Tertiary Creep

Abstract

In the tertiary stage, the creep rate is continously increasing eventually leading to rupture. Many mechanisms can contribute to the increasing creep rate such as particle coarsening, substructure coarsening, cavitation, changes in the dislocation density and necking. A large number of empirical models exist for the description of tertiary creep and the development of creep damage not least in the context of continuum damage mechanics (CDM). However, there are also basic models. An equation is presented that can describe the whole creep strain versus time curve. Only parameters that are already defined for secondary creep are needed. During the tertiary stage the true applied stress increases rapidly and faster than the counteracting dislocation strength, which is one main reason for the increase in the creep rate during the tertiary stage. Cavitation is of importance, but the cavitation is often local and therefore gives a modest contribution to the creep rate. According Hart’s criterion, necking starts right at the beginning of the tertiary stage. But the necking is not fully developed until close to rupture. This is demonstrated both by uniaxial and multiaxial models and it is also consistent with available experimental data.

12.1 General

Most creep tests are performed as tensile tests at constant load or stress. In a creep strain test the strain is recorded as a function of test time, and the result is referred to as a creep strain curve. As discussed in detail in Chap. 4, the common form of the creep strain curves is that the slope decreases in the primary stage, reaches a minimum in the secondary stage and increases in the tertiary stage. In the secondary stage the microstructure is assumed to be essentially constant. Changes in the microstructure contribute to the increase in the creep rate that is observed in the tertiary stage. These changes will be analyzed in this chapter.

In the scientific literature, much focus has been on secondary creep, primarily because that data have been used to identify the operating creep mechanisms. However, primary and tertiary creep are technically also of major importance, but the number of systematic studies is much more limited. A significant fraction of the creep strain in fcc alloys occurs in the primary stage. The key to the understanding of creep rupture is the behavior during the tertiary stage.

The changes in the microstructure that give rise to the acceleration of the creep rate in the tertiary stage are traditionally referred to as creep damage [1]. Many mechanisms are known that contribute to the creep damage [2]. The most discussed ones are particle coarsening and dissolution, formation of creep cavities, recovery of dislocations and subgrain growth. In creep resistant martensitic steels microstructure degradation has often been observed to induce creep rupture. Fine carbonitrides (e.g. MX) coarsen and dissolve during long-term creep. New brittle phases can be created (e.g. Z-phase, Laves phase, M₆X carbides). The fine particles give a significant contribution to the creep strength which is reduced when their number decreases. The creep strength and in particular the creep ductility are lowered further when new coarse phases are present and act as crack nucleation sites [3,4,5].

Basic models for particle coarsening [6] and subgrain growth have been available for a long time. This includes the effect of Zener pinning of subboundaries [7]. This is of importance for stabilizing the substructure in martensitic creep resistant steels. For cavitation the situation has been less satisfactory. A basic model for cavity nucleation has only appeared recently [8, 9]. The models for cavity growth needed improvement to describe experimental data, Chap. 10 [10]. Also the dislocation models were necessary to extend and take substructure into account to understand why essentially the same creep strain behavior is observed at low and high temperatures, Sect. 8.4 [11, 12].

There is extensive literature on modeling of creep damage, not least in the connection of continuum damage mechanics (CDM) where the models are used to analyze the behavior of components. Reviews are given in [13, 14]. Practically all commonly used models are empirical or partially based on physical principles with a number of adjustable parameters involved. It is important to recognize the limitations with the use of adjustable parameters. Empirical models for tertiary creep were analyzed in [15]. Only two adjustable parameters were needed for some models to represent tertiary creep data in a satisfactory way and with three or four parameters almost any of the available models can give a good fit, Sect. 4.2. The important conclusion is that a good description with an empirical model does not ensure physical significance and does not make the model predictable. An empirical model can be used to identify operating mechanisms only if the same parameter values are used to get a good fit for a number of curves that is much larger than the number of adjustable parameters. A brief summary of CDM is given in Sect. 12.2.

Spent nuclear fuel in Finland and Sweden will be placed in copper canisters 500 m down in the bedrock for permanent disposal. Oxygen free copper alloyed with 50 ppm P has been selected as canister material because of its creep properties and its corrosion resistance in the environment. The copper will be exposed to creep deformation due to the hydrostatic pressure and the swelling pressure from the surrounding bentonite clay at temperatures up to 100 °C. The canisters should stay intact for 100000 years until radiation has declined to a low level. The creep properties of the canister must be possible to predict for such an extended time. This is only possible with fundamental creep models. It has been verified that the creep model discussed in this book actually can cope with the required extended extrapolation, Sect. 14.4.6 [15].

Changes in the dislocation structure could generate microstructure degradation. A reduction in the dislocation density due to accelerated recovery has been found both experimentally [4, 16] and by computation [17]. Creep failure can also be induced by nucleation and growth of cavities [9, 10], Chap. 10. Also small defects can induce localized deformation if the material is plastically unstable which can result in necking [18]. During tensile creep testing at some stage, localized straining and necking occur. The role of the size of the initial defect has been studied [19,20,21,22].

Creep rupture is divided according to the type of failure. It is referred to ductile and brittle rupture if the creep ductility is high and low, respectively. There is no clear separation between these two types, but a rupture elongation above 30% is in general considered as high and an elongation below 10% as low. Brittle rupture is more crack sensitive since the creep cavities are more readily created. For brittle rupture the failure is primarily controlled by the formation of cavities, and when the cavitation is extensive enough, rupture occurs. Ductility exhaustion is assumed to initiate failure during ductile rupture. When the creep strain is sufficiently high, a plastic instability takes place and the component collapses. From a design point of view ductile rupture is preferred since more straining can take place before failure. The material is said to be more forgiving.

It will be assumed that tertiary creep is primarily controlled by the dislocation structure. The main mechanism for the increase in the creep rate in the tertiary stage has only recently been established [23]. The changes in the dislocation structure during a creep test can be quite complex. Modern 9Cr steel is a good example of that. To describe the creep deformation no less than three types of dislocation densities must be considered: mobile and immobile dislocations in the subgrain interiors and dislocations in the subgrain walls [24]. These complications will not be covered here. However, the general principles are expected to be the same. Instead, the analysis will be restricted to cases where only one type of dislocation controls the main behavior that is typical for example for fcc alloys. In this type of material there is generally a rapid increase in the dislocation density during primary creep. Contrary to what is stated in many places, there is slow continuous increase in the secondary stage in load controlled tests. During a creep test, there is a gradual reduction in the specimen cross section and for tests at constant load, this means that the true stress is increasing. During the secondary creep, this increase is matched by a corresponding increase in the dislocation density and thereby also the dislocation stress, Sect. 8.4. The rate of increase in the dislocation density is continuously reduced and in the tertiary stage this increase cannot keep up with the increase in the true stress and the creep rate is raised, Sect. 12.4.

