Primary Creep is Typified by a Continuously Decreasing Creep Rate Give Reasons Chegg

Creep fatigue behaviour and crack growth of steels

C. Berger , ... M. Schwienheer , in Creep-Resistant Steels, 2008

16.7 Concluding remarks

The multi-stage creep–fatigue behaviour of conventional and modern heat-resistant steels was investigated by service-type experiments and a numerical simulation. Knowledge of cyclic deformation and creep–fatigue damage assessment has been obtained. The industrial benefit can be summarized as follows:

•

The development of a creep–fatigue interaction concept covers physical interpretations of deformation and failure mechanisms.

•

Stress–strain path and creep–fatigue life can be predicted by user program SARA on the basis of rules for deformation, relaxation and cyclic stress– strain behaviour including internal stress and mean stress.

•

A creep–fatigue life estimation procedure was developed for multi-stage loading, which covers cycle counting methods. The procedure was established in power plant applications. An extension to automotive and aero engine applications is of future interest. For verification purposes long-term service-type creep–fatigue data up to 70000   h were generated.

•

In order to characterize creep and creep fatigue crack behaviour, different methods exist, like the two-criteria-diagram for creep crack initiation. The methods consider the linear elastic-parameter K₁ as well as the creep fracture mechanics parameter C*. These methods are validated in long term regime on numerous materials, mainly on type 1CrMoNiV steels and in recent years on type 10CrMoWVNbN steels.

•

Long-term experiments, both uniaxial as well as multiaxial, are necessary in order to verify advanced life prediction methods for components.

•

Modelling creep–fatigue behaviour in multiaxial and multi-stage loading with constitutive models is a current challenge.

•

Advanced lifing methods and knowledge of materials as well as methodologies enables a reduction design efforts, an increase in component loading and an increase in design quality linked with a reduction in technical risk and an increase in efficiency and economic benefits.

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MECHANICAL PROPERTIES OF SINGLE-PHASE CRYSTALLINE MEDIA: DEFORMATION IN THE PRESENCE OF DIFFUSION

A.S. ARGON , in Physical Metallurgy (Fourth Edition), 1996

6.1. Overview

As stated, earlier in sections 2 and 5 above, in the secondary creep stage of pure metals and Class II alloys a dynamic balance is reached between strain hardening and diffusion controlled thermal recovery associated with the establishment of a statistically stationary dislocation structure. Unlike the relatively simple set of processes established in the creep response of Class I solid solution alloys discussed in section 3, the processes that lead to steady state in pure metals and Class II alloys are complex and have resisted efforts to provide simple and straightforward models. In this section we take account of these processes and furnish some insights into possible mechanisms.

The principal distinguishing characteristic that differentiates the behavior of the pure metals and Class II alloys from Class I alloys is the establishment of well formed subgrains that take the part of the cells that form in the course of plastic deformation of these materials in the low temperature range. In fact, the similarity in the phenomenology of evolving dislocation structures indicates that the processes of intersecting slip, production of slip obstacles, fluctuations in dislocation fluxes, etc., which are important in cell formation in Stage III and Stage IV deformation at low temperatures, continue to be active in creep. The presence of effective diffusional processes of obstacle removal as described in section 5, however, result in substantial modifications of the low-temperature processes of cell formation, giving rise to much better defined subgrains which have a far lower ratio of redundant to geometrically necessary dislocation density in their boundaries. In what follows, we describe the most prominent characteristics of the evolving dislocation microstructure associated with the creep deformation which were thoroughly reviewed by Takeuchi and Argon [1976b], and by Martin and Argon [1986].

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Dynamic systems for ultrahigh temperature energy conversion

Adam Robinson , ... James Young , in Ultra-High Temperature Thermal Energy Storage, Transfer and Conversion, 2021

9.5.2 Creep

Creep is slow and continuous deformation with time and is a function of both stress and temperature. Secondary stage creep follows Arrhenius's law and is commonly described by Eq. (9.3) [61].

