Micromechanical modeling for the thermo-elasto-plastic behavior of functionally graded composites

Abstract

By gradually changing the compositions, functionally graded materials (FGMs) are constructed and possess comprehensive performance. In this paper, based on the micromechanics, the thermo-elasto-plastic behaviors of FGMs are studied with consideration of the pairwise particle interaction, where the graded microstructures of the FGMs are represented by employing a particular representative volume element (RVE). Based on the assumption that the matrix dominates the plastic behavior of the FGMs while the particles stay in their linearly elastic state, the von-Mises yield function is extended to solve FGMs problems. By employing the backward Euler’s method, the overall effective thermo-elasto-plastic behavior of FGMs can be numerically obtained. When eliminating the plastic or thermal effect, the proposed model can be downgraded to the thermoelastic model or the elastoplastic model of the FGMs, respectively. In addition, the proposed model is validated with available experimental results. Finally, the effects of temperature changes, particle distributions, volume fractions, and material properties on the effective thermo-elasto-plastic properties of FGMs are studied.

Appendix

Due to the existed of the particles (which is also called inhomogeneities), the local strain field ${{\varvec{\upvarepsilon}}}$ is equal to the superposition of the far field strain ${{\varvec{\upvarepsilon}}}^{0}$ and the disturbed strain ${\varvec{\upvarepsilon}}^{\prime}$, which is given as

$${{\varvec{\upvarepsilon}}} = {{\varvec{\upvarepsilon}}}^{0} + {\varvec{\upvarepsilon}}^{\prime},$$

(32)

where the far field strain ${{\varvec{\upvarepsilon}}}^{0} = \left( {{\mathbf{C}}_{0} } \right)^{ - 1} :{{\varvec{\upsigma}}}^{0} + \alpha_{0} T{{\varvec{\updelta}}}$ is caused by the external far field stress ${{\varvec{\upsigma}}}^{0}$ and the temperature change $T$. In Eq. (42), ${\mathbf{C}}_{0}$ is the elastic material coefficients tensor of the matrix; $\alpha_{0}$ is the CTE of the matrix; ${{\varvec{\updelta}}}$ is the Kronecker Delta tensor. The disturbed strain ${\varvec{\upvarepsilon}}^{\prime}$ is caused by the material mismatch. Using the Green’s function technique, the disturbed strain ${\varvec{\upvarepsilon}}^{\prime}$ is given as

$${\varvec{\upvarepsilon}}^{\prime} = - \mathop \int \limits_{{{\varvec{\Omega}}}}^{{}} {\mathbf{G}}\left( {{\mathbf{x}},{\mathbf{x^{\prime}}}} \right) \cdot {\mathbf{C}}_{0} :\left[ {{{\varvec{\upvarepsilon}}}^{*} \left( {{\mathbf{x^{\prime}}}} \right) + {{\varvec{\upvarepsilon}}}^{{\text{T}}} } \right]{\mathbf{dx^{\prime}}} = - {\mathbf{D}}^{{\Omega }} :{\mathbf{C}}_{0} :\left[ {{{\varvec{\upvarepsilon}}}^{*} \left( {{\mathbf{x^{\prime}}}} \right) + {{\varvec{\upvarepsilon}}}^{{\text{T}}} } \right],$$

(33)

where “$\cdot$” means the tensor contraction between two fourth-rank tensors, “$:$” represents the tensor contraction between fourth-rank and second rank tensors; ${{\varvec{\upvarepsilon}}}^{*}$ is the induced virtual elastic eigenstrain to refer the material mismatch of the elastic field; ${{\varvec{\upvarepsilon}}}^{{\text{T}}} = \left( {\alpha_{1} - \alpha_{0} } \right)T{{\varvec{\updelta}}}$ is the thermal eigenstrain caused by the thermal properties mismatch, where $\alpha_{1}$ is the CTE of the particle; ${\mathbf{G}}\left( {{\mathbf{x}},{\mathbf{x}}^{\prime}} \right)$ is the modified Green’s function; ${{\varvec{\Omega}}}$ accounts for the certain particle domain; the fourth rank tensor ${\mathbf{D}}^{{{\varvec{\Omega}}}}$ which is the integral of ${\mathbf{G}}\left( {{\mathbf{x}},{\mathbf{x}}^{\prime}} \right)$ over the spherical particle domain, which can be found in Yin’s work (2007). According to EIM and equivalent condition in particle domain ${{\varvec{\Omega}}}$, the following equation is obtained as

$${\mathbf{C}}_{1} :\left[ {{{\varvec{\upvarepsilon}}}^{0} + {\varvec{\upvarepsilon}}^{\prime} - \alpha_{1} T{{\varvec{\updelta}}}} \right] = {\mathbf{C}}_{0} :\left[ {{{\varvec{\upvarepsilon}}}^{0} + {\varvec{\upvarepsilon}}^{\prime} - \alpha_{1} T{{\varvec{\updelta}}} - {{\varvec{\upvarepsilon}}}^{*} } \right],$$