There are other possible contributions to the increase in the creep rate in the tertiary stage. Cavitation plays a role in particular in creep brittle materials. However, extensive cavitation is typically strongly localized and does not appear over the whole material [25]. This means that cavitation gives a modest contribution to the increase in the creep rate. Particle coarsening is another effect. If particle coarsening takes place the internal stress from the particles is reduced and thereby an additional increase in the creep rate is obtained [26]. Particle coarsening is considered in Sect. 12.3. Coarsening of the substructure also takes place, but that is considered to be a part of the changes in the dislocation structure mentioned above, Sect. 12.4. Finally, necking and other forms of plastic instabilities are of importance. The few systematic studies on necking in creep specimens that exist suggest that necking takes place close to rupture. This will be further discussed in Sect. 12.5 [23, 27].

12.2 Empirical Models for Tertiary Creep and Continuum Damage Mechanics

12.2.1 Models for Tertiary Creep

In this section a division is made between models that only aim to describe tertiary creep and those to attempt to model the development of the creep damage as well. The first type of models was analyzed in Sect. 4.2, and the analysis will not be repeated here. It was found that the Omega model [28, 29] could represent the creep rate in tertiary creep quite well with only two adjustable parameters in the considered cases for modified 9Cr steel, Fig. 4.4, and austenitic stainless steel, Fig. 4.8. If more adjustable parameters are involved, many models can be used to describe the tertiary stage, for a review, see [30, 31].

12.2.2 Continuum Damage Mechanics (CDM)

In continuum damage mechanics (CDM), the changes in the microstructure during creep is described with one or more damage parameters representing cavitation, particles coarsening, etc. Equations for the development of the damage parameters are formulated. When one of the damage parameters has reached the value of unity, crack initiation is assumed to take place and failure is close. Most models consider in reality cavitation to be the main damage mechanism. This is natural since extensive cavitation is closely related to crack initiation. There is a vast literature on CDM. For reviews, see [14, 32,33,34]. The models in general start with some basic concepts that are combined with empirical approaches. The number of adjustable parameters is often as high as 6–8 [13].

The work of CDM was initiated by Kachanov and Rabotnov. They considered the following types of model

$$ \dot{\varepsilon } = A\left( {\frac{\sigma }{{1 - \omega_{1} }}} \right)^{n} \;\;\;\;\;\;\;\;\;\;\dot{\omega }_{1} = B\frac{{\sigma^{m} }}{{(1 - \omega_{1} )^{\chi } }} $$

(12.1)

where $\dot{\varepsilon }$ is the creep rate, σ the applied stress, and ω₁ a damage parameter. A, B, m, n and χ are adjustable parameters. The new idea was the introduction of the damage parameter. It was assumed to have the value 0 at the start of the creep test and 1 at rupture. Although it is rarely stated in the literature, it is obvious that the equations are based on the assumption that the damage is due to cavitation. If the cavitation gives a reduction in the load bearing area, the modified Norton equation in (12.1) is obtained. In Eq. (12.1) an equation for the development of the creep damage is also given. With 5 adjustable parameters there is no difficulty in describing almost any creep curve in the tertiary stage.

As mentioned above there are numerous empirical CDM models in the literature. Two examples will be mentioned here because they seem to be still used frequently. The first one is due to Othman et al. [35]. The creep rate is given by

$$ \dot{\varepsilon } = \frac{A}{{(1 - \omega_{1} )(1 - \omega_{2} )^{n} }}\sinh (B\sigma )\;\;\;\;\;\;\dot{\omega }_{1} = C(1 - \omega_{1} )^{2} \dot{\varepsilon } $$

(12.2)

$$ \omega_{1} = 1 - \frac{{\rho_{1} }}{\rho }\;\;\;\;\;\dot{\omega }_{2} = \frac{{\dot{\varepsilon }}}{{3\varepsilon_{{\text{u}}} }}\left( {\frac{{\sigma_{1} }}{{\sigma_{{\text{e}}} }}} \right)^{\nu } $$

(12.3)

In Eqs. (12.2) and (12.3) there are two damage parameters ω₁ and ω₂. ω₁ takes into account the role of the increasing dislocation density ρ during creep. ρ₁ is the initial mobile dislocation density. Equation (12.2) gives a very rapid increase in the dislocation density close to rupture. The damage due to the cavitation is described by ω₂. It is supposed to take both the effect of nucleation and growth into account. ω₂ is proportional to the creep strain that is well established for nucleation, Eq. (10.8) and at least for some materials can describe growth as well, Eq. (10.24). σ₁ is the maximum principal stress and σ_e the effective stress. A, B, C, n, ν and ε_u are constants.

The second model that will be mentioned is due to Perrin et al. [36]. They give the following equations

$$ \dot{\varepsilon }_{ij} = \frac{{3s_{ij} }}{{2\sigma_{{\text{e}}} }}A\sinh \left( {\frac{{B\sigma_{{\text{e}}} (1 - H)}}{{(1 - \Phi )(1 - \omega_{2} )}}} \right)\;\;\;\;\;\;\dot{H} = \left( {\frac{{h\dot{\varepsilon }_{{\text{e}}} }}{{\sigma_{{\text{e}}} }}} \right)\left( {1 - \frac{H}{{H^{*} }}} \right) $$

(12.4)

$$ \dot{\Phi } = \left( {\frac{{K_{{\text{c}}} }}{3}} \right)(1 - \Phi )^{4} \;\;\;\;\;\;\;\Phi = (1 - \lambda_{{\text{i}}} /\lambda ) $$

(12.5)

To show that the CDM equations are straight forward to transfer to multiaxial stress states, Eqs. (12.4) and (12.5) are given in this form. $\dot{\varepsilon }_{ij}$ is the strain tensor and s_ij the stress deviator, and $\dot{\varepsilon }_{e}$ the effective strain rate. H is a damage parameter that is intended to take into account primary creep. Φ considers the effect of particle coarsening. λ is the particle spacing and λ_i the corresponding initial value. A, B, h, H*, K_c are constants. Including the temperature dependence, no less than 12 constants is listed in [36].