(9.3) ${\stackrel{·}{ϵ}}_{ss}=A{\sigma }^n{e}^{-Q/\overline{R}T}$

where the creep rate ${\stackrel{·}{ϵ}}_{ss}$ is determined by shear stress ${\sigma }^n$ and $T$ along with the material dependent constants $Q$ , $\overline{R}$ , $A$ . A deformation mechanism map can be created to plot the creep mechanism depending on the shear stress and temperature. Fig. 9.22 displays such a map for β-silicon carbide.

Figure 9.22. Deformation mechanism map for β-silicon carbide [62].

A threshold stress for diffusion creep, ${\sigma }_t$ , is visible on this map [63], which can be calculated using Eq. (9.4)

(9.4) ${\sigma }_t=\frac{0.3\mu b_b}{d}$

where $\mu$ is the shear modulus, $b_b$ is the average extrinsic grain boundary dislocation Burgers vector (about b/3), and $d$ is the grain size. This indicates that a stressed component may have an indefinite life. For a honeycomb heat exchanger, there are two sources of load; self-weight and aerodynamic. As no significant pressure difference exists between ducts and flow velocity (6.7   m/s) is minimal as a consequence of keeping pressure losses low [64], self-weight is the main source of load to drive creep deformation.

It would be highly desirable for a heat exchanger in the thermal storage context to have a life in excess of 25 years (7.88×10⁸ seconds). Therefore if the operating temperature is 1527°C, the melting point is 2730°C, and the maximum allowable deformation is 1%, the maximum allowable shear stress is 0.39   MPa ( $\mu$ =32   GPa). Calculating the stress caused by self-weight in a honeycomb is complex and requires a numerical tool like Abaqus [65]. When this result is computed for the geometry shown in Fig. 9.12 with the aforementioned conditions, the maximum shear stress was 0.08   MPa indicating that a heat exchanger life of more than 25 years is reasonable.

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Stress and reliability analysis for interconnects

Hengyun Zhang , ... Wensheng Zhao , in Modeling, Analysis, Design, and Tests for Electronics Packaging beyond Moore, 2020

4.1.3.3.2.2 Creep strain fatigue model

Syed [25] proposed creep strain fatigue models for Sn–Pb and Sn–Ag–Cu solder by partitioning creep strain into two parts corresponding to transient creep and steady-state creep stages:

(4.1.42a) $\text{For Sn}-\text{Pb}:N_{\mathrm{f}}={\left(0.02{\epsilon }_{\mathrm{cr},i}+0.063{\epsilon }_{\mathrm{cr},ii}\right)}^{-1}$

(4.1.42b) $\text{For Sn}-\text{Ag}-\text{Cu}:N_{\mathrm{f}}={(0.106{\epsilon }_{\mathrm{cr},i}+0.045{\epsilon }_{\mathrm{cr},ii})}^{-1}$

Syed [25] also presented creep strain fatigue model for Sn–Ag–Cu using total accumulated creep strain by considering two different creep constitutive models of power law and hyperbolic sine creep model:

(4.1.43a) $\text{Power law}:N_{\mathrm{f}}={\left(0.0468{\epsilon }_{\mathrm{cr}}\right)}^{-1}$

(4.1.43b) $\text{Hyperbolic sine}:N_{\mathrm{f}}={\left(0.0513{\epsilon }_{\mathrm{cr}}\right)}^{-1}$

Schubert et al. [26] proposed creep strain fatigue model for Sn–Pb and Sn–Ag–Cu using total accumulated creep strain by experiments and simulations:

(4.1.44a) $\text{For Sn}-\text{Pb}:N_{\mathrm{f}}=0.69{\epsilon }_{\mathrm{cr}}^{\left(-1.80\right)}$

(4.1.44b) $\text{For Sn}-\text{Ag}-\text{Cu}:N_{\mathrm{f}}=4.5{\epsilon }_{\mathrm{cr}}^{\left(-1.295\right)}$

where ε _cr from Eqs. (4.1.42) to (4.1.44) is the accumulated creep strain per cycle calculated by volume-averaging technique [27]:

(4.1.45) $\Delta {\epsilon }_{\mathrm{cr}}=\frac{\sum _i^n{\epsilon }_{\mathrm{cr}2,i}·v_{2,i}}{\sum _i^n{v}_{2,i}}-\frac{\sum _i^n{\epsilon }_{\mathrm{cr}1,i}·v_{1,i}}{\sum _i^n{v}_{1,i}}$

where ε _cr2,i and ε _cr1,i are the total accumulated creep strain in one element at the end point and start point of the ith cycle, respectively, v _2,i and v _1,i are the element volume at the end point and start point of the ith cycle, respectively, and n is the amount of selected elements to calculate averaged creep strain.

