**3. Creep model: experimental input and simulation setup**

The input parameters needed to simulate creep over a range of stresses can be divided into three groups/types:


Group (i) is applicable to a wider range of materials and has already been collected in [5]. **Table 2** gives an overview on the findings.

Group (ii) is the parameter for a specific material badge. This group accounts for the different creep behavior stemming from specific processing routines, e.g.


#### **Table 2.**

*Input parameters for the material group. Input parameters type (i).*

chemical composition and heat treatment. Result from the processing routine is the asreceived microstructure, which also acts as an input for our simulation. In our case, we directly measured the subgrain size *R*sgb and the boundary dislocation density *ρb*via electron backscatter diffraction (EBSD) [5]. The typical as-received mobile dislocation density *ρm*was taken from Panait [19]. Simulations reveal that *ρ<sup>m</sup>* quickly converges toward generic values during creep, so the results are not very sensitive to the initial value of this parameter. The same is true for*ρs*, which has been adopted from [11]. The precipitate data have to be split into particles at subgrain boundaries (SGBs) and particles within subgrains (VN, NbC, and AlN). Both have been calculated by the thermodynamic software MatCalc [20] with the chemical composition of the heat, the actual heat treatment, and matrix data [5]. The results of the particle simulation, coarsening of particles from as-received condition onward, have been mimicked by a simple coarsening law, as shown in Eqs. (20) and (21), which is now used as input for the creep simulation. **Table 3** summarizes the microstructure-specific input data for the as-received condition, which are the starting values of our creep simulation.

After these parameters have been set, only the variables *a*1, *A*, and *β* remain (group (iii) of input parameters), whereas *A* is of phenomenological nature and thus cannot be measured directly as a matter of principle, and the parameters *a*<sup>1</sup> and *β* have physical interpretations but are extremely inconvenient for direct assessment. We therefore have to set up one single (!) creep experiment at set temperature *T* and stress *σapp*,0 to fit these parameters.

*Microstructurally Based Modeling of Creep Deformation and Damage in Martensitic Steels DOI: http://dx.doi.org/10.5772/intechopen.104381*


#### **Table 3.**

*Microstructure-specific input data for the as-received state. Input parameters type (ii).*

We conducted/performed an instrumented creep test at 650°C and a nominal stress of 70 MPa which led to a rupture time of 8740 h [5]. The simulated creep curve has been validated against the experimental results and the missing parameters *a*1, *A*, and *β* were optimized using a least-squares fit of the time-dependent creep deformation. **Table 4** summarizes the results which are also used for all other creep simulations presented in the following text.

Since the creep model already contains the impact of stress and temperature explicitly within its network of equations, the found input parameters are stress- and temperature-independent. We can thus use the same input for other stresses to produce time-to-rupture (TTR) diagrams. To do so, we have left all input parameters unchanged as indicated in **Tables 2**–**4** and generated creep simulations in the stress range of 50–120 MPa. The next section comprises the result of the simulated master creep curve including the calculated microstructural evolution, as well as the changes of the creep behavior with altering stresses, leading to the construction of the TTR diagram.


**Table 4.**

*Parameters found by fitting against one creep curve. Input parameters type (iii).*

## **4. Creep model: simulation results and discussion**

First simulated result is the master-creep curve at 650°C and 70 MPa, indicating the creep deformation and deformation rate. **Figures 2** and **3** demonstrate the agreement between the simulated and the experimental result of the creep deformation: the simulated creep curve is very close to the experiment including primary, secondary, and tertiary creep stage, and also the final fracture of the sample. The experimental minimum of the creep strain rate is about 2.5 <sup>10</sup><sup>6</sup> <sup>h</sup><sup>1</sup> in the range between 1.000 and 3.000 h, whereas the simulated result is 3.0 <sup>10</sup><sup>6</sup> <sup>h</sup><sup>1</sup> at 900 h, suggesting good agreement as well.