The two models in Eqs. (12.2) and (12.3) as well as in (12.4) and (12.5) represented the state of the art when the papers were written. Unfortunately, not much of the modeling can be considered basic today.

12.3 Particle Coarsening

During later stages of the creep life, precipitates are often coarsening due to Ostwald ripening. The driving force for the coarsening is the reduction of the surface energy of the particles. For a given volume fraction, a distribution of coarser particles has a lower total surface energy than one with smaller particles. The coarsening takes place by diffusion of elements between the particles. Coarsening is believed to degrade creep properties in 9–12%Cr steels. Typically, there are two main types of particles in these steels: M₂₃C₆ and MX. In M₂₃C₆, M represents Cr, Fe, Mo or W. For MX, M stands for V and Nb and X for C or N. The distribution of the two types of particles is different. M₂₃C₆ are mainly found in the subgrain boundaries where they are nucleated during tempering. MX particles on other hand are more homogeneously distributed in the steel. The distribution of M₂₃C₆ is generally coarser than that of MX. As a consequence the two particle types have different roles during creep. M₂₃C₆ slows down or prevents the coarsening of the substructure. In this way the total dislocation density can be kept at a high level, which is most important for the creep strength. The MX particles on the other hand give precipitation hardening and in this way contribute directly to the creep strength.

If only a single particle type is involved and the coarsening is controlled by lattice diffusion, the coarsening can be described by Ostwald ripening

$$ r_{j}^{3} = r_{0,j}^{3} + k_{j} t $$

(12.6)

where r_j is the average particle radius for type j and r_0,j the corresponding initial value. k_j is the coarsening rate constant. For a system with N elements, k_j is given by [6]

$$ k_{j} = \frac{8}{9}\frac{{\Gamma_{{{\text{surf}}}} V_{{\text{m}}}^{j} }}{{\sum\nolimits_{k = 1}^{N} {\frac{{(x_{k}^{j} - x_{k}^{\alpha /j} )^{2} }}{{x_{k}^{\alpha /j} D_{k}^{\alpha } /R_{{\text{G}}} T}}} }} $$

(12.7)

where Γ_surf is the particle interfacial energy per unit area, $V_{{\text{m}}}^{j}$ is the molar volume of the particle type j, R_G the gas constant, and T the absolute temperature. The denominator is a sum in k over the elements involved in the diffusion, $D_{k}^{\alpha }$ is the diffusion coefficient, as well as $x_{k}^{j}$ and $x_{k}^{\alpha /j}$ the equilibrium mole fraction in the particle respectively the matrix at particle/matrix interface. Equation (12.7) is complex but the value of k_j can often be obtained directly from thermodynamic software.

From Eq. (12.6), it is evident that the average volume of the particles increases linearly with time, so it is quite a simple dependence. The presence of particle in the subgrain walls slows down the growth of the subgrain and sets a limit to the maximum subgrain size. This is described by the following equation [7]

$$ \frac{{dd_{{{\text{sub}}}} }}{dt} = \frac{{3M_{{{\text{climb}}}} \tau_{{\text{L}}} }}{{2d_{{{\text{sub}}}} }} \cdot \left[ {1 - \left( {\frac{{d_{{{\text{sub}}}} }}{{d_{{{\text{sub}}\,{\text{lim}}}} }}} \right)^{2} } \right]^{2} $$

(12.8)

d_sub is the subgrain diameter and d_{sub lim} the limiting subgrain size due to the retarding force from the particles, which is referred to as Zener drag. The limiting subgrain size is given by [7]

$$ d_{{{\text{sub}}\,{\text{lim}}}} = \frac{{\pi r_{{\text{p}}} }}{{\gamma f_{{\text{p}}} }} $$

(12.9)

r_p is the radius of particles at subgrain boundaries, and f_p their volume fraction. The constant γ has a value of about 0.5 [37].

As was described in Sect. 2.6, creep strain can generate a large number of vacancies. This means that phenomena that are diffusion controlled can also be strain controlled. Diffusion requires the presence of vacancies. An equilibrium amount of vacancies C_Va is formed due to thermal activation. C_Va can be represented with an Arrhenius expression

$$ C_{{{\text{Va}}}} = c_{0} = \exp \left( {\frac{{S_{{{\text{Va}}}}^{{\text{F}}} }}{{k_{{\text{B}}} }} - \frac{{H_{{{\text{Va}}}}^{{\text{F}}} }}{{k_{{\text{B}}} T}}} \right) $$

(12.10)

where $S_{{{\text{Va}}}}^{{\text{F}}}$ and $H_{{{\text{Va}}}}^{{\text{F}}}$ are the entropy and the enthalpy for vacancy formation, and k_B Boltzmann’s constant. The corresponding formula for the diffusion coefficient D_Va is

$$ D_{{{\text{Va}}}} = A_{{{\text{Va}}}}^{{\text{D}}} \exp \left( { - \frac{{H_{{{\text{Va}}}}^{{\text{F}}} + H_{{{\text{Va}}}}^{{\text{M}}} }}{{k_{{\text{B}}} T}}} \right) $$

(12.11)

where $H_{{{\text{Va}}}}^{{\text{M}}}$ is the enthalpy for vacancy migration and $A_{{{\text{Va}}}}^{{\text{D}}}$ a constant. Diffusion depends on both formation and migration of vacancies, but the vacancy concentration is only a function of the formation energy. The part of the diffusion coefficient that depends on the vacancy concentration can be extracted

$$ D_{{{\text{Va}}}} = A_{{{\text{Va}}}}^{{\text{D}}} \exp \left( { - \frac{{H_{{{\text{Va}}}}^{{\text{M}}} }}{{k_{{\text{B}}} T}} - \frac{{S_{{{\text{Va}}}}^{{\text{F}}} }}{{k_{{\text{B}}} }}} \right)c_{0} $$

(12.12)

The corresponding amount of vacancies generated by straining is given by Eq. (2.37)

$$ \Delta c = 0.5c_{0} \frac{{\sqrt 2 K_{sub}^{2} \dot{\varepsilon }b^{2} }}{{D_{self} }}\frac{G}{\sigma } $$

(12.13)

It can be assumed that straining gives rise to coarsening in the same way as Ostwald ripening in Eq. (12.6)

$$ \frac{{dV_{{\text{p}}} }}{d\varepsilon } = k_{\varepsilon } $$

(12.14)

where V_p is the average particle volume and k_ε a constant. The value of k_ε can be derived from Eq. (12.7) by replacing c₀ in the expression for the diffusion coefficient by Δc using Eq. (12.12).

