Deformation of Elastically Inhomogeneous Solid Modelled with Diffuse Interface Approach

Abstract

Diffuse interfaces are useful tools for simulating evolution of material phases, and they are fundamentally important part of the phase field method. Since, in solid state, different phases can have different elastic constants, it is important to be able to incorporate them into the numerical simulations. In current study, the theory of deformation of elastically inhomogeneous solid with diffuse interface approach and application of relatively simple numerical method for solving the problem are described. Deformation of a solid containing an elastical inhomogeneity is simulated for two cases for compression in two-dimensional plane strain condition: a harder inclusion and a softer inclusion.

1 Introduction

The elastic problem of material containing inhomogeneities is almost as old as the continuum elasticity theory itself: the foundations of the continuum elasticity theory were laid in the first part of the nineteenth century, and by the end of the century it had become obvious that it is necessary to consider the effect of inhomogeneities in the material when analysing the actual performance of a solid structure [1].

Although there exist some important analytic solutions to certain problems involving elastic inhomogeneities, most notably the Eshelby inclusion problem, there exist serious mathematical difficulties which hinder the development of general analytic solutions [2].

The problems associated with the analytical theory can be circumvented by the usage of numerical methods, which have become useful standard tools for analysis. Although the usage of numerical methods has their own difficulties associated with accuracy, stability and performance, they provide reasonably straightforward general application to many different types of problems.

A useful approximation for simulating a system containing elastical and structural inhomogenieity is provided by the so-called diffuse interface approach, where the phase boundary is modelled as smooth but rapid transition between phases [2,3,4,5]. This approach allows for straightforward calculation of the phase boundary evolution, merging and disappearance of the field regions, and it was used in the previous study to investigate phase region evolution [6].

In the current article, the aim is to investigate an elastic inhomogeneity located inside of strained surrounding material using the diffuse phase boundary approach. The current study complements the work presented in [6] by considering numerical experiments with phases that have different elastic constants.

2 Theory

Current approach is based on the formalism presented in [3] with the extension for different elastic constants for different phases described in [6]. For completeness, the mathematical description is derived from first principles of linear elasticity, including the description for transformation strain (i.e. stress-free strain of a phase, eigenstrain).

Assume that there exist two phases, $\alpha$ and $\gamma$, in a solid material. The function $\phi (\vec{r})$ describes the fraction of $\alpha$ phase at material position $\vec{r}(t)$. When $\phi =0$, the local transformation strain (eigenstrain) is 0. The formation of the $\alpha$ phase causes the local structure of the material to change, so that when $\phi =1$, the local stress-free strain (the eigenstrain) is $\epsilon _{kl}^{00}$.

The initial material position of unstrained $\gamma$ phase is described by the coordinates $\vec{w}=(w_1,w_2,w_3)$ and the current material position is $\vec{r}=(r_1,r_2,r_3)$. The displacement of the material is $\vec{u}=\vec{r}-\vec{w}$. The total strain of the material is $\epsilon _{kl}^{\textrm{tot}}=\frac{1}{2}\left( \frac{\partial u_k}{\partial w_l}+\frac{\partial u_l}{\partial w_k}\right)$. The elastic strain field of the $\gamma$ phase is $\epsilon _{kl}^{e\gamma }=\epsilon _{kl}^{\textrm{tot}}$. On the other hand, the elastic strain field of the alpha phase takes in to account the stress-free transformation strain so that $\epsilon _{kl}^{e\alpha }=\epsilon _{kl}^{\textrm{tot}}-\epsilon _{kl}^{00}$.

To describe the strain and stress partitioning within the diffuse interface, a phase fraction weighed mixing model is adopted. The calculated quantities are volume fraction weighted averages, where a grid point value represents the local average. The local weighed average is obtained by replacing the volume integral with the product of the average and the local volume represented by the gridpoint. It is assumed that the local volume $V_L$ can be separated in the volumes occupied by the $\alpha$ and $\gamma$ phases, denoted by $V_L^\alpha$ and $V_L^\gamma$, respectively. In the case of strain $\int _{V_L} \epsilon _{ij}=\int _{V_{L}^\alpha } \epsilon _{ij}^\alpha +\int _{V_{L}^\gamma } \epsilon _{ij}^\gamma \Rightarrow V_L\langle \epsilon _{ij}\rangle =V_L^\alpha \langle \epsilon _{ij}^\alpha \rangle +V_L^\gamma \langle \epsilon _{ij}^\gamma \rangle$, where $\langle \epsilon _{ij}\rangle$ is average strain in volume $V_L$, $\langle \epsilon _{ij}^\alpha \rangle$ and $\langle \epsilon _{ij}^\gamma \rangle$ are the average local stresses in the $\alpha$ and $\gamma$ regions $V_L^\alpha$ and $V_L^\gamma$, respectively. Dividing both sides with $V_L$ yields $\langle \epsilon _{ij}\rangle =\frac{V_L^\alpha }{V_L}\langle \epsilon _{ij}^\alpha \rangle +\frac{V_L^\gamma }{V_L} \langle \epsilon _{ij}^\gamma \rangle$. Since the local volume fraction of $\alpha$ phase is $\phi =\frac{V_L^\alpha }{V_L}$ and only two phases are present, the local volume fraction of $\gamma$ phase is $\frac{V_L^\gamma }{V_L}=1-\phi$ and $\langle \epsilon _{ij}\rangle =\phi \langle \epsilon _{ij}^\alpha \rangle +(1-\phi ) \langle \epsilon _{ij}^\gamma \rangle$. The volume fraction weighted averages approach the local values at the material point when the grid spacing approaches zero (and consequently the local volume $V_L \rightarrow 0$). In this limit $\epsilon _{ij}=\phi \epsilon _{ij}^\alpha +(1-\phi )\epsilon _{ij}^\gamma$. Similarly, stress is represented by the volume fraction weighted average value at the grid point $\langle \sigma _{ij}\rangle =\phi \langle \sigma _{ij}^\alpha \rangle +(1-\phi ) \langle \sigma _{ij}^\gamma \rangle$ and as the grid point spacing approaches zero, the local stress approaches $\sigma _{ij}=\phi \sigma _{ij}^\alpha +(1-\phi ) \sigma _{ij}^\gamma$. The grid point value in the numerical calculations represents the volume fraction weighted local average value.

