Fourier-based strength homogenization of porous media

Abstract

An efficient numerical method is proposed to upscale the strength properties of heterogeneous media with periodic boundary conditions. The method relies on a formal analogy between strength homogenization and non-linear elasticity homogenization. The non-linear problems are solved on a regular discretization grid using the Augmented Lagrangian version of Fast Fourier Transform based schemes initially introduced for elasticity upscaling. The method is implemented for microstructures with local strength properties governed either by a Green criterion or a Von Mises criterion, including pores or rigid inclusions. A thorough comparison with available analytical results or finite element elasto-plastic simulations is proposed to validate the method on simple microstructures. As an application, the strength of complex microstructures such as the random Boolean model of spheres is then studied, including a comparison to closed-form Gurson and Eshelby based strength estimates. The effects of the microstructure morphology and the third invariant of the macroscopic stress tensor on the homogenized strength are quantitatively discussed.

Appendices

Appendix 1: Green operator

1.1 (a) Continuous operator

Consider the following problem of prestressed linear elasticity:

$$\begin{aligned} \begin{aligned}&\text {find } {\varvec{\xi }} \in \mathcal {K}({\varvec{E}}), \, {\varvec{\sigma }} \in \mathcal {S} \text { such that } \forall {\varvec{z}} \in \varOmega , \\&\qquad {\varvec{\sigma }}({\varvec{z}}) = \mathbb {C}_0 : {\varvec{\epsilon }}({\varvec{z}}) + {\varvec{\tau }}({\varvec{z}}) \text { with } {\varvec{\epsilon }} = {\varvec{{{\mathrm{grad}}}}}^s {\varvec{\xi }} \end{aligned} \end{aligned}$$

(54)

In this problem, ${\varvec{\xi }}$ is the displacement field, ${\varvec{\epsilon }}$ the associated strain field and ${\varvec{\sigma }}$ the stress field. The set $\mathcal {K}({\varvec{E}})$ (resp. $\mathcal {S}$) is defined by (8) (resp. (12)). The stiffness $\mathbb {C}_0$ is homogeneous, ${\varvec{E}}$ is the imposed macroscopic deformation and ${\varvec{\tau }}$ is the so-called polarization field which is an imposed arbitrary loading parameter corresponding to a prestress field.

The displacement field ${\varvec{\xi }}$ solution to the prestressed problem (54) minimizes the potential energy among kinematically admissible displacement fields ${\varvec{\xi }}^\prime \in \mathcal {K}({\varvec{E}})$:

$$\begin{aligned} {\varvec{\xi }} = \text {arg } \min _{{\varvec{\xi }}^\prime \in \mathcal {K}({\varvec{E}})} \int _{\varOmega } \dfrac{1}{2} {\varvec{\epsilon }}^\prime : \mathbb {C}_0 : {\varvec{\epsilon }}^\prime + {\varvec{\epsilon }}^\prime : {\varvec{\tau }} {{\mathrm{d}}}V \end{aligned}$$

(55)

By definition, the Green operator ${\varvec{\varGamma }}_0$ is the operator such that the strain field solution to (54) is:

$$\begin{aligned} \begin{aligned}&{\varvec{\epsilon }}({\varvec{z}}) = {\varvec{E}} - \left( {\varvec{\varGamma }}_0 *{\varvec{\tau }}\right) ({\varvec{z}})\\&\text {where } \left( {\varvec{\varGamma }}_0 *{\varvec{\tau }}\right) ({\varvec{z}}) = \int _{\varOmega } {\varvec{\varGamma }}_0({\varvec{x}},{\varvec{y}}) : {\varvec{\tau }}({\varvec{y}}) {{\mathrm{d}}}V_y \end{aligned} \end{aligned}$$

(56)

In the Fourier space, let ${\varvec{k}}$ denote the wave vector, $k =|{\varvec{k}}|$ its norm and ${\varvec{n}} = {\varvec{k}}/k$ its normalized direction. The Fourier transform of a $\varOmega$-periodic function f is denoted $\hat{f}$ and is computed using the following convention:

$$\begin{aligned} \hat{f}({\varvec{k}}) = \dfrac{1}{|\varOmega |} \int _{\varOmega }{ f({\varvec{z}}) \exp \left( - \imath {\varvec{k}} \cdot {\varvec{z}} \right) {{\mathrm{d}}}V_z}. \end{aligned}$$

(57)

The FFT-based method benefits from the property on the Fourier transform of a convolution product:

$$\begin{aligned} \widehat{{\varvec{\varGamma }}_0 *{\varvec{\tau }}} = {\varvec{\hat{\varGamma }}}_{0} : {\varvec{\hat{\tau }}} \end{aligned}$$

(58)

