Thermal stresses and related phenomena in composite ceramics

Abstract

The paper deals with elastic thermal stresses in an isotropic multi-particle-matrix system consisted of periodically distributed spherical particles in an infinite matrix, imaginarily divided into cubic cells containing a central spherical particle. Originating during a cooling process as a consequence of the difference in thermal expansion coefficients between the matrix and the particle, and investigated within the cubic cell, the thermal stresses, as functions of the particle volume fraction v, being transformed for v = 0 to those of an isotropic one-particle-matrix system, are maximal at the critical particle volume fraction, representing a considerable value related to maximal resistance of the thermal-stress strengthened multi-particle-matrix system against mechanical loading. The thermal stresses are derived for such temperature range within which the multi-particle-matrix system exhibits elastic deformations, considering the yield stress and the particle-matrix boundary adhesion strength. With regard to a curve integral of the thermal-stress induced elastic energy density, critical particle radii related to crack initiation in ideal-brittle particle and matrix, functions describing crack shapes in a plane perpendicular to the direction of crack formation in the particle and in the matrix, and consequently dimensions of a crack in the particle and in the matrix are derived along with the condition concerning the direction of the crack formation. Additionally, derived by two equivalent mathematical techniques, the elastic energy gradient within the cubic cell, representing a surface integral of the thermal-stress induced elastic energy density, is presented to derive the thermal-stress induced strengthening in the spherical particle and in the cubic cell matrix. The former parameters for v = 0 are derived using the model of a spherical cell with the radius ${R_c\rightarrow\infty}$. Derived formulae are applied to the SiC–Si₃N₄ multi-particle-matrix system, and calculated values of investigated parameters are in a good agreement with those from published experimental results.

Appendix

The elastic moduli of the particle (q = p) and the matrix (q = m), s _q , have the forms

$$ s_{11q}=\frac{1}{E_q}, $$

(114)

$$ s_{12q}=-\frac{\mu_q}{E_q}, $$

(115)

where E _q and μ _q are the Young’s modulus and Poisson’s number, respectively.

The coefficient c is derived as

$$ c_1=\frac{3\pi\left(s_{11m}+s_{12m}\right)}{2\left(\pi-6v\right)}, $$

(116)

$$ c_2=c_1+s_{11p}+2s_{12p}-\left(s_{11m}+2s_{12m}\right), $$

(117)

$$ c_3=\int_0^{\pi/4} \left[ \frac{\cos^4\varphi}{\sqrt{1+\cos^2\varphi}} -\frac{1}{3} \left( \frac{\cos^2\varphi}{\sqrt{1+\cos^2\varphi}} \right)^3 \right]\,\hbox{d}\varphi, $$

(118)

where the coefficient c ₃ can be derived by the Taylor series for the integrated function. Consequently, after integration of the Taylor series in terms of ${\left(\varphi-\varphi_0\right)^n}$ about the point ${\varphi_0=0}$, the coefficient ${c_3=0.337497}$ for n = 100;

$$ \begin{aligned} c_4&= 2v\left(s_{11p}+2s_{12p}\right)\left(\frac{c_1}{c_2}\right)^2 \\ &+ \frac{1}{\left(\pi-6v\right)^2} \left[ 2\pi^2\left(1-v\right)\left(s_{11m}+2s_{12m}\right) \left(1-\frac{6vc_1}{\pi c_2}\right)^2 \right. \\ &\left.\quad\quad\quad\quad + v\left(s_{11m}-s_{12m}\right) \left(\pi^2-36vc_3\right) \left(1-\frac{c_1}{c_2}\right)^2 \right], \end{aligned} $$

(119)

$$ \begin{aligned} c_5&= \frac{2c_1\left(s_{11p}+2s_{12p}\right)}{c_2} \\ &+ \frac{1}{\left(\pi-6v\right)^2} \left[ 12\pi\left(1-v\right)\left(s_{11m}+2s_{12m}\right) \left(1-\frac{6vc_1}{\pi c_2}\right) \right. \\ & \left.\quad\quad\quad\quad + \left(s_{11m}-s_{12m}\right) \left(\pi^2-36vc_3\right) \left(1-\frac{c_1}{c_2}\right) \right], \end{aligned} $$

(120)

$$ c_6=\frac{\pi\left[c_2c_4+vc_5\left(c_1-c_2\right)\right]} {\left(\pi-6v\right)\left(c_2c_4+vc_1c_5\right)}, $$