For homogeneously distributed particles, coarsening of the particles implies that their contribution to the creep strength is reduced. This is described by Eq. (7.17). When coarsening takes place, the critical spacing between particles λ_crit increases.

12.4 Dislocation Strengthening During Tertiary Creep

12.4.1 The Role of Substructure During Tertiary Creep

During deformation a cell structure is formed in most alloys. Already after 10% strain the majority of the dislocations can be found in the cell boundaries [38]. After 20% deformation practically all dislocation are located in the cell boundaries [39]. As will be demonstrated below the cell structure plays an important role during tertiary creep. To simplify the analysis the role of dislocations in the cell interiors will be neglected. This has been justified experimentally. For example, Straub et al. showed with X-ray techniques that the dislocations in cell interiors in copper only contributed 10 MPa to the strength [40].

As was introduced in Sect. 8.1, there are two sets of dislocations in the cell walls: balanced and unbalanced. This is a direct consequence of the basic nature of dislocations. The dislocations are initially randomly distributed in the cells. If there are dislocations in the cell interior with burgers vector b and opposite burgers vector −b on a given slip plane, they would move in opposite directions under stress. The effect is that dislocations with one sign end up at one side of the cells and the ones with the opposite sign on the other side. This means that the dislocations have different signs on the two sides of a cell wall in the stress direction. Such a set of dislocations are called polarized or unbalanced. The term unbalanced is due to the fact that all the dislocations in the neighborhood have the same sign. At other regions of the cells, the dislocations with both types of burgers vectors are present. These sets of dislocations are referred to as balanced.

That the dislocations in the cell walls are divided into two sets, balanced and unbalanced is therefore natural. It is well documented experimentally that the dislocations in cell walls can be statistically distributed and polarized [41]. The central and main part of the cell walls is found to consist of balanced dislocations, whereas the outer layers are polarized. The polarized dislocations cannot move through the walls due to the large number of dislocations in the cell walls [12]. Argon has proposed that the balanced dislocations in the cell walls to a significant extent are dislocation locks [42]. To understand tertiary creep and some other properties, the distinction between balanced and unbalanced dislocation is of importance.

Equations for the dislocation densities of the balanced and unbalanced types as well as for locks were given in Eqs. (8.17)–(8.19). The dislocation densities satisfy the following equations [11, 12]

$$ \frac{{d\rho_{{{\text{bnd}}}} }}{d\varepsilon } = k_{{{\text{bnd}}}} \frac{{m\rho_{{{\text{bnd}}}}^{1/2} }}{{bc_{L} }} - \omega \rho_{{{\text{bnd}}}} - 2\tau_{{\text{L}}} M\rho_{{{\text{bnd}}}}^{{2}} /\dot{\varepsilon } $$

(12.15)

$$ \frac{{d\rho_{{{\text{bnde}}}} }}{d\varepsilon } = k_{{{\text{bnde}}}} \frac{{m(\rho_{{{\text{bnd}}}}^{1/2} + \rho_{{{\text{bnde}}}}^{1/2} )}}{{bc_{L} }} - \omega \rho_{{{\text{bnde}}}} $$

(12.16)

$$ \frac{{d\rho_{{{\text{lock}}}} }}{d\varepsilon } = k_{{{\text{lock}}}} \omega (\rho_{{{\text{bnd}}}} + \rho_{{{\text{bnde}}}} ) - \omega \rho_{{{\text{lock}}}} - 2\tau_{{\text{L}}} M\rho_{{{\text{lock}}}}^{2} /\dot{\varepsilon } $$

(12.17)

ρ_bnd, ρ_bnde and ρ_lock are the balanced, unbalanced and lock dislocation density in the cell walls, which are defined as the total length of the dislocations divided by the cell volume. ε is the strain, m_T the Taylor factor, b Burger’s vector, c_L, k_bnd and k_bnde are work hardening constants, ω the dynamic recovery constant, τ_L the dislocation line tension, $\dot{\varepsilon }$ the strain rate and M the creep climb mobility. In Eq. (12.15) the three terms on the right hand side represent work hardening, dynamic recovery and static recovery in the same way as in the basic Eq. (2.17). Since the unbalanced dislocations cannot meet a dislocation with opposite sign, there is no static recovery term in Eq. (12.16). Both unbalanced and balanced dislocations are subjected to dynamic recovery. The dislocation locks cannot generate dislocations, but instead they obtain an input of dislocation due to dynamic recovery of balanced and unbalanced dislocations. As a consequence the first term in Eq. (12.17) has a different appearance in comparison to Eqs. (12.15) and (12.16). It is important to understand the difference between static and dynamic recovery. Static recovery occurs when dislocations of opposite signs meet and annihilate. Dynamic recovery takes place through the formation of dislocation configurations with lower energy [43]. A contributing factor has been suggested by Argon [42]. It is well documented that when dislocations are released during plastic straining they move through one or more cell boundaries. When this happens a fraction of the dislocation in the boundaries is removed, giving rise to a recovery effect. Both dynamic recovery and static recovery should be considered when describing tertiary creep as will be discussed below. Dynamic recovery requires straining [44] while static recovery is a time dependent process [45]. In the analysis below the dislocation locks will be considered to be a part of the balanced density in the cell walls.

