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Open Access 2024 | OriginalPaper | Buchkapitel

13. Creep Ductility

verfasst von : Rolf Sandström

Erschienen in: Basic Modeling and Theory of Creep of Metallic Materials

Verlag: Springer Nature Switzerland

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Abstract

For a number of creep resistant steels, the creep ductiliy decreases with increasing temperature and time. As a function of stress, the ductiity is often describe with an S-shaped curve with an upper and a lower shelf level. As a function of time, the S-shape is inverted. If the ductility is high, the rupture is referred to as ductile, and for low ductility levels as brittle. Ductile rupture is believed to be due to a plastic instability such as necking. Brittle rupture on the other hand is controlled by the nucleation, growth and linkage of creep cavities. With the help of the basic models for creep deformation and cavitation, the rupture stress and ductility can be predicted. Several models exist for the influence of multiaxiality on the creep ductility. Although the models are based on different principles, they predict approximately the same behavior, which is verified by comparison to rupture data for notched bars.

13.1 Introduction

In creep tests, the ductility is commonly measured in two ways: as creep elongation and reduction of area, both at rupture. The creep ductility influences several properties. With a low ductility cracks are more easily formed and the risk for failure is higher than when the ductility is high. This is not least the case for cyclic loading. The reason is that creep cavities are more readily formed when the ductility is low. Cavities grow and link which results in initiation sites for cracks. For materials with low ductility, the risk is obviously higher that the strain allowance is exceeded in notches and at inhomogenieties like in welds. Materials with high creep ductility are considered to be more forgiving.

As a consequence it is desirable to select a material with high creep ductility. Unfortunately that is not easy. Most creep resistant alloys loose ductility with increasing rupture time. In addition, there can be a large cast to cast difference in the ductility. It was demonstrated early on that the 17Cr–12Ni–2Mo steel 316 showed such a variation [1]. The rupture elongation was observed to vary between 2 and 120%. Also for the martensitic 9Cr1Mo steel P91, low ductility is often found. In a larger investigation it was recorded that about 10% of the casts had a reduction of area below 20% [2]. It can be concluded that also for materials that have a long successful operation record, the ductility can frequently be low.

There are many mechanisms that are known to influence the ductility. A coarse grain size often reduces the ductility. This is natural since the amount of grain boundary sliding and thereby the cavitation increases with the grain size. This is evident from Eq. (9.12). Particles that are present in the grain boundaries act as nucleation sites for cavitation. With increasing number of particles, more cavities are formed, see Eq. (10.8). It is known that the presence of coarse particles in the grain boundaries increases the risk for crack initiation. Impurity elements P and S can lower the ductility in steels. It is suspected that the presence of impurity elements is the cause of low creep ductility in many casts but the number of systematic studies is limited [2]. It is a common experience for steels that if the creep strength is raised, the ductility is often reduced. Some of the mentioned effects can be modeled but not all of them. In particular, the observed cast to cast variation is difficult to explain. One reason is that some mechanisms can be both positive and negative. One example is particles in grain boundaries. As mentioned they can act as nucleation sites for cavitation but they can probably also limit the amount of grain boundary sliding and thereby resist cavitation, but that does not seem to have been verified. The limitation concerning the understanding of the controlling mechanisms must be considered when modeling creep ductility. In most cases only a general description of influencing factors can be obtained, not a detailed computation.

To illustrate the influence of parameters on the creep ductility, schematic diagrams are often used. Such diagrams are shown in Fig. 13.1.

At high stresses and short rupture times the ductility is high and approximately constant. This is referred to as the upper shelf or regime I. When the stress is reduced, the ductility drops to much lower values over a fairly narrow range of stresses and rupture times. This is regime II. At low stresses or long rupture times the ductility takes very low values. In this range the ductility is again approximately constant. It is called the lower shelf or regime III. In some materials for example 9%Cr steel [3] and CrMoV steels [4], the ductility can increase again at very long rupture times. The curves in Fig. 13.1 move to the left with increasing temperature.

To illustrate how the schematic curves in Fig. 13.1a, look for observed values, results for 17Cr12Ni2Mo (316H) are given in Fig. 13.2 for the reduction of area and for the elongation at rupture.

Although the tests have been performed under well controlled conditions there is a large scatter in the data. In fact, the scatter in creep ductility values is typically much larger than for creep strength values. A difference in comparison with Fig. 13.1 is that the upper shelf ductility in Fig. 13.2 varies somewhat with temperature in particular for the creep elongation. It should be noticed that the ductility can take very low values.

It is also instructive to plot the creep ductility as a function of rupture time. This is shown in Fig. 13.3.

Both the reduction in area and the creep elongation decrease with increasing rupture time and increasing temperature. For reduction of area an upper shelf is apparent at shorter rupture times. This is only evident for the elongation at lower temperatures. Except at the lowest temperature, the ductility values at long times can be quite low. This demonstrates that a low ductility shelf is present. The ductility versus rupture time can also be represented with an S-shape curve as in Fig. 13.1 but with inverted S curves. Nice S-shaped curves can be found in the literature with much less scatter than in Figs. 13.2 and 13.3 [6].

