Macroscopic response and microstructure evolution in viscoplastic polycrystals with pressurized pores

Abstract

This paper presents a finite-strain homogenization model for the macroscopic behavior of porous polycrystals containing pressurized pores that are randomly distributed in a polycrystalline matrix. The porous polycrystal is modeled as a three-scale composite, where the pore size is taken to be much larger than the grain size, and the grains are described by single-crystal viscoplasticity. The instantaneous macroscopic response and corresponding field statistics in the material are determined using a generalization of the recently developed iterated second-order homogenization method, which employs the effective behavior of a linear comparison composite to estimate that of the nonlinear composite by means of a suitably designed variational approximation. Moreover, consistent evolution laws are derived for the pore pressure, pore geometry, and the underlying texture for the polycrystalline matrix. The model is then used to investigate porous ice polycrystals under a wide range of loading conditions. It is found that the pore pressure evolution has a strong effect on the material’s response under compressive loadings. More specifically, the macroscopic response of the porous polycrystals can be categorized into three different regimes: (i) a texture-controlled regime at low triaxialities, where the materials behave like solid polycrystals; (ii) a porosity-controlled regime at high triaxialities, where the materials behave like porous untextured materials; and (iii) a transition regime at intermediate triaxialities, where the materials exhibit a more complex behavior. This work highlights the importance of accounting for the interplay between porosity and matrix texture evolution in describing the constitutive response of porous polycrystals undergoing finite deformations.

Appendices

Homogenization problems for a porous composite with pressurized pores and a porous composite with vacuous pores

In this appendix, we show that the effective behavior and field statistics for porous polycrystals with pressurized pores can be determined from those for porous polycrystals with vacuous pores ($P=0$), provided that the macroscopic stress is appropriately modified, as discussed in Sect. 3.1. As will be seen below, this is a direct consequence of the incompressibility of the matrix phase. Thus, we will provide a demonstration in the more general context of porous materials with an incompressible viscoplastic matrix.

Here we consider an RVE ${\varOmega }$ consisting of pressurized pores randomly distributed in an incompressible, viscoplastic matrix. We assume that the matrix and pores occupy, respectively, the spatial domains ${\varOmega }^{(1)}$ and ${\varOmega }^{(2)}$, and that the pores are subjected to an internal pressure P. In the following, we will make use of the symbols $\left\langle \cdot \right\rangle$ and $\langle \cdot \rangle ^{(q)}$ ($q=1,2$) to denote the volume averages of fields over ${\varOmega }$ and ${\varOmega }^{(q)}$, respectively.

The constitutive response for the matrix can be described by a stress potential $u^{(1)}({{\varvec{\sigma }}})\equiv \psi ({\varvec{\sigma }}_{d})$ (${\varvec{\sigma }}_{d}$ denotes the deviatoric stress), so that

$$\begin{aligned} \mathbf{D }=\frac{\partial u^{(1)}(\varvec{\sigma })}{\partial \varvec{\sigma }}=\frac{\partial \psi (\varvec{\sigma }_d)}{\partial \varvec{\sigma }}, \end{aligned}$$

(39)

where $\mathbf{D }$ is the strain rate satisfying the incompressibility constraint $\mathrm{tr}(\mathbf{D })=0$. On the other hand, the constitutive behavior for the pressurized pores can be characterized by a stress potential $u^{(2)}({\varvec{\sigma }})$, such that $u^{(2)}({\varvec{\sigma }})=0$ if ${\varvec{\sigma }}=-P\mathbf{I }$, while $u^{(2)}({\varvec{\sigma }})=\infty$ otherwise, as has been discussed in Section 2.2.2.

In order to determine the homogenized response of the porous composite for a prescribed macroscopic stress ${\overline{{\varvec{\sigma }}}}$ (note that ${\overline{{\varvec{\sigma }}}}=\langle {\varvec{\sigma }}\rangle$), we need to solve the following boundary value problem for the velocity field $\mathbf{v}$:

$$\begin{aligned} \begin{aligned}&\nabla \cdot \varvec{\sigma }={\mathbf {0}}~\text {in} ~{\varOmega }^{(1)}, \quad \mathrm{with} \\&\varvec{\sigma }{\mathbf {n}}=\overline{\varvec{\sigma }}{\mathbf {n}}~\text {on}~ \partial {\varOmega }, \quad \mathrm{and}\\&\varvec{\sigma }{\mathbf {n}}=-P{\mathbf {n}}~\text {on}~\partial {\varOmega }^{(2)}, \end{aligned} \end{aligned}$$

