Shape and packing effects in particulate composites: micromechanical modelling and numerical verification

Abstract

The aim of this study is to analyse the joint effect of reinforcement shape and packing on the effective behaviour of particulate composites. The proposed semi-analytical modelling method combines the Replacement Mori–Tanaka scheme, by means of which the concentration tensors for non-ellipsoidal inhomogeneities are found numerically, and the analytical morphologically representative pattern approach to account for particle packing. Five shapes of inhomogeneities are selected for the analysis: a sphere, a prolate ellipsoid, a sphere with cavities, an oblate spheroid with a cavity as well as an inhomogeneity created by three prolate spheroids crossing at right angles. Semi-analytical estimates are compared with the results of numerical simulations performed using the finite element method and with the outcomes of classical mean-field models based on the Eshelby solution, e.g. the Mori–Tanaka model or the self-consistent scheme.

Appendices

Appendix A: Stiffness tensor decompositions: transverse isotropy and cubic symmetry

In the paper, five different shapes are studied (see Fig. 3a–e). Let us assume that the spheroid is prolate (Fig. 3c) or oblate and drilled (Fig. 3e) in the $\mathbf {m}_1$ direction. Then the stiffness tensor $\overline{\mathbb {L}}^{dil }_{i }$ of a tiny spheroid embedded in a large cube (Fig. 4) and the effective stiffness tensor $\overline{\mathbb {L}}_{NDil }$ in Eq. 14 exhibit transverse isotropy and can be written in the following form [43]:

$$\begin{aligned} \overline{\mathbb {L}}^{Trans }= 3 \overline{K} \mathbb {I}^{P }+ 2 \overline{G}_{1} \mathbb {P}_{1}+ 2 \overline{G}_{2} \mathbb {P}_{2}+ 2 \overline{G}_{3} \mathbb {P}_{3}+ 2 \overline{G}_{12} \frac{1}{\sqrt{6}}\left( \mathbf {d}\otimes \mathbf {I} + \mathbf {I}\otimes \mathbf {d} \right) \,. \end{aligned}$$

(A.1)

The orthogonal projectors $\mathbb {P}_K$ ($K=1,2,3$) define:

• 1D space of isochoric tension/compression along $\mathbf {m}_1$

  $$\begin{aligned} \mathbb {P}_{1}= \mathbf {d}\otimes \mathbf {d}\,,\quad \mathbf {d}=\frac{1}{\sqrt{6}}\left( 3\mathbf {m}_1\otimes \mathbf {m}_1-\mathbf {I}\right) \,, \end{aligned}$$

  (A.2)

• 2D space of in-plane shears

  $$\begin{aligned} \mathbb {P}_{2}= \frac{1}{2} \left[ \left( \mathbf {m}_2\otimes \mathbf {m}_3+\mathbf {m}_3\otimes \mathbf {m}_2\right) \otimes \left( \mathbf {m}_2\otimes \mathbf {m}_3+\mathbf {m}_3\otimes \mathbf {m}_2\right) + \left( \mathbf {m}_2\otimes \mathbf {m}_2-\mathbf {m}_3\otimes \mathbf {m}_3\right) \otimes \left( \mathbf {m}_3\otimes \mathbf {m}_3-\mathbf {m}_2\otimes \mathbf {m}_2\right) \right] \,, \end{aligned}$$

  (A.3)

• 2D space of out-of-plane shears where

  $$\begin{aligned} \mathbb {P}_{3}= \frac{1}{2} \sum _{k=2,3} \left( \mathbf {m}_1\otimes \mathbf {m}_k+\mathbf {m}_k\otimes \mathbf {m}_1\right) \otimes \left( \mathbf {m}_1\otimes \mathbf {m}_k+\mathbf {m}_k\otimes \mathbf {m}_1\right) \,. \end{aligned}$$

  (A.4)

As it was mentioned, to calculate the five independent components: $3\overline{K}$, $2\overline{G}_1$, $2 \overline{G}_{12}$, $2 \overline{G}_2$, $2 \overline{G}_3$, a set of four analyses with micro-periodic displacement boundary conditions (Eq. 16)   were performed for a unit volume element. The four strain tensors $\mathbf {E}^{(n)}$ ($n=0,1,2,3$) imposed in these analyses are given by Eq. (19).

