**4. Cavity growth**

#### **4.1. Unconstrained cavity growth model**

After the cavities have been nucleated, they start to grow if they exceed a critical size. The main mechanism for the growth is diffusion. Vacancies are transported away from the surfaces of the cavities. The grain boundaries are good sinks for the vacancies. The first model for

**Figure 9.** Comparison of experimental number of cavities and modelling number of particles that initiate cavities [24]. Experimental data for austenitic stainless steels from Refs. [27, 45, 46, 49, 50].

diffusion controlled growth was presented by Hull and Rimmer [15]. A much more elegant formulation was later given by Beere and Speight [16] and this is the model that has been used since. Their growth equation can be expressed as

where λ is the interpaticle spacing, γ<sup>s</sup>

30 Study of Grain Boundary Character

types of austenitic stainless steels [24].

**4.1. Unconstrained cavity growth model**

**4. Cavity growth**

The experimental data clearly support Harris' model.

observed ones for austenitic stainless steels in **Figure 9.**

the surface energy, δ the grain boundary width, *D*GB the

grain boundary self-diffusion coefficient, Ω the atomic volume, *k*B Boltzmann's constant and *T* the absolute temperature. The application of Eq. (11) is illustrated in **Figure 8** for different

**Figure 8.** Comparison of experimental and modelling GBS velocity as a function of particle radius [24]. Experimental

data for 304, 321 and 347 austenitic stainless steels from Refs. [45, 46, 49, 50]. Modelling results from Eq. (11).

In **Figure 8** the particle parameters are taken from the experimental references. The minimum particles that nucleated cavities in the experiments are chosen for the critical particle radius.

From the particle size distributions [24] the number of nucleated cavities can be estimated if the critical particle size is known. The computed number of nuclei is compared with the

After the cavities have been nucleated, they start to grow if they exceed a critical size. The main mechanism for the growth is diffusion. Vacancies are transported away from the surfaces of the cavities. The grain boundaries are good sinks for the vacancies. The first model for

$$\frac{d\mathbb{R}}{dt} = \mathcal{D}\_0 K\_\mathbb{I} (\sigma\_{\text{uppl}} - \sigma\_0) \frac{1}{R^2} \,\,\,\,\tag{12}$$

where *R* the cavity radius in the grain boundary plane, *dR*/*dt* its growth rate, σ<sup>0</sup> the sintering stress 2γ<sup>s</sup> sin(α)/*R*, where γ<sup>s</sup> is the surface energy of the cavity per unit area and α the cavity tip angle. *D*<sup>0</sup> is a grain boundary diffusion parameter, *D*<sup>0</sup> = δ*D*GBΩ/*k*B*T*, where δ is the grain boundary width, *D*GB the grain boundary self-diffusion coefficient, Ω the atomic volume, *k*<sup>B</sup> Boltzmann's constant and *T* the absolute temperature. The factor *K*<sup>f</sup> was introduced in [16]. It is a function of the cavitated grain boundary area fraction *f* <sup>a</sup> = (2*R*/*L*)2

$$K\_{\mathbf{i}} = -1\left| \left[ 2\log f\_{\mathbf{i}} + (1 - f\_{\mathbf{i}})(3 - f\_{\mathbf{i}}) \right] \right.\tag{13}$$

From the number of cavities per unit grain boundary area *n*cav, the cavity spacing *L* can be determined

$$L = 1/\sqrt{n\_{\text{cov}}}.\tag{14}$$

*n*cav can be found from the nucleation model, Eq. (10). Plastic deformation can also contribute to the growth rate. Danavan and Solomon have given an expression for that [51]

$$\frac{dR}{dt} = \frac{\sin^2(\alpha)}{\alpha - \sin(\alpha)\cos(\alpha)} \frac{R}{3} \,\mathrm{\dot{\varepsilon}}\,. \tag{15}$$

#### **4.2. Constrained cavity growth**

When diffusion controlled growth models were compared with experimental data, it was evident that the models often strongly exaggerated the growth rate. Dyson found that the predicted growth rate of the cavities many times exceeded the deformation rate of the surrounding material which he considered as unphysical [17]. He suggested that the cavity growth rate should not be larger than the creep rate of the material. This was referred to as constrained growth. Based on this assumption, Rice developed a quantitative model [18]. The result is that in the growth equation, the applied stress is replaced by a reduced stress

$$\frac{d\mathbb{R}}{dt} = \mathcal{Z}\,D\_0\,\mathrm{K}\_\mathbb{I}(\sigma\_{\mathrm{mod}} - \sigma\_0)\,\frac{1}{\mathbb{R}^2}.\tag{16}$$

The reduced stress is given by

$$
\sigma\_{\rm rad} = \sigma\_0 + \frac{1}{\frac{1}{\sigma\_{\rm opt}} + \frac{32D\_oK\_\gamma}{L^2 \, d\beta \, \mathcal{E}\{\sigma\_{\rm opt}\}}} \,\tag{17}
$$

where β is a material constant (β = 1.8 for homogeneous materials) and *d* the grain diameter. With this approach a growth model that fulfils Dyson's criterion has been achieved.