Creep strain fatigue model provides a more comprehensive approach because it can consider dwell time and strain amplitude in TC load. However, one limitation of creep strain fatigue model is the absence of plastic strain effects. Plastic strain effects can be neglected only if the strain rate is low enough, thus resulting in a constant stress situation and the strain is indeed time dependent.

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Properties of high-strength steels at and after elevated temperature

Guo-Qiang Li , ... Xin-Xin Wang , in Behavior and Design of High-Strength Constructional Steel, 2021

5.3.5.5 Three-stage creep model

Based on previous creep studies, a three-stage creep model of steel at high temperature is proposed according to the test results of HSSs and creep behavior of the three stages (transient, steady-state and tertiary creep stages) [26]. This model fits for the creep behavior of steels at temperatures ranging from 400°C to 800°C. The effect of phase changes in microscopic structure of steels at 800°C on creep is considered as well. It is expressed by the following formulas.

(5.14) $\mathrm{Stage}1:{\epsilon }_{c\mathrm{p}}{=c}_1\cdot {\alpha }^{c_2}\cdot {(T-T_{\mathrm{a}})}^{c_3}\cdot {\mathrm{t}}^{\frac{1}{2}}(0\le t\le t_1)$

(5.15) $\mathrm{Stage}2:{\epsilon }_{c\mathrm{p}}={\int }_{t_1}^t\left[{\mathrm{c}}_4\cdot {\alpha }^{c_5}\cdot \exp \left(-\frac{c_6}{T-T_b}\right)\right]dt+{\epsilon }_1(t_1<t\le t_2)$

(5.16) $\mathrm{Stage}3:{\epsilon }_{c\mathrm{p}}=c_7\cdot {\alpha }^{c_8}\cdot \exp \left(-\frac{c_9}{T}-\frac{c_{10}}{t-t_2+c_{11}}\right)+{\epsilon }_2(t_2<t)$

where ε _cp is creep strain, α is stress ratio [at 800°C and should be adjusted due to the phase change in steel; $m=0.9\alpha +0.02$ (Q550); $m=0.75\alpha +0.0833$ (Q690); $m=0.625\alpha +0.1117$ (Q890)], T is temperature (°C), t is creep time (min), t ₁ is demarcation time between stage 1 and 2, t ₂ is demarcation time between stage 2 and 3, ε ₁,ε ₂ are creep strains at the end of stage 1 and stage 2 respectively, and T _a,T _b, c ₁~c ₁₁ are creep parameters needed to be calibrated.

The set of three-stage creep model curves for Q550, Q690, and Q890 steels calibrated with the creep test results are shown in Figs. 5.31–5.33, respectively. The calibrated three-stage creep model parameters at various elevated temperatures for Q550, Q690, and Q890 steels are listed in Tables 5.14–5.16, respectively.

Figure 5.31. Three-stage creep model curves for Q550 steel [26]. (A) T=400°C. (B) T=550°C. (C) T=700°C. (D) T=800°C.

Figure 5.32. Three-stage creep model curves for Q690 steel [26]. (A) T=400°C. (B) T=550°C. (C) T=700°C. (D) T=800°C.

Figure 5.33. Three-stage creep model curves for Q890 steel [26]. (A) T=400°C. (B) T=550°C. (C) T=700°C. (D) T=800°C.