Many attempts have been made in the past decades to model creep ductility. With few exceptions, empirical approaches have been used. One important method to assess the remaining life of plants operating at high temperature, where creep has been the life controlling mechanism, has involved ductility exhaustion. With the help of continuum damage mechanics (CDM) [7‐9], the creep strain in critical components is computed to ensure that it does not exceed the ductility values. In this process some observed microstructural changes have been recorded and then been the basis in the modeling. In the literature there are a large number of papers discussing this type of analysis. The mentioned volumes on CDM can serve as a starting point in this respect.

In the present chapter empirical models for the creep ductility will be discussed in Sect. 13.2. These models are mainly statistical. In statistical methods mathematical expressions are chosen and fitted to the experimental data. The choice of expression is merely for numerical convenience to get a good fit to the data. The approaches involve a number of adjustable parameters that are fitted to the data. The reason for developing these models has in general been to use them in design or in residual life assessment. To meet this aim, it must be possible to generalize the data for example to longer times. This requires that the models are trained against a large set of data. In particular, the number of independent experimental data points must be very much larger than the number of adjustable parameters involved.

To avoid these limitations basic models based on physical principles and without the use of adjustable parameters have been developed. Such models will be presented in Sect. 13.3.

Failures that are associated with low and high creep ductility are referred to as brittle and ductile rupture, respectively. As was illustrated in Figs. 13.1, 13.2 and 13.3, brittle rupture occurs primarily at low stress, long rupture times and high temperatures, whereas the conditions for ductile rupture are opposite, i.e. it takes place at high stresses, low temperatures and short rupture times. Brittle rupture is assumed to be initiated by the nucleation, growth and linkage of creep cavities. Cavitation models for the creep ductility will be presented in Sect. 13.3.1. Such models have turned out to give successful results in a number of cases. Much less work has been carried out for ductile rupture. For a number of steels and copper it has been demonstrated that necking controls the failure during ductile rupture. This approach will be discussed in Sect. 13.3.2. To study the necking, creep strain data must be available. However, also a ductility corresponding to the upper shelf in Fig. 13.1 can be used to predict ductile rupture.

13.2 Empirical Ductility Models

Creep strength data have been analyzed with statistical methods for a long time. The European Collaborative Creep Committee (ECCC) has developed suitable procedures that ensure that the statistical analysis is performed in a good way and that the results behave in a physically correct way. There is a large number of methods available analyzing and extrapolating creep strength data and practically all are empirical. In spite of this, the methods can provide quite valuable information due to the large number of data points.

ECCC have also proposed procedures for assessment of creep ductility values [10]. A number of different expressions for the ductility are suggested [11]. Various combinations of constants, stress and temperature dependencies are used such as

$$ \log (\varepsilon_{{\text{R}}} ) = \log (\beta_{0} ) + \beta_{1} \log (T) + \beta_{2} \log (\sigma ) + \beta_{3} /T + \beta_{4} \sigma /T $$

(13.1)

where ε_R is the rupture ductility, T the absolute temperature, σ the stress and β₀–β₄ constants. Analyses with such expressions have for example been performed by Spindler for austenitic stainless steels [12], by Payten et al. for 9Cr1Mo steels [13], and by Holdsworth and co-workers for 1CrMoV rotor steel [4, 14]. Many alternative expressions have also been considered; see for example [15, 16].

Other types of analyses have also been performed. Lai collected a large database for the austenitic stainless steel 316H. He made a regression analysis to determine the influence of the composition and some microstructure parameters on the ductility [1, 17]. Wilshire used creep data from NIMS to generate master curves for high Cr-steels. The principle was to use an activation energy to make ductility values at different temperatures merge to a single curve [18]. Xu and Hayhurst used continuum damage mechanics (CDM) to assess the creep ductility of 316H [19]. Low alloy rotor steels were studied by Singh and Kamaraj, again with a CDM approach [20].

The complexity and variability of creep ductility data were illustrated in Figs. 13.2 and 13.3. The cast to cast differences have rarely been possible to model. One notable exception is the paper by Binda and Holdsworth [14], where the influence of composition on 1CrMoV steel was analyzed. However, in most cases it is beneficial to concentrate the modeling to the most essential features. From a technical point, the start and level of the lower shelf are the most important aspects in general. For this purpose, the variation of the ductility can be described by a step function. A suitable step function is the sigmoid function that has a characteristic S-shape.

$$ f_{{{\text{sigm}}}} (x) = 1/(1 + \exp ( - x)) $$

(13.2)

With the help of the sigmoid function, curves of the form in Fig. 13.1 can be generated

$$ \varepsilon_{{{\text{f}}\sigma }} = L_{{{\text{shelf}}}} + (U_{{{\text{shelf}}}} - L_{{{\text{shelf}}}} )f_{{{\text{sigm}}}} ((\log \sigma e^{{Q/R_{{\text{G}}} T}} - \log \sigma_{0} )/\log \sigma_{{{\text{rng}}}} ) $$

