Continuous damage parameter calculation under thermo-mechanical random loading

Graphical abstract

The paper presents a method on how the mean stress effect on fatigue damage can be taken into account under an arbitrary low cycle thermo-mechanical loading. From known stress, elastoplastic strain and temperature histories the cycle amplitudes and cycle mean values are extracted and the damage parameter is computed. In contrast to the existing methods the proposed method enables continuous damage parameter computation without the need of waiting for the cycles to close. The limitations of the standardized damage parameters are thus surpassed. The damage parameters derived initially for closed and isothermal cycles assuming that the elastoplastic stress-strain response follows the Masing and memory rules can now be used to take the mean stress effect into account under an arbitrary low cycle thermo-mechanical loading. The method includes: stress and elastoplastic strain history transformation into the corresponding amplitude and mean values; stress and elastoplastic strain amplitude and mean value transformation into the damage parameter amplitude history; damage parameter amplitude history transformation into the damage parameter history.

Introduction
It is well known that mean stress affects fatigue life significantly and can therefore not be neglected under low cycle isothermal mechanical loading [1][2][3][4]. The mean stress correction formulae [5][6][7] are applied to transfer the extracted closed cycles with known stress amplitudes s a , mean stresses s m and elastoplastic strain amplitudes e ep a into the closed cycles with equivalent damage parameter amplitudes P a . Moreover, closed cycles can be extracted by, e.g., the rainflow counting method [8].
However, under low cycle non-isothermal mechanical loading a cycle closure problem may appear due to variable temperature or strain rate or both, which results in a more difficult determination of the stress and strain amplitudes and mean stresses [3]. The cycle closure problem leads to a more difficult determination of s a , s m and e ep a because the cycle counting methods [8] cannot count the cycles before they close. This is particularly important if damage is calculated continuously (at any moment without the need of 'waiting' for the cycle to finish), which is the case in the damage operator approach (DOA) [4].
The aim of the method is to extend the usage of damage parameters derived initially for closed cycles and isothermal mechanical loading to arbitrary cycles and non-isothermal mechanical loading.

Requirements
The stress and strain tensor histories are gained by elastoplastic models from structural finite element analyses (FEA) and are converted into equivalent uniaxial stress s(t i ) and equivalent uniaxial elastoplastic strain e ep ðt i Þ histories for i = 1, . . ., n. Thermal FEA are required to assess the corresponding temperature history T(t i ). Test stand tests can replace FEA. Temperature history T(t i ) influences the elastoplastic stress-strain response and material parameters, e.g., Young modulus E(t i ) = E(T i ), where T i = T(t i ) but does not appear in the algorithm directly. . Logical operator s enables the identification of rainflow cycles that hold nested rainflow cycles. It is set to true if the rainflow cycle is identified and j > 2. Otherwise it is false.

Algorithm flow
The procedure of working out the damage parameter is given in Fig. 1. Here an overview of the pseudo code is provided. In line 1, counters i, j and maximum absolute strain e ep max are initiated. Superscript ep stands for elastoplastic. Next, the outer loop begins. Index i runs over n available times t i . If the number of strain reversal points in the residuum j > 2, the algorithm in line 3 first checks if residuum strains e ep;res jÀ2 ; e ep;res jÀ1 , e ep;res j and strain e ep ðt i Þ form a rainflow [8] cycle. The three residuum strains representing strain reversal points as well as e ep ðt i Þ are required for the standardized four point rainflow counting algorithm [8]. The Clormann-Seeger [9] cycle is searched for in line 11. If and mean stress referring to time t i are calculated. Similarly, mean strain is given by Maximum stress is obtained in line 21 as follows Otherwise, P a ðt i Þ ¼ 0 in line 25. Damage parameter in line 27 referring to time t i is then

Example
For simplicity reasons and to enable the point-to-point validation of the pseudo code, the isothermal linear elastic stress-strain response is studied.
Let elastoplastic strain e ep ðt i Þ, stress sðt i Þ ¼ Ee ep ðt i Þ, where E = 210,000 MPa, and temperature Tðt i Þ ¼ 293 K histories be known (see Fig. 2 and Table 1). From e ep ðt i Þ history one Clormann-Seeger cycle (strain reversal points for the four point cycle counting algorithm at times t 1 , t 5 , t 11 , t 25 ) and three rainflow cycles can be identified. The first appears within the Clormann-Seeger cycle (t 11 , t 15 , t 18 , t 25 ), the second within times t 25 , t 29 , t 31 , t 39 and the third compressive cycle within t 31 , t 39 , t 42 , t 48 . The memory [5] M1 rule applies at time t 23 , memory M2 rule at t 21 , t 33 and at t 45 and memory M3 rule at t 47 . As temperature is kept constant in this simple case, the strain, stress and damage parameter reversal points coincide.
Unlike the existing strain life approaches, where amplitude and mean values cannot be determined before cycle closure at times t 21 , t 23 , t 33 and t 45 , the presented algorithm enables the determination of all load parameters at any time t i . It can be noted that the third compressive cycle starting at t 39 results in P a ðt 41 Þ ¼ P a ðt 42 Þ ¼ P a ðt 43 Þ ¼ P a ðt 44 Þ ¼ 0, which is correct as P a ðt i Þ 6 ¼ 0 only if s max ðt i Þ > 0 (see line 22 in Fig. 1). The complete calculation is listed in Table 1.
Although damage parameter P(t i ) is not a monotonic function, it still results in monotonically increasing fatigue damage D f (t i ) as it is expressed in the form of total variation introduced in Ref. [10]. Since the damage parameter history is now known, fatigue damage D f (t i ) can be calculated as explained in detail in Section 5.3 of Ref. [11].

Results
The strain-life approach is standardized and widely accepted for determining fatigue damage under strain-controlled low cycle fatigue loading. It is a standard practise to use damage parameters to account for the mean stress correction here. However, to perform the conversion of the load parameters into the damage parameter, temperature should be kept constant.
The presented method generalizes the damage parameters by continuous calculation of the stress and strain amplitude and mean stress instead of waiting for the cycles to close. Not only the discussed Smith-Watson-Topper damage parameter but also other parameters [5,7] can now be used identically in the strain-life predictions under thermo-mechanical cyclic loading. The method will enable an application of widely used strain-life mean stress correction techniques in energy based approaches such as in Refs. [12][13][14].

Background
Traditionally, after identifying a closed cycle the value of damage parameter is computed. There are several definitions of damage parameters, such as Morrow, Smith-Watson-Topper (SWT) or Nihei [5] that have been derived for closed and isothermal cycles assuming that the elastoplastic stress-strain response follows the Masing [15] and memory [5] rules. In this paper the well-known and widely used Smith-Watson-Topper [6] damage parameter is addressed. Other damage parameters [7] could be considered equally. The SWT damage parameter for a closed cycle is given as follows The corresponding SWT damage parameter amplitude at arbitrary times 0 t 1 t 2 Á Á Á t i Á Á Á is given by [  points increases or decreases monotonically. Consequently, stress and strain reversal points coincide [5]. An elastoplastic stress-strain response reflecting an arbitrary thermo-mechanical loading is sketched in Fig. 3. The stress reversal points in the black circles and strain reversal points in the white ones do not coincide. This comes from variable temperature.
The existing strain-life approaches have been derived for stress-strain responses with coincidental stress and strain reversal points for which s a , s m and e ep a can be determined in a simple and straightforward way [8]. However, for the elastoplastic stress-strain response as depicted in Fig. 3 it has not been possible to uniquely identify the three values yet. If elastoplastic stress-strain response is interpreted in terms of dissipated energy where the integration boundaries are the strain reversal points