Since the unbalanced dislocations in the cell walls are not exposed to static recovery they give rise to an extra hardening that is referred to as a back stress [11]. The increase in the true applied stress $\sigma = \sigma_{0} e^{\varepsilon }$ during creep under constant load is compensated by the back stress, where σ₀ is the applied nominal stress. The magnitude of the back stress equals the dislocation stress plus other strengthening contributions σ_i minus the nominal applied stress

$$ \sigma_{{{\text{back}}}} = \sigma_{{{\text{disl}}}} + \sigma_{{\text{i}}} - \sigma_{0} $$

(12.18)

where σ_disl is given by (cf. Eq. (8.3))

$$ \sigma_{{{\text{disl}}}} = \frac{{m_{{\text{T}}} \alpha Gb}}{2}\sqrt {\rho_{{{\text{bnd}}}} + \rho_{{{\text{bnde}}}} } $$

(12.19)

where α is a constant in Taylor’s equation, and G the shear modulus. The effective creep stress is obtained as the true applied stress minus the back stress

$$ \sigma_{{{\text{creep}}}} = \sigma - \sigma_{{{\text{back}}}} $$

(12.20)

From Eq. (12.15), an expression for the secondary creep rate can be obtained

$$ \dot{\varepsilon }_{\sec } = 2\tau_{{\text{L}}} M(T,\sigma_{{{\text{creep}}}} )\rho_{{{\text{bnd}}}}^{{2}} /\left( {k_{{{\text{bnd}}}} \frac{{m\rho_{{{\text{bnd}}}}^{1/2} }}{{bc_{L} }} - \omega \rho_{{{\text{bnd}}}} } \right) $$

(12.21)

where the effective creep is inserted. It is now assumed that Eq. (12.21) it is not just applicable to secondary creep but it describes the influence of the changes of the dislocation density on the whole creep curve [12]

$$ \dot{\varepsilon } = 2\tau_{{\text{L}}} M(T,\sigma_{{{\text{creep}}}} )\rho_{{{\text{bnd}}}}^{{2}} /\left( {k_{{{\text{bnd}}}} \frac{{m\rho_{{{\text{bnd}}}}^{1/2} }}{{bc_{L} }} - \omega \rho_{{{\text{bnd}}}} } \right) $$

(12.22)

This model suggests that if the stress dependence of the secondary creep rate is known, the influence of the dislocation density on the whole creep curve can be derived. Primary creep was dealt with in Sect. 8.2.

The variation of the stress components is illustrated for copper in Fig. 12.1. At the start of the creep test, the dislocation is low and the effective creep stress is high. But already after a short time they are of about the same magnitude. The dislocation stress then balances the true applied stress quite well in the secondary stage giving almost overlapping curves. This means that the creep stress is approximately constant in the secondary stage. Finally in the tertiary stage the increase of the applied stress is faster than that of the dislocation stress. Thus, there is an increase in the creep stress and thereby in the creep rate.

Fig. 12.1

Evolution of dis-location stress, true applied stress and effective creep stress as a function of time for Cu-OFP at 75 °C with an applied stress of 175 MPa. Redrawn from [23] with permission of Springer

By integrating Eq. (12.22) together with (12.15) and (12.16), creep strain versus time curves can be obtained. Two examples for Cu-OFP at 75 °C are given in Fig. 12.2. The experimental curves show distinct primary, secondary and tertiary creep in spite of the low temperature. An extended secondary stage is found in spite of a stress exponent that exceeds 50. How this is possible was explained in Sect. 8.4. The substructure plays an important role in this respect and that is taken into account in Eq. (12.22). There are steps in the experimental curves due to the necessity of reloading the creep when a certain creep strain was reached. No attempts have however been made to try to compensate for the reloading of the creep machine. The experimental creep curves can be reproduced in a reasonable way. Some of the differences can be accounted for by taking necking into account which is analyzed in Sect. 12.5.

Fig. 12.2

Comparison of experimental creep curves with the model in Eq. (12.22) for Cu-OFP, a 75 °C, 175 MPa; b 75 °C, 180 MPa. Redrawn from [23] with permission of Springer

12.4.2 Accelerated Recovery Model

During later stages of creep there is often a degradation of the creep strength. This degradation is often referred to as creep damage. Important examples of creep damage are cavitation and particle coarsening. The latter effect was analyzed in Sect. 12.3. Particle coarsening gives rise to a reduction of the precipitation hardening. In addition, subgrain coarsening can take place if the distance between particles in the boundaries is increased.

Particle coarsening is usually represented with a time dependence described by Ostwald ripening in Eq. (12.6). However, tertiary creep shows more typically strain dependence. This is illustrated in Sect. 4.2. The strain rate in the tertiary stage increases exponentially with strain. This is referred to as the Omega model, which was discovered during work with estimates of residual life time of fossil fired power plants. For this reason it is natural to consider strain dependent processes. One such process is the strain dependent coarsening given in Eq. (12.14). A strain dependent process is also present for static recovery. Time dependent static recovery has been the basis of many derivations in this book. Its basic form is given in Eq. (2.17)

$$ \frac{d\rho }{{dt}} = - 2\tau_{{\text{L}}} M\rho^{2} $$

(12.23)

However, strain dependence is also possible to consider in the same way as for particle coarsening

$$ \frac{d\rho }{{d\varepsilon }} = - 2\tau_{{\text{L}}} M_{\varepsilon } \rho^{2} $$

(12.24)

In the modified version of the climb mobility M_ε in Eq. (12.24), the diffusion constant has to be replaced by the expression given in Eq. (12.12). The principles for the derivation of Eqs. (12.16) and (12.24) are straightforward, but the expressions have not yet been verified experimentally. By taking also Eq. (12.24) into account, Eq. (2.17) takes the form

$$ \frac{d\rho }{{d\varepsilon }} = \frac{{m_{{\text{T}}} }}{{bc_{L} }}\rho^{1/2} - \omega \rho - 2\tau_{{\text{L}}} (M/\dot{\varepsilon } + M_{\varepsilon } )\rho^{2} $$

(12.25)

From Eq. (12.25), an expression for the creep rate can be derived in the same way as in Sect. 8.2.2

$$ h(\sigma ) = 2\tau_{{\text{L}}} M(T,\sigma )\frac{{\sigma^{3} }}{{(\alpha m_{{\text{T}}} Gb)^{3} }}/\left( {\frac{{m_{{\text{T}}} }}{{bc_{{\text{L}}} }} - \omega \frac{\sigma }{{\alpha m_{{\text{T}}} Gb}} - 2\tau_{{\text{L}}} M_{\varepsilon } \frac{{\sigma^{3} }}{{(\alpha m_{{\text{T}}} Gb)^{3} }}} \right) $$

(12.26)

$$ \dot{\varepsilon } = h(\sigma_{{{\text{creep}}}} ) $$

(12.27)

The effective creep stress σ_creep is given by Eq. (12.20). With the help of Taylor’s Eq. (2.29), Eq. (12.25) can be expressed in term of the dislocation stress σ_disl

$$ \frac{{d\sigma_{{{\text{disl}}}} }}{d\varepsilon } = \frac{{\alpha m_{{\text{T}}}^{2} Gb}}{{2bc_{{\text{L}}} }} - \frac{\omega }{2}\sigma_{{{\text{disl}}}} - 2\tau_{{\text{L}}} (f_{{\text{M}}} M/h(\sigma_{{{\text{creep}}}} ) + M_{\varepsilon } )\frac{{\sigma_{{{\text{disl}}}}^{3} }}{{(\alpha m_{{\text{T}}} Gb)^{2} }} $$

(12.28)

Since the value of M_ε is unknown, it has been assumed that its stationary value is that of $M/h(\sigma_{{{\text{creep}}}} )$, i.e. that the two types of static recovery give about the same contribution.