(13.3)

$$ \varepsilon_{{{\text{f}}t_{{\text{R}}} }} = L_{{{\text{shelf}}}} + \left( {U_{{{\text{shelf}}}} - L_{{{\text{shelf}}}} } \right)f_{{{\text{sigm}}}} \left( { - \left( {\log t_{{\text{R}}} e^{{Q/R_{G} T}} - \log t_{R0} } \right)/\log t_{{R{\text{rng}}}} } \right) $$

(13.4)

where σ and t_R are the creep stress and rupture time and ε_fσ and ε_ftR rupture ductilities. L_shelf and U_shelf are the lower and upper shelf energies. The parameters with index 0 and rng indicate the central position of the curve and the size of the transition range, respectively. An activation Q is introduced to represent curves at different temperatures. With the help of Eqs. (13.3) and (13.4), it should be straightforward to generate curves describing the influence of parameters on the ductility. Ductility curves as a function of stress and rupture time have been presented above. In the literature, other parameters are also considered. Use of the strain rate or normalized stress are often applied, see for example [6, 11].

13.3 Basic Ductility Methods

13.3.1 Brittle Rupture

The main mechanism for brittle rupture is the nucleation and growth of cavities. When a critical cavitated area fraction in the grain boundaries is reached failure occurs. A combination of nucleation and growth of cavities must take place. A model for nucleation of cavitation based on grain boundary sliding was presented in Eq. (10.8)

(13.5)

where n_cav is the number of cavities, d_sub the subgrain size, $\dot{\varepsilon }$ the creep strain rate, λ the interparticle spacing in the grain boundary and C_s a constant. The factors g_sub and g_part are the fraction of active nucleation sites at sub-boundary junctions and particles.

Diffusion controlled growth of cavities is described by Eqs. (10.15) and (10.18)

$$ \frac{{dR_{{{\text{cav}}}} }}{dt} = 2D_{0} K_{{\text{f}}} (\sigma_{{{\text{red}}}} - \sigma_{0} )\frac{1}{{R_{{{\text{cav}}}}^{2} }} $$

(13.6)

$$ 2\pi D_{0} K_{{\text{f}}} (\sigma_{{{\text{red}}}} - \sigma_{0} )n_{{{\text{cav}}}} R_{{{\text{cav}}}} + \dot{\varepsilon }(\sigma_{{{\text{red}}}} ) = \dot{\varepsilon }(\sigma ) $$

(13.7)

where R_cav the cavity radius in the grain boundary plane, σ₀ the sintering stress. The grain boundary diffusion parameter D₀ is equal to δD_GBΩ/k_BT where δ is the grain boundary width, D_GB the grain boundary self-diffusion coefficient, and Ω the atomic volume. k_B is the Boltzmann’s constant and T the absolute temperature. The factor K_f is given in Eq. (10.12).

Equations (13.6) and (13.7) model constrained cavity growth. Constrained growth is essential to take into account to avoid overestimating the growth rate. σ_red is a reduced stress that is lower than the applied stress σ. $\dot{\varepsilon }(\sigma_{{{\text{red}}}} )$ and $\dot{\varepsilon }(\sigma )$ are the creep rates at the reduced and applied stress, respectively. σ_red is found by solving Eq. (13.7).

Grain boundary decohesion is the main mechanism for brittle rupture. Due to both cavity nucleation and growth, there is a gradual increase of the cavitated grain boundary area fraction during creep. When this fraction reaches a critical value, failure takes place. It is fairly well established that this critical area fraction is about 0.25 [21]. The cavitated grain boundary area fraction A_cav can be calculated from [22]

$$ A_{{{\text{cav}}}} = \int\limits_{0}^{t} {\frac{{dn_{{{\text{cav}}}} }}{{dt_{1} }}(t_{1} )} \pi R_{{{\text{cav}}}}^{{2}} (t,t_{1} )dt_{1} $$

(13.8)

The nucleation rate and the cavity radius in Eq. (13.8) are given by Eqs. (13.5) and (13.6). The resulting time dependence of these quantities is illustrated in Fig. 13.4. In this Figure the number of cavities, the average cavity radius, the cavitated area fraction, and the creep strain are given as a function of time in a common diagram. Ductile rupture occurs when the creep strain reaches a fixed elongation value of 0.2. Brittle rupture takes place when the cavitated area fraction gets a value of 0.25. These levels are marked in the diagram. The condition that is first satisfied controls the type of failure. Thus, if the creep strain reaches 0.2 before the cavitated area fraction is 0.25, the rupture is ductile but if cavitated area fraction takes its critical value first, the rupture is brittle.

Two cases for the austenitic stainless steel 316H are considered in Fig. 13.4a at a relatively high stress and low temperature and in Fig. 13.4b at a low stress and high temperature. The number of cavities increases at the same rate as the creep strain. This is a direct consequence of Eq. (13.5). Since a constant creep rate is assumed, both the creep strain and the number of nucleated cavities are linear in time. For unconstrained growth the cavity volume is linear in time, cf. Eq. (10.11). This means that the cavity radius is proportional to t^1/3 where t is the time. For constrained growth, the growth rate is lower. Since the cavitated area fraction in the grain boundaries increases both with the number of cavities and the cavity radius, it shows a faster increase than the two contributing processes. In Fig. 13.4a, the strain criterion is met first. Consequently the rupture is ductile. On the other hand in Fig. 13.4b, the criterion for the cavitated area fraction is satisfied first. The rupture is brittle.

