A perspective on the Wilshire creep equations

Abstract

The Wilshire creep equations were introduced around fifteen years ago. Their aim was to address the non-physical extrapolation of power-law models, especially to high stresses, and the unrealistic values for activation energy and stress exponent that often arise from simple fits to data. In application they have met with some success, also with some difficulties which have largely been addressed empirically. No detailed mathematical analysis of the model seems to have been performed. This paper considers the fundamental characteristics of the Wilshire equations, as originally given, commencing with their internal consistency. It is found that the strain-time equation is incompatible with those for minimum creep-rate and rupture life. A consistent rate equation is derived, enabling the model to address the creep process rather than merely its results. Predictions made using the original and developed models are compared with actual materials behaviour; this reveals aspects of the approach which require reconsideration. The upper limit imposed by the ultimate tensile strength, and departures from a simple power law emerge as the key characteristics to be preserved and considered further.

Keywords

Wilshire equations creep models creep rate rupture life ductility Monkman–Grant ultimate tensile stress activation energy stress exponent form factor scale factor failure criterion

1. Introduction

In the last of many contributions to the study of creep, the late Professor Wilshire proposed a set of equations intended to overcome some of the perceived weaknesses of power law approaches [29–31]. Issues addressed were the non-physical extrapolation of power-law equations, especially to high stresses, and the unrealistic values for activation energy and stress exponent that often arise from simple fits to data. An essential feature of the Wilshire approach is the normalisation of applied stress to ultimate tensile strength. Evolution of the equations has been described by Whittaker and Harrison [28] and comparisons between the Wilshire approach and other methods have been made by Abdallah et al. [2], amongst others. The equations have engendered considerable interest and lively discussion.

Effort has been devoted to application of the model to a wide variety of engineering alloys. It has been found that, in general, a single calibration to the entire data range for an alloy is unsatisfactory; the fit is poor, and the values of activation energy obtained are unrealistic. Considerable improvement is obtained by partitioning the data into two or more ranges of normalised stress [2,16,27]. This allows changes in activation energy at the range boundaries; changes to which mechanistic significance has been qualitatively attached [12,27]. Methods have been proposed for non-subjective determination of the optimum partitioning [14,15]. Whilst these developments are interesting, the Wilshire equations remain, essentially, bare parametric descriptions of some of the results of the creep process – and therefore lie in a domain that is already well-populated. They do not describe or explain the creep process itself. Thus, they do not yet have the sophistication necessary for advanced component life assessment – they cannot readily address evolving plant operating conditions, nor interacting mechanisms such as fatigue and corrosion, nor can they be integrated with the formulations of creep fracture mechanics.

This paper focuses on the basic characteristics of the original Wilshire equations. It considers their formulation, import, and internal consistency. An expression for creep rate, compatible with the existing Wilshire model, is derived. The characteristics of this developed equation set are explored, and their implications and potential are discussed – taking data for $2\frac{1}{4}$ CrMo steel and titanium alloy IMI 834 as illustrations.

2. Formulation

2.1. Original form

In its original form, the Wilshire model consists of three equations. These are implicit equations, relating the independent variable, stress, to the dependent variables of rupture life, minimum creep rate and time to a given strain. They are usually expressed as: $\begin{eqnarray}\displaystyle \frac{{\sigma}}{{\sigma}_{\text{UTS}}}\text{} & =\text{} & \displaystyle \exp \left(-k_{1}\left[t_{r}\exp \left(\frac{-Q}{RT}\right)\right]^{u^{\prime }}\right)\end{eqnarray}$ (1a) $\begin{eqnarray}\displaystyle \frac{{\sigma}}{{\sigma}_{\text{UTS}}}\text{} & =\text{} & \displaystyle \exp \left(-k_{2}\left[\dot{{\varepsilon}}_{\min }\exp \left(\frac{Q}{RT}\right)\right]^{v^{\prime }}\right)\end{eqnarray}$ (1b) $\begin{eqnarray}\displaystyle \frac{{\sigma}}{{\sigma}_{\text{UTS}}}\text{} & =\text{} & \displaystyle \exp \left(-k_{3}\left[t_{{\varepsilon}}\exp \left(\frac{-Q}{RT}\right)\right]^{w^{\prime }}\right)\end{eqnarray}$ (1c) Symbols have their conventional meanings, given in Table 1.

Table 1

Notation

𝜀	Strain^∗	k ₁ , k ₂ , k ₃	Constants in the original formulation
𝜀_r	True rupture ductility	u ^′, v ^′, w ^′	Exponents in the original formulation
e _r	Engineering ductility	k _r , k _min , k _𝜀	Constants in the developed formulation
$\dot{{\varepsilon}}$	Creep strain rate	k (𝜀)	A function of strain
$\dot{{\varepsilon}}_{\min }$	Minimum creep rate	u, v, w	Exponents in the developed formulation
𝜎	Stress	f _i (x)	A function of x
𝜎₀	Initial stress	$f_{i}^{\prime }(x)$	The derivative of f _i (x)
𝜎_UTS	Ultimate tensile strength	K	Constant of integration
T	Absolute temperature	n	Power-law stress exponent
t _r	Time to rupture	Q	Activation energy
t _𝜀	Time to a given strain	R	Gas constant

^∗Standard expositions of the Wilshire model do not specify whether true or engineering values of stress and strain are considered. Most papers fit the rupture life equation to constant load data. However, Abdallah et al. [2] use constant load rupture data for $2\frac{1}{4}$ CrMo steel but constant stress creep strain data for the IMI 834 titanium alloy. The model derivation in this paper uses true values, with uniform deformation assumed. These are converted to engineering values where needed for comparison with experimental data.