Based on the discussion above, the total elastic strain at a material point is described by Eq. (1).

$$\begin{aligned} \epsilon _{kl}^\textrm{e}=\phi \epsilon _{kl}^{e\alpha }+(1-\phi )\epsilon _{kl}^{e\gamma }=\epsilon _{kl}^{\textrm{tot}}-\phi \epsilon _{kl}^{00} \end{aligned}$$

(1)

The stress of the $\gamma$ phase is $\sigma _{ij}^{\gamma }=c_{ijkl}^\gamma \epsilon _{kl}^\gamma$ and the stress of the $\alpha$ phase is $\sigma _{ij}^\alpha =c_{ijkl}\epsilon _{kl}^{e\alpha }$, where Einstein summation convention is applied. The total stress is given by Eq. (2).

$$\begin{aligned} \sigma _{ij}=\phi \sigma _{ij}^\alpha +(1-\phi )\sigma _{ij}^\gamma \end{aligned}$$

(2)

The force $\vec{F}=(F_1,F_2,F_3)$ exerted on a material point is obtained from Eq. (3) (see Refs. [3, 7, 8]).

$$\begin{aligned} F_i=\sum _j \frac{\partial \sigma _{ij}}{\partial r_j} \end{aligned}$$

(3)

In the static case $\vec{F}=0$.

To obtain the elastostatic solution, we mimic the dynamics of viscously damped material, which removes the vibrations from the solution. The evolution equation for viscously damped material is $\partial _{tt} \vec{u} = \vec{F}/m - \eta \partial _t \vec{u}$, where $\eta$ is the viscous damping coefficient. When the accelerations vanish, the local velocity approaches $\partial _t \vec{u}=\vec{F}/\eta .$ For obtaining the elastostatic solution, it is assumed that this condition is reached instantaneously, and the numerical solution procedure operates by moving the material points with small steps to the direction of the local force until static equilibrium is reached. The spatial derivatives were calculated using the method described in [9]. In the numerical solution, the elastostatic solution is obtained by moving the material point in the direction of the force, $\frac{du_i}{dt_p}=\lambda F_i \mathrm {\Delta } t_p$, where $\lambda = \eta ^{-1} =5 \times 10^{-12}$ was chosen and $t_p$ is artificial time-like coordinate. The positions are updated as $u_i(t_p+\mathrm {\Delta } t_p)=u_i(t_p)+\langle \frac{du_i}{dt_p}\rangle \mathrm {\Delta } t_p$, where the average value $\langle \frac{du_i}{dt_p}\rangle =\frac{1}{2} \left( \frac{du_i}{dt_p}|_{t_p=t_{p0}}+\frac{du_i}{dt_p}|_{t_p=t_{p0}+\mathrm {\Delta } t_p} \right)$ is applied. The calculation procedure is applied until the total sum of the absolute value of the forces acting on the all material points was below a threshold value. The threshold value controls the accuracy of the solution. Lowering the value yields higher accuracy, but increases the required computational time. It was checked in the previous publication [6] by comparison to an analytic solution that this numerical procedure yielded the correct elastostatic solution for Eshelby inclusion problem with transformation strains included. In [6], stress state of system containing two elastically similar phases was considered in numerical experiments.

The material starts to yield, when the von Mises equivalent stress reaches or exceeds the material yield stress $\sigma _y$ [3], i.e. Eq. (4) holds

$$\begin{aligned} \frac{1}{2}(\sigma _{xx}^\theta -\sigma _{yy}^\theta )^2+\frac{1}{2}(\sigma _{yy}^\theta -\sigma _{zz}^\theta )^2+\frac{1}{2}(\sigma _{xx}^\theta -\sigma _{zz}^\theta )^2+3({\sigma _{xy}^\theta }^2+{\sigma _{yz}^\theta }^2+{\sigma _{xz}^\theta }^2) \ge {\sigma _y^\theta }^2 \end{aligned}$$

(4)

where the yield condition is defined for both phases separately, i.e. the equation is applied for $\theta =\gamma$ and $\theta =\alpha$.