Regarding the microstructural evolution, the simulation predicts a quick recovery of the mobile dislocation density *<sup>ρ</sup>m*from 4.5 <sup>10</sup><sup>14</sup> <sup>m</sup><sup>2</sup> to 1.5 <sup>10</sup><sup>13</sup> <sup>m</sup><sup>2</sup> within the first 500 h of creep, which is exactly mirroring the continuous decrease in the creep rate during the primary creep regime. According to the (modified) Orowan equation (Eq. (1)), two potential reasons can be responsible for the creep rate: the mobile dislocation density and the effective velocity of the dislocations. Our model demonstrates the dominating role of the dislocation density in this regard. After reaching its minimum, *<sup>ρ</sup><sup>m</sup>* increases slowly up to about 2 <sup>10</sup><sup>13</sup> <sup>m</sup><sup>2</sup> at the time of fracture. This level agrees well with several literature data on similar material under comparable conditions, indicating dislocation densities at time of fracture of 2 <sup>10</sup><sup>13</sup> <sup>m</sup><sup>2</sup> [21] or 2.7 <sup>10</sup><sup>13</sup> <sup>m</sup><sup>2</sup> to 3.5 <sup>10</sup><sup>13</sup> <sup>m</sup><sup>2</sup> [22].

*ρ<sup>b</sup>* and *R*sgb were verified using EBSD on the fractured sample of the master-creep experiment, showing very good agreement with the simulated data, whereas simulation indicates a subgrain radius of 0.95 μm at the time of fracture and experimental verifications reveal a mean value of 0.7 μm [5]. The experimental value of boundary dislocation density of 3.4 <sup>10</sup><sup>14</sup> <sup>m</sup><sup>2</sup> also compares very well to the simulated result of 3.0 <sup>10</sup><sup>14</sup> <sup>m</sup><sup>2</sup> (**Figure 4**) [5].

**Figure 2.** *Experimental and simulated master-creep curve at 650°C, 70 MPa.*

*Microstructurally Based Modeling of Creep Deformation and Damage in Martensitic Steels DOI: http://dx.doi.org/10.5772/intechopen.104381*

**Figure 3.** *Experimental and simulated creep-strain rates at 650°C, 70 MPa.*

**Figure 4.** *Simulated microstructural evolution at 650°C, 70 MPa.*

In summary, the simulation results of the test case of the master-creep experiment are good enough to apply the model to multiple stresses. We carried out creep simulation in the stress range of 50–120 MPa in steps of 10 MPa, with a resulting creep

curve and accompanying microstructural evolution in each simulation. One side result, the rupture time *t*R, could then be used for reconstructing a time-torupture (TTR) diagram. In the simulation, the sample ruptures when the damage parameter *D*cav reaches a level of 1, which is basically a result of extensive creep strain and/or strain rate [5]. This typically occurs at a strain between 3% (low stresses) and 10% (high stresses). Please note that the fracture elongation is not an input, but a result of the simulation. Please also note that the simulation does not consider local necking but only deals with sample areas of uniform cross sections. **Figure 5** shows the individual simulated creep curves within the investigated stress range.

Please note that the creep curves appear to look different from **Figure 2** because the logarithmic time-scale is used in order to simultaneously show all results. Each of the creep curves feature primary, secondary, and tertiary creep regimes. **Figure 6** now finally shows the constructed TTR stemming from the simulated creep data and compares them against the standard literature data from European Creep Collaborative Committee (ECCC) [23], ASME [24], and NIMS [25].

The agreement is excellent, and the simulated curve lies right between the data from the three standard literature sources for creep rupture data of P91. Once again, it is important to mention that all model input data (except for the system stress) were identical for all creep simulations. This detail is very important, because the simulation allows for a predictive extrapolation from a single-creep experiment carried out for 8.740 h to up to six times longer creep times. As it appears, the simulation also allows for extrapolating to shorter running times by a factor of about 40 in our case. We thus motivate to use and test our model for even shorter reference experiments in order to extrapolate for long running times.

**Figure 5.** *Simulated microstructural evolution at 650°C, 70 MPa.*

*Microstructurally Based Modeling of Creep Deformation and Damage in Martensitic Steels DOI: http://dx.doi.org/10.5772/intechopen.104381*

**Figure 6.** *Simulated TTR diagram compared against standard literature data.*

## **5. Pore formation model: introduction**

It has been long established [26] and is now well accepted [27, 28] that failure during creep loading results mainly due to intergranular rupture. Cavities nucleate predominantly at grain boundaries, grow during creep exposure, and coalesce to form microcracks. In tertiary creep, these cracks are so numerous that they significantly weaken the microstructure, and the remaining available cross section is put under more stress which further promotes damage and accelerates the strain rate.

In some cases, the remaining creep life can be directly correlated with the degree of cavitation [29, 30].