(34)

where ${\mathbf{C}}_{1}$ is the elastic material coefficients tensor of the particle. Substituting Eq. (33) into Eq. (34), the eigenstrain ${{\varvec{\upvarepsilon}}}^{*}$ is derived as

$${{\varvec{\upvarepsilon}}}^{*} = {\mathbf{C}}_{0}^{ - 1} \cdot \left( {{\mathbf{D}}^{{\Omega }} - {\Delta }{\mathbf{C}}^{ - 1} } \right)^{ - 1} :\left( {{{\varvec{\upvarepsilon}}}^{0} - \alpha_{1} T{{\varvec{\updelta}}} - {\mathbf{D}}^{{\Omega }} \cdot {\mathbf{C}}_{0} :{{\varvec{\upvarepsilon}}}^{{\text{T}}} } \right),$$

(35)

where ${\Delta }{\mathbf{C}} = {\mathbf{C}}_{1} - {\mathbf{C}}_{0}$ is the stiffness difference.

By substituting the disturbed strain ${\varvec{\upvarepsilon}}^{\prime}$ and eigenstrain ${{\varvec{\upvarepsilon}}}^{*}$ in Eq. (33) and Eq. (35) into Eq. (32), the strain field in the particle domain induced by a single particle can be obtained as

$${\overline{\varvec{\upvarepsilon}}} = \alpha_{1} T{{\varvec{\updelta}}} + \left( {{\mathbf{I}} - {\mathbf{D}}^{{\Omega }} \cdot {\Delta }{\mathbf{C}}} \right)^{ - 1} :\left( {{{\varvec{\upvarepsilon}}}^{0} - \alpha_{1} T{{\varvec{\updelta}}} - {\mathbf{D}}^{{\Omega }} \cdot {\mathbf{C}}_{0} :{{\varvec{\upvarepsilon}}}^{{\text{T}}} } \right),$$

(36)

where ${\mathbf{I}}$ is the standard fourth rank unit tensor, the overbar of ${\overline{\varvec{\upvarepsilon}}}$ represents the presence of only one particle in the infinite matrix domain. To conduct the pairwise interaction, Yin et al. [49] obtained the averaged strain field induced by two particles as

$$\begin{aligned} \overline{\overline{{{\varvec{\upvarepsilon}}}}} & = \alpha_{1} T{{\varvec{\updelta}}} + \left\{ {{\mathbf{I}} - \left[ {{\mathbf{D}}^{{\Omega }} + {\mathbf{D}}\left( {{\mathbf{x}}_{1} } \right)} \right] \cdot {\Delta }{\mathbf{C}}} \right\}^{ - 1} \\ & \quad :\left\{ {{{\varvec{\upvarepsilon}}}^{0} - \alpha_{1} T{{\varvec{\updelta}}} - \left[ {{\mathbf{D}}^{{\Omega }} + {\mathbf{D}}\left( {{\mathbf{x}}_{1} } \right)} \right] \cdot {\mathbf{C}}_{0} :{{\varvec{\upvarepsilon}}}^{{\text{T}}} } \right\} + O\left( {\rho^{8} } \right). \\ \end{aligned}$$

(37)

The double overbar indicates the presence of two particles in the infinite matrix domain, $\overline{\overline{{{\varvec{\upvarepsilon}}}}}$ is uniform strain inside the particle domain considering the interaction of the other particles. Subtracting Eq. (36) from Eq. (37), the average interaction between two particles is

$${\mathbf{d}}\left( {0,{\mathbf{x}}_{1} } \right) = \overline{\overline{{{\varvec{\upvarepsilon}}}}} - {\overline{\varvec{\upvarepsilon}}} = {\Delta }{\mathbf{C}}^{ - 1} \cdot {\mathbf{L}}\left( {0,{\mathbf{x}}_{1} } \right):\left( {{{\varvec{\upvarepsilon}}}^{0} - \alpha_{1} T{{\varvec{\updelta}}} - {\Delta }{\mathbf{C}}^{ - 1} \cdot {\mathbf{C}}_{0} :{{\varvec{\upvarepsilon}}}^{{\text{T}}} } \right) + O\left( {\rho^{8} } \right),$$