The use of Eqs. (12.27) and (12.28) in Fig. 12.3 is illustrated for Sanicro 25. A comparison is made with the same experimental data set as in Fig. 4.8. In Fig. 12.3a, the dislocation stress is shown as a function of creep strain. There is initially a rapid increase in the dislocation density. When the stress touches the true stress curve and the secondary stage is reached, the increase in the stress continues but at a much lower rate. The difference between the sum of the true stress and the nominal stress on one hand and the dislocation stress on the other is the effective creep stress, Eq. (8.29). Considering the difference between the true stress and any of the dislocation stresses, it is evident from Fig. 12.3a that the creep rate is higher in the tertiary stage than in the secondary stage but lower than in the primary stage.

Fig. 12.3

a Dislocation stress according Eq. (12.28) with (M_cl + M_ε) and only with (M_cl). M_ε and M_cl are the strain and time dependent climb mobility, respectively. The true stress is also shown; b creep rate from Eq. (12.27) for the same cases as in a). A comparison with experiment is included. Sanicro 25 at 700 °C, 200 MPa

In Fig. 12.3a, the dislocation stress is compared with the true applied stress. Most creep models are based on the nominal stress, so also in this book. However, to describe tertiary creep, the true stress plays an important role, and Eqs. (12.26)–(12.28) are based on the true stress. To take the true stress into account instead of the nominal one, some adjustments of the model must be made. For example, the secondary stress to give the creep rate in the secondary stress is slightly higher when the true stress is used. This is covered with the constant f_M in Eq. (12.28). f_M is close to unity. This can also be seen in Fig. 12.3a, where the stresses in the secondary stage are slightly higher than the nominal value of 200 MPa.

In Fig. 12.3, a comparison is made between the case when only the time dependent static recovery and the case when both the time and strain dependent types are included. In the first case only the time dependent climb mobility M is taken into account and in the second case both M and M_ε. The difference between the two cases is clearly observed in Fig. 12.3b where the creep rate is plotted versus strain. The creep rate drops rapidly in the primary stage. The position of the minimum creep rate does not agree with the observations. The reason is most likely that the strain on loading is included in the modeling but not in the experiments. In the tertiary stage the logarithm of the creep rate increases linearly as a function of strain. This is the characteristic feature of the Omega model as was discussed in detail in Sect. 4.2. The slope of the experimental curve in the tertiary stage Fig. 12.3b is n_Ω = 14. The creep stress exponent at 700 °C for the material is n_N = 7. At 725 and 750 °C the corresponding values are n_Ω = 11 and 9 and n_N = 6 and 5, respectively. If only M is taken into account in the model the slope is close to the n_N value whereas a higher value close to n_Ω is obtained when also M_ε is involved.

For P91 that is also studied in Sect. 4.2, the difference between n_Ω and n_N is even larger. The experimental data in [46] is for 600 °C. The n_N value at this temperature is 12 whereas n_Ω takes the values 28, 39, 65 and 95 for applied stresses of 180, 150, 130 and 110 MPa, respectively. That n_Ω increases with decreasing applied stress is also found for Sanicro 25 although the effect is less dramatic. The fact that n_Ω is significantly larger than n_N clearly shows that degradation of the microstructure takes place. It is natural that the degradation increases with decreasing stress since there is more time for microstructural changes to take place. The degradation is larger for P91 than for Sanicro 25. This is also expected since P91 has a martensitic structure that is prone to changes at high temperatures. The model including M_ε describes how the recovery is accelerated during the tertiary stage, which is a direct consequence of microstructural degradation.

12.5 Necking

12.5.1 Hart’s Criterion

During tensile creep testing a plastic instability develops towards the end of the experiment. A waist is formed around the specimen that grows until the specimen fails. This phenomenon is usually referred to as necking. It is assumed that it is initiated due to the presence of a geometric imperfection or a material inhomogeneity. The continued growth of the waist does not depend on how it was initiated.

Hart proposed a criterion for the initiation of necking during creep [47]. The initiation is assumed to be due to area fluctuations. The deformation is stable provided that the variation at a particle point is larger than zero. From the relation between the area reduction and the strain a stability criterion can be derived

$$ \ddot{\varepsilon }/\dot{\varepsilon }^{2} \le 1 $$

(12.29)

where $\dot{\varepsilon }$ is the strain rate, and $\ddot{\varepsilon }$ is strain acceleration, the second time derivative of the strain. One can expect that when this stability criterion fails, necking would be initiated in the same way as for Considère’s criterion during tensile testing at ambient temperatures.

12.5.2 Use of Omega Model

The implications of Hart’s criterion can be illustrated with the help of the Omega (Ω) model. It was illustrated in Sect. 4.2 that primary creep often follows the phi (ϕ) model and tertiary creep the Omega model. As was demonstrated in Sect. 4.2, it is not necessary to have a separate term for secondary creep. Then according to Table 4.1, the creep rate can be expressed as

$$ \dot{\varepsilon } = \phi_{1} \varepsilon^{{ - \phi_{2} }} + \Omega_{3} \exp (n_{\Omega } \varepsilon ) $$

(12.30)

where the first and second term on the RHS refers to primary and tertiary creep. ϕ₁, ϕ₂, Ω₃ and n_Ω are adjustable parameters. Ω₄ in Table 4.1 has been replaced by n_Ω since that designation was used above. According to Sect. 4.2, tertiary creep can be represented by the second term in Eq. (12.30) over a fair strain range. Consequently, it is of interest to analyze that term separately.