Calibration to test data is achieved by linearisation. Thus, for example, Eq. (1a) becomes: $\begin{eqnarray}\displaystyle \ln (t_{r})=\frac{1}{u^{\prime }}\ln \left(\frac{1}{k_{1}}\right)+\frac{1}{u^{\prime }}\ln \left(-\ln \left(\frac{{\sigma}}{{\sigma}_{\text{UTS}}}\right)\right)+\left(\frac{Q}{RT}\right) & & \displaystyle\end{eqnarray}$ (1d) Multiple linear regression of ln(t _r) against ln(−ln(𝜎∕𝜎_UTS)) and 1∕T generates values for 1∕u ^′,1∕k ₁ and Q∕R. Many authors perform this operation correctly. Some, however, are seduced by the implicit form of Eq. (1a) into regressing the independent variable ln(−ln(𝜎∕𝜎_UTS)) against the dependent variable ln(t _r). This returns incorrect values for the constants.

2.2. Suggested rearrangement

A rearrangement is suggested here; it introduces no essential change to the model but facilitates further analysis. If the constants k ₁, k ₂, k ₃ are brought, in reciprocal form, inside their associated [square-bracketed] terms, then their dimensionalities and normalising roles become clear, and their values can be meaningfully compared between materials. It is also helpful to reverse the sign of the exponent v ^′ to match the alternation in sign of Q.

Assuming the stress to be the initial stress, the equations become: $\begin{eqnarray}\displaystyle \frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\text{} & =\text{} & \displaystyle \exp \left(-\left[\frac{t_{r}}{k_{r}}\exp \left(\frac{-Q}{RT}\right)\right]^{u}\right)\end{eqnarray}$ (2a) $\begin{eqnarray}\displaystyle \frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\text{} & =\text{} & \displaystyle \exp \left(-\left[\frac{\dot{{\varepsilon}}_{\min }}{k_{\min }}\exp \left(\frac{Q}{RT}\right)\right]^{-v}\right)\end{eqnarray}$ (2b) $\begin{eqnarray}\displaystyle \frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\text{} & =\text{} & \displaystyle \exp \left(-\left[\frac{t_{{\varepsilon}}}{k_{{\varepsilon}}}\exp \left(\frac{-Q}{RT}\right)\right]^{w}\right)\end{eqnarray}$ (2c) The constants k _r, k _min, k _𝜀 are seen to have dimensions of time, time⁻¹, and time, respectively.

2.3. Import

Whilst this implicit form is elegant, the meaning and import of these equations become more apparent if they are rendered explicitly: $\begin{eqnarray}\displaystyle t_{r}\text{} & =\text{} & \displaystyle k_{r}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{1/u}\exp \left(\frac{Q}{RT}\right)\end{eqnarray}$ (3a) $\begin{eqnarray}\displaystyle \dot{{\varepsilon}}_{\min }\text{} & =\text{} & \displaystyle k_{\min }\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{-1/v}\exp \left(\frac{-Q}{RT}\right)\end{eqnarray}$ (3b) $\begin{eqnarray}\displaystyle t_{{\varepsilon}}\text{} & =\text{} & \displaystyle k_{{\varepsilon}}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{1/w}\exp \left(\frac{Q}{RT}\right)\end{eqnarray}$ (3c) Equations (3a), (3b) determine, for a specific material, the variation of rupture life and minimum creep rate with stress and temperature – they thus describe, and for calibration require, the results of multiple tests, or component histories. The desired dependent variable is expressed as the product of a constant term and apparently separable functions of stress and temperature. If k _𝜀 is interpreted as a constant, as are k _r and k _min, then Eq. (3c) provides an equivalent determination of the time to a given strain over a series of distinct tests or histories. However, if k _𝜀 is interpreted as a function of strain, as exemplified in Abdallah et al. [2], and thus more correctly written k (𝜀), then Eq. (3c) describes the evolution of strain with time through a single test, or history – that is, at a single stress and temperature. This strain function encodes the entire creep curve relative to a normalised time axis; the stress and temperature terms act as simple multiplicative factors, scaling the time axis to generate specific creep curves for specific conditions. Despite the identity of form, Eq. (3c) is thus of a different character to Eqs (3a), (3b). What is lacking is any definition of where this creep curve terminates, that is, a correlation with the rupture life equation and a definition of the ductility.

3. Compatibility

3.1. Minimum creep rate and time to rupture

Equating the right-hand sides of Eqs (2a), (2b) yields: $\begin{eqnarray}\displaystyle \left[\frac{t_{r}}{k_{r}}\exp \left(\frac{-Q}{RT}\right)\right]^{u}=\left[\frac{\dot{{\varepsilon}}_{\min }}{k_{\min }}\exp \left(\frac{Q}{RT}\right)\right]^{-v} & & \displaystyle\end{eqnarray}$ (4a) This does not conform to the simple Monkman–Grant relationship [21]; the product of minimum creep rate and time to rupture varies with both normalised stress and temperature.