3 Numerical Experiments

3.1 2D Elastic Deformation of Solid Containing Circular Inhomogeneity

In the current study, numerical experiments are conducted for a simple two-dimensional case, assuming isotropic $\alpha$ and $\gamma$ regions with distinct elastic constants. The numerical model was implemented using MATLAB language [10]. Plane strain conditions are applied. The material is subjected to compressive strain in horizontal x-direction, and the boundaries in the vertical y-direction are assumed to move freely. The strain in z-direction is zero for the plane strain condition. The boundary conditions and the $\gamma$ and $\alpha$ regions are schematically depicted in Fig. 1 (with exaggerated strains).

Fig. 1

Schematic showing the boundary conditions and the $\alpha$ and $\gamma$ regions. The arrows schematically indicate the (exaggerated) movement of the material at the interface relative to the position in the absence of external strain

Consider first the condition where the $\alpha$ region is not present, i.e. the material consists only of isotropic gamma region. From Hooke’s law it follows that in this case $\sigma _{yy}^\gamma =C_{yyxx}^\gamma \epsilon _{xx}+C_{yyyy}^\gamma \epsilon _{yy}$. Since the boundary conditions were defined so that material is free to expand in the vertical y-direction, $\sigma _{yy}^\gamma =0$, and it follows that $\epsilon _{yy}=-\frac{C_{yyxx}^\gamma }{C_{yyyy}^\gamma } \epsilon _{xx}$.[11] In the following, this strain condition will be applied as a boundary condition in the vertical y-direction.

Consider now that in the middle of the $\gamma$ region, far away the boundaries, there exists $\alpha$ region, where the elastic constants are different from the $\gamma$ region, i.e. the $\alpha$ region is elastic inhomogeneity in the $\gamma$ matrix. Far away from the $\alpha$ region, the strains are not affected. However, near the $\alpha$ region, the elastic inhomogeneity changes the strain and stress states, causing a potent site for stress concentration that can lead to material failure.

3.1.1 Hard Inclusion, $E_\alpha > E_\gamma$

As a first case, we consider the case where the inclusion, represented by the $\alpha$ region, is harder than the surrounding $\gamma$ region under the vertical compression. The ratio $E_\alpha =2 E_\gamma$ was used in the numerical calculation.

The calculated von Mises stress of the gamma phase in a system containing a hard inclusion is shown in Fig. 2. It can be seen in a and b that in the $\gamma$ region the stress is concentrated near the interface, but within the diffuse interface the value of $\sigma _\textrm{VM}^\gamma$ drops. The reason for this is that in the diffuse interface the harder $\alpha$ constituent shields the $\gamma$ region, and this effect becomes more pronounced when the fraction of $\alpha$ increases within the diffuse interface region. The von Mises stress of the $\alpha$ region is highest near the boundaries, as shown in c and d. Comparing b and c it can be seen that the von Mises stress of the $\alpha$ phase $\sigma _\textrm{VM}^\gamma$ is higher everywhere inside of the inclusion than the corresponding far-field value of the $\gamma$ region.

Fig. 2

Hard inclusion ($E_\alpha > E_\gamma$). a The von Mises stress of the $\gamma$ region $\sigma _\textrm{VM}^\gamma$, b Plotted values along the lines shown in a, c the von Mises stress $\sigma _\textrm{VM}^\alpha$ of the $\alpha$ region, d plotted values along the lines shown in c

3.1.2 Soft Inclusion, $E_\alpha < E_\gamma$

As a second case, we consider the case where the inclusion, represented by the $\alpha$ region, is softer than the surrounding $\gamma$ region under the vertical compression. In this case the value $E_\alpha =E_\gamma /2$ was used.

The calculated von Mises stress of the system containing the soft inclusion is shown in Fig. 3. It can be seen that the result is opposite of the one observed for the hard inclusion. In this case, the softer alpha region gives into the surrounding stress when analysed along the horizontal and vertical lines passing through the centre of the inclusion.

Fig. 3

Soft inclusion ($E_\alpha < E_\gamma$). a The von Mises stress of the $\gamma$ region $\sigma _\textrm{VM}^\gamma$, b Plotted values along the lines shown in a, c the von Mises stress $\sigma _\textrm{VM}^\alpha$ of the $\alpha$ region, d plotted values along the lines shown in c

4 Conclusions

The description for the deformation of elastically inhomogeneous material with diffuse interface approach was derived from the principles of linear elasticity. Two numerical experiments were conducted for a two-dimensional solid containing the inhomogeneity under horizontal compression in plane strain condition: a softer inhomogeneity and a harder inhomogeneity. The numerical experiments of the current study complement the earlier results, where transformation strains were considered [6].