**7.2 Application of energetic approaches**

An advantage of energy criteria is the significant reduction of parameters in comparison to total strain life curve approaches according to Manson-Coffin-Basquin (Manson, 1954; Coffin, 1954; Basquin, 1910), as energy criteria are able to describe several influences due to the interaction of stress and strain variables. Energy criteria are representative for the cyclic behaviour of materials. They are sound damage indicators which are linked to macroscopic crack initiation and allow for a generalisation to multiaxial loading. The input parameters

Most models for TMF fatigue life calculation are based on linear damage accumulation

Nonlinear cumulative approaches mainly come from Chaboche and Lemaitre (Chaboche &

The methods of calculation are mostly based upon local loading parameters such as stress, strain and temperature, which are calculated in complex components by means of elaborate material models. Thus the classification of models describing the fatigue life behaviour under TMF and LCF loading shows the same subdivision as the models describing the cyclic

Empirical models represent the major group of this classification. With these models, usually the fatigue life and the parameters of the load cycles are linked. These models are

Furthermore they are effective only for a quite narrow load spectrum as there is no distinction between the individual effects. The empirical models can be set up for various levels of complexity and are divided into approaches based on strain life curves and damage parameters, methods describing creep damage, energy-based approaches, and approaches

Damage mechanics models usually describe the development of damage by means of methods from continuum mechanics. The origin of these models is found with Kachanov (Kachanov, 1986) and Rabotnov (Rabotnov, 1969) who were concerned with creep damage. Damage mechanics approaches see damage being caused by creep and plastification. As the damage rates are linked to the current damage value, damage accumulation is nonlinear

In general the physically based models play a minor role for practical application, which is due to their complexity and the difficulties in determining their input parameters experimentally. They attempt to characterise the damage development on the basis of atom, vacancy and dislocation movement. At the current state of knowledge and application, physically based models for fatigue life computations are primarily relevant for depicting

Fracture mechanics models are linked to the local plastic strains at the crack tip, which can be described, for example, by a Δ*J* integral or a modified Δ*J* integral, respectively. A link to

An advantage of energy criteria is the significant reduction of parameters in comparison to total strain life curve approaches according to Manson-Coffin-Basquin (Manson, 1954; Coffin, 1954; Basquin, 1910), as energy criteria are able to describe several influences due to the interaction of stress and strain variables. Energy criteria are representative for the cyclic behaviour of materials. They are sound damage indicators which are linked to macroscopic crack initiation and allow for a generalisation to multiaxial loading. The input parameters

the physical background of empirical and damage mechanics methods, respectively.

the physically based models exists with models for micro-crack propagation.

according to Palmgren-Miner (Palmgren, 1924; Miner, 1945).

deformation behaviour and shall be briefly explained in the following.

and requires the damage variable to be integrated cycle by cycle.

simple, however the partly lacking physical interpretability is a disadvantage.

Lesne, 1988; Lemaitre & Chaboche, 1985).

**Empirical models** 

for partial damage accumulation. **Damage mechanics models** 

**Physically based models** 

**Fracture mechanics models** 

**7.2 Application of energetic approaches** 

have to be known, i.e., the local load parameters in consideration of the cyclic deformation behaviour have to be determined in advance. Especially for aluminium alloys the combination of stress and strain variables yields an adequate parameter as the cyclic deformation behaviour is dominantly affected by the ageing effect. If the temperature dependent ageing reduces the stress, the plastic strain increases in a similar way.

Whilst a single-parameter plastic energy approach is an adequate criterion for the highly ductile aluminium alloy AlCuBiPb, for ductile gravity die casting alloys (AlSi7MgCu0.5 and AlSi8Cu3) it is a total strain based energy criterion. For brittle materials such as AlSi6Cu4 lost foam, a fracture mechanics based energy criterion with a cyclic *J* integral provides the smallest standard deviation (Riedler et al., 2005).