For u = v, the temperature and stress dependencies disappear, and Eq. (4a) simplifies to: $\begin{eqnarray}\displaystyle t_{r}\dot{{\varepsilon}}_{\min }=k_{r}k_{\min } & & \displaystyle\end{eqnarray}$ (4b) which is the standard Monkman–Grant relationship with a logarithmic gradient of −1. The product k _r ⋅ k _min represents the intercepts of Eq. (4b) with the axes $\dot{{\varepsilon}}_{\min }∼=∼1,t_{r}∼=∼1$ .

Equations (2a), (2b) are thus seen to be compatible, in the sense that no inconsistencies arise from their combination. This conclusion carries back to Eqs (1a), (1b).

3.2. Times to given strains

Similarly, combination of Eq. (2a) with three instances of Eq. (2c) produces: $\begin{eqnarray}\displaystyle \frac{t_{{\varepsilon}(i)}}{k_{{\varepsilon}(i)}}=\frac{t_{{\varepsilon}(j)}}{k_{{\varepsilon}(j)}}=\ldots =\frac{t_{{\varepsilon}(r)}}{k_{{\varepsilon}(r)}}=\left(\frac{t_{r}}{k_{r}}\right)^{\frac{u}{w}}\left(\exp \left(\frac{-Q}{RT}\right)\right)^{\frac{u}{w}-1} & & \displaystyle\end{eqnarray}$ (5a) where 𝜀(i), 𝜀(j) are general strains and 𝜀(r) represents the rupture condition.

At rupture, t _𝜀(r) = t _r and k _𝜀(r) = k _r, requiring that u = w. Equation (5a) then becomes: $\begin{eqnarray}\displaystyle \frac{t_{{\varepsilon}(i)}}{k_{{\varepsilon}(i)}}=\frac{t_{{\varepsilon}(j)}}{k_{{\varepsilon}(j)}}=\ldots =\frac{t_{{\varepsilon}(r)}}{k_{{\varepsilon}(r)}}=\frac{t_{r}}{k_{r}} & & \displaystyle\end{eqnarray}$ (5b) This confirms the above prediction of creep curves which differ only by linear scaling of the time axis; it also implies a constant ductility. Accordingly, the minimum creep rate varies inversely with rupture life, complying with the simple Monkman–Grant relationship.

In the previous section it was shown that the first two Wilshire equations are generally compatible. Adding the third equation forces the restriction that u = v = w.

4. The creep process

Creep is a process, characterised by a rate equation (or a coupled set of rate equations for interacting processes). Its development depends on the characteristics of the material, and on the conditions (stresses, temperature…) and constraints (geometry, loading mode…) under which this process operates.

Physically, the creep process is manifest as a progressive increase in deformation (and in any other forms of degradation and damage) which, in certain loading modes, leads to failure. Any useful model of the creep process will be based on the necessary rate equations, integration of which will define the evolution of strain – and other forms of degradation and damage – with time. It will identify the minimum in creep rate and define the final failure point, i.e. the rupture life and ductility.

In their original form, the Wilshire equations do not describe the creep process, only certain of its results. The coherence and validity of the equation set would best be demonstrated by the development of a consistent creep rate equation. This would clarify the physical significance of the model and potentially enhance its applicability. Such a rate equation should conform to the essential characteristics of the Wilshire approach, which are:

No degradation or damage processes other than deformation are explicitly considered.

Stress is normalised to the ultimate tensile strength (UTS).

An Arrhenius [3,4] time-temperature relationship is assumed.

5. A consistent creep rate equation for the Wilshire model

5.1. Derivation of a creep rate equation

In a constant load tensile creep test, the instantaneous stress increases with strain. Combining this mechanical constraint with the above materials characteristics of the Wilshire model suggests that the simplest candidate creep rate equation is of the form: $\begin{eqnarray}\displaystyle \dot{{\varepsilon}}=f_{1}^{^{\prime }}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon})\right)\exp \left(\frac{-Q}{RT}\right) & & \displaystyle\end{eqnarray}$ (6a) which may conveniently be inverted and then integrated to give: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle f_{2}^{^{\prime }}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon})\right)\dot{{\varepsilon}}=\exp \left(\frac{-Q}{RT}\right)\end{eqnarray}$ (6b) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle f_{2}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon})\right)=\exp \left(\frac{-Q}{RT}\right)t+K\end{eqnarray}$ (6c) The constant of integration is obtained from the initial condition 𝜀 = 0 at t = 0: $\begin{eqnarray}\displaystyle f_{2}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)=K & & \displaystyle\end{eqnarray}$ (6d) Substituting the final condition 𝜀 = 𝜀_r at t = t _r then gives: $\begin{eqnarray}\displaystyle f_{2}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon}_{r})\right)-f_{2}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)=\exp \left(\frac{-Q}{RT}\right)t_{r} & & \displaystyle\end{eqnarray}$ (6e) The natural choice of a failure criterion consistent with the Wilshire model is for rupture to occur when the instantaneous stress reaches the UTS [20,25], thus: $\begin{eqnarray}\displaystyle {\sigma}_{0}\exp ({\varepsilon}_{r})={\sigma}_{\text{UTS}} & & \displaystyle\end{eqnarray}$ (7a) This is consistent with the limit case of Eqs (1a), (2a); i.e. t _r = 0 at 𝜎₀ = 𝜎_UTS. Substitution into the first term of Eq. (6e) yields: $\begin{eqnarray}\displaystyle f_{2}(1)-f_{2}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)=\exp \left(\frac{-Q}{RT}\right)t_{r} & & \displaystyle\end{eqnarray}$ (7b) Comparison with the Wilshire equations requires inversion of Eq. (2a): $\begin{eqnarray}\displaystyle k_{r}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{1/u}=\exp \left(\frac{-Q}{RT}\right)t_{r} & & \displaystyle\end{eqnarray}$ (8a) This defines the form of the function f ₂(x) such that Eq. (7b) becomes: $\begin{eqnarray}\displaystyle -f_{2}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)=k_{r}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{1/u}=\exp \left(\frac{-Q}{RT}\right)t_{r} & & \displaystyle\end{eqnarray}$ (8b) with the incidental result that f ₂(1) = 0.