Table 5.14. Three-stage creep model creep parameters for Q550 steel.

c 1 (°C−1  min−0.5)	c 2	c 3	c 4 (min−1)	c 5	c 6 (°C)	c 7
1.31E-3	3.053	3.28E-1	2.46E-3	4.191	−2.527	3.400
c 8	c 9 (°C)	c 10 (min)	c 11 (min)	T a (°C)	T b (°C)
−5.474	−5.52E2	8.50E2	1.05E2	399.97	400.542

Table 5.15. Steel Q690 – three-stage creep model creep parameters for Q690 steel.

c 1 (°C−1  min−0.5)	c 2	c 3	c 4 (min−1)	c 5	c 6 (°C）	c 7
1.1E-6	2.64	1.52	0.03	4.34	832.41	1.96E10
c 8	c 9 (°C)	c 10 (min)	c 11 (min)	T a (°C)	T b (°C)
4.81	3545.6	10,064	466.23	362.32	295.51

Table 5.16. Three-stage creep model creep parameters for Q890 steel.

c 1 (°C−1  min−0.5)	c 2	c 3	c 4 (min−1)	c 5	c 6 (°C）	c 7
2.225E-5	2.571	1.025	2.286	3.973	4.287E3	5.149E3
c 8	c 9 (°C)	c 10 (min)	c 11 (min)	T a (°C)	T b (°C)
3.546	1.996E3	1.078E3	117.849	391.078	43.226

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Development of creep-resistant steels and alloys for use in power plants

F. Abe , in Structural Alloys for Power Plants, 2014

9.1.1 Overview of creep

Creep is a time-dependent plastic deformation of materials under load at elevated temperatures, often higher than roughly 0.4T _m, where T _m is the absolute melting temperature of the material. Creep in metals occurs as a result of two diffusion-controlled mechanisms, dislocation climb and stress-induced vacancy flow at elevated temperatures. A basic description of the major creep processes is given below. However, the reader is advised to refer to standard textbooks for detailed descriptions of creep mechanisms, governing equations, microstructural evolution, and creep-induced damage and fracture. ^1–7

Creep tests can be conducted under either a constant load or constant stress. For experimental convenience, most creep tests of engineering steels are conducted under a constant tensile load and at a constant temperature. The test results are normally plotted as series of curves, which represent the strain or strain rate of the material as a function of time, all measured over a certain gauge length. Figure 9.1 shows idealized creep and creep rate curves when the tests are carried out under a constant load and at a constant temperature. However, in practice, creep-resistant steels often exhibit complicated behaviour, especially under low stress and over long periods of time, reflecting the occurrence of very complex microstructural evolutions during creep.

9.1. (a), (b) and (c) Creep curves of engineering steels under constant tensile load and constant temperature and (d), (e) and (f) their creep rate curves as a function of time.

Textbooks generally describe three stages of creep, consisting of primary or transient, secondary or steady-state, and tertiary or acceleration creep, which appear after initial strain ε ₀ upon loading as shown in Fig. 9.1(a), when the test or homologous temperature is high enough. (Homologous temperature is defined as the ratio T/T _m, where T is the test temperature in kelvin and T _m the absolute melting temperature.)

The initial strain ε ₀ contains elastic strain and possibly plastic strain depending on the magnitude of applied stress. In the primary creep stage, i.e. between ε ₀ and ε ₁ in Fig. 9.1(d), the creep rate decreases with the time. The decreasing creep rate in the primary creep stage has been attributed to strain hardening or reduction in the density of free or mobile dislocations.

In the secondary or steady-state region, the creep rate remains almost constant. The secondary stage is attributed to a state of balance between the rate of dislocation generation and rate of recovery, where the former contributes to hardening and the latter to softening of the test material.

In the tertiary or acceleration stage, the creep rate increases rapidly until the material ruptures after undergoing a total of strain ε _r within a time of t _r. It should be pointed out that during a creep test under a constant tensile load, the actual stress increases continuously due to the reduction in the cross-section (i.e. necking). The strain increases in the tertiary creep stage is much faster than in the previous stages once the necking commences and this leads to a relatively fast failure of the material. The increase in creep rate in the tertiary creep stage can also be attributed to the microstructure evolution such as dynamic recovery, dynamic recrystallization, coarsening of precipitates and other phenomena, which cause softening and hence reduction in creep life. The creep-induced microstructural damage, such as formation of voids and cracks, often occurs along grain boundaries, resulting in failure of the material.

The extent and shape of the three creep stages described above can vary substantially depending on test conditions, as shown schematically in Fig. 9.2, where the final point on each curve represents the final rupture. With increasing stress and/or temperature, the time to rupture and the extent of secondary stage usually decrease whereas the total elongation increases.