The strains $\mathbf {E}^{\left( n\right) }$ result in the overall stresses, calculated as the local stresses averaged over the RVE’s volume, $\mathbf {\Sigma }^{(n)}=1/V\int _V\varvec{\sigma }dV$, which allow one to derive each independent component of $\overline{\mathbb {L}}^{Trans }$ according to the relations:

$$\begin{aligned}&3 \overline{K}=\frac{{\Sigma }^{\left( 0\right) }_{11}+2{\Sigma }^{\left( 0\right) }_{22}}{3d}\,,\quad 2\overline{G}_{12}=\frac{{\Sigma }^{\left( 0\right) }_{11}-{\Sigma }^{\left( 0\right) }_{22}+{\Sigma }^{\left( 1\right) }_{11}+2{\Sigma }^{\left( 1\right) }_{22}}{3d}\,,\quad 2 \overline{G}_1=\frac{2\left( {\Sigma }^{\left( 1\right) }_{11}-{\Sigma }^{\left( 1\right) }_{22}\right) }{3d}\,,\quad \nonumber \\ 2\overline{G}_2=\frac{{\Sigma }^{\left( 2\right) }_{23}}{d}\,,\quad 2 \overline{G}_3=\frac{{\Sigma }^{\left( 3\right) }_{12}}{d}\,,\quad \end{aligned}$$

(A.5)

where ${\Sigma }^{\left( n\right) }_{i\,j}$ denotes the ij component of the averaged stress tensor $\mathbf {\Sigma }^{\left( n\right) }$ obtained as a material response to the strain $\mathbf {E}^{\left( n\right) }$.

For the other shapes, i.e. the drilled sphere and crossed spheroids (Fig. 3c and d), the effective elastic stiffness tensor $\overline{\mathbb {L}}^{Cub }$ of the unit cells is the anisotropic fourth-order tensor relevant for cubic symmetry with three Kelvin moduli $3\overline{K}$, $2\overline{G}_1$, and $2\overline{G}_2$, namely [38]:

$$\begin{aligned} \overline{\mathbb {L}}^{Cub }= 3\overline{K}\mathbb {I}^{\mathrm{P}}+ 2\overline{G}_1(\mathbb {K}-\mathbb {I}^{P })+ 2\overline{G}_2(\mathbb {I}-\mathbb {K})\,, \end{aligned}$$

(A.6)

where

$$\begin{aligned} \mathbb {K}=\sum _{k=1}^3\mathbf {m}_k\otimes \mathbf {m}_k\otimes \mathbf {m}_k\otimes \mathbf {m}_k\,,\quad \mathbb {I}^{P }=\frac{1}{3}\mathbf {I}\otimes \mathbf {I}\,, \end{aligned}$$

(A.7)

and $\mathbf {m}_k$ are the main symmetry axes of the unit cell. The eigensubspaces corresponding to $3\overline{K}$, $2\overline{G}_1$, and $2\overline{G}_2$ are one dimensional (the space of hydrostatic states), two dimensional, and three dimensional (two orthogonal subspaces of deviatoric states), respectively.

Only three FE analyses with strains $\mathbf {E}^{\left( 0\right) }$, $\mathbf {E}^{\left( 1\right) }$, and $\mathbf {E}^{\left( 2\right) }$ imposed by periodic boundary conditions (Eq. 19) are needed to derive each Kelvin modulus independently, according to the relations:

$$\begin{aligned} 3 \overline{K}=\frac{\mathbf {\Sigma }^{\left( 0\right) }_{11}}{d}\,,\quad 2 \overline{G}_1=\frac{\mathbf {\Sigma }^{\left( 1\right) }_{11}}{d}\,,\quad 2 \overline{G}_2=\frac{\mathbf {\Sigma }^{\left( 2\right) }_{23}}{d}\,.\quad \end{aligned}$$

(A.8)

It can be seen that the above three strains $\mathbf {E}^{\left( 0\right) }$, $\mathbf {E}^{\left( 1\right) }$, and $\mathbf {E}^{\left( 2\right) }$ belong to three distinct eigensubspaces of the tensor $\overline{\mathbb {L}}^{Cub }$ and result in the overall stresses $\mathbf {\Sigma }^{\left( 0\right) }$, $\mathbf {\Sigma }^{\left( 1\right) }$, and $\mathbf {\Sigma }^{\left( 2\right) }$ of the same structure as strains. Finally, the response of a unit volume element reinforced with a single tiny spherical inhomogeneity (Fig. 4) is isotropic.