Equations (6b), (6a) can then be particularised as: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle f_{2}^{^{\prime }}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon})\right)\dot{{\varepsilon}}=\frac{k_{r}}{u}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon})\right)\right\}^{\left(\frac{1-u}{u}\right)}\dot{{\varepsilon}}=\exp \left(\frac{-Q}{RT}\right)\end{eqnarray}$ (9a) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle \dot{{\varepsilon}}=f_{1}^{^{\prime }}\left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon})\right)\exp \left(\frac{-Q}{RT}\right)=\frac{u}{k_{r}}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon})\right)\right\}^{\frac{u-1}{u}}\exp \left(\frac{-Q}{RT}\right)\end{eqnarray}$ (9b) Equation (9b) provides the desired creep rate equation, fully consistent with the original Wilshire model. The creep rate is seen to increase monotonically with strain, so, in conventional terms, this represents a secondary-tertiary creep model with no primary element. The minimum creep rate occurs at zero strain (and zero time) and is given by: $\begin{eqnarray}\displaystyle \dot{{\varepsilon}}_{\min }=\frac{u}{k_{r}}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{\frac{u-1}{u}}\exp \left(\frac{-Q}{RT}\right) & & \displaystyle\end{eqnarray}$ (10a) This can be rearranged to match the implicit form of Eq. (2b): $\begin{eqnarray}\displaystyle \frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}=\exp \left(-\left[\frac{k_{r}}{u}\dot{{\varepsilon}}_{\min }\exp \left(\frac{Q}{RT}\right)\right]^{\frac{u}{u-1}}\right) & & \displaystyle\end{eqnarray}$ (10b) From this, the constants in Eqs (2a), (2b) are seen to relate as follows: $\begin{eqnarray}\displaystyle v=\frac{u}{1-u}\quad \text{and}\quad k_{\min }=\frac{u}{k_{r}} & & \displaystyle\end{eqnarray}$ (10c) Thus the number of model-specific constants required has naturally reduced from six to two: u, which is a dimensionless form factor defining the shape of the creep curve, and k _r, which is a scale factor with dimension of time. The activation energy, Q, is common to most creep models, rather than specific to the Wilshire equations.

This relationship between u and v does not conflict with the general compatibility of Eqs (2a), (2b) demonstrated earlier. However, it does imply that Eq. (2c) (and therefore Eq. (1c)) cannot be correct, as that would require the contradictory relation u = v = w. Given the identification of f ₂(x), Eqs (6c), (6d) together yield the correct expression for the time to a given strain: $\begin{eqnarray}\displaystyle k_{r}\left(\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{1/u}-\left\{-\ln \left(\frac{{\sigma}_{0}\exp ({\varepsilon})}{{\sigma}_{\text{UTS}}}\right)\right\}^{1/u}\right)\exp \left(\frac{Q}{RT}\right)=t_{{\varepsilon}} & & \displaystyle\end{eqnarray}$ (11a) This relates time to strain explicitly, rather than via a series of time factors, k _𝜀(i) or an independently determined function k (𝜀). It has the typical form: $\begin{eqnarray}\displaystyle f(\text{Current state})=f(\text{Initial state})+\text{Rate}\times \text{Time} & & \displaystyle\end{eqnarray}$ (11b) This structure shows why the construction of Eq. (1c) by analogy with Eqs (1a), (1b) cannot be correct and why Eq. (11a) cannot be reduced to the same implicit form as Eq. (2c).