An early example of the application of Eqs. (13.5), (13.6) and (13.8) was for pure oxygen free copper with 50 ppm P (Cu-OFP) and without P. The difference in cavitation between these two types of alloys has been possible to model in detail, see Sect. 10.4. The predicted rupture elongation as a function of temperature is shown in Fig. 13.5.

The influence of P in Fig. 13.5 is striking. With 50 ppm P the rupture is ductile and this is modeled with ductility exhaustion. In contrast, Cu without P can have very low creep ductility values (brittle rupture) that can be fully described by the model.

With the help of Eqs. (13.5), (13.6) and (13.8), predicted creep ductility values for brittle rupture are given as a function of rupture time for two austenitic stainless steel 321H and 316H in Fig. 13.6.

The computed ductility values decrease with increasing temperature and rupture time. This is regime II in Fig. 13.1. The upper shelf regime I appears at higher ductility values, see Fig. 13.3. In Fig. 13.6 there is no direct evidence of a lower shelf regime III. The modeling can only describe the general behaviour of the ductility, not the cast to cast variation as explained in Sect. 13.1.

13.3.2 Ductile Rupture

Ductile creep rupture of tensile creep specimen is believed to be initiated by necking, i.e. the plastic instability that forms a waist around the specimen. That was demonstrated for creep of copper in Sect. 12.4.1. In addition, when tertiary creep can be described with the Omega model, the necking takes place very close to the rupture and can be assumed to start the rupture, Sect. 12.5.2. For a number of steels that follow the Omega model including many low alloy steels, 9 and 12%Cr steels and austenitic stainless steels, ductile rupture can be considered to be controlled by necking. Since necking and the associated ductility values are fully described in Sect. 12.5, the results will not be repeated here.

13.4 The Role of Multiaxiality

In the literature there has been considerable interest in the effect of multiaxiality on the creep ductility. One of the main reasons is that creep ductility exhaustion is an important method for residual life assessment of components operating at high temperatures. For surveys on the role of multiaxiality, see [6, 11].

13.4.1 Diffusion Controlled Growth

Giessen and Tvergaard has proposed that Eq. (10.8) for cavity nucleation should be modified by including the ratio between the stress normal to the grain boundary σ_n and the effective stress σ_e to take into account the effect of the stress state [25]

$$ \frac{{dn_{{{\text{cav}}}} }}{dt} = \frac{{0.9C_{s} }}{{d_{{{\text{sub}}}} }}\left( {\frac{{g_{{{\text{sub}}}} }}{{d_{{{\text{sub}}}}^{2} }} + \frac{{g_{{{\text{part}}}} }}{{\lambda^{2} }}} \right)\left( {\frac{{\sigma_{{\text{n}}} }}{{\sigma_{{\text{e}}} }}} \right)^{2} \dot{\varepsilon } = B_{{\text{s}}} \left( {\frac{{\sigma_{{\text{n}}} }}{{\sigma_{{\text{e}}} }}} \right)^{2} \dot{\varepsilon } $$

(13.9)

Gonzales and Cocks have taken the average of this expression over all contributing grain boundaries and found the following result for the multiaxiality factor [26]

$$ \frac{{dn_{{{\text{cav}}}} }}{{dt}} = Bf_{{{\text{MA}}}} \dot{\varepsilon}\,\text{where}\;\;f_{{{\text{MA}}}} = \frac{4}{9} + 5\left( {\frac{{\sigma _{{\text{h}}} }}{{\sigma _{{\text{e}}} }}} \right)^{2} $$

(13.10)

σ_h is the hydrostatic stress and σ_e the effective stress.

$$ \sigma_{{\text{h}}} = (\sigma_{1} + \sigma_{2} + \sigma_{3} )/3;\;\;\;\sigma_{{\text{e}}} = \sqrt {((\sigma_{1} - \sigma_{2} )^{2} + (\sigma_{2} - \sigma_{3} )^{2} + (\sigma_{3} - \sigma_{1} )^{2} )/2} $$

(13.11)

σ₁, σ₂ and σ₃ are the principal stresses.