The so-called *unified energy approach* is derived based on this knowledge of materialdependent TMF fatigue life criteria.

$$
\Delta\text{VV}\_u = \boldsymbol{\varepsilon}\_u \cdot \Delta\text{VV}\_{u,e} + \Delta\text{VV}\_{u,p} = \boldsymbol{\varepsilon}\_u \left(\boldsymbol{\sigma}\_o \cdot \boldsymbol{\varepsilon}\_{a,e}\right) + \left(\boldsymbol{\sigma}\_a \cdot \boldsymbol{\varepsilon}\_{a,p}\right) \tag{1}
$$

$$\mathbf{N}\_{B} = \mathbf{A}\_{u} \cdot \Delta \mathbf{V} \mathbf{V}\_{u}^{-B\_{u}} \tag{2}$$

The specific hysteresis energy for a representative cycle consists of an elastic and a plastic portion. Whereas the elastic portion is formed by the maximum stress and the elastic strain amplitude, the plastic portion is formed by the amplitude values of stress and plastic strain. The material parameter *cu* takes values near 1. The fatigue life is determined by a power law approach according to (2). The quality of computing OP-TMF fatigue life for the examined cylinder head alloy made of aluminium and cast iron by means of the *unified energy approach* can be seen in figure 9. It shows 95% of the data points of the six examined alloys influenced by maximum temperature, average and local strain, pre-ageing as well as ageing in operation lying in a fatigue life scatter band of 2.5, and two-thirds of the data points lying in a scatter band of 1.6.

Fig. 9. Quality of the TMF fatigue life simulation by means of the *unified energy approach*

Comparison of Energy-Based and Damage-Related

0

φ∫ mit

*dt*

1 *Ct creep creep C*

*t*

corresponding model parameters were determined.

effect of creep damage, see fig. 10 and 11.

Φ =

factor is introduced:

break-up is also measured.

Sehitoglu damage model

Fatigue Life Models for Aluminium Components Under TMF Loading 343

It is made use of the internal variables of the material models according to Slavik-Sehitoglu (Slavik & Sehitoglu, 1987). Creep damage is highest under IP-TMF loading, if the maximum temperature coincides with tensile stress. Isothermal LCF experiments show low creep damage, and for OP-TMF loading it is almost zero. Thus another phase

*creep*

In (Neu & Sehitoglu, 1989) the parameters are determined by means of elaborate experiments. In doing so, also experiments for measuring the oxide layer growth at different temperatures are conducted. Furthermore the growth of the oxide layer under repeated

The parameter adjustment in (Minichmayr, 2005)] is carried out solely by manual parameter variation and automatic parameter optimisation. In addition to the OP-TMF experiments (classical cylinder head applications, basis of energetic approaches) also LCF experiments with different strain rates, LCF experiments in argon atmosphere as well as in-phase TMF experiments were necessary. By means of non-linear parameter optimisation the

The fatigue life computation by means of the Sehitoglu damage model is slightly more accurate in comparison to the energy criteria; 90% of the data points for the cast alloy AlSi7MgCu0.5 lie within a scatter band of 1.85. The biggest advantage results from several damage mechanisms being active at the same time. Likewise it is possible, for example, to predict the in-phase TMF experiments correctly, which are characterised by a dominant

Fig. 10. Fatigue life computing under LCF, OP-TMF, and IP-TMF loading according to the

φ 2

(10)

1( / 1 exp <sup>2</sup> *th mech*

<sup>⎡</sup> <sup>⎤</sup> ⎛ ⎞ <sup>−</sup> = −⎢ <sup>⎥</sup> ⎜ ⎟ <sup>⎢</sup> <sup>⎥</sup> ⎝ ⎠ <sup>⎣</sup> <sup>⎦</sup>

ε ε

*creep*

ξ

#### **7.3 Application of the damage rate model according to Sehitoglu**

The model of Neu-Sehitoglu (Neu & Sehitoglu, 1989) is based on the assumption that overall damage is caused by fatigue, oxidation, and creep:

$$D^{total} = D^{fat} + D^{ox} + D^{creep} \tag{3}$$

or expressed as an equation using the cycles to fracture:

$$\frac{1}{N\_B^{total}} = \frac{1}{N\_B^{fat}} + \frac{1}{N\_B^{ox}} + \frac{1}{N\_B^{crecp}}\tag{4}$$