Rice based his analysis on a linear viscoplastic model of an opening crack. He and Sandstrom reanalyzed the model and avoided the assumption of linearity [52]. A grain structure with a pillar of height *h* and width corresponding to the grain size *d* was set up. In this pillar the creep deformation in the axial (*z*) direction is given by

$$\frac{dz}{dt} = 4\pi \, D\_0 K\_l(\sigma\_{\rm rad} - \sigma\_0) \, n\_{\rm av} + h \, \dot{\varepsilon}(\sigma\_{\rm rad}) = h \, \dot{\varepsilon}(\sigma\_{\rm appl}).\tag{18}$$

*ε* . (*σ*red) and *ε* . (*σ*appl) are the creep rates at the reduced and applied stress, respectively. The first term in the middle part of Eq. (17) is the volume growth rate of a cavity multiplied by the number of cavities per unit grain boundary area. The creep displacement of the pillar at the reduced stress is the second term. The final term on the right hand is the displacement in the surrounding material. A finite element analysis was performed to determine the size of

*<sup>L</sup>* <sup>=</sup> <sup>1</sup>/ <sup>√</sup>

\_\_\_ *dR*

\_\_\_ *dR*

*<sup>σ</sup>*red <sup>=</sup> *<sup>σ</sup>*<sup>0</sup> <sup>+</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_ 1 \_\_\_\_ <sup>1</sup>

creep deformation in the axial (*z*) direction is given by

The reduced stress is given by

\_\_\_ *dz*

.

*ε* .

(*σ*red) and *ε*

**4.2. Constrained cavity growth**

32 Study of Grain Boundary Character

stress

\_

\_\_ *R* 3 *ε* .

\_\_1

*n*cav can be found from the nucleation model, Eq. (10). Plastic deformation can also contribute

*dt* <sup>=</sup> sin2 \_\_\_\_\_\_\_\_\_\_\_\_\_ (*α*) *<sup>α</sup>* <sup>−</sup> sin(*α*) cos(*α*)

When diffusion controlled growth models were compared with experimental data, it was evident that the models often strongly exaggerated the growth rate. Dyson found that the predicted growth rate of the cavities many times exceeded the deformation rate of the surrounding material which he considered as unphysical [17]. He suggested that the cavity growth rate should not be larger than the creep rate of the material. This was referred to as constrained growth. Based on this assumption, Rice developed a quantitative model [18]. The result is that in the growth equation, the applied stress is replaced by a reduced

*dt* <sup>=</sup> <sup>2</sup> *<sup>D</sup>*<sup>0</sup> *<sup>K</sup>*f(*σ*red <sup>−</sup> *<sup>σ</sup>*0)

*σ*appl

where β is a material constant (β = 1.8 for homogeneous materials) and *d* the grain diameter.

Rice based his analysis on a linear viscoplastic model of an opening crack. He and Sandstrom reanalyzed the model and avoided the assumption of linearity [52]. A grain structure with a pillar of height *h* and width corresponding to the grain size *d* was set up. In this pillar the

term in the middle part of Eq. (17) is the volume growth rate of a cavity multiplied by the number of cavities per unit grain boundary area. The creep displacement of the pillar at the reduced stress is the second term. The final term on the right hand is the displacement in the surrounding material. A finite element analysis was performed to determine the size of

With this approach a growth model that fulfils Dyson's criterion has been achieved.

*dt* <sup>=</sup> <sup>4</sup>*<sup>π</sup> <sup>D</sup>*<sup>0</sup> *<sup>K</sup>*f(*σ*red <sup>−</sup> *<sup>σ</sup>*0) *<sup>n</sup>*cav <sup>+</sup> *<sup>h</sup> <sup>ε</sup>*

<sup>+</sup> <sup>32</sup> *<sup>D</sup>*<sup>0</sup> *<sup>K</sup>* \_\_\_\_\_\_\_\_\_\_ <sup>f</sup> *L*<sup>2</sup> *dβ ε* . (*σ*appl)

.

(*σ*appl) are the creep rates at the reduced and applied stress, respectively. The first

(*σ*red) = *h ε*

.

to the growth rate. Danavan and Solomon have given an expression for that [51]

*n*cav . (14)

. (15)

*<sup>R</sup>*<sup>2</sup> . (16)

, (17)

(*σ*appl). (18)

**Figure 10.** Reduced stress according to Eq. (19) versus time [52]. The result is compared with the model of Rice in Eq. (17) [18]. Cavity growth for 18Cr10Ni at 727°C and 100 MPa [45].

**Figure 11.** Cavity radius as a function of creep time for 18Cr10Ni without or with Nb (347) or Ti (321) austenitic stainless steels. Model according to Eq. (19) and experimental data from Refs. [27, 46, 49]. The creep tests were performed at temperatures in the interval of 650–812°C.

the height *h*. It was found that *h* ≈ 2*R* in the investigated cases [52]. If this value for *h* is used and *n*cav is replaced by 1/*L*<sup>2</sup> according to Eq. (14), the following equation is obtained

$$2\pi \, D\_0 \mathcal{K}\_{\text{fl}}(\sigma\_{\text{rad}} - \sigma\_0) \Big| L^2 \, \mathcal{R} + \dot{\varepsilon}(\sigma\_{\text{rad}}) = \dot{\varepsilon}(\sigma\_{\text{appl}}).\tag{19}$$

In general Eq. (19) has to be solved by iteration to find the new value of σred. This new value for σred is lower than that given by Eq. (17). This is illustrated in **Figure 10**: Both the absolute and relative difference increase with time.

The new constrained growth model is compared to experimental data for austenitic stainless steels in [52]. Some examples are given here in **Figure 11**. Growth data for 18Cr10Ni steel with and without Nb or Ti are shown. It can be seen that the growth data can be described with fair accuracy. The lower growth rate according Eq. (19) is important in this respect.