5.2. Summary of the Wilshire model with consistent creep rate equation

The original Wilshire model has been developed here by the derivation of a consistent creep rate equation. This leads naturally to the original implicit expressions for rupture life and minimum creep rate, to explicit versions of these, and to a correct strain-time relationship. Thus, in summary: $\begin{eqnarray}\displaystyle \dot{{\varepsilon}}\text{} & =\text{} & \displaystyle \frac{u}{k_{r}}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon})\right)\right\}^{\frac{u-1}{u}}\exp \left(\frac{-Q}{RT}\right)\end{eqnarray}$ (12a) $\begin{eqnarray}\displaystyle \frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\text{} & =\text{} & \displaystyle \exp \left(-\left[\frac{t_{r}}{k_{r}}\exp \left(\frac{-Q}{RT}\right)\right]^{u}\right)\end{eqnarray}$ (12b) $\begin{eqnarray}\displaystyle \frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\text{} & =\text{} & \displaystyle \exp \left(-\left[\frac{k_{r}}{u}\dot{{\varepsilon}}_{\min }\exp \left(\frac{Q}{RT}\right)\right]^{\frac{u}{u-1}}\right)\end{eqnarray}$ (12c) $\begin{eqnarray}\displaystyle t_{r}\text{} & =\text{} & \displaystyle k_{r}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{1/u}\exp \left(\frac{Q}{RT}\right)\end{eqnarray}$ (12d) $\begin{eqnarray}\displaystyle \dot{{\varepsilon}}_{\min }\text{} & =\text{} & \displaystyle \frac{u}{k_{r}}\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{\frac{u-1}{u}}\exp \left(\frac{-Q}{RT}\right)\end{eqnarray}$ (12e) $\begin{eqnarray}\displaystyle t_{{\varepsilon}}\text{} & =\text{} & \displaystyle k_{r}\left(\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\}^{1/u}-\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\exp ({\varepsilon})\right)\right\}^{1/u}\right)\exp \left(\frac{Q}{RT}\right)\end{eqnarray}$ (12f) The only additional assumption introduced to this derivation is the rupture criterion: $\begin{eqnarray}\displaystyle {\varepsilon}_{r}=\ln \left(\frac{{\sigma}_{\text{UTS}}}{{\sigma}_{0}}\right) & & \displaystyle\end{eqnarray}$ (12g) Equations (12d), (12e) combine to define a stress-dependent Monkman–Grant product: $\begin{eqnarray}\displaystyle \dot{{\varepsilon}}_{\min }t_{r}=u\left\{-\ln \left(\frac{{\sigma}_{0}}{{\sigma}_{\text{UTS}}}\right)\right\} & & \displaystyle\end{eqnarray}$ (12h) and, from Eq. (12g), (12h), it is seen that the ductility ratio is constant: $\begin{eqnarray}\displaystyle \frac{{\varepsilon}_{r}}{\dot{{\varepsilon}}_{\min }t_{r}}=\frac{1}{u} & & \displaystyle\end{eqnarray}$ (12i) This corresponds to the relationship proposed by Dunand et al. [13].

It is interesting to observe that, in terms of engineering strain, the rupture ductility is: $\begin{eqnarray}\displaystyle e_{r}=\left(\frac{{\sigma}_{\text{UTS}}}{{\sigma}_{0}}\right)-1 & & \displaystyle\end{eqnarray}$ (12j)

6. Illustration

It is believed that the above derivation of a Wilshire creep rate equation is rigorous and the solution unique. The purpose of this section is not to produce a definitive fit to data but to illustrate the characteristics of the Wilshire equations, particularly as revealed by this development.

6.1. Material

To allow comparison with earlier work, the material selected for much of this illustration is the $2\frac{1}{4}$ CrMo steel tube MAF. Tensile, creep and rupture data for this are contained in the NIMS Creep Data Sheets 30B, 50A [22,23], which provide full background details. Photographs in Sheet 30B show two features of interest: marked microstructural evolution, specifically spheroidization of the carbide precipitates, and significant loss of section by oxidation, particularly at longer times and higher temperatures. The former is inextricably linked to the creep process and has been investigated extensively by the present author and colleagues [8,10]. The latter is, in principle, separable and has also been studied [9]. Fitting parametric relationships is often complicated by inhomogeneous distribution of the data, and this has been discussed, with particular reference to the NIMS data, [5]. For these reasons, caution must be exercised when interpreting the rupture data for tube MAF.

6.2. The original Wilshire model – rupture lives

Several applications of the original Wilshire equations to $2\frac{1}{4}$ CrMo steel rupture data have been published [2,16,32]. It is observed that the data do not conform, over the full stress range, to the linearisation given in Eq. (1d); this seems equally the case for other engineering alloys. Faced with such divergence between a model and observation, the possible courses of action are:

to identify the causes of the divergence and to modify, enrich, or replace the model to address these; or,

to resort to empirical adjustments.

To date, the latter course has been the majority choice.

Evans [16] demonstrates the effect of partitioning the MAF rupture data. Figures 1–3 show plots of rupture life, transposed to a datum temperature of 550 °C, against the normalised stress for the original Wilshire model with zero, one and two partitions. Temperature transposition uses the Q values given by Evans and the model curves employ his values for k ₁ and u ^′. It should be noted that his fits are to data points with actual life less than 10,000 h, as he wished to demonstrate extrapolation to longer times. Fitting without partitioning (Fig. 1) produces a poor result; improvement is obtained by partitioning once (Fig. 2), then twice (Fig. 3), but at the expense of creating discontinuities in the overall rupture curve and of multiplying fitting constants. It has been suggested by Evans [16] and others [27,32] that the partitions coincide with changes in creep mechanism; however, no quantitative correlations are made, nor are any functional relationships suggested.

Fig. 1.

Original Wilshire model – Tube MAF – no partitions, after Evans [16].