(40)

keeping in mind that ${\varvec{\sigma }}$ depends on $\mathbf{v}$ through the constitutive relation (39), while $\partial {\varOmega }$ and $\partial {\varOmega }^{(2)}$ denote, respectively, the boundaries of ${\varOmega }$ and ${\varOmega }^{(2)}$. After solving (40), the effective stress potential ${\widetilde{u}}({\overline{{\varvec{\sigma }}}};P)$ for the porous composite can be determined by computing the average of the local stress potential via

$$\begin{aligned} {\widetilde{u}}({\overline{{\varvec{\sigma }}}};P) = (1-f) \langle u^{(1)}({\varvec{\sigma }})\rangle ^{(1)}=(1-f) \langle \psi ({\varvec{\sigma }}_{d})\rangle ^{(1)},\nonumber \\ \end{aligned}$$

(41)

where $f=V^{(2)}/V$ is the porosity, with $V^{(2)}=|{\varOmega }^{(2)}|$ and $V=|{\varOmega }|$ denoting the total volumes of the pores and the RVE, respectively.

Making use of the change of variables (Vincent et al. 2009a)

$$\begin{aligned} {\varvec{\sigma }}^{*} = {\varvec{\sigma }}+ P\mathbf{I }, \end{aligned}$$

(42)

we have from (40) that

$$\begin{aligned} \begin{aligned}&\nabla \cdot \varvec{\sigma ^{*}}={\mathbf {0}}~\text {in}~{\varOmega }^{(1)}, \quad \mathrm{with} \\&\varvec{\sigma ^{*}}{\mathbf {n}}=(\overline{\varvec{\sigma }}+P\mathbf{I }){\mathbf {n}}~\text {on}~ \partial {\varOmega }, \quad \mathrm{and}\\&\varvec{\sigma ^{*}}{\mathbf {n}}=\mathbf{0} ~\text {on}~\partial {\varOmega }^{(2)}. \end{aligned} \end{aligned}$$

(43)

On closer inspection, we realize that (43) defines the corresponding boundary value problem for a porous composite with the exact same substructure and matrix, but with vacuous pores, for a prescribed macroscopic stress ${\overline{{\varvec{\sigma }}}}^{*}=\langle {\varvec{\sigma }}^{*}\rangle ={\overline{{\varvec{\sigma }}}}+P\mathbf{I }$. Making use of the fact that $u^{(1)}({\varvec{\sigma }}) = u^{(1)}({\varvec{\sigma }}^{*})$, together with expression (41), it is straightforward to verify that ${\widetilde{u}}({\overline{{\varvec{\sigma }}}};P) = {\widetilde{u}}({\overline{{\varvec{\sigma }}}}^*,0)$, from which equations (10) and (11) in Section 3.1 follow. In addition, it can be easily shown that the field statistics in the porous composite with pressurized pores can be obtained from those in the porous composite with vacuous pores. In particular, we have that

$$\begin{aligned} \begin{aligned}&\langle {\varvec{\sigma }}\rangle ^{(1)}= \langle {\varvec{\sigma }}^{*}\rangle ^{(1)}-P{\mathbf {I}}, \\&\langle {\varvec{\sigma }}\otimes {\varvec{\sigma }}\rangle ^{(1)} = \langle {\varvec{\sigma }}^{*}\otimes {\varvec{\sigma }}^{*}\rangle ^{(1)}\\&\quad -P\langle {\varvec{\sigma }}^{*}\rangle ^{(1)}\otimes \mathbf{I }-P \mathbf{I }\otimes \langle {\varvec{\sigma }}^{*}\rangle ^{(1)} + P^2\mathbf{I }\otimes \mathbf{I }, \quad \text {and} \\&{\mathbb {C}}^{(1)}({\varvec{\sigma }})={\mathbb {C}}^{(1)}({\varvec{\sigma }}^{*}), \end{aligned} \end{aligned}$$

(44)

where use has been made of (42). In the above equation, ${\mathbb {C}}^{(1)}(\cdot )$ denotes the fluctuation covariance tensor for a given quantity in the matrix phase, e.g., ${\mathbb {C}}^{(1)}({\varvec{\sigma }})=\langle {\varvec{\sigma }}\otimes {\varvec{\sigma }}\rangle ^{(1)}-\langle {\varvec{\sigma }}\rangle ^{(1)}\otimes \langle {\varvec{\sigma }}\rangle ^{(1)}$.