The pure fatigue damage portion is described by means of the Manson-Coffin-Basquin approach with the mechanical strain range Δ*εmech*:

$$\frac{\Delta \boldsymbol{\varepsilon}^{\rm mech}}{\mathbf{2}} = \frac{\sigma\_f^{\cdot}}{E} \left( \mathbf{N}\_{\rm B}^{\rm fatt} \right)^{b} + \boldsymbol{\varepsilon}\_f^{\cdot} \left( \mathbf{N}\_{\rm B}^{\rm fatt} \right)^{c} \tag{5}$$

The parameters *E*, *σf*', *b*, *εf*' and *c* are determined from isothermal fatigue experiments at room temperature. Thereby it is assumed that all experiments at elevated temperature show a similar or shorter fatigue life than at room temperature and furthermore that in these cases the fatigue life reduction is due to the oxidation and creep damage portions.

The oxidation damage portion describes the repeated formation and destruction of an oxide layer at the crack tip as a function of mechanical strain rate, mechanical strain amplitude, temperature and phasing between mechanical strain and temperature:

$$\frac{1}{N\_B^{\alpha\alpha}} = \left[\frac{h\_{cr}\delta\_0}{B\Phi^{\alpha\alpha}K\_p^{\text{eff}}}\right]^{-\frac{1}{\beta}}\frac{\mathfrak{Z}(\Delta\varepsilon^{m\text{cch}})^{(2/\beta+1)}}{\dot{\varepsilon}^{1-(\alpha/\beta)}}\tag{6}$$

The temperature dependency of the oxidation is described by means of an Arrhenius approach. The effective oxidation constant is obtained by integration over a complete cycle:

$$K\_p^{eff} = \frac{1}{t\_C} \int\_0^{t\_C} D\_0 \exp\left(-\frac{Q}{RT(t)}\right) dt\tag{7}$$

The phase factor takes into account that the oxidation damage portion of an OP-TMF load is higher than that of an IP-TMF load:

$$\mathbf{p}^{\alpha\alpha} = \frac{1}{t\_C} \int\_0^{t\_C} \boldsymbol{\phi}^{\alpha\alpha} dt \text{ mit } \boldsymbol{\phi}^{\alpha\alpha} = \exp\left[ -\frac{1}{2} \left( \frac{\boldsymbol{\dot{\varepsilon}}^{\text{fl}} \boldsymbol{\prime} \, \boldsymbol{\dot{\varepsilon}}^{\text{mech}} + 1}{\boldsymbol{\xi}^{\text{ox}}} \right)^2 \right] \tag{8}$$

The creep damage portion describes the damage due to pore and intergranular crack formation. The creep damage portion is defined as a function of temperature, equivalent stress, hydrostatic stress and *drag stress*:

$$D^{crep} = \Phi^{crep} \int\_0^{t\_C} A \exp\left(-\frac{\Delta H}{RT(t)}\right) \cdot \left(\frac{a\_1 \overline{\sigma} + a\_2 \sigma\_H}{K}\right)^m dt \tag{9}$$

The model of Neu-Sehitoglu (Neu & Sehitoglu, 1989) is based on the assumption that overall

1 11 1 *total fat ox creep N N B B N N B B*

The pure fatigue damage portion is described by means of the Manson-Coffin-Basquin

*mech b c <sup>f</sup> fat fat N N B B <sup>f</sup> <sup>E</sup>*

The parameters *E*, *σf*', *b*, *εf*' and *c* are determined from isothermal fatigue experiments at room temperature. Thereby it is assumed that all experiments at elevated temperature show a similar or shorter fatigue life than at room temperature and furthermore that in these cases

The oxidation damage portion describes the repeated formation and destruction of an oxide layer at the crack tip as a function of mechanical strain rate, mechanical strain amplitude,

1

β

The temperature dependency of the oxidation is described by means of an Arrhenius approach. The effective oxidation constant is obtained by integration over a complete cycle:

> 0 0

*<sup>Q</sup> K D dt t RT t* ⎛ ⎞ <sup>=</sup> ⎜ ⎟ <sup>−</sup>

The phase factor takes into account that the oxidation damage portion of an OP-TMF load is