When the stiffness tensor $\mathbb {C}_0$ is isotropic with shear modulus $\mu _0$ and Poisson’s ratio $\nu _0$, the Fourier transform of the Green operator is:

$$\begin{aligned} {\varvec{\hat{\varGamma }}}_{0}({\varvec{k}}) = {\varvec{k}} \mathop {\otimes }\limits ^{s} {\varvec{\hat{G}}}_{0}({\varvec{k}}) \mathop {\otimes }\limits ^{s} {\varvec{k}} . \end{aligned}$$

(59)

where ${\varvec{\hat{G}}}_{0}$ is the Fourier transform of the Green function:

$$\begin{aligned} {\varvec{\hat{G}}}_{0}({\varvec{k}}) = {\left\{ \begin{array}{ll} \dfrac{1}{\mu _{0}k^{2}} \left( {\varvec{1}}- \dfrac{1}{2 (1- \nu _{0})} {\varvec{n}} \otimes {\varvec{n}} \right) &{} \text {if } {\varvec{k}} \ne {\varvec{0}},\\ {\varvec{0}} &{}\text {if } {\varvec{k}} = {\varvec{0}}. \end{array}\right. } \end{aligned}$$

(60)

The Green operator is null for ${\varvec{k}} = {\varvec{0}}$ consistently with the property ${\varvec{\varGamma }}_0 *{\varvec{\tau }} \in \mathcal {K}({\varvec{0}})$.

1.2 (b) Discretized operators

The discretization of the fictitious non-linear problem carried out in Sect. 3.2 leads to compute in (27) the average over any voxel of the strain rate ${\varvec{d}}$ solution to (54) for a voxel-wise constant polarization field. As shown in [7], the term involving the Green operator can be rigorously computed using the Discrete Fourier Transform as detailed in Sect. 3.3 provided that the consistent discrete Green operator $\hat{{\varvec{\varGamma }}}_0^{\text {c}}$ is used, defined by:

$$\begin{aligned} \begin{aligned} \forall {\varvec{b}} \in \mathcal {I} , \, \hat{{\varvec{\varGamma }}}^{\text {c}}_{0,{\varvec{b}}} = \sum _{{\varvec{n}} \in \mathbb {Z}^{d}}&\left[ \prod _{i \in (1,..,d)} {{\mathrm{sinc}}}\left( \dfrac{\pi b_{i}}{N_{i}} \right) \right] ^{2} \\&\times \hat{{\varvec{\varGamma }}}_0({\varvec{k}}_{b_{1}+n_{1}N_{1}, ..., b_{d}+n_{d}N_{d}} ), \end{aligned} \end{aligned}$$

(61)

where for any multi-index ${\varvec{b}} = b_1,...,b_d$ of $\mathbb {Z}^d$ the wave vector ${\varvec{k}}_{{\varvec{b}}}$ is defined by:

$$\begin{aligned} {\varvec{k}}_{{\varvec{b}}} = \dfrac{2\pi b_1}{L_1}{\varvec{e}}_1 + ... + \dfrac{2\pi b_d}{L_d}{\varvec{e}}_d. \end{aligned}$$

(62)

with the notations introduced in Sect. 3.2.

As the consistent discrete Green operator $\hat{{\varvec{\varGamma }}}_0^{\text {c}}$ involves infinite sums, it is uneasy to implement. As an alternative, a so-called filtered, non consistent Green operator $\hat{{\varvec{\varGamma }}}_0^{\text {fnc}}$ has been introduced in [8] as a good approximation to the consistent discrete Green operator:

$$\begin{aligned} \begin{aligned} \forall {\varvec{b}} \in \mathcal {I} , \, \hat{{\varvec{\varGamma }}}^{\text {fnc}}_{0,{\varvec{b}}} = \sum _{{\varvec{n}} \in \{0,1\}^{d}}&\left[ \prod _{i \in (1,..,d)} \cos \left( \dfrac{\pi b_{i}}{2 N_{i}} \right) \right] ^{2} \\&\times \hat{{\varvec{\varGamma }}}_0({\varvec{k}}_{b_{1}+n_{1}N_{1}, ..., b_{d}+n_{d}N_{d}} ), \end{aligned} \end{aligned}$$

(63)

This approximate operator can be readily computed at each iteration and does not require to be stored in memory.

Appendix 2: Estimates of the homogenized strength of porous material with Von Mises solid phase

1.1 (a) Criteria based on an exterior kinematic approach of yield design on a hollow sphere

Gurson criterion The Gurson criterion corresponds to an exterior kinematic approach of the yield design theory on a hollow sphere with a porosity $\phi$ and a Von Mises solid matrix [21]. Using the notations (40) for the Von Mises criterion, the Gurson criterion reads:

$$\begin{aligned} \dfrac{\varSigma _d^2}{b(1+\phi ^2)} + \dfrac{2 \phi }{1+\phi ^2} \cosh \left( \sqrt{\dfrac{3}{2b}}\varSigma _m\right) - 1 \le 0 \end{aligned}$$

(64)

As this criterion does not take into account interaction between pore, [55] has proposed to replace $\phi$ by the correcting term $q_1 \phi$ with $q_1$ around 1.5.