The derivation of the expression for diffusion controlled growth Eq. (10.11) has the form that it is natural to assume the following effect of the stress state [27]

$$ \frac{{dR_{{{\text{cav}}}} }}{dt} = 2D_{0} K_{{\text{f}}} \sigma_{{\text{e}}} \frac{1}{{R_{{{\text{cav}}}}^{2} }}\frac{{\sigma_{{\text{n}}} }}{{\sigma_{{\text{e}}} }} $$

(13.12)

where σ_n and σ_e have the same interpretation as in Eq. (13.9). For simplicity the role of the sintering stress is neglected since it is small anyway. Thus, the influence of the stress state has a similar form as for the nucleation rate in Eq. (13.9). If the averaging over grain boundary orientation is made in the same way as in Eq. (13.10), the following result is obtained

$$ \frac{{dR_{{{\text{cav}}}} }}{dt} = 2D_{0} K_{f} \sigma_{{\text{e}}} \frac{1}{{R_{{{\text{cav}}}}^{2} }}f_{{{\text{MA}}}}^{1/2} $$

(13.13)

If a constant stress is assumed, Eqs. (13.10) and (13.12) can be integrated directly

$$ n_{{{\text{cav}}}} = B_\text{s}f_{{{\text{MA}}}} \dot{\varepsilon }t;\;R_{{{\text{cav}}}} = (R_{{{\text{cav0}}}}^{3} + 6D_{0} K_{f} \sigma_{{\text{e}}} f_{{{\text{MA}}}}^{1/2} (t - t_{0} ))^{1/3} $$

(13.14)

where t is the time. These expressions can be inserted in Eq. (13.8) for the cavitated area fraction

$$ A_{{{\text{cav}}}} = \pi B_\text{s}f_{{{\text{MA}}}}^{4/3} \dot{\varepsilon }(6D_{0} K_{f} \sigma_{{\text{e}}} )^{2/3} \int\limits_{0}^{t} {(t - t^{\prime})^{2/3} } dt^{\prime};\;\int\limits_{0}^{t} {(t - t^{\prime})^{2/3} } dt^{\prime} = \frac{3}{5}t^{5/3} $$

(13.15)

In this equation the rupture time t_R for t is introduced. With a constant stress and only secondary creep $\varepsilon_{{\text{R}}} = \dot{\varepsilon }t_{{\text{R}}}$.

$$ A_{{{\text{cav}}}} = \frac{3\pi }{5}B_\text{s}f_{{{\text{MA}}}}^{4/3} \dot{\varepsilon }^{ - 2/3} (6D_{0} K_{f} \sigma_{{\text{e}}} )^{2/3} \varepsilon_{R}^{5/3} $$

(13.16)

From Eq. (13.16), the rupture ductility can be obtained if A_cav is taken as its critical value A_crit

$$ \varepsilon_{R} = (A_{{{\text{crit}}}} \dot{\varepsilon }^{2/3} / \left(\frac{3\pi }{5}B_\text{s}f_{{{\text{MA}}}}^{4/3} (6D_{0} K_{f} \sigma_{{\text{e}}} )^{2/3} \right)^{3/5}) \;\;\;\;\left( {\text{Diffusion control}} \right) $$

(13.17)

Since the expression is based on unconstrained rather than constrained growth is cannot be used to predict the uniaxial ductility. However, the effect of the stress state is expected to be the same for unconstrained and constrained growth. The influence of multiaxiality can now be extracted

$$ \varepsilon_{{\text{R}}} = \varepsilon_{{\text{R}}}^{0} /f_{{{\text{MA}}}}^{4/5} \;\;\;\;\left( {\text{Diffusion control}} \right) $$

(13.18)

where $\varepsilon_{{\text{R}}}^{0}$ is the uniaxial ductility.

13.4.2 Strain Controlled Growth

There are several expressions for strain control of cavity growth that are properly derived. The one due to Cocks and Ashby [28] was discussed in Sect. 10.5.3. Wen and Tu has improved one expression in Cocks and Ashby’s derivation and proposed a new formula [29]. Another result was derived by Rice and Tracey [30]. It gives a cavity growth rate of the form

$$ \frac{1}{{R_{{{\text{cav}}}} }}\frac{{dR_{{{\text{cav}}}} }}{dt} = \dot{\varepsilon }\frac{1}{2}\sinh \frac{{3\sigma_{{\text{h}}} }}{{2\sigma_{{\text{e}}} }} $$

(13.19)

where σ_h is the hydrostatic stress and σ_e the effective stress. This equation was derived for plastic deformation. Hellan transferred the equation to creep conditions [31]

$$ \frac{1}{{R_{{{\text{cav}}}} }}\frac{{dR_{{{\text{cav}}}} }}{dt} = \alpha_{{\text{H}}} \dot{\varepsilon }\sinh \frac{{\beta_{{\text{H}}} \sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }} $$

(13.20)

where

$$ \alpha_{{\text{H}}} = 3{\text{arcsinh(2(}}n_{{\text{N}}} - 1)/n_{{\text{N}}} );\;\;\;\beta_{{\text{H}}} = {2(}n_{{\text{N}}} - 1)/n_{{\text{N}}} ) $$

(13.21)

n_N is the creep rate stress exponent. Equation (13.20) can be combined with Eq. (13.10) for the nucleation rate to derive the cavitated area fraction. But Eq. (13.8) must first be transformed from time to strain dependence

$$ A_{{{\text{cav}}}} = \int\limits_{0}^{\varepsilon } {\frac{{dn_{{{\text{cav}}}} }}{{d\varepsilon_{1} }}(\varepsilon_{1} )\pi R_{{{\text{cav}}}}^{{2}} (\varepsilon ,\varepsilon_{1} )d\varepsilon_{1} } $$