The creep damage portion describes the damage due to pore and intergranular crack formation. The creep damage portion is defined as a function of temperature, equivalent

exp ( )

*D A creep creep <sup>H</sup> <sup>H</sup> dt*

0

⎡ ⎤ <sup>Δ</sup> <sup>=</sup> ⎢ ⎥

δ

1

*C*

φ

*eff p*

*dt*

0

0

φ∫ mit

1 *Ct ox ox C*

*t* Φ =

higher than that of an IP-TMF load:

stress, hydrostatic stress and *drag stress*:

*Ct*

1 2( ) *mech cr ox ox eff B p h N B K*

'

σ

the fatigue life reduction is due to the oxidation and creep damage portions.

temperature and phasing between mechanical strain and temperature:

2

ε

Δ

() ()

ε

'

*total fat ox creep D D DD* = ++ (3)

= ++ (4)

= + (5)

(2/ 1)

β

⎢ ⎥ <sup>Φ</sup> ⎣ ⎦ (6)

⎝ ⎠ <sup>∫</sup> (7)

2

(8)

1( / )

1( / 1 exp <sup>2</sup> *th mech ox*

<sup>⎡</sup> <sup>⎤</sup> ⎛ ⎞ <sup>+</sup> = −⎢ <sup>⎥</sup> ⎜ ⎟ <sup>⎢</sup> <sup>⎥</sup> ⎝ ⎠ <sup>⎣</sup> <sup>⎦</sup>

ε ε

*ox*

ξ

1 2

α β

ε

<sup>−</sup> <sup>+</sup> −

ε

exp ( )

*Ct m*

*RT t K* ⎛ ⎞ Δ + ⎛ ⎞ ασ ασ= Φ ⎜ ⎟ − ⋅⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∫ (9)

**7.3 Application of the damage rate model according to Sehitoglu** 

damage is caused by fatigue, oxidation, and creep:

approach with the mechanical strain range Δ*εmech*:

or expressed as an equation using the cycles to fracture:

It is made use of the internal variables of the material models according to Slavik-Sehitoglu (Slavik & Sehitoglu, 1987). Creep damage is highest under IP-TMF loading, if the maximum temperature coincides with tensile stress. Isothermal LCF experiments show low creep damage, and for OP-TMF loading it is almost zero. Thus another phase factor is introduced:

$$\boldsymbol{\Phi}^{crep} = \frac{1}{t\_{\mathbb{C}}} \int\_{0}^{t\_{\mathbb{C}}} \boldsymbol{\phi}^{crep} dt \text{ mit } \boldsymbol{\phi}^{crep} = \exp\left[ -\frac{1}{2} \left( \frac{\boldsymbol{\varepsilon}^{th} \boldsymbol{/} \boldsymbol{\varepsilon}^{mech} - 1}{\boldsymbol{\xi}^{crep}} \right)^{2} \right] \tag{10}$$

In (Neu & Sehitoglu, 1989) the parameters are determined by means of elaborate experiments. In doing so, also experiments for measuring the oxide layer growth at different temperatures are conducted. Furthermore the growth of the oxide layer under repeated break-up is also measured.

The parameter adjustment in (Minichmayr, 2005)] is carried out solely by manual parameter variation and automatic parameter optimisation. In addition to the OP-TMF experiments (classical cylinder head applications, basis of energetic approaches) also LCF experiments with different strain rates, LCF experiments in argon atmosphere as well as in-phase TMF experiments were necessary. By means of non-linear parameter optimisation the corresponding model parameters were determined.

The fatigue life computation by means of the Sehitoglu damage model is slightly more accurate in comparison to the energy criteria; 90% of the data points for the cast alloy AlSi7MgCu0.5 lie within a scatter band of 1.85. The biggest advantage results from several damage mechanisms being active at the same time. Likewise it is possible, for example, to predict the in-phase TMF experiments correctly, which are characterised by a dominant effect of creep damage, see fig. 10 and 11.

