2.2.3 Explicit creep cavity damage fracture model

The explicit creep fracture model can be derived with the given values of creep cavitation constants [12].

For the given values of α ¼ 1, β ¼ 2, and γ =1 for P91 [12], Eq. (5), w, is simplified as:

$$w = \pi \times \frac{\mathfrak{Z}}{\mathfrak{F}} \times \mathfrak{F}^{\frac{2}{3}} \times U^{\frac{\mathfrak{z}}{3}} \times A\_1^{2/3} A\_2 \times t^{1+\gamma} \tag{11}$$

And further,

$$
\omega = U' \times \mathbf{t}^{1+\gamma} \tag{12}
$$

where <sup>U</sup><sup>0</sup> <sup>¼</sup> <sup>π</sup> � <sup>3</sup> <sup>5</sup> � 3 2 <sup>3</sup> � <sup>U</sup><sup>5</sup> � <sup>A</sup><sup>1</sup> 2=3 A2; it is termed as creep lifetime coefficient. Insert γ =1 in Eq. (12); the creep cavity damage function is:

$$
\omega = U' \times t^2 \tag{13}
$$

$$
\omega\_f = U' \times \mathbf{t}\_f^{\;>} \tag{14}
$$

The creep fracture is assumed to occur when the area coverage attains a critical value, denoted by wf . The critical value, wf , will be will be chosen for as wf <sup>¼</sup> <sup>π</sup> 4, since regularly spaced round cavities touch each other if wf <sup>¼</sup> <sup>π</sup> <sup>4</sup> [1].

#### 2.2.4 Application for lifetime prediction over a wider stress range

This section investigates such dependence on stress. The method is stated as:


#### 2.3 Mesoscopic composite approach modelling

	- 1.The current creep continuum damage mechanics operates at a macroscopic level with ambiguity in the depicting of the creep deformation and creep damage; hence a mesoscopic level composite model is necessary.

This micro-mechanical-based smeared-out grain boundary element for of copper-antimony alloy [23, 24] has been chosen in the current development, as there is not that much choice. The main contents are:


#### 2.3.2 Grain boundary element

The GB displacement jump at a normal direction can be obtained by the model which is developed by Markus Vöse [30]. It takes into account nucleation, growth, coalescence, and sintering of multiple cavities and can be written as (Figure 10) [30, 31]:

$$\frac{d\beta}{d\dot{t}} = \frac{\Im\beta}{2\dot{\beta}} \left(\dot{a}\_p - \acute{a}\_d\right) + \sqrt{\dot{\beta}} \sqrt[3]{3\Theta h(\wp)\pi\beta^2} \frac{d\dot{a}}{d\dot{t}},\tag{15}$$

$$\frac{d\rho}{dt} = \acute{a}\_p(1 - f) - \acute{a}\_a \tag{16}$$

$$
\acute{a}\_a = \varkappa\_3 \cdot 8\pi \acute{\rho}^2 \acute{a} \frac{d\acute{a}}{d\acute{t}},\tag{17}
$$

$$\frac{d\boldsymbol{d}}{d\boldsymbol{\acute{t}}} = \boldsymbol{\infty}\_{1} \cdot \frac{2\boldsymbol{\acute{D}}\_{gb}}{h(\boldsymbol{\wp})} \frac{\left[1 - \acute{\boldsymbol{a}}\_{tip}(\boldsymbol{\acute{a}}) \cdot (\mathbbm{1} - \boldsymbol{\varkappa}\_{2}\boldsymbol{\alpha})\right]}{\boldsymbol{a}^{2} \cdot q(\boldsymbol{\varkappa}\_{2}\boldsymbol{\alpha})},\tag{18}$$

$$
\rho = \sqrt[3]{\frac{9\pi\rho^2}{16h^2(\mu)}} \mathbf{\hat{a}} = \frac{1}{\sqrt{\rho}} \sqrt[3]{\frac{3}{4} \frac{\rho}{h(\mu)\pi}},\tag{19}
$$

$$f = \frac{(\eta - 1)\alpha}{1 - \alpha},\tag{20}$$

$$\eta = \exp\left( \left[ \varkappa\_4 \cdot 2\pi \dot{D}\_{gb} \left( \dot{a}\_{tip}(\dot{a} = 1) - \dot{a}\_{tip}(\dot{a}) \right) \dot{\rho} \left( \frac{d\dot{\rho}^p}{d\dot{t}} \right)^{-1} \right] \right) \tag{21}$$

$$\frac{d\mu^p}{dt} = \frac{\beta}{\sqrt{\rho^5}} \left(\dot{a}\_p - \acute{a}\_a\right) + \sqrt[3]{3\Theta h(\wp)\pi\rho^2} \frac{d\acute{a}}{dt},\tag{22}$$

where

$$q(\boldsymbol{\alpha}) = -2ln\boldsymbol{\alpha} - (\mathfrak{Z} - \boldsymbol{\alpha})(\mathfrak{1} - \boldsymbol{\alpha}); \acute{a}\_{tip}(\acute{a}) = \mathfrak{Z} \acute{\boldsymbol{\jmath}}\_{\boldsymbol{s}} \boldsymbol{\sin}\boldsymbol{\mu}/\acute{a} \tag{23}$$

In this equation, β is the damage variable, ρ is the cavity density, a is the average radius of the cavity, α�<sup>p</sup> is the stress-dependent nucleation rate, α�<sup>a</sup> is the annihilation rate, ψ is the dihedral angle of the cavity (70° ), D�gb is the GB diffusion coefficient, and ω is the damaged area fraction. The creep degradation of GB is calibrated by three variables: ρ, β and a. These three parameters not only determine the failure degree of GB but also determine the amount of the creep nonlinear deformation. Therefore, ρ, β, and a are the three indicators for the benchmark.

#### 2.3.3 Computational platform development

The well-known displacement-based creep damage algorithm will be adopted here directly [31], which were used and reported, for example, research [32] by Richard Hall.

The main further work is the calculation of the stiffness matrix for the grain boundary element and the integration of the grain boundary element, parallel to that for grain element.

#### 3. Materials


The parameters for the grain boundary cavity model is <sup>D</sup>~gb <sup>¼</sup> <sup>10</sup>�<sup>14</sup>mm<sup>5</sup>N�<sup>1</sup> s �1, ap <sup>¼</sup> <sup>2</sup> � <sup>10</sup><sup>2</sup> mm�<sup>2</sup>s �1, and bp <sup>¼</sup> 1. The power law creep is used to describe the creep mechanism of the grain part.

The material parameters of copper power law for grain [24] are (400–700°C): A: 38.8 MPanS<sup>m</sup><sup>1</sup> , Q: 197 KJmol<sup>1</sup> , n:4.8, and m:0.

A 1 mm<sup>2</sup> squire geometry is chosen, 20 grains and 60 grain boundary were meshed by 909 triangle pane strain element; 152 interface elements, using Neper free software [34]. A finer mesh would be desirable; however, it is a compromise to accept this size to proceed.

A tensile load of 10 MPa was applied uniformly on the top side; and the left side and the bottom side was pinned in X direction and Y direction, respectively.