(13.22)

The nucleation rate Eq. (13.10) can directly be expressed as

$$ \frac{{dn_{{{\text{cav}}}} }}{d\varepsilon } = B_\text{s}f_{{{\text{MA}}}} $$

(13.23)

Equation (13.20) can also easily be transformed to strain dependence and integrated

$$ \frac{1}{{R_{{{\text{cav}}}} }}\frac{{dR_{{{\text{cav}}}} }}{d\varepsilon } = \alpha_{{\text{H}}} \sinh \frac{{\beta_{{\text{H}}} \sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }};\;R_{{{\text{cav}}}} = R_{{{\text{cav}}}}^{0} \exp (\alpha_{{\text{H}}} \sinh \frac{{\beta_{{\text{H}}} \sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }}(\varepsilon - \varepsilon_{0} )) $$

(13.24)

where $R_{{{\text{cav}}}}^{0}$ is the initial cavity radius when a cavity starts to grow at the strain ε₀. Inserting Eqs. (13.23) and (13.24) into (13.22) gives

$$ A_{{{\text{cav}}}} = B_\text{s}f_{{{\text{MA}}}} \pi (R_{{{\text{cav}}}}^{0} )^{2} \int\limits_{0}^{\varepsilon } {\exp (2\alpha_{{\text{H}}} \sinh \frac{{\beta_{{\text{H}}} \sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }}(\varepsilon - \varepsilon_{1} ))} d\varepsilon_{1} $$

(13.25)

The integral in Eq. (13.25) is elementary

$$ A_{{{\text{cav}}}} = B_\text{s}f_{{{\text{MA}}}} \pi (R_{{{\text{cav}}}}^{0} )^{2} \frac{{\exp \left( {2\alpha_{{\text{H}}} \sinh \frac{{\beta_{{\text{H}}} \sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }}\varepsilon } \right)}}{{2\alpha_{{\text{H}}} \sinh \frac{{\beta_{{\text{H}}} \sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }}}} $$

(13.26)

By replacing A_cav by its critical value A_crit the rupture ductility is obtained

$$ \varepsilon_{{\text{R}}} = \frac{1}{{2\alpha_{{\text{H}}} \sinh \frac{{\beta_{{\text{H}}} \sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }}}}\log \left( {\frac{{A_{{{\text{crit}}}} 2\alpha_{{\text{H}}} \sinh \frac{{\beta_{{\text{H}}} \sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }}}}{{B_\text{s}f_{{{\text{MA}}}} \pi (R_{{{\text{cav}}}}^{0} )^{2} }}} \right)\;\;\;\left( {{\text{Hellan}}} \right) $$

(13.27)

One uncertainty in Eq. (13.27) is the initial value of the cavity radius. Another limitation is the absence of significant temperature dependence. These problems seem to be common for many strain controlled growth mechanisms. The equation is consequently difficult to use to predict the uniaxial rupture ductility. The main influence of the stress state is in the sinh function outside the logarithm.

13.4.3 Growth Due to Grain Boundary Sliding (GBS)

In Sect. 10.5.3 a model for cavity growth due to GBS was presented in Eq. (10.24)

$$ R_{{{\text{cav}}}} = C_{{\text{s}}} \varepsilon $$

(13.28)

If this expression together with Eq. (13.10) for the nucleation rate are inserted in Eq. (13.22) the cavitated area fraction in the grain boundaries is found

$$ A_{{{\text{cav}}}} = Bf_{{{\text{MA}}}} \pi C_{{\text{s}}}^{2} \varepsilon^{2} /2 $$

(13.29)

This gives the following rupture ductility

$$ \varepsilon_{{\text{R}}} = (2A_{{{\text{crit}}}} /Bf_{{{\text{MA}}}} \pi C_{{\text{s}}}^{2} )^{1/2} \;\;\;\left( {\text{GBS growth}} \right) $$

(13.30)

13.4.4 Comparison of Models

In Fig. 13.7 the multiaxial creep ductility factor for the diffusion controlled growth model, Eq. (13.18), is shown as a function the stress triaxiality ratio σ_h/σ_e. Also results for Hellan’s model in Eq. (13.27) are illustrated. A comparison is made to experimental data for notched bars. Values for Durehete 1055 (1Cr1Mo), 2.25Cr1Mo (P22), 9Cr1Mo, 9Cr1Mo mod. (P91) and the stainless steel 316H (17Cr12Ni2Mo) are included.

