Damage and damage models

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2.8 Fundamentals of thermo-mechanical failures .1 Overview of thermo-mechanical structural

2.8.3 Damage and damage models

Damage is defined as the ratio of the actual number of cycles (or time) to the required number of cycles (or time) to reach a pre-defined macroscopic failure (e.g., crack length due to LCF or HCF; wear amount). The concept of damage in structural analysis has usually referred to the failures of thermo- mechanical fatigue, but it can easily be extended to cover the damage caused by other mechanical failures such as wear and cavitation. Fatigue damage mainly includes three mechanisms: fatigue, creep, and oxidation. The damage can be caused by a single type of cyclic loading or multiple types of cyclic loading. For complex stress time histories of driving/usage profiles (e.g., obtained from thermo-mechanical simulation or test data), the commonly used approach to decompose them into a set of simple fatigue cycles is the

‘rainflow decomposition’ method (Masuishi and Endo, 1968).

The simplest way to estimate the fatigue damage under a combination of multiple types of loading cycle is to assume a linear accumulation of the

2.2 Stress–strain property of steel.

Stress Breaking strength

Ultimate strength

Breaking strength Yield strength Yield strength

s

e

The hysteresis loop Strain hardening region Engineering (nominal)

stress–strain curve

Necking region

Strain Rupture Rupture True stress–strain curve

Ultimate strength

fatigue damage due to the contributions from each type of cyclic loading as

D n

N

i

m li

= fi = 1

S

=1 2.1

where D is the total accumulated damage over the entire stress–strain history, N_fi is the number of cycles required to reach the pre-defi ned failure criterion under the loading of the ith type of decomposed simple fatigue cycles, n_li is the actual number of loading cycles under the ith type of loading, and m is the number of types of the loading cycle. When the damage D = 1, it is assumed that fatigue failure occurs. In fact, D can be greater than or less than 1 in reality.

The above damage model can be extended to include the independent superposition of various damage mechanisms in order to account for the combined effect of fatigue, creep, oxidation, etc. The following model has been widely used:

D = D₁ + D₂ + D₃ + … + D₁D₂ + …

= +

=1 ,1

,1 =1

,2

,2 =1

1 2

1 ,1 2

S

,1

S S

⁺

S S

^{,1 +}

S

S

^,1

S S

i

m

S S

¹¹ lili^li

S S

²²

fi i

1 mm2

1 2 li

fi i

n m

1 n 2

1 2

S

ⁿ

S S

_NN

S S

_N

S S S

_TT^t

Tfi_fi

333 ,3 ,3

t T

li

Tfi_fi

T

+ … + _1,1,22

=1 ,1

,1 =1 ,2

,2

1 ,1 2

1 ,1 2

C

C ¹¹ nn ²² N

t T

i

m¹¹ lili ²²

fi i

1 mm2

1 2 li

Tfi_fi

S S

T

S

^·

S S

^{,1 ·}

S S

^{n ,1}

S S

_Nⁿ

S S

_N^li

S S

^li

S

ÊÊÊ Ë ËÁ ÊÊÊÁ ÊÊÊ ËÁË ËÁ Ë

ˆ

¯˜

ˆ˜ ˆ

¯˜¯ + … 2.2

where D is the total damage, D₁ is the damage due to fatigue, D₂ is the damage due to creep, D₃ is the damage due to oxidation or degradation, etc. D₁D₂ is the interaction term of damage between the two mechanisms.

C_1,2 is a constant characterizing the importance of the interaction effect (e.g., C_1,2 = –0.5 ~ 0.5). T_fi is the required time to reach creep failure for the ith type of temperature or mechanical loading for creep, t_i is the actual time of creep under the ith type of loading for creep. The value of D can be calculated by using equation 2.2 based on experimental data or numerical simulation models. When C_1,2 = 0 and D = 1, it means the accumulation of the damages due to different failure mechanisms follow the simplest model of linear accumulation of damage. Experimental evidence showed that the total damage due to the combined effects of fatigue, creep, and oxidation does not always give D = 1. Instead, the actual measured data of D fl uctuate around the line of D = 1 (Fig. 2.3a).

The stress–strength model has traditionally been used in the probabilistic reliability evaluation for structural durability. Since many thermo-mechanical fatigue problems require using plastic strain (e.g., for low cycle fatigue problems) instead of stress as an evaluation criterion, the damage parameter

has been commonly used to replace the stress in the stress–strength model in this scenario. For each material, component design and a usage loading profile at a given vehicle mileage or engine hours, one damage indicator can be calculated. Such a damage calculation can be repeated with the Monte Carlo simulation for the entire production population including all the variabilities in order to construct the stress–strength curve.