(121)

$$ c_7=\frac{c_2c_4+vc_1c_5}{c_2^2c_4}, $$

(122)

$$ c_8=\frac{v\left[\pi c_2c_5-6\left(c_2c_4+vc_1c_5\right)\right]} {\left(\pi-6v\right)\left(c_2c_4+vc_1c_5\right)}, $$

(123)

$$ \begin{aligned} c_9&= \left\{ 5v^2\left(s_{11m}+2s_{12m}\right) \left[c_5\left(6vc_1-\pi c_2\right)+6c_2c_4\right]^2\right. \\ & \left.\quad - \left(s_{11p}+2s_{12p}\right)\left(\pi-6v\right)^2 \left(c_2c_4+vc_1c_5\right)^2 \right\} \\ &+ 16\pi^2\left(s_{11m}-s_{12m}\right) \left[vc_5\left(c_1-c_2\right)+c_2c_4\right]^2 \\ & \quad\times \left(\frac{3v}{4\pi}\right)^{5/3}\left\{ 1- \left[ \frac{\left(4\pi\right)^{1/3}} {\sqrt{\left(4\pi\right)^{2/3}+4\left(3v\right)^{2/3}}} \right]^5 \right\}, \end{aligned} $$

(124)

$$ c_{10}=s_{11p}+2s_{12p}+\frac{1}{2}\left(s_{11m}-s_{12m}\right), $$

(125)

$$ c_{11}=\frac{3\pi+8}{2R^4}\left(\frac{3v}{4\pi}\right)^{4/3},\quad x_1=0, $$

(126)

$$ c_{11}=\int_0^{\pi/4} \left[ \frac{4\left(3v\right)^{2/3}\cos^2\xi} {R^2\left(4\pi\right)^{2/3}+4x_1\left(3v\right)^{2/3}\cos^2\xi} \right]^2\,\hbox{d}\xi,\quad x_1\in\left(0,\frac{d}{2}\right\rangle, $$

(127)

where the coefficient c ₁₁ for ${x_1\in\left(0,d/2\right\rangle}$ can be derived for a concrete isotropic particle-matrix system by the Taylor series for the integrated function in terms of ${\left(\xi-\xi_0\right)^n}$ about the point ${\xi_0=0}$;

$$ c_{12}=3\varphi_{73} +\frac{2x_1R\left(12\pi v\right)^{1/3} \left[3R^2\left(4\pi\right)^{2/3}+20x_1^2\left(3v\right)^{2/3}\right]} {\left[R^2\left(4\pi\right)^{2/3}+4x_1^2\left(3v\right)^{2/3}\right]^2}, $$

(128)

$$ c_{13}=\int_0^{\varphi_{73}}\nu_6\cos^4\varphi \,\hbox{d}\varphi, $$

(129)

$$ \begin{aligned} c_{14}&= \int_0^{\varphi_{22}} \frac{R^4\left(4\pi\right)^{4/3}\cos^5\varphi} {\left[ R^2\left(4\pi\right)^{2/3}\cos^2\varphi+4x_1^2\left(3v\right)^{2/3} \right]^2}\,\hbox{d}\varphi \\ &= \frac{R\left(4\pi\right)^{1/3}} {\sqrt{R^2\left(4\pi\right)^{2/3}+4x_1^2\left(3v\right)^{2/3}}} \left\{ \frac{R^2\left(4\pi\right)^{2/3}+3x_1^2\left(3v\right)^{2/3}} {R^2\left(4\pi\right)^{2/3}+2x_1^2\left(3v\right)^{2/3}} \right. \\ &\quad- \left( \frac{2x_1}{R}\right)^2\left(\frac{3v}{4\pi}\right)^{2/3} \frac{R^2\left(4\pi\right)^{2/3}+3x_1^2\left(3v\right)^{2/3}} {R^2\left(4\pi\right)^{2/3}+4x_1^2\left(3v\right)^{2/3}} \\ & \left.\quad\quad \times\ln\left[ 1+ \frac{1}{2} \left(\frac{R}{x_1}\right)^2 \left(\frac{4\pi}{3v}\right)^{2/3} \right] \right\}, \end{aligned} $$