9.2. Shift in creep curves as applied stress and temperature increases (shown by arrows).

Under certain conditions, the secondary or steady-state creep stage may be absent, hence after the primary creep stage the tertiary creep stage begins at t _m, as shown in Figs 9.1(b) and (e). In this case, the minimum creep rate ${\stackrel{.}{\mathit{\epsilon }}}_{\text{min}}$ can be defined instead of the steady-stage creep rate ${\stackrel{.}{\mathit{\epsilon }}}_{\mathrm{S}}$ . In many creep-resistant steels and alloys, there is no steady-state stage because of the continuous change in their microstructure under the service condition. This suggests that no dynamic microstructural equilibrium is reached during creep. Therefore, the term 'minimum creep rate' has been favoured by engineers when estimating the creep life of the heat-resistant steels and alloys used in power plant.

At homologous temperatures (T/T _m) less than about 0.3, where diffusion is sluggish, only the primary stage appears and usually only limited strains well below 1% occur without leading to a final rupture, as shown in Figs 9.1(c) and (f). This deformation process is designated as logarithmic creep.

There are several model equations available for characterizing the primary, secondary and tertiary creep stage characteristics, ranging in complexity from simple phenomenological to physically based constitutive ones. Recent progress on the modelling of creep constitutive equations and their application for service life prediction has been reviewed by Holdsworth. ⁸

Complicated creep deformation is normally demonstrated by strain/creep rate curves rather than absolute strain values. Figure 9.3 shows an example of the complicated creep rate curves of 1Cr-0.5Mo steel at 550   °C. ⁹ At high stresses above 108   MPa, the creep rate curves are relatively simple and consist of the primary and tertiary stages with almost no steady-state stage, similar to that shown in Fig. 9.1(e). The shape of creep rate curves becomes more complicated with decreasing stress. At low stresses, below 88   MPa, two minima appear in the creep rate curves. This suggests that additional strengthening effects such as the precipitation of new phases are operating after an extended period, causing a reduction in creep rate after the first acceleration creep. The subsequent loss of these existing strengthening effects due to further microstructural evolution, such as the coarsening of new phases, results in the second fast creep rate. Thus the creep rate versus time curves exhibit oscillating shapes under low stress and long periods of time, reflecting complex microstructural evolution. Similar oscillating creep rate curves have been reported in some low alloy steels.

9.3. Creep rate versus time curves of 1Cr-0.5Mo steel at 550   °C (823   K).

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MECHANICAL PROPERTIES OF MULTIPHASE ALLOYS

Jean-Loup STRUDEL , in Physical Metallurgy (Fourth Edition), 1996

7.1. Creep curves

The general aspect of high-temperature creep curves is radically different from that of the matrix alone (fig. 47). Primary creep is replaced either by a short sigmoidal stage at 850°C or by an incubation period ( Leverant et al. [1973] . The steady-state creep stage is short compared with an overwhelming third stage which either starts very early in the lower part of the temperature domain (850–1000°C) or contains most of the creep strain at higher temperature (1000–1100°C) and is soon followed by rupture. Comparing the forms of tensile (fig. 37) and creep (fig. 47) curves strongly points to the conclusion that two-phase alloys hardened by soluble quasi-coherent precipitates are mechanically unstable structures. The lack of initially mobile dislocations is the corner-stone of their mechanical resistance. The onset of dislocation movement, especially when it is dominated by climb, is the beginning of their degradation.

Fig. 47. High-temperature creep curves of cube-oriented single crystals of the alloy CM SX2.

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CREEP BUCKLING STUDY ON CYLINDRICAL SPECIMEN SUBJECTED TO AXIAL COMPRESSION WITH CYCLED TEMPERATURE AND ITS APPLICATION

Xin Wang , Fuzhong Shen , in Advances in Engineering Plasticity and its Applications, 1993

3.2 Analytical calculation

It is obvious that the results calculated at constant high temperature cann't explain the phenomenon at cycled temperature. We must seek other means to explain the testing data at cycled temperature.