Fig. 2.

Original Wilshire model – Tube MAF – one partition, after Evans [16].

In this context, it must be noted that the partitions in Fig. 3 are of opposite character. The lower one, at a normalised stress value of 0.2, is consistent with the usual expectation that on either side of a mechanism boundary it is the process with the lower activation (stress or temperature) that predominates. The upper partition, at a normalised stress just below 0.5, is quite contrary (as is its near equivalent, the single partition in Fig. 2). Either side of this, the predominant process is that with the higher activation, slower rate, and longer life. Such behaviour is thermodynamically improbable. This partition cannot, therefore, represent a transition between physically plausible mechanisms.

Fig. 3.

Original Wilshire model – Tube MAF – two partitions, after Evans [16].

The partitioning method is thus cumbersome empirically and unsustainable physically.

Evans concludes by introducing a non-algebraic fitting method. This generates a smooth curve but – being non-algebraic – it is difficult to see how this result might be applied in engineering practice.

This need for piecewise fitting indicates that the functional form of the original Wilshire equations does not adequately reflect the true rupture behaviour over the stress and temperature range of the data. But a more fundamental issue is apparent. Most noticeable in Fig. 1, but persistent even with the improved fits on partitioning, is the temperature stratification of the datapoints. This indicates that the assumption of a simple Arrhenius relationship is inadequate. The problem would seem to lie in the stress normalisation. Usual practice has been to treat the UTS as a temperature dependent constant, transforming the stress values prior to data fitting. This conceals the temperature dependence of the UTS, which should be incorporated explicitly in the Wilshire equations if a true optimum fit is to be obtained. Figure 4 shows the tensile data for tube MAF [22]; they do not conform to a simple Arrhenius relationship (neither in general, nor particularly in the creep range).

Fig. 4.

Tensile data for tube MAF, from NIMS Creep Data Sheet 3B.

A practical consequence of this stratification is that it complicates any curve fitting method that treats the data as a homogeneous set. It is notable that the partitioning of the data proposed by Evans coincides more-or-less exactly with stratification steps, particularly the bounds of the data at 450 and 550 °C.

In contrast, a simple Arrhenius time-temperature relationship (equivalent to the Orr-Sherby-Dorn parameter [24]) yields a close, unstratified correlation with stress, as seen in Fig. 5. Apparently high values for Q and n result, but these can be accounted for by considering primary creep effects [6,7].

Fig. 5.

Arrhenius temperature dependence of tube MAF stress-rupture data [22].

Whilst it must be agreed that the UTS constitutes a real limit and that no creep model should predict positive lives at stresses above it, it seems that normalisation of stress by the UTS is not the ideal way to model this limit. It introduces an additional temperature dependence into the equations, which confounds the assumptions underlying the fitting methods.

Figure 5 shows no separation of the data by temperature at high stresses, up to 0.8 of the UTS, indicating that the influence of the UTS limit is more local than that implied by the Wilshire normalisation. (The separation at low stress is fully accounted for by the oxidation effects noted above.)

6.3. The original Wilshire model – creep curves

The NIMS data for tube MAF do not include full creep curves. Accordingly, Abdallah et al. [2] illustrate Eq. (1c) of the original Wilshire model using data from constant stress tests on the titanium alloy IMI 834. They report, for ‘high’ and ‘low’ stress regimes respectively, the following values of w ^′ and corresponding dependencies of k ₃ on strain: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle w^{\prime }=0.8\quad k_{3}=3\times 10^{9}{\varepsilon}^{-1}+2\times 10^{10}\end{eqnarray}$ (13a) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle w^{\prime }=0.21\quad k_{3}=3.815484{\varepsilon}^{-0.8}+350\end{eqnarray}$ (13b) The units of k ₃ are seconds^{−w
^′}. The time axis of their Fig. 16 is clearly mis-printed, but can be corrected from Fig. 8 of the related work [1] in order to estimate a value for Q∕R.

Figure 6 shows creep curves calculated using these parameters at 923 K and 3 values of normalised stress. As expected, a constant curve shape is seen for each parameter set. This inevitably results in a discontinuity in behaviour at the partition between high and low normalised stresses. The transition point is not defined in the source reference, but analysis of the minimum creep rates suggests a normalised stress value of 0.69. Equations (13a), (13b) give no definition of the terminal point, either in terms of life or of ductility.

Fig. 6.

Predicted creep curves for IMI 834 at 923 K, after Abdallah et al. [2], Fig. 16.

Whilst equations like Eqs (13a), (13b) can be developed empirically – in conjunction with Eq. (1c) – to fit any experimental creep curve to an arbitrary level of precision (an equation of the same form is applied to a Nickel-based superalloy by Harrison et al. [19]), such fits do not convey physical meaning. In this instance there is nothing to explain the small, but significant, primary, nor the extensive tertiary – which cannot result from loss of section as these were constant stress tests. Finally, it must be noted that substitution of Eqs (13a), (13b) into (1c) and differentiating to determine the minimum creep rate does not yield expressions consistent with Eq. (1b). This finding is not unexpected given the earlier analysis.

