Thermal shock

Thermal shock resistance measures can be used for material selection in applications subject to rapid temperature changes. The maximum temperature jump, ${\displaystyle \Delta T}$, sustainable by a material can be defined for strength-controlled models by:[4][3]

$${\displaystyle B\Delta T={\frac {\sigma _{f}}{\alpha E}}}$$

where ${\displaystyle \sigma _{f}}$ is the failure stress (which can be yield or fracture stress), ${\displaystyle \alpha }$ is the coefficient of thermal expansion, ${\displaystyle E}$ is the Young's modulus, and ${\displaystyle B}$ is a constant depending upon the part constraint, material properties, and thickness.

$${\displaystyle B={\frac {C}{A}}}$$

where ${\displaystyle C}$ is a system constrain constant dependent upon the Poisson's ratio, ${\displaystyle \nu }$, and ${\displaystyle A}$ is a non-dimensional parameter dependent upon the Biot number, ${\displaystyle \mathrm {Bi} }$.

$${\displaystyle C={\begin{cases}1&{\text{axial stress}}\\(1-\nu )&{\text{biaxial constraint}}\\(1-2\nu )&{\text{triaxial constraint}}\end{cases}}}$$

${\displaystyle A}$ may be approximated by:

$${\displaystyle A={\frac {Hh/k}{1+Hh/k}}={\frac {\mathrm {Bi} }{1+\mathrm {Bi} }}}$$

where ${\displaystyle H}$ is the thickness, ${\displaystyle h}$ is the heat transfer coefficient, and ${\displaystyle k}$ is the thermal conductivity.

If perfect heat transfer (${\displaystyle \mathrm {Bi} =\infty }$) is assumed, the maximum heat transfer supported by the material is:[4][5]

$${\displaystyle \Delta T=A_{1}{\frac {\sigma _{f}}{E\alpha }}}$$

• ${\displaystyle A_{1}\approx 1}$ for cold shock in plates
• ${\displaystyle A_{1}\approx 3.2}$ for hot shock in plates

A material index for material selection according to thermal shock resistance in the fracture stress derived perfect heat transfer case is therefore:

$${\displaystyle {\frac {\sigma _{f}}{E\alpha }}}$$

For cases with poor heat transfer (${\displaystyle \mathrm {Bi} <1}$), the maximum heat differential supported by the material is:[4][5]

$${\displaystyle \Delta T=A_{2}{\frac {\sigma _{f}}{E\alpha }}{\frac {1}{\mathrm {Bi} }}=A_{2}{\frac {\sigma _{f}}{E\alpha }}{\frac {k}{hH}}}$$

• ${\displaystyle A_{2}\approx 3.2}$ for cold shock
• ${\displaystyle A_{2}\approx 6.5}$ for hot shock

In the poor heat transfer case, a higher heat transfer coefficient is beneficial for thermal shock resistance. The material index for the poor heat transfer case is often taken as:

$${\displaystyle {\frac {k\sigma _{f}}{E\alpha }}}$$

According to both the perfect and poor heat transfer models, larger temperature differentials can be tolerated for hot shock than for cold shock.

In addition to thermal shock resistance defined by material fracture strength, models have also been defined within the fracture mechanics framework. Lu and Fleck produced criteria for thermal shock cracking based on fracture toughness controlled cracking. The models were based on thermal shock in ceramics (generally brittle materials). Assuming an infinite plate and mode I cracking, the crack was predicted to start from the edge for cold shock, but the center of the plate for hot shock.[4] Cases were divided into perfect and poor heat transfer to further simplify the models.

The sustainable temperature jump decreases, with increasing convective heat transfer (and therefore larger Biot number). This is represented in the model shown below for perfect heat transfer (${\displaystyle \mathrm {Bi} =\infty }$).[4][5]

$${\displaystyle \Delta T=A_{3}{\frac {K_{Ic}}{E\alpha {\sqrt {\pi H}}}}}$$

where ${\displaystyle K_{Ic}}$ is the mode I fracture toughness, ${\displaystyle E}$ is the Young's modulus, ${\displaystyle \alpha }$ is the thermal expansion coefficient, and ${\displaystyle H}$ is half the thickness of the plate.

• ${\displaystyle A_{3}\approx 4.5}$ for cold shock
• ${\displaystyle A_{4}\approx 5.6}$ for hot shock

A material index for material selection in the fracture mechanics derived perfect heat transfer case is therefore:

$${\displaystyle {\frac {K_{Ic}}{E\alpha }}}$$

For cases with poor heat transfer, the Biot number is an important factor in the sustainable temperature jump.[4][5]

$${\displaystyle \Delta T=A_{4}{\frac {K_{Ic}}{E\alpha {\sqrt {\pi H}}}}{\frac {k}{hH}}}$$

Critically, for poor heat transfer cases, materials with higher thermal conductivity, k, have higher thermal shock resistance. As a result, a commonly chosen material index for thermal shock resistance in the poor heat transfer case is:

$${\displaystyle {\frac {kK_{Ic}}{E\alpha }}}$$

The temperature difference to initiate fracture has been described by William David Kingery to be:[6][7]

$${\displaystyle \Delta T_{c}=S{\frac {k\sigma ^{*}(1-\nu )}{E\alpha }}{\frac {1}{h}}={\frac {S}{hR^{'}}}}$$

where ${\displaystyle S}$ is a shape factor, ${\displaystyle \sigma ^{*}}$ is the fracture stress, ${\displaystyle k}$ is the thermal conductivity, ${\displaystyle E}$ is the Young's modulus, ${\displaystyle \alpha }$ is the coefficient of thermal expansion, ${\displaystyle h}$ is the heat transfer coefficient, and ${\displaystyle R'}$ is a fracture resistance parameter. The fracture resistance parameter is a common metric used to define the thermal shock tolerance of materials.[1]

$${\displaystyle R'={\frac {k\sigma ^{*}(1-v)}{E\alpha }}}$$

The formulas were derived for ceramic materials, and make the assumptions of a homogeneous body with material properties independent of temperature, but can be well applied to other brittle materials.[7]