(38)

where ${\mathbf{L}}\left( {0,{\mathbf{x}}_{1} } \right)$ is the particle pairwise interaction tensor as

$${\mathbf{L}}\left( {0,{\mathbf{x}}_{1} } \right) = \left[ {\Delta {\mathbf{C}}^{ - 1} - {\mathbf{D}}^{{\Omega }} - {\mathbf{D}}\left( {{\mathbf{x}}_{1} } \right)} \right]^{ - 1} - \left( {\Delta {\mathbf{C}}^{ - 1} - {\mathbf{D}}^{{\Omega }} } \right)^{ - 1} .$$

(39)

When there are multiple particles $P_{i} { }\left( {i = 2,3, \ldots } \right)$, each of them will generate an additional interaction on the central particle ${\Omega }_{0}$ as Eq. (38) dominated. Thus, the average strain of the central particle considering pairwise particle interaction is

$$\left\langle {{\varvec{\upvarepsilon}}} \right\rangle^{1} \left( 0 \right) = \alpha_{1} T{{\varvec{\updelta}}} + \left( {{\mathbf{I}} - {\mathbf{D}}^{{\Omega }} \cdot {\Delta }{\mathbf{C}}} \right)^{ - 1} :\left[ {\left\langle {{\varvec{\upvarepsilon}}} \right\rangle^{0} \left( 0 \right) - \alpha_{1} T{{\varvec{\updelta}}} - {\mathbf{D}}^{{\Omega }} \cdot {\mathbf{C}}_{0} :{{\varvec{\upvarepsilon}}}^{{\text{T}}} } \right] + \mathop \sum \limits_{i = 1}^{\infty } {\mathbf{d}}\left( {0,{\mathbf{x}}_{i} } \right),$$

(40)

where $\left\langle {*} \right\rangle$ denotes the volume average over the material phase. By integrating pairwise interaction over all particles, the total particle interaction is obtained as

$$\begin{aligned} \left\langle {\mathbf{d}} \right\rangle \left( 0 \right) & = \mathop \sum \limits_{i = 1}^{\infty } {\Delta }{\mathbf{C}}^{ - 1} \cdot {\mathbf{L}}\left( {0,{\mathbf{x}}_{i} } \right):\left[ {\left\langle {{\varvec{\upvarepsilon}}} \right\rangle^{0} \left( {x_{3}^{i} } \right) - \alpha_{1} T{{\varvec{\updelta}}} - {\Delta }{\mathbf{C}}^{ - 1} \cdot {\mathbf{C}}_{0} :{{\varvec{\upvarepsilon}}}^{{\text{T}}} } \right] \\ & = \mathop \int \limits_{D}^{{}} P\left( {{\mathbf{x}}{|}0} \right){\Delta }{\mathbf{C}}^{ - 1} \cdot {\mathbf{L}}\left( {0,{\mathbf{x}}} \right):\left[ {\left\langle {{\varvec{\upvarepsilon}}} \right\rangle^{0} \left( 0 \right) + \left\langle {{\varvec{\upvarepsilon}}} \right\rangle_{,3}^{0} \left( 0 \right)x_{3} - \alpha_{1} T{{\varvec{\updelta}}} - {\Delta }{\mathbf{C}}^{ - 1} \cdot {\mathbf{C}}_{0} :{{\varvec{\upvarepsilon}}}^{{\text{T}}} } \right]d{\mathbf{x}}, \\ \end{aligned}$$

(41)

where $\left\langle {{\varvec{\upvarepsilon}}} \right\rangle^{0} \left( {x_{3}^{i} } \right) = \left\langle {{\varvec{\upvarepsilon}}} \right\rangle^{0} \left( 0 \right) + \left\langle {{\varvec{\upvarepsilon}}} \right\rangle_{,3}^{0} \left( 0 \right)x_{3}$. And $P\left( {{\mathbf{x}}{|}0} \right) = \frac{3g\left( x \right)}{{4\pi a^{3} }}\left[ {\phi \left( {X_{3} } \right) + e^{{ - \frac{{\mathbf{x}}}{\delta }}} \phi_{,3} \left( {X_{3} } \right)x_{3} } \right]$ is the particle number density function, where g(x) = 1 to insure the density of local field is unchanged [49], the $X_{3}$ denotes the gradation direction, $\phi \left( {X_{3} } \right)$ is the average volume fraction of particle at $X_{3}$, $\delta = \frac{e}{{\phi_{,3} \left( {X_{3}^{0} } \right)}}{\text{min}}\left( {\phi ,\phi^{c} - \phi } \right)$ denotes the attenuating rate of the gradation of the particle volume fraction in the far field, and $\phi^{c}$ represents the maximum volume fraction of particles. Thus, the effect of particle distribution at the microscale can be integrated to reflect the effective material properties at the macroscale. Specifically, by plugging Eq. (41) into Eq. (40), the average particle strain considering pairwise interaction at $X_{1} - X_{2}$ layer along the graded direction is obtained as