There have been several attempts to address these issues. In Eq. (1c) Evans and Williams [17] normalise the strain to the observed ductility. This prevents the model predicting a constant ductility, but at the expense of requiring a further experimental input for every individual condition, rather than allowing the ductility to arise naturally from the model formulation. They constrain Eqs. (1a), (1b) by imposing a Monkman–Grant relation (or similar) and attempt to resolve the mismatch between Eqs (1c) and (1a) by replacing Eqs (13a), (13b) with forms where k ₃ → k ₁ as t → t _r. The results are not altogether satisfactory.

Gray and Whittaker [18] depart from the Wilshire model; they retain the stress normalisation, but work with the normalised stress directly, rather than its logarithm. Nonetheless, this remains an empirical treatment.

Cano [11] considers the difficulties experienced when fitting creep curves to Eq. (1c) other than by making k ₃ a purely empirical function of strain. His solution is to consider Eq. (1b) as a general rate equation and to derive Eq. (1a) from it by assuming a Monkman–Grant relationship. He adds a damage term, derived from a hyperbolic-sine continuum damage formulation. Integration of this leads to an expression replacing Eq. (1c). This hybrid approach meets with some success but is not fully self-consistent.

6.4. The developed Wilshire model – creep curves

To explore the developed model, data for tube MAF were considered. Optimisation was performed over both rupture life and creep rate data, in the range between Evans’s two partitions. This yielded a slightly higher value for Q than that obtained by Evans solely on rupture data below 10,000 h (253,600 against 223,700 kJ/mol). Values for the constants u, k _r were established from each of Eqs (12d), (12e) independently (Table 2).

Table 2
Values for the constants in the developed Wilshire model, for tube MAF

From rupture lives From minimum creep rates

Form factor, u 0.1690 0.1466

Scale factor, k _r , hours 1.088 × 10⁻⁹ 3.858 × 10⁻⁹

Activation energy, Q, kJ/mol 253,600

	From rupture lives	From minimum creep rates
Form factor, u	0.1690	0.1466
Scale factor, k _r , hours	1.088 × 10⁻⁹	3.858 × 10⁻⁹
Activation energy, Q, kJ/mol	253,600

Fig. 7.

Predicted creep rates for tube MAF at 550 °C – developed Wilshire model.

Fig. 8.

Predicted and observed engineering creep rates for tube MAF at 550 °C.

Fig. 9.

Predicted creep curves for tube MAF at 550 °C – developed Wilshire model.

Figure 7 shows, on logarithmic axes, the dependence of creep rate on strain described by Eq. (12a). Curves are plotted for each of the stresses at which tests were performed on this material at 550 °C; they are identified by their stress to UTS ratio. The variation in ductility with normalised stress is apparent, as is the gentle evolution of curve shape. A near-horizontal line shows, for comparison, the strain rate at which the UTS data were obtained [22]. Figure 8 shows the low strain area of Fig. 7, with the experimental data added (simple curves: model; curves with datapoints: experiment). Observed behaviour includes a marked primary, the extent of which is stress dependent, as discussed by Sakamoto et al. [26]. Accordingly, the minimum creep rate does not occur at zero time. These features are not reflected in the Wilshire model. Figures 9, 10 show the corresponding strain-time curves, Eq. (12f), compared with the experimental data. The same effects are seen, as is the difference in ductility trend between model and observation. This difference results from the action of degradation and damage processes additional to the simple strain accumulation represented in the Wilshire model.

Fig. 10.

Predicted and observed engineering creep curves for tube MAF at 550 °C.

Fig. 11.

Predicted and observed rupture lives for tube MAF – developed Wilshire model.

Fig. 12.

Predicted and observed creep rates for tube MAF – developed Wilshire model.

Fig. 13.

Predicted and observed ductilities for tube MAF – developed Wilshire model.

6.5. The developed Wilshire model – rupture life, minimum creep rate, ductility

Figures 11, 12 show the experimental rupture lives and minimum creep rates for tube MAF [22,23], together with the model predictions. In each case two curves are given, one using the constants obtained from fitting the rupture lives, the other using those obtained from the minimum creep rates (Table 2). In each case, better agreement is shown by the former. This reflects that the set of rupture lives is larger and more widely distributed than that of creep rates. For the same reason, the fit to the rupture data is better than that to the minimum creep rates. The latter diverge widely at lower stresses, due to the oxidation effects at 600 and 650 °C noted earlier.

Figure 13 shows the corresponding ductility and Monkman–Grant product data (respectively right and left of the broken diagonal line), together with the modelled relationships. Equations (12g), (12h) predict that both decrease with increasing normalised stress, with no temperature effect. Actual data display more complex behaviour. Ductility passes through a maximum, while the trend in Monkman–Grant product is sigmoidal. Both increase with temperature. These phenomena are more evident when plotted against stress without normalisation to the UTS, Fig. 14. The ductility maximum occurs at around 185 MPa, as does the trough in Monkman–Grant product. This pattern of behaviour is close to that predicted by a primary-modified continuum damage model [7].

Fig. 14.

Predicted and observed ductilities for tube MAF – simple stress dependence.

7. Concluding observations

Two key issues have been identified.

First, the original Wilshire equations are not mutually compatible; those for rupture life and minimum creep rate are inconsistent with that for times to strain. The diverse attempts to model complete creep curves exemplify the consequences of this.