Additio nally, because the matrix is incompressible and insensitive to the hydrostatic stress (see (39)), it is easy to verify that the local velocity field, along with the local strain-rate and spin fields, obtained by solving (40) are identical to those obtained by solving (43), that is $\mathbf{v} = \mathbf{v}^{*}$, $\mathbf{D} = \mathbf{D}^{*}$ and $\mathbf{W} = \mathbf{W}^{*}$. It then follows that

$$\begin{aligned} \begin{aligned}&\langle \mathbf{D }\rangle ^{(1)}= \langle \mathbf{D }^{*}\rangle ^{(1)}, \\&\langle \mathbf{D }\otimes \mathbf{D }\rangle ^{(1)} = \langle \mathbf{D }^{*}\otimes \mathbf{D }^{*}\rangle ^{(1)}, \quad \text {and} \\&{\mathbb {C}}^{(1)}(\mathbf{D })={\mathbb {C}}^{(1)}(\mathbf{D }^{*}). \end{aligned} \end{aligned}$$

(45)

Similarly, the corresponding relations for the spin field and the slip rates over different slip systems can be written in forms completely analogous to (45).

Furthermore, making use of the divergence theorem, it can be shown that the average strain-rate and spin fields over the pores can be determined from the velocity field on the pore boundaries (Gurson 1977). For example, we have that

$$\begin{aligned} \begin{aligned}&\langle \mathbf{D }\rangle ^{(2)}= \frac{1}{V^{(2)}}\int _{\partial {\varOmega }^{(2)}}\frac{1}{2}\left( \mathbf{v}\otimes \mathbf{n}+\mathbf{n}\otimes \mathbf{v} \right) d\mathbf{x}, \quad \text {and} \\&\langle \mathbf{W }\rangle ^{(2)}=\frac{1}{V^{(2)}}\int _{\partial {\varOmega }^{(2)}}\frac{1}{2}\left( \mathbf{v}\otimes \mathbf{n}-\mathbf{n}\otimes \mathbf{v} \right) d\mathbf{x}. \end{aligned} \end{aligned}$$

(46)

Since the velocity fields obtained by solving (40) and (43) are identical, we also have the result that

$$\begin{aligned} \begin{aligned}&\langle \mathbf{D }\rangle ^{(2)}= \langle \mathbf{D }^{*}\rangle ^{(2)}, \quad \text {and} \\&\langle \mathbf{W }\rangle ^{(2)}= \langle \mathbf{W }^{*}\rangle ^{(2)}. \end{aligned} \end{aligned}$$

(47)

Evolution law for the pore pressure

In this Appendix, we derive the evolution law (22) for the average pore pressure. Let us denote the initial volumes of the RVE and pores by $V_{I}$ and $V_{I}^{(2)}$, respectively, while denote the current volumes of the RVE and pores by V and $V^{(2)}$, respectively. Furthermore, we denote the initial and current (average) pore pressures by $P_{I}$ and P, respectively.

As discussed in Section 3.2.1, we assume that the product of the average pore pressure with the total pore volume remains unchanged, so that for any given state

$$\begin{aligned} \begin{aligned} \frac{P}{P_I}=\frac{V^{(2)}_I}{V^{(2)}}. \end{aligned} \end{aligned}$$

(48)

Taking the derivative of (48) with respect to time, we have that

$$\begin{aligned} \begin{aligned} \frac{{\dot{P}}}{P_I}=-\frac{V^{(2)}_I}{V^{(2)}} \frac{{\dot{V}}^{(2)}}{V^{(2)}}. \end{aligned} \end{aligned}$$

(49)

Making use of (48) in (49), and after some manipulations, we arrive at

$$\begin{aligned} \begin{aligned} {\dot{P}}=-P \frac{{\dot{V}}^{(2)}}{V^{(2)}}. \end{aligned} \end{aligned}$$

(50)

Because the matrix phase of the porous composite is incompressible, the volume change of the composite equals the volume change of the pores (i.e., ${\dot{V}}={\dot{V}}^{(2)}$). Thus, we have that

$$\begin{aligned} \begin{aligned} \frac{{\dot{V}}^{(2)}}{V^{(2)}}=\frac{{\dot{V}}}{V^{(2)}}=\frac{{\dot{V}}}{fV}=\frac{{\overline{D}}_{kk}}{f}, \end{aligned} \end{aligned}$$

(51)

where use has been made of the facts that ${\dot{V}}/V={\overline{D}}_{kk}$, and $V^{(2)} = fV$ (f is the porosity in the current state of the RVE). Substituting (51) into (50), we arrive at equation (22).