Criterion based on trial fields from linear elasticity While using the kinematic fields solution to the same hollow sphere with an incompressible linear elastic matrix which are richer than the trial fields used by Gurson, the derived criterion improves the Gurson criterion (see e.g. [3]). Although the expression are no longer known in closed-form, they only require a numerical volume integration and feature effects of the third invariant of the macroscopic stress tensor.

1.2 (b) Eshelby-based criteria

Non-linear techniques relying on Eshelby-based homogenization schemes have been used to estimate the solution to the fictitious problem (17).

In case of heterogeneous material comprising a Von Mises solid phase and pores, these techniques rely on the estimate of the homogenized stiffness $\mathbb {C}^{\text {hom}}$ of a linear elastic composite with pores and incompressible solid phase with shear modulus $\mu _s$. When $\mathbb {C}^{\text {hom}}$ is isotropic,

$$\begin{aligned} \mathbb {C}^{\text {hom}} = 3k^{\text {hom}} \mathbb {J} + 2 \mu ^{\text {hom}} \mathbb {K} \quad \text {with} \quad {\left\{ \begin{array}{ll} k^{\text {hom}} = 2 \mu _s x&{} \\ 2\mu ^{\text {hom}} = 2\mu _s y&{} \\ \end{array}\right. } \end{aligned}$$

(65)

where x and y are functions depending on morphological parameters (porosity, aspect ratio of the constituents, ...).

Based on the modified secant method for non-linear homogenization techniques [52], the homogenized strength properties may then be estimated by a Green strength criterion [14]:

$$\begin{aligned} {\varvec{\varSigma }} \in G^{\text {hom}} \Leftrightarrow \dfrac{\varSigma _m^2}{A} + \dfrac{\varSigma _d^2}{B} - 1 \le 0 \quad \text {with} \quad {\left\{ \begin{array}{ll} A = (1-\phi ) x b&{} \\ B = (1-\phi ) y b&{} \\ \end{array}\right. } \end{aligned}$$

(66)

Fig. 14

Schematic representation of the interpolation of octahedral cuts by the conic (70) for Lode angles $\theta \in [0;\pi /3]$ based on the three values $R_1=\varSigma _d(\theta =0)$, $R_2=\varSigma _d(\theta =\pi /6)$ and $R_3=\varSigma _d(\theta =\pi /3)$ and its symmetric and periodic reproductions. Yellow area admissible values of $R_2$ to ensure convexity for given values of $R_1$ and $R_3$ [see (73)]. (Color figure online)

Values of x and y are recalled below for classical homogenization schemes (see [14] for details):

• Mori–Tanaka scheme [36] with spherical pores

  $$\begin{aligned} x_{\text {mt}} = 2\dfrac{1-\phi }{3\phi }\quad ; \quad y_{\text {mt}} = \dfrac{1-\phi }{1+2\phi /3} \end{aligned}$$

  (67)

• Differential scheme [40] with spherical pores

  $$\begin{aligned} x_{\text {ds}} = \dfrac{2y_{\text {ds}}}{3(1- y_{\text {ds}}^{3/5})} \quad ; \quad \dfrac{y_{\text {ds}}^3}{2-y_{\text {ds}}^{3/5}}=(1-\phi )^6 \end{aligned}$$

  (68)

• Self-consistent scheme [9, 23] with spherical pores and solid grains

  $$\begin{aligned} x_{\text {sc}} = 2\dfrac{(1-2\phi )(1-\phi )}{\phi (3-\phi )}\quad ; \quad y_{\text {sc}} = 3\dfrac{1-2\phi }{3-\phi } \end{aligned}$$

  (69)

The expressions of x and y for the self-consistent scheme with spheroidal solid grains with aspect ratio $\omega _s$ and spheroidal pores with aspect ratio $\omega _p$ having both an isotropic distribution of orientation (see e.g. [50]) are implicit functions of $\phi$, $\omega _s$ and $\omega _p$, too long to be reproduced here.