(130)

$$ \begin{aligned} c_{15} =\; &\int_0^{\varphi_{73}} \frac{R^2\left(4\pi\right)^{2/3}\cos^5\varphi} {R^2\left(4\pi\right)^{2/3}\cos^2\varphi+4x_1^2\left(3v\right)^{2/3}} \,\hbox{d}\varphi \\ =\;&\frac{\left(3v\right)^{1/3}} {\sqrt{R^2\left(4\pi\right)^{2/3}+4x_1^2\left(3v\right)^{2/3}}} \left\{ \frac{R^2\left(4\pi\right)^{2/3}-4x_1^2\left(3v\right)^{2/3}} {R\left(12\pi v\right)^{1/3}}\right. \\ &-\ \frac{4\pi R^3} {3\left(3v\right)^{1/3} \left[R^2\left(4\pi\right)^{2/3}+4x_1^2\left(3v\right)^{2/3}\right]} \\ &\left.+\ \frac{6vx_1^4}{\pi R^3} \ln \left[ 1+ \frac{1}{2} \left(\frac{R}{x_1}\right)^2 \left(\frac{4\pi}{3v}\right)^{2/3} \right]\right\}, \end{aligned} $$

(131)

$$ \begin{aligned} c_{16}=\;\;& \int_0^{\varphi_{73}} \frac{\left(4\pi\right)^{4/3}R^4\cos^7\varphi} {\left[ R^2\left(4\pi\right)^{2/3}\cos^2\varphi+4x_1^2\left(3v\right)^{2/3} \right]^2}\,\hbox{d}\varphi \\ =\;\;&\frac{1}{\sqrt{R^2\left(4\pi\right)^{2/3}+4x_1^2\left(3v\right)^{2/3}}} \left\{ \frac{R^2\left(4\pi\right)^{2/3}-8x_1^2\left(3v\right)^{1/3}} {R\left(4\pi\right)^{1/3}} \right. \\ &\left.+\ \frac{3R^4\left(4\pi\right)^{4/3}-16x_1^4\left(3v\right)^{4/3}} {4R\left(4\pi\right)^{1/3} \left[R^2\left(4\pi\right)^{2/3}+2x_1^2\left(3v\right)^{2/3}\right]} \right. \\ &- \frac{4\pi R^3} {3\left[R^2\left(4\pi\right)^{2/3}+4x_1^2\left(3v\right)^{2/3}\right]} \\ &+ \frac{2R^5\left(4\pi\right)^{5/3} \left[R^4\left(4\pi\right)^{4/3}-6x_1^4\left(3v\right)^{4/3}\right]} {\left[R^2\left(4\pi\right)^{2/3}+4x_1^2\left(3v\right)^{2/3}\right]^4} \\ &\left.\quad \times\ln\left[ 1+ \frac{1}{2} \left(\frac{R}{x_1}\right)^2 \left(\frac{4\pi}{3v}\right)^{2/3} \right] \right\}, \end{aligned} $$

(132)

$$ \begin{aligned} c_{17}&= \frac{2x_1}{R} \left(\frac{3v}{4\pi}\right)^{1/3} \left[ c_{14}\left(\frac{2x_1}{R}\right)^2\left(\frac{3v}{4\pi}\right)^{2/3} +4c_{15}-c_{16}\right] \\ &+ \frac{3\pi c_{12}}{16}-3c_{13}, \end{aligned} $$

(133)

where the coefficient c ₁₃ can be derived for a concrete isotropic particle-matrix system by the Taylor series for the integrated function in terms of ${\left(\varphi-\varphi_0\right)^n}$ about the point ${\varphi_0=0}$;

$$ \varphi_{73}= \arctan\left[\frac{R}{2x_1}\left(\frac{4\pi}{3v}\right)^{1/3}\right], $$

(134)

$$ \nu_6=\arctan \left[\frac{2x_1}{R\cos\varphi}\left(\frac{3v}{4\pi}\right)^{1/3} \right], $$

(135)

$$ c_{18}=\frac{s_{11m}}{2} \left[R^2\left(\frac{4\pi}{3v}\right)^{2/3}-\pi\left(R^2-x_1^2\right)\right], \quad x_1\in\left\langle 0,R\right\rangle,$$

(136)

$$ c_{19}=\frac{s_{11m}R^2}{2}\left(\frac{4\pi}{3v}\right)^{2/3}. $$

(137)