The situation changes in the case of unit cells with the FCC placement of inhomogeneities (Fig. 9), which for spherical inclusions are of cubic symmetry. If the composite is reinforced with prolate (Fig. 9c) or oblate (Fig. 9e) spheroids, the effective stiffness tensor $\overline{\mathbb {L}}$ has tetragonal symmetry and can be written in the following form:

$$\begin{aligned} \overline{\mathbb {L}}^{Tetra }= 3 \overline{K} \mathbb {I}^{P }+ 2 \overline{G}_{1} \mathbb {P}_{1}+ 2 \overline{G}_{2} \hat{\mathbb {P}}_{2}+ 2 \overline{G}_{3} \mathbb {P}_{3}+ 2 \overline{G}_{4} \hat{\mathbb {P}}_{4}+ 2 \overline{G}_{12} \frac{1}{\sqrt{6}}\left( \mathbf {d}\otimes \mathbf {I} + \mathbf {I}\otimes \mathbf {d} \right) \,, \end{aligned}$$

(A.9)

where $3\overline{K}$, $2\overline{G}_1$, $2 \overline{G}_{12}$, $2 \overline{G}_2$, $2 \overline{G}_3$, $2 \overline{G}_4$ are six independent components. Note that it is assumed that the main axis of a prolate or oblate spheroid, or a drilled sphere, is coaxial with the $\mathbf {m}_1$ direction, which is at the same time coaxial with one of the unit cell’s edges $\mathbf {m}_k$. Under such conditions, the projectors $\mathbb {P}_K$ (K=1,3) for tetragonal symmetry are the same as for the transversal one. The projectors $\hat{\mathbb {P}}_2$ and $\hat{\mathbb {P}}_4$ sum up to $\mathbb {P}_2$ for transverse isotropy. They define two 1D orthogonal subspaces of shearing in the 2–3 plane, namely:

• pure shear along the $\mathbf {m}_2$ and $\mathbf {m}_3$ directions,

  $$\begin{aligned} \hat{\mathbb {P}}_{2}= \frac{1}{2} \left[ \left( \mathbf {m}_2\otimes \mathbf {m}_3+\mathbf {m}_3\otimes \mathbf {m}_2\right) \otimes \left( \mathbf {m}_2\otimes \mathbf {m}_3+\mathbf {m}_3\otimes \mathbf {m}_2\right) \right] \,, \end{aligned}$$

  (A.10)

• pure shear along directions inclined by $45^o$ with respect to the $\mathbf {m}_2$ and $\mathbf {m}_3$ directions,

  $$\begin{aligned} \hat{\mathbb {P}}_{4}= \frac{1}{2} \left[ \left( \mathbf {m}_2\otimes \mathbf {m}_2-\mathbf {m}_3\otimes \mathbf {m}_3\right) \otimes \left( \mathbf {m}_3\otimes \mathbf {m}_3-\mathbf {m}_2\otimes \mathbf {m}_2\right) \right] \,. \end{aligned}$$

  (A.11)

To find the six independent components of $\bar{\mathbb {L}}^{Tetra }$, the four analyses with the overall strains $\mathbf {E}^{(n)}$ (n = 0, 1, 2, 3) given by Eqs. (19), performed in the case of the overall transverse isotropy, are completed with a fifth analysis with an imposed strain of the following representation in the basis $\{\mathbf {m}_i\}$:

$$\begin{aligned} E^{\left( 4\right) }_{ij}= \left( \begin{array}{ccc} 0 &{} 0&{} 0 \\ 0 &{} d &{} 0 \\ 0 &{} 0 &{} -d \end{array}\right) \,, \end{aligned}$$

(A.12)

where d specifies the strain magnitude.

The imposed strains $\mathbf {E}^{\left( n\right) }$ result in the overall stresses $\mathbf {\Sigma }^{\left( n\right) }$, which allow one to derive each independent component of $\overline{\mathbb {L}}^{Tetra }$ according to Eq. A.5 with the addition of $\overline{G}_4$:

$$\begin{aligned} 2 \overline{G}_4=\frac{\mathbf {\Sigma }^{\left( 4\right) }_{22}}{d}\,. \end{aligned}$$

(A.13)

The effective stiffness of a composite with the FCC arrangement of inhomogeneities shown in Fig. 9a, b and d is of cubic symmetry as long as the main axes of ingomogeneities coincide with the axes of the unit cell.