Appendix 3: Interpolation of octahedral projections of $\partial G^{\text {hom}}$ by combinations of a conic

Projections of the boundary of the macroscopic strength criteria by octahedral planes $\varSigma _m=$ constant can be excellently approximated by a combination of conics (ellipse or hyperbola). In such $(x=\varSigma _{d1},y=\varSigma _{d2})$ plane [refering to (50)], the equation of an arbitrary ellipse or hyperbola of center $(x_0,y_0)$ is:

$$\begin{aligned} \dfrac{(x-x_0)^2}{A}+\dfrac{(y-y_0)^2}{B}+\dfrac{2(x-x_0)(y-y_0)}{C} = 1 \end{aligned}$$

(70)

The conic (70) is an ellipse if $AB-C^2<0$ and $AB>0$. The parameters A, B, C, $x_0$ and $y_0$ are deduced from the values of the stress deviator $\varSigma _d$ at three Lode angles $\theta$: $R_1=\varSigma _d(\theta =0)$, $R_2=\varSigma _d(\theta =\pi /6)$ and $R_3=\varSigma _d(\theta =\pi /3)$ (see Fig. 14). Imposing that the tangent to the conic in $\theta =0$ and $\pi /3$ is orthogonal to the radial vector in polar coordinates, geometrical considerations lead to the unique expression of the conic parameters given in equation (71).

$$\begin{aligned} \begin{aligned} \varDelta&=-27(R_1-R_3)^2R_2^2+2\sqrt{3}(8R_3^2-17R_1R_3+8R_1^2)(R_3+R_1)R_2 -R_1R_3(32R_1^2+32R_3^2-71R_1R_3)\\ B&=\dfrac{(R_1-2R_3)^2(-3R_1R_3+\sqrt{3}R_2R_1+R_2\sqrt{3}R_3)^2}{3 \varDelta }\\ A&=B\dfrac{(R_2\sqrt{3}-2R_1)(2R_3-R_2\sqrt{3})(2R_1-R_3)^2}{(4R_1R_3+2R_1^2-7R_3^2)R_2^2-2\sqrt{3}(R_3+R_1)(2R_1-3R_3)R_3R_2+6R_1R_3^2(R_1-2R_3)}\\ C&=B \dfrac{(-R_2\sqrt{3}+2R_1)(2R_3-R_2\sqrt{3})(2R_1-R_3)^2}{\sqrt{3}(-(R_1-2R_3)^2R_2^2-2\sqrt{3}(R_1-R_3)(R_3+R_1)R_3R_2+R_1R_3^2(-4R_3+5R_1))}\\ x_0&=\dfrac{1}{\varDelta } \times \left[ (18R_1^2R_3-6R_3^3-15R_1^3)R_2^2+2\sqrt{3}(-2R_3^2-R_1R_3+4R_1^2)(R_3+R_1)(R_1-R_3)R_2\right. \\&\qquad \qquad \left. -R_1R_3(16R_1^3-4R_1R_3^2-23R_1^2R_3+8R_3^3) \right] \\ y_0&= \dfrac{\sqrt{3}\left[ (R_1-2R_3)^2R_2^2+2\sqrt{3}(R_1-R_3)(R_3+R_1)R_3R_2-R_1R_3^2(-4R_3+5R_1)\right] (R_1-2R_3)}{\varDelta }\\ \end{aligned} \end{aligned}$$

(71)

For Lode angles $\theta \in [0;\pi /3]$, the interpolation of the macroscopic stress deviator is then explicitly given by equation (72).

(72)

where

$$\begin{aligned} \epsilon = {\left\{ \begin{array}{ll} -1 &{} \text {if } \left( \text {the conic~(70) is an ellipse and } C<0 \right) \\ &{} \text {or } \left( \text {the conic~(70) is a hyperbola and } A<0 \right) \\ 1 &{} \text {otherwise}\\ \end{array}\right. } \end{aligned}$$

Due to the symmetry properties exposed in Sect. 4.1.4, the whole cut of the boundary of the macroscopic strength criteria by the octahedral plane $\varSigma _m=$ constant is constructed first by symmetry with respect to the $\varSigma _{d2}$ axis and then by periodic reproduction with a $2\pi /3$ period.

The convexity of the strength domain imposes the conditions:

$$\begin{aligned} \dfrac{R_3}{2} \le R_1 \le 2R_3 \quad \text {and} \quad \dfrac{\sqrt{3}R_1R_3}{R_1+R_3} \le R_2 \le \dfrac{2}{\sqrt{3}} \min (R_1,R_3) \end{aligned}$$

(73)

In the case where $R_2$ is chosen as a function of $R_1$ and $R_3$ such that $y_0$ vanishes and $C\rightarrow \infty$, the expression (72) corresponds to the Lode angle dependence of the Willam-Warnke criterion [59]. In this case, a circle is obtained if $R_1=R_3$ or an equilateral triangle (as for the Rankine criterion) is obtained if $R_3=2R_1$.

In the case $R_1=R_3=2R_2/\sqrt{3}$, the expression (72) corresponds to a regular hexagon and hence to the Lode angle dependence of the Tresca and Mohr-Coulomb criteria. The case $R_1=R_2=R_3$ corresponds to a circle.

Excepted in the limit cases $R_1=R_3=2R_2/\sqrt{3}$ (hexagonal shape) and $2R_1=\sqrt{3}R_2 = R_3$ (triangular shape), the obtained criterion is smooth: the tangent is continuously varying.