Creep curves of metal materials have three stages, known as deceleration (the first), constant (the second) and acceleration(the third) stages respectively. The strain rates of the second stage is constant and relatively small. Therefore the second stage is named as minimum creep rate stage or steady creep stage. Each point in the specimen is under uniaxial tensile and compressed status. But its stress is different from each other and increases with the increasing of the maximum deflection. Except for the primary stage of testing, the specimen is in the second creep stage if temperature is constant. Thus the maximum deflection changing pattern of creep buckling is somewhat similar to that of creep curve, as shown in Fig. 4. But the first stage is not very obvious due to a relatively small initial deflection. The shut—down follow by start up will cause the presence of the first stage again (Fig. 5a). Because at that time it has a relatively larger deflection than the initial stage, thus causing a greater stress and then a greater deflection rate. The deflection changing of the specimen is similar to the creep curve at the effect of the first stage (Fig. 5b).

Fig. 5. The influence of cycled temperature

The influence of the first stage of creep on the creep buckling must be taken into account in order to quantitatively study the changing relation of the maximum deflection vs. time in the prsence of cycle temperature. The fitting equation of the first stage of HK—40 alloy at 871°C is obtained by experiment

(10) ${\dot{\text{ɛ}}}^c=6.8839\times {10}^{-7}{\text{σ}}^{5.076}{t}^{-0.46}$

where t is time. Replacing Eq. 1 by Eq. 10 and then combined with the second stage equation, the calculated critical time t _th of creep buckling of the specimen subjected to cycled high temperature can be obtained, as listed in table 2. The changing relation of maximum deflection of the specimem 2C11 vs. time is shown in Fig. 6. The table 2 and Fig. 6 show the closeness of calculated (theoretical) values to test data.

Fig. 6. The maximum deflection changing of the specimem 2C11

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Thermal and mechanical tests for packages and materials

Hengyun Zhang , ... Wensheng Zhao , in Modeling, Analysis, Design, and Tests for Electronics Packaging beyond Moore, 2020

6.2.1.2 Creep test and analysis

The study of creep behavior and associated characteristics is important in the reliability tests of solder joints. Generally, creep behavior is difficult to describe because it depends on a large number of variables. However, for technological applications, it is sufficient to describe the steady-state creep behavior. A steady-state creep is reached after a constant stress has been applied and the material has already passed through a transient phase of creep. Creep characteristics of solder alloys can be studied by different experimental techniques such as lap shear creep [23,24], uniaxial tensile [24,25], stress relaxation [26], and indentation testing methods [27]. In this section, the tensile creep behavior of the bulk solder alloy of SAC105Ni0.02 was investigated at different temperatures and stress levels.

The same specimen used in tensile test is also applied in creep test. Creep tests were carried out at four isothermal conditions including −35°C, 25°C, 75°C, and 125°C. For each testing temperature, several stress levels were designed from 5 to 45   MPa. For testing conditions at high and low temperatures, a thermal chamber was used to enclose the tested specimen to provide the designed temperature condition. The specimens were held at the designed temperature for 10   min before loading. At the beginning, tensile force is applied and increases gradually to the designed value. The designed stress is held to the end of the test, and creep strain is recorded during the whole test. For the creep test at a high stress level or high temperature, the creep test continues to be sample broken. While for the creep test at a low stress level or low temperature, the test is stopped for saving time when the constant creep strain rate is observed. Tests are repeated for three samples at the same condition and the averaged results are used.

Fig. 6.2.11 shows a typical creep strain curve of the solder. Creep has three stages including primary, secondary, and tertiary stages. Primary creep happens at the beginning of the test and very short. The secondary creep stage, also called steady-state creep, is a long creep stage, and it occupies most of the creep lifetime. The tertiary creep stage occurs before creep rupture, which has fast creep strain rate. Creep strain rate is calculated by taking the derivative of creep strain with time. The minimum rate was taken as the creep strain rate of the steady-state creep stage. Creep strain rate at steady-state creep stage is an important parameter that usually determines lifetime of solder joints. Stress and testing temperature have a significant effect on creep strain rate.

Figure 6.2.11. Typical tensile creep curve of solder (test conditions: 25°C and 20   MPa load).

