ( x^{2}-1 ) }, \end{aligned}$$

(32)

in the case of spheroidal inclusion defined by the aspect ratio $x=\frac{a_{3}}{a_{1}}=\frac{a_{3}}{a_{2}}$ with $a_{i}$ half of the length of the principal axes. In that case, Eq. (27) shows that $\tilde{\mathbf{{L}}}_{\mathit{ve}}^{(\mathit{HS})}(p)$ is transversely isotropic too and can be written

$$ \tilde{\mathbf{{L}}}_{\mathit{ve}}^{(\mathit{HS})}(p)=\frac{3}{2}\alpha _{L}^{(\mathit{HS})}(p) \mathbf{{K}}_{E}+\delta _{L}^{(\mathit{HS})}(p) \mathbf{{K}}_{T} $$

(33)

with the modulus $\alpha _{L}^{(\mathit{HS})}(p)$, $\delta _{L}^{(\mathit{HS})}(p)$ and ${\gamma }_{L}^{(\mathit{HS})}(p)$ given by Eqs. (27) to (33).

The harmonic effective complex moduli tensor is then given by

$$ \tilde{\mathbf{{L}}}^{*(\mathit{HS})}(2\pi f )=\tilde{\mathbf{{L}}}_{\mathit{ve}}^{(\mathit{HS})}(i 2\pi f )=\tilde{\mathbf{{L}}}^{\prime \,(\mathit{HS})}(2\pi f )+i \tilde{\mathbf{{L}}} ^{\prime\prime \,(\mathit{HS})}(2\pi f ), $$

(34)

with $f$ the frequency, $i$ the imaginary unit, $\tilde{\mathbf{{L}}}^{\prime \,(\mathit{HS})}(2\pi f )$ the storage moduli tensor and $\tilde{\mathbf{{L}}}^{\prime\prime \,(\mathit{HS})}(2\pi f )$ the loss moduli tensor.

Appendix B: Meta-model function values

The different identified functions are specific to the exact case of a material consisting in a short glass fiber reinforced PC matrix, and more specifically when the matrix is supposed to behave like a simple spring dashpot model. Thus these should be used with caution. They are all related in Table 1. The linear functions are written as

$$ f(c_{1}) = Ax+B $$

(35)

and the exponential functions are

$$ f(c_{1}) = Ae^{Bx}. $$

(36)

Table 1 Values of the meta-model functions