$$\begin{aligned} \left\langle {\upvarepsilon } \right\rangle^{1} \left( {{\text{X}}_{3} } \right) & = {\upalpha }_{1} {\text{T}}\updelta + \left( {{\text{I}} - {\text{D}}^{{\Omega }} \cdot \Delta {\text{C}}} \right)^{ - 1} :\left[ {\left\langle {\upvarepsilon } \right\rangle^{0} \left( {{\text{X}}_{3} } \right) - {\upalpha }_{1} {\text{T}}\updelta - {\text{D}}^{{\Omega }} \cdot {\text{C}}_{0} :{\upvarepsilon }^{{\text{T}}} } \right] \\ & \quad + \phi \Delta {\text{C}}^{ - 1} \cdot {\mathbf{\mathcal{D}}}:\left[ {\left\langle {\upvarepsilon } \right\rangle^{0} \left( {{\text{X}}_{3} } \right) - {\upalpha }_{1} {\text{T}}\updelta - \Delta {\text{C}}^{ - 1} \cdot {\text{C}}_{0} :{\upvarepsilon }^{{\text{T}}} } \right] \\ & \quad + \phi_{,3} \Delta {\text{C}}^{ - 1} \cdot {\mathbf{\mathcal{F}}}:\left\langle {\upvarepsilon } \right\rangle_{,3}^{0} \left( {{\text{X}}_{3} } \right), \\ \end{aligned}$$

(42)

where tensor ${\mathbf{\mathcal{D}}}$ represents the pairwise interaction, and tensor ${\mathbf{\mathcal{F}}}$ addresses the coupling effect of layers along the $X_{3}$ direction [49].

$${\mathbf{\mathcal{D}}} = \mathop \int \limits_{D}^{{}} \frac{3}{{4\pi a^{3} }}{\mathbf{L}}\left( {0,{\mathbf{x}}} \right)d{\mathbf{x}},$$

(43)

$${\mathbf{\mathcal{F}}} = \mathop \int \limits_{D}^{{}} e^{{ - \frac{{\mathbf{x}}}{\delta }}} \frac{3}{{4\pi a^{3} }}{\mathbf{L}}\left( {0,{\mathbf{x}}} \right)x_{3}^{2} d{\mathbf{x}}.$$

(44)

Thus, at a certain $X_{1} - X_{2}$ layer, the overall averaged stress and strain are expressed as

$$\langle{{\varvec{\upsigma}}}\left\rangle( {X_{3} } \right) = \phi \left( {X_{3} } \right)\langle{{\varvec{\upsigma}}}\rangle^{1} \left( {X_{3} } \right) + \left[ {1 - \phi \left( {X_{3} } \right)} \right]\langle{{\varvec{\upsigma}}}\rangle^{0} \left( {X_{3} } \right),$$

(45)

$$\langle{{\varvec{\upvarepsilon}}}\left\rangle( {X_{3} } \right) = \phi \left( {X_{3} } \right)\langle{{\varvec{\upvarepsilon}}}\rangle^{1} \left( {X_{3} } \right) + \left[ {1 - \phi \left( {X_{3} } \right)} \right]\langle{{\varvec{\upvarepsilon}}}\rangle^{0} \left( {X_{3} } \right),$$

(46)

where the averaged stress $\langle{{\varvec{\upsigma}}}\rangle^{1}$ and $\langle{{\varvec{\upsigma}}}\rangle^{0}$ for particle and matrix are

$$\langle{{\varvec{\upsigma}}}\rangle^{1} \left( {X_{3} } \right) = {\mathbf{C}}_{1} :\left[ \langle{{{\varvec{\upvarepsilon}}}\rangle^{1} \left( {X_{3} } \right) - \alpha_{1} T{{\varvec{\updelta}}}} \right],$$

(47)

$$\langle{{\varvec{\upsigma}}}\rangle^{0} \left( {X_{3} } \right) = {\mathbf{C}}_{0} :\left[ \langle{{{\varvec{\upvarepsilon}}}\rangle^{0} \left( {X_{3} } \right) - \alpha_{0} T{{\varvec{\updelta}}}} \right].$$

(48)