A comparison is also made to the models of Cocks and Ashby [28] and of Wen and Tu [29]

$$ \varepsilon_{{\text{R}}} = \sinh \left( {\frac{2}{3}\frac{{(n_{{\text{N}}} - 0.5)}}{{(n_{{\text{N}}} + 0.5)}}} \right)/\sinh \left( {\frac{{2(n_{{\text{N}}} - 0.5)}}{{(n_{{\text{N}}} + 0.5)}}\frac{{\sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }}} \right)\;\;\;\left( {\text{Cocks and Ashby}} \right) $$

(13.31)

$$ \varepsilon_{{\text{R}}} = \exp \left( {\frac{2}{3}\frac{{(n_{{\text{N}}} - 0.5)}}{{(n_{{\text{N}}} + 0.5)}}} \right)/\exp \left( {\frac{{2(n_{{\text{N}}} - 0.5)}}{{(n_{{\text{N}}} + 0.5)}}\frac{{\sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }}} \right)\;\;\;\left( {\text{Wen and Tu}} \right) $$

(13.32)

The five models in Fig. 13.7a are in reasonable agreement with the observations. The GBS growth model is in the upper end of the data range. For small stress triaxiality values, the models fall in two groups. The diffusion controlled and GBS growth models are close to that of Wen and Tu [29]. They derived a model that corrected an approximation in the Cocks and Ashby model improving the model at low triaxiality stresses [6]. Hellan’s and Cocks and Ashby’s models give results that are quite similar. The parts in their expressions for the influence of the stress states are also close.

In spite of the fact that the models tend to give similar results when compared to observations, they are based on quite different principles. The diffusion control and GBS growth models are based on expressions for nucleation and growth of creep cavities that are verified experimentally. For both models the nucleation plays an important role. For the strain governed models only the derivation of Hellan’s model takes nucleation into account but the nucleation has only a smaller effect on the results. The other two models for strain controlled growth, Cocks and Ashby as well as Wen and Tu do not involve nucleation. All the strain controlled models suffer from the limitation that there is no direct temperature or stress dependence in the models, which is not consistent with data for uniaxial creep ductility. The starting value of the cavity radius for growth in these models is not defined except for the GBS growth models. These facts imply that it is difficult to make direct comparisons to observed growth rates. If these limitations affect the multiaxial ductility factor is not known.

The derivation (13.27) based on Hellan’s model is new. In the past, most researches have started directly from Rice and Tracey’s Eq. (13.19) ignoring the transformation to creep. In the literature the multiaxial ductility in Eq. (13.33) is assumed to be derived from Rice and Tracey’s equation.

$$ \varepsilon_{{\text{R}}} = \varepsilon_{{\text{R}}}^{0} \exp \left( {\frac{1}{2} - \frac{3}{2}\frac{{\sigma_{{\text{h}}} }}{{\sigma_{{\text{e}}} }}} \right)\;\;\;\;\left( {{\text{Rice and Tracey orig}}.} \right) $$

(13.33)

Further comparisons between models are given in Fig. 13.7b. The transformation to creep in Hellan’s equation seems to have only a modest effect. This can be seen by comparing the curves for Hellan’s and Rice and Tracey’s models. However, the usual expression referred to as Rice and Tracey’s original equation is quite different to that of Hellan.

A number of empirical models for the multiaxial ductility factor exist. Wen et al. have given of survey of them [6]. Since several models derived from basic principles are available and they give results that are often not very different from the empirical models, the incentive to use the latter type must be limited in particular since the expressions for the basic models are not very complex.

13.5 Summary

For a number of creep resistant steels the creep ductility decreases with increasing rupture time and temperature. Sometimes an upper shelf level is observed at short rupture times and a lower shelf level at longer rupture times. For martensitic steels an increase in the ductility can be found at still longer times.
Creep ductility is traditionally modelled with empirical mathematical expressions describing an S or an inverted S-shaped curve depending on the variables used.
Basic expressions for cavity nucleation and diffusion controlled growth can be used to describe the ductility during brittle failure. For steels only general predictions are possible due to the complex cast to cast variation that is not fully understood at present.
The ductility during ductile rupture has been demonstrated to be controlled by necking for the investigated steels and copper alloys. Since necking occurs very close to the rupture, modeling of necking can be used to predict the creep ductility.
Several derivations for the influence of multiaxiality on the creep ductility are presented. In spite of the fact that they are based on many principles, the results are in general close to observations for notched bars.

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Vorheriges Kapitel Tertiary Creep

Nächstes Kapitel Extrapolation

J.K.L. Lai, Creep rupture ductility variations in AISI type 316 stainless steel. Mater. Sci. Eng. A 104, L7–L9 (1988)CrossRef

J.A. Siefert, J.D. Parker, Evaluation of the creep cavitation behavior in Grade 91 steels. Int. J. Pres. Ves. Pip. 138, 31–44 (2016)CrossRef

F. Abe, Influence of boron nitride inclusions and degradation in creep rupture strength on creep rupture ductility of Gr.122. Mater. High Temp. 38, 197–210 (2021)

S. Holdsworth, Creep ductility of 1CrMoV rotor steel. Mater. High Temp. 34, 99–108 (2017)CrossRef

Data Sheet on the Elevated-Temperature Properties of 18Cr–8Ni–Mo Stainless Steel Tubes for Boiler and Heat Exchangers (SUS 316H TB), National Research Institute for Metals Tokyo, Japan, Report No. 6B (2000)

J.-F. Wen, S.-T. Tu, F.-Z. Xuan, X.-W. Zhang, X.-L. Gao, Effects of stress level and stress state on creep ductility: evaluation of different models. J. Mater. Sci. Technol. 32, 695–704 (2016)CrossRef

D. Krajcinovic, Damage mechanics. Mech. Mater. 8, 117–197 (1989)CrossRef

T.H. Hyde, W. Sun, J.A. Williams, Creep analysis of pressurized circumferential pipe weldments—A review. J. Strain Anal. Eng. Des. 38, 1–30 (2003)CrossRef

H.-T. Yao, F.-Z. Xuan, Z. Wang, S.-T. Tu, A review of creep analysis and design under multi-axial stress states. Nucl. Eng. Des. 237, 1969–1986 (2007)CrossRef

10.