$$ \dot{\varepsilon } = \Omega_{3} \exp (n_{\Omega } \varepsilon )\;\;\;\;\;\ddot{\varepsilon } = \Omega_{3} n_{\Omega } \dot{\varepsilon }\exp (n_{\Omega } \varepsilon ) $$

(12.31)

By combing the two equations in (12.31), we find that

$$ \ddot{\varepsilon }/\dot{\varepsilon }^{2} = n_{\Omega } $$

(12.32)

Since n_Ω is typically much larger than unity, the stability criterion in (12.29) is very far from satisfied in the tertiary stage. In the primary stage $\ddot{\varepsilon }$ is negative and in the secondary stage zero, so in these stages the stability criterion is fulfilled. To satisfy it also in the tertiary stage, there must be a large contribution from the primary creep term in (12.30). This is only possible at the start of the tertiary stage. One can conclude that Eq. (12.29) implies that an instability is formed when tertiary creep is initiated. Similar results have been found when testing the criterion on experimental creep curves that do not follow the Ω model.

Burke and Nix [48] studied necking by analyzing the deformation in a cylindrical bar with an imperfection. They assumed a cross section that varied with a smooth sinusoidal function

$$ A(x,0) = A_{0} - \frac{\Delta A}{2}\cos \frac{2\pi x}{{L_{i} }}\;\;\;\;\;(0 \le x \le L_{i} ) $$

(12.33)

where A₀ is the original cross section area of specimen, and L_i the length of the specimen with the defected part. ΔA represents the changes of the initial cross section area. They considered that the development of the imperfection could be described by a uniaxial model. The shape of the initial imperfection is then not so important, only the initial reduction of the cross section.

It is possible to estimate the growth of the imperfection. As was discussed above, Eq. (12.29) suggests that an instability is formed shortly after the start of tertiary. Only then an imperfection can grow. It is thus possible to use the tertiary part of Eq. (12.30) to estimate the amount of necking.

$$ \dot{\varepsilon } = \Omega_{3} \exp (n_{\Omega } \varepsilon )\;\;\;\;\;\varepsilon_{{{\text{bar}}}} = - \frac{1}{{n_{\Omega } }}\log (\exp ( - n_{\Omega } \varepsilon_{0} ) - n_{\Omega } \Omega_{3} t) $$

(12.34)

The integrated solution is given in the second member. ε₀ is the starting strain of tertiary and t is the time. Following Eq. (12.33), the imperfection has a reduced cross section by a factor f_red. The solution of the equation in the presence of an imperfection is

$$ \varepsilon_{{{\text{waist}}}} = - \frac{1}{{n_{\Omega } }}\log \left( {\exp ( - n_{\Omega } \varepsilon_{0} ) - \frac{{n_{\Omega } \Omega_{3} t}}{{f_{{{\text{red}}}}^{{n_{{\text{N}}} }} }}} \right)\;\;\;\;f_{{{\text{red}}}} = 1 - \Delta A/A_{0} $$

(12.35)

The solutions of Eqs. (12.34) and (12.35) are illustrated in Fig. 12.4 for Sanicro 25 and P91. The cases are the same as the ones in Figs. 4.4 and 4.8.

Fig. 12.4

Strain versus time for the solutions in Eqs. (12.34) and (12.35) for a Sanicro 25, 750 °C, 200 MPa and b P91, 600 °C, 150 MPa. a Redrawn from [49] with permission of Taylor & Francis

Solutions with and without waist are given in (12.34) and (12.35). The difference between the strain in the waist and in the unaffected bar ε_diff gives the depth of the waist and this difference is shown in the Figure. Thus, we have the following simple relation

$$ \varepsilon_{{{\text{waist}}}} = \varepsilon_{{{\text{bar}}}} + \varepsilon_{{{\text{diff}}}} $$

(12.36)

The influence of the waist is only significant late in the creep life. It is evident that when this strain difference exceeds 0.2, rupture is close. By assuming an ε_diff value larger than 0.2 a criterion for rupture can be obtained. By combing the equations for ε_bar (12.34), for ε_waist (12.35) and Eq. (12.36), and expression for the rupture time can be derived

$$ t_{{\text{R}}} = \frac{{e^{{ - n_{\Omega } \varepsilon_{0} }} }}{{\Omega_{3} n_{\Omega } }}\frac{{1 - e^{{ - n_{\Omega } \varepsilon_{{{\text{diff}}}} }} }}{{(1/f_{{{\text{red}}}}^{{n_{{\text{N}}} }} - e^{{ - n_{\Omega } \varepsilon_{{{\text{diff}}}} }} )}}\,\,\;\;\;\;\Omega_{3} = \dot{\varepsilon }_{\min } \exp (\varepsilon_{0} ) $$

(12.37)

In Eq. (12.37), an expression for Ω₃ has been given in terms of the minimum creep rate. The latter quantity may be easier to find. Assuming f_red = 0.99 which is often done, Eq. (12.37) gives a rupture time of 51, 75, 144 and 244 h for Sanicro 25 for the four curves in Fig. 4.8 that are in reasonable accordance with the observed values 51, 63, 148 and 244 h. The corresponding values for P91 in Fig. 4.4 are 62, 357, 1787, 10800 h that should be compared with the observed ones 101, 546, 3650 and 13940 h. The predicted values clearly underestimate the experimental ones in this case. An equation for the rupture elongation ε_R can also be obtained. The same three Eqs. (12.34)–(12.36) are combined and this time the time t and ε_waist are eliminated

$$ \varepsilon_{{\text{R}}} = \frac{1}{{n_{\Omega } }}\frac{{(1/f_{{{\text{red}}}}^{{n_{{\text{N}}} }} - e^{{ - n_{\Omega } \varepsilon_{{{\text{diff}}}} }} )}}{{(1/f_{{{\text{red}}}}^{{n_{{\text{N}}} }} - 1)e^{{ - n_{\Omega } \varepsilon_{0} }} }}\,\, $$

(12.38)

For the same four curves in Fig. 4.8, Eq. (12.38) yields 0.31, 0.38, 0.31, and 0.34 that should be compared with the experimental values 0.24, 0.22, 0.36 and 0.43. The predicted values are of the right order of magnitude but they do not reproduce the observed values more precisely. For the curves for P91 in Fig. 4.4, the predicted values are 0.095, 0.078, 0.053, 0.034 and the experimental ones 0.17, 0.15, 0.19 and 0.072. The predicted values are about half the observed ones. The reason why the values for the martensitic steel are underpredicted is not known.