Appendix B: The form of concentration tensors: transverse isotropy and cubic symmetry

The numerical strain concentration tensor $\mathbb {A}_{i }^{NDil }$ of an inhomogeneity has the same symmetry group as the effective stiffness tensor of a large unit cell (Fig. 4) with a tiny inhomogeneity. Thus, for inhomogeneities having the shape of the prolate spheroid or drilled oblate spheroid (Fig. 3b and e) its symmetry group is that for transverse isotropy:

$$\begin{aligned} \mathbb {A}^{Trans }= A^{Trans }_0 \mathbb {I}^{P }+ A^{Trans }_1 \mathbb {P}_{1}+ A^{Trans }_2 \mathbb {P}_{2}+ A^{Trans }_3 \mathbb {P}_{3}+ \frac{1}{\sqrt{6}}\left( A^{Trans }_{12} \mathbf {d}\otimes \mathbf {I} +A^{Trans }_{21} \mathbf {I}\otimes \mathbf {d} \right) \,, \end{aligned}$$

(B.1)

and for the other shapes: drilled sphere and crossed spheroids (Fig. 3c and d), it has cubic symmetry, viz.

$$\begin{aligned} \mathbb {A}^{Cub }= A^{Cub }_0\mathbb {I}^{\mathrm{P}}+ A^{Cub }_1(\mathbb {K}-\mathbb {I}^{P })+ A^{Cub }_2(\mathbb {I}-\mathbb {K})\,, \end{aligned}$$

(B.2)

where the projectors $\mathbb {P}_K$ and the remaining tensorial quantities, e.g. $\mathbb {K}$, are listed above; see Eqs. A.2, A.3, and A.7. Finally, if a particle is spherical then the numerical strain concentration tensor is isotropic.

For the set of micro-periodic displacement boundary conditions specified by the strains $\mathbf {E}^{(n)}$ from Eq. (19), the components $A^{Cub }$ and $A^{Trans }$, established from Eq. (15), are:

$$\begin{aligned}&A^{Trans }_0=\frac{\langle \varvec{\varepsilon }\rangle ^{\left( 0\right) }_{11}+2\langle \varvec{\varepsilon }\rangle ^{\left( 0\right) }_{22}}{3d}\,,\quad A^{Trans }_{12}\approx A^{Trans }_{21}=\frac{\langle \varvec{\varepsilon }\rangle ^{\left( 0\right) }_{11}-\langle \varvec{\varepsilon }\rangle ^{\left( 0\right) }_{22}+\langle \varvec{\varepsilon }\rangle ^{\left( 1\right) }_{11}+2\langle \varvec{\varepsilon }\rangle ^{\left( 1\right) }_{22}}{3d}\,,\quad A^{Trans }_1=\frac{2\left( \langle \varvec{\varepsilon }\rangle ^{\left( 1\right) }_{11}-\langle \varvec{\varepsilon }\rangle ^{\left( 1\right) }_{22}\right) }{3d}\,,\quad \nonumber \\&A^{Trans }_2=\frac{\langle \varvec{\varepsilon }\rangle ^{\left( 2\right) }_{23}}{d}\,,\quad A^{Trans }_3=\frac{\langle \varvec{\varepsilon }\rangle ^{\left( 3\right) }_{12}}{d}\,,\quad \nonumber \\&A^{Cub }_0=\frac{\langle \varvec{\varepsilon }\rangle ^{\left( 0\right) }_{11}}{d}\,,\quad A^{Cub }_1=\frac{\langle \varvec{\varepsilon }\rangle ^{\left( 1\right) }_{11}}{d}\,,\quad A^{Cub }_2=\frac{\langle \varvec{\varepsilon }\rangle ^{\left( 2\right) }_{23}}{d}\,,\quad \end{aligned}$$

(B.3)

where $\langle \cdot \rangle$ is the volume averaging operation defined as $1/V \int _V (\cdot )dV$, $\varvec{\varepsilon }$ is the local strain tensor in the inhomogeneity domain, $\langle \varvec{\varepsilon }\rangle ^{\left( n\right) }_{ij}$ is the component ij of the inhomogeneity’s average strain in response to the displacement BC given by $\mathbf {E}^{\left( n\right) }$. The simplification $A^{Trans }_{12}\approx A^{Trans }_{21}$ was assumed taking into account very small values of this component compared to the remaining ones (see Table 2).

