**Abstract**

This chapter deals with modeling the microstructural evolution, creep deformation, and pore formation in creep-resistant martensitic 9–12% Cr steels. Apart from the stress and temperature exposure of the material, the input parameters for the models are as-received microstructure and one single-creep experiment of moderate duration. The models provide predictive results on deformation rates and microstructure degradation over a wide stress range. Due to their link to the underlying fundamental physical processes such as classical nucleation theory, Gibbs energy dissipation, climb, and glide of dislocations, etc., the models are applicable to any martensitic steel with similar microstructure to the presented case study. Note that we section the chapter into part 1: creep deformation and part 2: pore formation.

**Keywords:** microstructure, dislocations, precipitates, subgrains, pores, deformation rate, damage

## **1. Introduction**

When trying to simulate deformation rates during creep, one can basically follow two different approaches: (i) phenomenologically based—and or (ii) physically inspired models. In phenomenological models, the deformation (rate) is usually stated as analytical function with system parameters such as temperature and stress as input [1–4]. These kinds of models are easily and quickly employed; however, the approaches carry some disadvantages as well: (1) they give little or no insight into the actual underlying physical processes governing the creep rate, and (2) the model parameters cannot be usually determined independently from the creep experiment the model is actually aiming to predict. For these reasons, we choose to focus on a physically based model instead [5], which is a reviewed, corrected, and extended version of the seminal work of Ghoniem [6]. In addition to avoiding the mentioned drawbacks of phenomenological approaches, our physical model has the advantage of including a variety of microstructural elements such as dislocations and subgrain

boundaries (SGBs), and their interactions. This allows for getting a deeper understanding of the creep process and opens the opportunity of rating individual material badges by taking their as-received microstructure as starting condition for a creep simulation. Finally, we end up with an assembly of rate equations for the microstructural elements along with some side equations modeling the physical processes. In summary, the model gives us insights into the specific reasons why a material badge features good or bad creep behavior, as long as its microstructure can be considered homogeneous. The model also has the potential to rate the impact of individual microstructural phenomena.

Our model includes the microstructure by mean values of specific microstructural elements (e.g. dislocation density, grain boundary precipitates, etc.) instead of a spatially resolved features. This allows for a simpler construction of a "representative volume element," which speeds up computation and is more easily compared to microstructural investigations.

In our work, we focus on martensitic 9–12% Cr steels. We select the material P91 to demonstrate the validity of the model due to its widespread use and industrial significance. Nevertheless, the concept can be adapted to other material groups by including their specific microstructural elements and their interactions.

Please note that all symbols used in the equations are explained in **Table 6** at the end of the chapter.

## **2. Creep model: our microstructurally based approach**

The aim of our creep model is to predict the creep rate and microstructural evolution, based on the initial microstructure and the system parameters stress and temperature. A very simple, yet useful approximation of the creep rate *ε*\_ has been introduced by Orowan [7] and later extended by Yadav [8] to include damage. While Yadav considers damage due to cavities *Dcav* and precipitates *Dppt*, damage in our model is based on cavities alone, since the effect of precipitates is dealt with other equations in the framework in a physical manner. The resulting modified Orowan equation calculates the creep rate from the physical inputs of mobile dislocation density *ρ<sup>m</sup>* and an effective dislocation velocity *υeff* ; see Eq. (1). Latter term has been introduced by Riedlsperger [5] incorporating glide and climb processes. Since both quantities, *ρ<sup>m</sup>* and *υeff* , result from interactions within the microstructure, we take a closer look at those, see **Figure 1**. Within **Figure 1**, we indicate following interactions:


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

#### **Figure 1.**

*Microstructural interactions within a subgrain. See text for individual interactions. "X" signifies annihilation of dislocations and arrows mark transformations.*

These interactions (a–g) are also integrated into the rate equations for the microstructural evolution of mobile dislocations *ρm*, static dislocations *ρs*, dislocations within subgrain boundaries *ρb*, and subgrain boundaries *Rsgb*; see Eqs. (2)–(5) in **Table 1**. **Table 1** also indicates the original source for each equation.

In addition to the "rate equations" of the microstructural evolution of the material, **Table 1** also assembles the framework of the underlying physical phenomena. Within this paragraph, we only give a brief overview. Detailed discussions can be found in the cited sources.

The effective subgrain growth pressure *P*eff, Eq. (6), is rewritten, but it is equivalent to Ghoniem's work [6] and includes dissipation of grain boundary energy as well as Zener pinning from precipitates. The mobility of the subgrains, *M*sgb, Eqs (7) and (8), is governed mostly by diffusion coefficients and the boundary misorientation [6]. Dislocation spacing within a subgrain boundary, *h*b, Eq. (9), has been deducted by geometrical means [6]. The effective velocity of mobile dislocations *veff* includes glide processes as well as climbing of dislocations over precipitates within the subgrain interior [5]. The glide velocity of mobile dislocations *υg*considers forward and potential backward movements according to their jump probabilities, which are linked to mechanical and thermal activation [9]. The corrected applied stress *σapp*, as shown in Eq. (12), considers the reduced cross section of a creep sample due to its poisson ratio [5]. Since the climb of mobile dislocations depends on local diffusion, the climb velocity, *υc*, as shown in Eq. (13) [6], is split into a lattice diffusion share, *υcl*, as shown in Eq. (14) [5], and a pipe diffusion share, *υcp*, as shown in Eq. (16) [5], with additional terms regarding the


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


#### **Table 1.**

*Creep model equations and their primary literature sources.*

vacancy-dislocation interactions from Hirth and Lothe [10]; see Eq. (17). The internal stress *σi*, which is an important input for the glide and climb velocities and considers interactions of mobile-mobile and mobile-static dislocations, is taken from Basirat [11]; see Eq. (18). Finally, the phenomenological factor "cavitation damage" *D*cav, as shown in Eq. (19), has been introduced by Basirat [11] as well but slightly adapted here for a better agreement with the observed master creep curve.