12.5.3 Basic Dislocation Model

More detailed comparisons will now be made with experimental data for P alloyed pure copper Cu-OFP [23]. Distinct necking was observed on the ruptured specimens, which emphasizes the importance of necking in tertiary creep. To describe the deformation, the dislocation model in Eq. (12.27) is used. Again uniaxial behavior will be assumed for assessing the influence of necking. Influence of multiaxial stress states will be considered in Sect. 12.5.4. The starting imperfection is given by Eq. (12.33) with a 1% reduced cross section (f_red = 0.99) and a half length of the imperfection of L_i = 5 mm. With the formulae in Chap. 10, the amount of cavitation can be estimated. For the cases considered in [23], the local values of the cavitated grain boundary area were found to be about 2% which is consistent with observations on the specimens. However, as discussed in Sect. 12.1, the average amount is much lower. As a consequence the influence of cavitation on the creep curves is small and is not noticeable at the scale of the Figures. The initiation of the instability is assumed to follow Hart’s criterion (12.29). In Fig. 12.5b the position of this initiation point is marked. It is again evident that the point appears very early in the tertiary stage.

Fig. 12.5

Comparison of experimental creep curves with necking model results for Cu-OFP, a 75 °C, 170 MPa; b 75 °C, 175 MPa, plus marker indicating necking starting point according to Hart’s criterion (12.29), c 75 °C, 180 MPa. Redrawn from [23] with permission of Springer

Modeled creep curves are compared with experiments for three stresses at 75 °C for Cu-OFP in Fig. 12.5. The dislocation model in Eq. (12.27) takes primary, secondary and tertiary creep into account. The dislocation model gives only a modest contribution to tertiary creep, but the effect is clearly visible. There are other cases where the influence is much more pronounced. Examples are shown in Fig. 8.12 for cold worked copper.

The rise of the creep strain at the end of the creep life is due to necking. Obviously the uniaxial creep model can reproduce the sharp increase quite well. The model clearly demonstrates that the necking takes place late in the creep life.

12.5.4 Multiaxial Stress States

The formation of a waist clearly takes place under multiaxial stress states. For this reason finite element analysis (FEM) has been performed [23]. Unfortunately, special FEM software is required that can handle creep deformation with large strains. It should also allow for a plastic instability to occur, which makes the analysis sensitive. At this stage it is difficult to consider it for routine applications.

The dislocation model in Eqs. (12.15), (12.16), (12.22), etc. is implemented in the FEM program. The parameters used are the same as in the uniaxial case in 12.5.3. A creep test of Cu-OFP at 75 °C and 175 MPa was simulated, i.e. the same case as the one shown in Fig. 12.5a. The necking appeared at a uniform strain of 0.27. Then all the subsequent strain took place in the waist. This is illustrated in Fig. 12.6.

Fig. 12.6

Total strain versus uniform strain for the FEM model in Fig. 12.7. Redrawn from [23] with permission of Springer

The strain in the neck is as high as 2. This is not shown in Fig. 12.6 but this is clearly evident in Fig. 12.7, where the final strain distribution and profile of the specimen is shown.

Fig. 12.7

FEM results of profile strain distribution along a creep specimen of Cu-OFP at 75 °C with an applied stress of 175 MPa. Redrawn from [23] with permission of Springer

That the high strain value is in accordance with the experiment that is illustrated in Fig. 12.8, where the observed and modeled specimen profiles are shown. The experimental values are reproduced within about 10%.

Fig. 12.8

Specimen radius versus axial coordinate at the necking position. Comparison of experimental necking profile with FEM results for Cu-OFP at 75 °C with an applied stress of 175 MPa. Redrawn from [23] with permission of Springer

According to the FEM modeling, the necking develops quite slowly and only appears close to rupture. This has also been observed for the martensitic 9Cr1Mo steel P91 [27]. Once the necking has started to form, the strain in the neck increases rapidly. The results of the uniaxial and the multiaxial computations are obviously fairly consistent. Also the uniaxial modeling of necking gives a significant necking strain only close to rupture.

12.6 Summary

Many mechanisms that contribute to tertiary creep are well known such as particle coarsening, substructure coarsening, cavitation, changes in the dislocation density and necking. In the literature these mechanisms have mainly been modeled with empirical approaches not least in the context of continuum damage mechanics. However, due to the complexity of the phenomena empirical methods give unsafe predictions.

• A basic dislocation model for the whole creep curve is described. The model extends results from previous chapters to tertiary creep. The model is formulated in such a way that if the stress dependence of the secondary creep rate is known, tertiary as well as primary creep rate can be computed. This is done by introducing an effective creep stress that takes the changes in the dislocation density into account.

• In the secondary stage there is a balance between the applied stress and the stress from the dislocations plus contributions from other strengthening mechanisms. In the tertiary stage the dislocation strength continues to increase but the true stress increases faster. This means that the effective stress is raised and thereby the creep rate. This simple concept is proposed to be the main mechanism behind tertiary creep.

• For copper it is essential to take the substructure into account when modeling tertiary creep. A distinction is made between balanced and unbalanced dislocations in the cell walls. The main difference between balanced and unbalanced dislocations is that the former type is exposed to static recovery but not the latter. The unbalanced dislocations provide a counteracting stress against the rapidly increasing true stress at lower temperatures, which makes it possible to explain the observed creep rates.

• In steels there is often a very rapid increase in the creep rate in the tertiary stage that can be represented by a linear increase in the logarithm of the creep rate with strain (Omega model). To explain this behavior, the degradation mechanisms in the microstructure must be strain dependent. This applies in particular to particle and substructure coarsening. Quantitative models have been proposed for these mechanisms but the models have not yet been verified experimentally.

• With models presented in Chap. 10, the influence of cavitation on tertiary creep can directly be derived. However, pronounced cavitation typically occurs quite locally. This means that the overall effect on tertiary creep is limited.

• Necking is assumed to be initiated when Hart’s stability criterion fails which takes place at the very beginning of tertiary creep. Although necking is initiated early in the creep process, both uniaxial and multiaxial models suggest that significant necking takes place only close to rupture but then necking is progressing very rapidly. These results are fully consistent with available experimental data.