Appendix C: The impact of inclusion shape on damage evolution

As it has been shown, the shape of inclusions may substantially influence the local strain and stress fields in the composite phases. These local fields have a prominent effect on the initiation of damage in the material [20]. Below, we analyze damage evolution in composites with the considered shapes of heterogeneities by means of numerical homogenization.

Material degradation is simulated by FEM using the concept of the damage parameter d and the framework of continuous damage mechanics [44]. Within the local constitutive model of the phases, the free energy density $\Pi$ is postulated in the form

$$\begin{aligned} \Pi = (1-d) \Pi _e(\varvec{\varepsilon })\,, \end{aligned}$$

(C.1)

where $\Pi _e$ is the elastic energy density specified as

$$\begin{aligned} \Pi _e = \frac{1}{2}\varvec{\varepsilon }\cdot \mathbb {L}_{i/m }\cdot \varvec{\varepsilon }\,. \end{aligned}$$

(C.2)

The damage parameter d depends on the history parameter $\omega$ which is related to the elastic energy density according to the exponential law [45]:

$$\begin{aligned} d=d_{max } \left( 1 - Exp [- H \omega ]\right) , \end{aligned}$$

(C.3)

where $d_{max }=0.8$ is taken as the maximum damage, $H=0.1$ is the damage ductility, and $\omega$ is equal to the maximum value of the elastic energy density $\Pi _e$ achieved during the deformation process up to the considered time step.

Figure 16a presents the average value of the von Mises stress, calculated from the effective stress $\bar{\sigma }$, versus the overall strain component $\varepsilon _{23}$ of the MMC (Table 1) subjected to periodic boundary conditions (16) with $\mathbf {E}=\mathbf {E}^{(2)}$. Similarly to [46], damage evolution is enabled only in the matrix phase. Figure 16b shows the average damage evolution $d_{m }=1/V_{m } \int d \, d V_{m }\,$ in the matrix phase for five different inclusion shapes in the FCC spatial arrangement. The studied microstructures are presented in Fig. 9.

Fig. 16

MMC with the FCC particle distribution with a volume fraction $f_{i }=0.3$, subjected to the shear specified by $\mathbf {E}^{\left( 2\right) }$ (Eq. 19). Damage development is enabled in the matrix phase only. a The overall equivalent von Mises stress $\overline{\sigma }_{Mises }$, and b the averaged matrix damage d vs the overall strain component $\overline{\varepsilon }_{23}$

The results indicate that the shape of inhomogeneities plays a critical role in the non-linear response of the particulate composite. As can be seen in Fig. 16a, the difference in the overall von Mises stress between the crossed spheroids and the prolate spheroids is almost 500[MPa]. In all cases we observe first an increase in the stresses accompanied by a smooth reduction of stiffness. Next, some stabilization of the stress level is seen, so the strain increases under an approximately constant stress. Finally, the overall stresses start to increase again. The reason for this behaviour is the damage evolution law (Eq. C.1) and the assumption of the maximum damage $d_{max }=0.8$ which is less than 1.0, so the elastic stiffness of the matrix never degrades to zero. When d reaches its maximum value $d_{max }$, the locally damaged matrix becomes elastic but with a smaller stiffness: $(1-d_{max })\mathbb {L}_{m }$. The contour maps shown in Fig. 17 present the distribution of the damage parameter d at $\overline{\varepsilon }_{23}=0.02$. Details of the distribution vary with the inclusion shape. In the MMC reinforced with prolate spheroids, wide and almost straight damage bands develop (see Fig. 17c), while for crossed spheroids damage bands are more curved (see Fig. 17d), and for drilled spheres damage develops in the matrix inside the inclusions (see Fig. 17b). In general, higher damage localization is observed in specimens with a smaller distance between the particles’ external surfaces and with more spherical shapes of inclusions.

Fig. 17

Distribution of the damage parameter d (contour maps) at the strain $\overline{\varepsilon }_{23}=0.02$ in the shear test specified by $\mathbf {E}^{(2)}$ (Eq. 19). The MMC is reinforced with ceramic particles of five different shapes: a spheres, b drilled spheres, c prolate spheroids, d crossed spheroids, and e drilled oblate spheroids (see Fig. 3) placed in the FCC arrangement (Fig. 9). The damage evolution law of the matrix phase is specified by Eq. C.1 (damage evolution is disabled in the inclusions)