The damage models for life prediction (i.e., to determine the total required number of cycles to failure) are either stress-based (e.g., the Chaboche method) or strain-based (e.g., the Sehitoglu method, and the total strain–train range partitioning method or also known as the TS–SRP method). Ogarevic et al. (2001) concluded in their review article of thermo-mechanical fatigue that the three most widely accepted life prediction methods are the TS–SRP method developed by Manson and Halford, the Sehitoglu method, and the Chaboche method.

Damage model

50% reliability

99% reliability Linear

accumulation damage model

Reliability probabilistic

distribution 1

0 1

nl/Nf for fatigue

t_l/T_f for creep (a)

N_f (b)

Log (N)

(c) Log (N)

(d) S

P–S–Nf curve

Log (De) Log (Dep)

Total strain range (De) vs. life (N) Strain range vs. life curve

Elastic strain range vs. life

Plastic strain

range vs. life High temperature

Low temperature Temperature effect on LCF life

2.3 Damage model and durability life analysis.

The TS–SRP method is one of the earliest methods capable of predicting thermo-mechanical fatigue life. The method is based on the observation that the strain range of any stress–strain hysteresis loop associated with thermo-mechanical fatigue can be partitioned into four basic strain-range elements in order to account for their different impact on the fatigue life.

The partitioning depends on whether creep occurs and whether the strain is in compression or tension. The four strain range components are:

1. plastic strain in tension and plastic in compression, Depp

2. plastic in tension and creep in compression, Depc

3. creep in tension and plastic in compression, De_cp 4. creep in tension and creep in compression, Decc.

The relationship between the strain range De and fatigue life N_f is introduced in Section 2.8.5. The fatigue lifetimes of the four components are ranked from low to high as: N_f,cp, N_f,cc, N_f,pc, and N_f,pp (Fig. 2.4). The total life N_f,fc due to fatigue and creep is given by the single-cycle damage model as:

1 = 1 + 1 + 1 + 1

, , , ,

N =N

N_{f ff f,,} =N _,, NN + +NN N N_{f f} N

N _,,cc N_ff,, N _c N_f N_{f fcf ff ff f,,,,c} N_ff,,,, N_{f f} N N _c N_f N_{f f} N

N N _pp NNNN_{f pf pf p,,,, ccc} NNNN_{fff,,,,cp f cf c,,c} 2.3 Equation 2.3 can be further modifi ed by including weighting factors or proportional coeffi cients in order to make the life prediction more accurate.

The total damage is the superposition of these four types of mechanisms.

The history and details of the TS–SRP method were reviewed by Ogarevic et al. (2001).

In the Sehitoglu method, the total damage of the thermo-mechanical fatigue cycle is the sum of the damage due to fatigue, oxidation, and creep. The

Depp

Depc

Decc

Decp

Log (De)

Log (Nf)

2.4 Total strain–strain range partitioning.

fatigue life is calculated by a strain-based approach. The Chaboche method is a stress-based approach. In fact, it is generally believed that a strain-based method is more suitable than a stress-based method for LCF problems because LCF is controlled mainly by the plastic strain rather than the elastic stress as encountered in the case of HCF. As an example, in the Chaboche’s model, the number of cycles to failure for each decomposed simple fatigue cycle due to pure fatigue, N_fi , is calculated based on stresses as below (for example, used by Morin et al. (2005) for a diesel cylinder head):

N s s

s s

s

fi ssu_u ssi_i C

l

= ssss_uu – ssss_ii, i

, C1

, _l C1

, l 1

, 1

s_, s ₁

s s

s – s

s_, s ₁

s – s

s s

max – max ma

si m_{i m} s s_{i m,,} s ₁₁

s_, s ₁

s_{i m} s

s_, s ₁

s_{,, ax} s ₁₁

s_, s ₁

s _ax s

s_, s ₁

s_,,,, s_{flflffllfll 1111} Ê

ËÁ ÊÁ Ê ËÁË

ˆ

¯

, ¯ 1

, ˜ˆ 1

˜ˆ

¯˜¯

, ¯ 1

, ˜ 1

, ¯ 1

, 1

ÊD Ë

, Ë 1

, ÁÊ 1

ÁÊ ËÁË

, Ë 1

, Á 1

, Ë 1

, 1

ˆ

¯˜

ˆ˜ ˆ

¯˜¯

C C C₂

2.4 where i indicates the ith decomposed simple fatigue cycle, s_u is the ultimate stress, s_i,max is the maximum stress, s_fl is the fatigue limit stress, Ds_i is the stress amplitude, and C₁ and C₂ are material constants.

These three prevailing models of life prediction, along with their corresponding material testing procedures, were elaborated by Ogarevic et al. (2001). It should be noted that the three methods are not interchangeable.

Therefore, once a method is chosen, it is not easy to change to another one because the empirical database is usually built with one method.