Fig. 10. Fatigue life computing under LCF, OP-TMF, and IP-TMF loading according to the Sehitoglu damage model

Comparison of Energy-Based and Damage-Related

Schiele & Schön, Berlin

151, Vol. 48, No. 4, Hanser, München

*ICEM12*, Paper No. 102, Bari

H., Hatanaka, K., DVM, 2004, Berlin

Herrmann, K., Vollertsen, F., Paderborn

*Betriebsfestigkeitstage*, Planneralm,

*Fatigue & Fracture*, Lisboa, Portugal

Materials, Philadelphia

1335-0803

method should be preferred.

**10. References** 

Fatigue Life Models for Aluminium Components Under TMF Loading 345

every specific material. Depending on the application, one specific lifetime calculation

Reichstein, S., Hofmann, L. & Kenningley, S. (2005). Entwicklung von Kolbenwerkstoffen

Fagschlunger, C., Pötter, K., & Eichlseder, W. (2006). Abschätzung der Schwingfestigkeit

Oberwinkler, C., Leitner, H., Eichlseder W., Schönfeld, F. & Schmidt, S. (2010).

Powazka, D., Leitner, H., Brune, M., Eichlseder, W. & Oppermann, H. (2010).

Riedler, M.; Eichlseder, W. & Minichmayr, R. (2004). Relationship between LCF and TMF:

Riedler, M. (2005). TMF von Aluminiumlegierungen – Methodikfindung zur Simulation von

Löhe, D., Beck, T. & Lang, K.-H. (2004). Important aspects of cyclic deformation, damage

Thalmair, S. (2009). Thermomechanische Ermüdung von Aluminium-Silizium-Gusslegierungen

Halford, G.R., McGaw, M.A.; Bill, R.C. & Fanti, P.D. (1988). Bithermal Fatigue: A Link

Riedler, M. & Eichlseder, W. (2004) Temperature control method in elevated and fluctuating

Minichmayr, R., Riedler, M. & Eichlseder, W. (2005). Thermomechanische Ermüdung von

Simon, C. & Santacreu, P.O. (2000). Life Time Prediction of Exhaust Manifolds, pp. 257-267,

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Ogata, T. & Yamamoto, M. (2001). Life Evaluation of IN738LC under Biaxial Thermo-

pp. 591-600, *MP Materialprüfung*, Vol. 47, No. 10, Hanser, München

unter ottomotorischen Beanspruchungen, *Dissertation* Univ. Karlsruhe

*Materials Testing,* pp. 513-519, Vol. 52, No. 7-8, Hanser, München

*Giesserei,* pp. 34-42, Vol. 97, No. 7, Gießerei-Verlag, Düsseldorf

*Fortschritt-Berichte VDI*, Reihe 5, ISBN 3-18-371805-7

für moderne Hochleistungsdieselmotoren, *Giesserei-Praxis,* pp. 380-384, No. 10,

von porenfreien Randschichten in Al-Gussbauteilen. *MP Materialprüfung*, pp. 142-

Schädigungstolerante Auslegung von Aluminium-Druckgusskomponenten, *MP* 

Fertigungsbedingte Einflüsse auf die Schwingfestigkeit von Al-Gussbauteilen,

Similiarities and Varities, *12th International Conference on Experimental Mechanics*,

thermomechanisch beanspruchten Motorbauteilen aus Aluminiumlegierungen,

and lifetime behaviour in thermomechanical fatigue of engineering alloys, pp. 161- 175, *Fifth International Conference on Low Cycle Fatigue*, Eds.: Portella, P.D., Sehitoglu,

between Isothermal and Thermomechanical Fatigue, Low Cycle Fatigue pp. 625- 637, *ASTM STP 942*, Eds.: Solomon et al., American Society for Testing and

temperature fatigue tests, *Materials Engineering*, pp.1-7, Vol. 11, 2004 No. 3, ISSN

Aluminiumlegierungen – Versuchstechnik und Methoden der Lebensdaueranalyse,

*Proc. CAMP2002 – High-Temperature Fatigue*, Eds.: Biallas, G., Maier, H.J., Hahn, O.,

an Rundproben auf thermomechanisch beanspruchte Bauteile, *1. Leobener* 

Mechanical Fatigue, pp. 839-847, *Sixth International Conference on Biaxial/Multiaxial* 

Fig. 11. Quality of the TMF fatigue life calculation using the Sehitoglu damage model