S. Holdsworth, Developments in the assessment of creep strain and ductility data. Mater. High Temp. 21, 25–32 (2004)CrossRef

11.

S. Holdsworth, Creep-ductility of high temperature steels: a review. Metals 9 (2019)

12.

M.W. Spindler, The multiaxial creep ductility of austenitic stainless steels. Fatigue Fract. Eng. Mater. Struct. 27, 273–281 (2004)CrossRef

13.

W.M. Payten, D. Dean, K.U. Snowden, Analysis of creep ductility in a ferritic martensitic steel using a hybrid strain energy technique. Mater. Des. 33, 491–495 (2012)CrossRef

14.

L. Binda, S.R. Holdsworth, E. Mazza, The exhaustion of creep ductility in 1CrMoV steel. Int. J. Pres. Ves. Pip. 87, 319–325 (2010)CrossRef

15.

Y. Takahashi, Modelling of rupture ductility of metallic materials over wide ranges of temperatures and loading conditions, part II: comparison with strain energy-based approach. Mater. High Temp. 37, 340–350 (2020)CrossRef

16.

Y. Takahashi, Modelling of rupture ductility of metallic materials for wide ranges of temperatures and loading conditions, part I: development of basic model. Mater. High Temp. 37, 357–369 (2020)CrossRef

17.

J.K. Lai, A set of master curves for the creep ductility of type 316 stainless steel. J. Nucl. Mater. 82, 123–128 (1979)CrossRef

18.

B. Wilshire, P.J. Scharning, Creep ductilities of 9–12% chromium steels. Scripta Mater. 56, 1023–1026 (2007)CrossRef

19.

Q. Xu, D.R. Hayhurst, The evaluation of high-stress creep ductility for 316 stainless steel at 550 °C by extrapolation of constitutive equations derived for lower stress levels. Int. J. Pres. Ves. Pip. 80, 689–694 (2003)CrossRef

20.

K. Singh, M. Kamaraj, Creep ductility of 1Cr1Mo1/4V low alloy forging and casting steels. Mater. Sci. Eng. A 510–511, 51–57 (2009)CrossRef

21.

J. He, R. Sandström, Creep cavity growth models for austenitic stainless steels. Mater. Sci. Eng. A 674, 328–334 (2016)CrossRef

22.

R. Sandström, R. Wu, Influence of phosphorus on the creep ductility of copper. J. Nucl. Mater. 441, 364–371 (2013)CrossRef

23.

Data Sheet on the Elevated-Temperature Properties of 18Cr–10Ni–Ti Stainless Steel for Boiler and Heat Exchanger Seamless Tubes (SUS 321H TB), National Research Institute for Metals Tokyo, Japan, Report No. 5B (1987)

24.

R. Sandström, J.-J. He, Prediction of creep ductility for austenitic stainless steels and copper. Mater. High Temp. 39, 427–435 (2022)

25.

E. van der Giessen, V. Tvergaard, Interaction of cavitating grain boundary facets in creeping polycrystals. Mech. Mater. 17, 47–69 (1994)CrossRef

26.

D. Gonzalez, A.C.F. Cocks, T. Fukahori, T. Igari, Y. Chuman, Creep failure of a P91 simulated heat affected zone material under multiaxial states of stress, in ECCC2014 3rd International ECCC Conference (2014)

27.

W. Beere, M.V. Speight, Creep cavitation by vacancy diffusion in plastically deforming solid. Metal Sci. 21, 172–176 (1978)CrossRef

28.

A.C.F. Cocks, M.F. Ashby, Intergranular fracture during power-law creep under multiaxial stresses. Metal Sci. 14, 395–402 (1980)CrossRef

29.

J.-F. Wen, S.-T. Tu, A multiaxial creep-damage model for creep crack growth considering cavity growth and microcrack interaction. Eng. Fract. Mech. 123, 197–210 (2014)CrossRef

30.

J.R. Rice, D.M. Tracey, On the ductile enlargement of voids in triaxial stress fields*. J. Mech. Phys. Solids 17, 201–217 (1969)CrossRef

31.

K. Hellan, An approximate study of void expansion by ductility or creep. Int. J. Mech. Sci. 17, 369–374 (1975)CrossRef

Titel: Creep Ductility
verfasst von: Rolf Sandström
Verlag: Springer Nature Switzerland
Buch: Basic Modeling and Theory of Creep of Metallic Materials
Print ISBN: 978-3-031-49506-9

Electronic ISBN: 978-3-031-49507-6

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-49507-6_13

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