Fig. 6.2.12 shows the effect of temperature on creep curve of solder. The creep strain rate increases and creep rupture time decreases significantly with increasing temperature at a certain stress level. Fig. 6.2.13 shows the effect of stress on creep curve of solder. The creep strain rate increases and creep rupture time decreases sharply when increasing the applied stress level. The creep rupture time as a function of stress at different temperatures is summarized in Fig. 6.2.14. It can be seen that the creep rupture time is sensitive to the stress level and temperature. Both high stress level and temperature accelerate creep rupture of solder due to the corresponding high creep strain rate.

Figure 6.2.12. Effect of temperature on creep curve of solder (test load: 15   MPa).

Figure 6.2.13. Effect of stress level on creep strain of solder (test at −35°C).

Figure 6.2.14. Relationship between creep rupture time and stress at different temperatures.

Fig. 6.2.15 shows the relationship between creep stress and steady-state creep strain rate at test temperature of 25°C. The creep strain rate changes from 10⁻⁹ to 10⁻² s⁻¹ when stress level changes from 5 to 30   MPa. The creep strain rate is less dependent on stress in the low stress range, while it becomes more dependent on stress in the high stress range. The power law constitutive model can be used to describe the relationship between creep strain rate and stress. The stress exponent n is larger in the high stress range than that in the low stress range. This phenomenon is referred as power law creep behavior breakdown. This power law breakdown behavior can be found at any given test temperature as shown in Fig. 6.2.16. The stress exponents in the low and high stress ranges can be determined based on creep strain rate data in this figure.

Figure 6.2.15. Relationship between creep stress and steady-state creep strain rate.

Figure 6.2.16. Steady-state creep strain rate at various load conditions.

Fig. 6.2.17 shows the stress exponents separately for high and low stress ranges at different temperatures. The stress exponent decreases with increasing temperature for both high and low stress ranges. Stress exponent can vary from 2 to 13 for different solders due to different creep mechanisms. At low stress, creep deformation is controlled mainly by grain boundary sliding in which stress exponent is 2–4 [28]. At high stress, creep deformation is mainly due to dislocation climb and glide with high stress exponent [24]. At high temperature, diffusion-controlled creep is a dominant mechanism.

Figure 6.2.17. Effect of temperature and stress level on stress exponent.

From the analysis mentioned above, the stress exponent and creep activation energy in the creep models are stress level and temperature dependent. The averaged values of stress exponent and creep activation energy can be determined by fitting all the creep experimental results under different stresses and temperatures. We conducted the curve fitting for the creep test data through iterative multivariable nonlinear regression to develop the creep constitutive models based on testing data shown in Fig. 6.2.16. The different types of creep constitutive models for SAC105Ni0.02 solder are presented below.

Norton's power law model (refer to Eq. 4.1.26):

(6.2.4) ${\dot{\epsilon }}_{\mathrm{cr}}=2.39\times {10}^{-7}{\sigma }^{8.90}\exp \left(-\frac{6282.7}{T}\right)$

Double power law model (refer to Eq. 4.1.27):

(6.2.5) ${\dot{\epsilon }}_{\mathrm{cr}}=9.87\times {10}^{-7}{\sigma }^7\exp \left(-\frac{4883.1}{T}\right)+5.01\times {10}^{-10}{\sigma }^{13}\exp \left(-\frac{8949.2}{T}\right)$

Garofalo hyperbolic sine model (refer to Eq. 4.1.28):

(6.2.6) ${\dot{\epsilon }}_{\mathrm{cr}}=3.94\times {10}^4{\left[\sinh \left(0.0607\sigma \right)\right]}^{6.32}\exp \left(-\frac{7024.2}{T}\right)$

Exponential model (refer to Eq. 4.1.29):

(6.2.7) ${\dot{\epsilon }}_{\mathrm{cr}}=277.4\exp \left(\frac{\sigma }{2.11}\right)\exp \left(-\frac{7744.3}{T}\right)$

These four creep constitutive models have been applied in FEA simulation for fine pitch ball grid ball package subjected to thermal cycling load. FEA simulation results showed that different models lead to similar creep strain and creep strain energy density, and similar fatigue life was predicted based on results from these four creep models [29]. Among steady-state creep models, hyperbolic sine model is commonly implemented for electronic solder materials. The creep activation energy is an important parameter used for measuring the creep resistance of the solder alloy. The larger the creep activation energy is, the higher the creep resistance is. For the purpose of using the creep model in FEA simulation, Eq. (4.1.28) is usually rewritten as the following format:

(6.2.8) ${\dot{\epsilon }}_{\mathrm{cr}}=C_1{\left[\sinh \left(C_2\sigma \right)\right]}^{C_3}\exp \left(\frac{-C_4}{T}\right)$

where C ₁, C ₂, C ₃, C ₄ are material constants, which are determined from tensile creep test results.

