Computational modeling of fiber flow during casting of fresh concrete

Abstract

The Folgar–Tucker fiber orientation model coupled with weakly compressible Smoothed Particle Hydrodynamics is used to simulate the process of casting of fiber reinforced concrete and to predict the spatial-temporal evolution of the probability density function of fiber orientation. The flowable concrete-fiber mix is modeled as a viscous Bingham-type fluid. Model predictions qualitatively agree with fiber orientations observed in an L-box test with fibers suspended in transparent gel. Important factors and assumptions regarding the fiber flow are reviewed and conclusions are drawn based on numerical experiments.

Appendices

Appendix A: Fitting coefficients for orthotropic closures

The coefficients for generating the orthotropic closure ORW3 according to Chung and Kwon [12] are presented below. The components $A_{11}$, $A_{22}$, and $A_{33}$ are calculated according to Eq. 38.

$$\begin{aligned} \begin{aligned} \bar{A}_{mm}\vert _{ORW3}&= {C_{m}}^{1} + {C_{m}}^{2}a_{1} + {C_{m}}^{3}[a_{1}]^{2} + {C_{m}}^{4}a_{2} \\&\quad + {C_{m}}^{5}[a_{2}]^{2} + {C_{m}}^{6}a_{1}a_{2} + {C_{m}}^{7}[a_{1}]^{2}a_{2} \\&\quad + {C_{m}}^{8}a_{1}[a_{2}]^{2} + {C_{m}}^{9}[a_{1}]^{3} + {C_{m}}^{10}[a_{2}]^{3}, \end{aligned} \end{aligned}$$

(38)

with $m=1,2,3$; no sum on m. Here, $a_{1}$, $a_{2}$, and $a_{3}$ are eigenvalues of the second order orientation tensor, and the coefficients ${C_{m}}^{k}$ are taken according to Table 2.

Table 2 Coefficients for ORW3 closure (Chung and Kwon [12])

Appendix B: Important matrices and vectors in Mandel notation

Formulas for the construction of Mandel vectors and Mandel matrices from 2nd and 4th order tensors are adopted here according to [9]. Note, that for symmetric tensors, due to the nature of Mandel notation, the gray entries in Eqs. 39 and 40 vanish, therefore it is sufficient to use only $6\times 1$ and $6\times 6$ vectors and matrices. For antisymmetric tensors, the black entries vanish.

(39)

(40)

where

and $a_{ij}$, $a_{ijkl}$ are components of 2nd and 4th order tensors, respectively. Reduced Strain Closure Folgar–Tucker equation for determination of the orientation state is solved for every integration point (SPH particle) at every time step.

$$\begin{aligned} \begin{aligned} \frac{d{\mathbf a}_2^{RSC}}{dt}&=- \frac{1}{2}(\dot{{{\varvec{\omega }}}}\cdot {\mathbf a}_2 -{\mathbf a}_2\cdot \dot{{{\varvec{\omega }}}})+\frac{1}{2}{\lambda }({\dot{{{\varvec{\varepsilon }}}}}\cdot {\mathbf a}_2+{\mathbf a}_2\cdot {\dot{{{\varvec{\varepsilon }}}}} \\&\quad - 2 [{\mathbf a}_4+(1-{\kappa })({\mathbf L}-{\mathbf M}: {\mathbf a}_4)] : {\dot{{{\varvec{\varepsilon }}}}}) + 2{\kappa }D_{r}({\mathbf I}-3{\mathbf a}_2), \end{aligned} \end{aligned}$$

(41)

where ${\kappa }$ is an empirical strain reduction factor ($0\le {\kappa }\le 1$),

$$\begin{aligned} {\mathbf L}= & {} \sum _{i=1}^{3} a_i {\mathbf e}_i {\mathbf e}_i {\mathbf e}_i {\mathbf e}_i = {\mathbf L}_1+{\mathbf L}_2+{\mathbf L}_3\nonumber \\= & {} a_1 {\mathbf M}_1 + a_2{\mathbf M}_2 + a_3{\mathbf M}_3, \end{aligned}$$

(42)

$$\begin{aligned} {\mathbf M}= & {} \sum _{i=1}^{3} {\mathbf e}_i {\mathbf e}_i {\mathbf e}_i {\mathbf e}_i = {\mathbf M}_1 + {\mathbf M}_2 + {\mathbf M}_3, \end{aligned}$$

(43)

with $a_{i}$ and ${\mathbf e}_i$ being eigenvalues and eigenvectors of 2nd order orientation tensor respectively. Matrices ${\mathbf M}_1$, ${\mathbf M}_2$, ${\mathbf M}_3$ are presented below (Eqs. 44, 45, 46).

(44)

(45)

(46)

For efficient computation, the terms $\dot{{{\varvec{\omega }}}} \cdot {\mathbf a}_2 - {\mathbf a}_2 \cdot \dot{{{\varvec{\omega }}}}$ and $\dot{{{\varvec{\varepsilon }}}} \cdot {\mathbf a}_2 + {\mathbf a}_2 \cdot \dot{{{\varvec{\varepsilon }}}}$ on right hand side of the Eq. 41 are recast in following form:

(47)

where $\dot{{{\omega }}}_{7}=\dot{{{\omega }}}_{\overline{32}}$, $\dot{{{\omega }}}_{8}=\dot{{{\omega }}}_{\overline{13}}$, $\dot{{{\omega }}}_{9}=\dot{{{\omega }}}_{\overline{21}}$.

(48)

where $\dot{{\varepsilon }}_1=\dot{{\varepsilon }}_{11}$, $\dot{{\varepsilon }}_2=\dot{{\varepsilon }}_{22}$, $\dot{{\varepsilon }}_3=\dot{{\varepsilon }}_{33}$, $\dot{{\varepsilon }}_4=\dot{{\varepsilon }}_{\underline{23}}$, $\dot{{\varepsilon }}_5=\dot{{\varepsilon }}_{\underline{31}}$, $\dot{{\varepsilon }}_6=\dot{{\varepsilon }}_{\underline{12}}$. Note, that $\dot{{{\varvec{\omega }}}}$ is an antisymmetric tensor and $\dot{{{\varvec{\varepsilon }}}}$ is symmetric and, consequently, many terms vanish due to the use of Mandel notation. The remaining terms of Eq. 41 do not need a special treatment and are just recast into Mandel notation according to Eqs. 39 and 40.

Appendix C: Rotation of fourth order tensor in Mandel notation

After the orthotropic closure tensor is generated according to Eq. 38, it needs to be rotated from principal into the initial coordinate system according to

$$\begin{aligned} {\mathbf A}={\mathbf R}: \tilde{{\mathbf A}}^{principal}:{\mathbf R}^T=R_{IP}\tilde{A}^{principal}_{PQ} R_{JQ} \end{aligned}$$

(49)

Rotation tensor is given below (Eq. 50).

(50)

with $e_{ij}$ being the components of three eigenvectors ${\mathbf e}_{i}$ of the second order orientation tensor.