Second, whilst a consistent creep rate equation can be derived – and from it a correct strain-time relationship, neither it nor the rupture life and minimum creep rate equations are particularly successful at describing observed behaviour without ad hoc empirical modification. The fundamental form of the original equations – a power-law in the logarithm of normalised stress – must, therefore, be reconsidered. Also, other terms should be added, to allow realistic inclusion of primary and tertiary creep processes.

That these issues have persisted through numerous applications of the Wilshire equations seems to be a consequence of the lack of formal analysis of the model.

7.1. Description, or explanation?

Whilst models might be based on deductions from first principles or from existing, accepted treatments of related processes, it is more common that they arise by inference from observed behaviour. For practical purposes, a simple description of the data may suffice, but for explanatory or interpretative purposes a physical significance must be carried both by the form of the model and by the values of its required constants. ‘Physical’ implies at least a phenomenological, if not a mechanistic, basis.

Application and development of the Wilshire model has largely emphasised its descriptive ability. Inferences as to mechanistic significance have been qualitative, with no formalisation. As has been shown here, suggestions made as to mechanism boundaries have not always been physically sound.

It is possible that development might continue in this direction, the approach serving simply as a more elaborate parametric description of rupture, minimum creep rate or strain-time behaviour, adjusting as required to improve the fit to data and without concern as to physical realism or internal consistency. However, there has been no widespread take-up of the Wilshire equations for this purpose; there is no real sign that they might displace the already numerous stress rupture parameters. Claims as to improvement over ‘conventionally employed creep lifing methods [2]’ appear exaggerated. It is telling that that reference does not consider the parametric model for tube MAF included in the NIMS Data Sheet [22] but rather takes an unsourced stress function as a basis for its ‘critical review’. The pathological mismatch between data and model demonstrated in that paper’s Fig. 3 arises entirely from this arbitrary choice and an error in the curve-fitting algorithm employed; it does not reflect on the conventional method at all.

A more productive future for the Wilshire approach is likely to lie in a reconsideration of its physical basis.

7.2. Creep and the ultimate tensile strength

A key characteristic of the Wilshire approach is normalisation of the stress to the UTS. However, this does not appear to achieve quite the desired effect. It is not a new concept; normalisation to modulus and to hardness [10] have been proposed previously. Modulus might be considered an invariant boundary condition, in the sense that it does not change with the processes that influence creep. Hardness, on the other hand, is expected to change through creep life due to thermal and strain effects.

The UTS lies between these; at low temperatures it parallels the modulus; as temperature rises, thermally activated processes cause it to fall in a manner that is strain-rate dependent. Which then should be the normalising parameter – the modulus or the rate-dependent tensile test result? The former seems the more stable option. In practice, the high-temperature UTS is a result of creep-related plasticity processes, rather than a more fundamental property. Normalisation to this effectively introduces a circular argument. It also introduces a further temperature dependence which leads to a cross-correlation between the stress and temperature terms of Eqs (1a)--(1c), (2a)--(2c), which are thus not separable, as they first appear.

Further, the UTS is an engineering stress; it would be more correct to consider the corresponding true value, as it is true stress which controls creep rate.

7.3. Essential Wilshire characteristics

The following are considered essential features of the Wilshire approach, which are of value, and which should be retained in future creep studies.

A limiting stress at which ‘time dependent’ creep plasticity is circumscribed by ‘instantaneous’ plasticity. However, is this more realistically addressed by a Wilshire normalisation or by a Rosenblum boundary condition [25]? Is this limit effectively invariant, or does it evolve with strain and time at temperature?

The power law is simple, and often a good approximation, but it is not the only possibility.

The Arrhenius temperature dependence [3,4] remains an empirical relation and potentially open to improvement. However, no specific issues arise from its use in the Wilshire model other than the interaction with the temperature dependence of the normalising stress.

Footnotes

Acknowledgements

The author recalls, with gratitude and pleasure, numerous conversations on creep with the late Brian Wilshire – invariably accompanied by his generous hospitality. Thanks are due to Mark Whittaker, Veronica Gray, Koichi Yagi, and John Williamson for many thought-provoking discussions on this and related subjects.

Conflict of interest

None to report.

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A perspective on the Wilshire creep equations

Abstract

Keywords

1. Introduction

2. Formulation

2.1. Original form

3.1. Minimum creep rate and time to rupture

5. A consistent creep rate equation for the Wilshire model

5.1. Derivation of a creep rate equation

6.1. Material

6.2. The original Wilshire model – rupture lives

Table 2 Values for the constants in the developed Wilshire model, for tube MAF From rupture lives From minimum creep rates Form factor, u 0.1690 0.1466 Scale factor, k r , hours 1.088 × 10−9 3.858 × 10−9 Activation energy, Q, kJ/mol 253,600

7.1. Description, or explanation?

7.2. Creep and the ultimate tensile strength

7.3. Essential Wilshire characteristics

Footnotes

Acknowledgements

Conflict of interest

References

Table 2
Values for the constants in the developed Wilshire model, for tube MAF

From rupture lives From minimum creep rates

Form factor, u 0.1690 0.1466

Scale factor, k _r , hours 1.088 × 10⁻⁹ 3.858 × 10⁻⁹

Activation energy, Q, kJ/mol 253,600