Table 6.2.3 lists material constants in Eq. (6.2.8) for SAC105Ni0.02 solder and other solder alloys for comparison. Creep activation energy of the SAC105Ni0.02 solder is comparable with that of high Ag content Sn–3.9Ag–0.6Cu solder, which indicates that SAC105Ni0.02 solder has similar creep resistance as the high Ag content SAC solder under TC condition. Ni addition in solder increases creep resistance and creep rupture lifetime due to refinement of IMC particles [20,24]. Compared with Sn–1.0Ag–0.5Cu solder as listed in Table 6.2.3 [30], adding 0.02%wt Ni improves creep resistance of the SAC105 solder. From other researchers' studies, SAC105Ni0.02 solder has similar mechanical properties as SAC105 solder, but the SAC105Ni0.02 solder improves drop performance compared with SAC105 solder [33,34].

Table 6.2.3. Material constants of hyperbolic sine creep model for different solders.

Solder alloys	C 1	C 1(1/MPa)	C 2	C 4(K)	Q (kJ/mol)	References
Sn–1.0Ag–0.5Cu–0.02Ni	3.94   ×   104	0.0607	6.32	7024.2	58.4	This study
Sn–1.0Ag–0.5Cu	2.8   ×   10−4	0.0581	7.1	5292.3	44	[30]
Sn–2.0Ag–0.5Cu	3.7   ×   10−4	0.0659	6.8	9165.3	76.2	[14]
Sn–2.0Ag–0.5Cu–0.05Ni	2.8   ×   10−4	0.0677	7.9	10,175.6	84.6	[14]
Sn–1.0Ag–0.3Cu	3.1   ×   10−4	0.0576	5.1	5412.6	45	[17]
Sn–3.9Ag–0.6Cu	0.184	0.221	2.89	7457	62.0	[31]
63Sn–37Pb	10	0.2	2.0	5400	44.9	[32]
Sn–3.5Ag	9   ×   105	0.0653	5.5	8690	72.2	[35]

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High Temperature Deformation of Intermetallic Compounds

In High Temperature Deformation and Fracture of Materials, 2010

8.4.3.1 Creep Behavior of Single Phase γ-TiAl

A typical creep curve for single phase γ-TiAl alloy is shown in Fig. 8.18 ^[26] . The creep behavior of the stoichiometric Ti-50A1 alloy varies with the applied stress. At high stress the creep curve appears as the usual form, showing a primary creep followed by the steady-state creep. At intermediate stress, the steady-state creep stage becomes very short but followed by an accelerated creep. In the low applied stress case, steady-state creep disappears, accelerated creep turns to occur immediately after the primary creep.

8.18. The creep curves of γ-TiAl single phase alloy ^[26] .

The curves in Fig. 8.19 show the relationship between the minimum creep rate and the applied stress for three γ-TiAl alloys with different Al content, namely, Ti-50A1, Ti-51A1, and Ti-53A1 ^[27] . Three regions of creep behavior are identified. The stress exponents are different in the three regions: n  =   4.7 in the high stress region (region I); n  =   7.5 in the intermediate stress region (region II); and n  =   3.5 in the low stress region (region III). For the three alloys, the slopes of their curves are nearly the same in each of the regions, indicating the mechanism controlling the creep rate is the same. In addition to this, the curves tend to show that at a constant applied stress, the steady-state creep rate increases with increasing Al content.

8.19. The minimum creep rate as a function of applied stress in single phase γ-TiAl alloys with different Al content ^{[27, 28]} .

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Source: https://www.sciencedirect.com/topics/engineering/creep-stage

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