Reduced order modelling and large time increment method 

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Continuum damage mechanics (CDM) theories

The mechanics of damage is the study, through mechanical variables, of the mechanisms involved in the deterioration when the materials are subjected to loading. The idea of degradation of material being quantified by internal variable in a thermodynamically consistent continuum framework was first developed by Kachanov (1958, 1986). This is a relatively new branch, which is an extension of the classical continuum mechanics. The first usage of CDM for fatigue computation was in Chaboche and Lesne (1988). The proposed damage law is of the form D = 1 − 􀀀 1 − r1/(1−α) 1/(1+β) , (1.21).
where, β is a material parameter, α is a function of the stress state, and r is the cycle ratio. Many notable works have followed since, and well-established continuum damage mechanics theories can be found in Lemaitre (1996), Lemaitre and Desmorat (2005), Murakami (2012). CDM will be dealt in details in Chap.3.

Fatigue crack propagation theories

Although many authors (see Fatemi and Yang, 1998) do not distinguish between fatigue damage theories and crack propagation theories, a distinction between damage mechanics and fracture mechanics concepts following the demarcation from Cui (2002), Suresh (2001) has been made here. However the counter argument in Fatemi and Yang (1998) is also valid as crack is initiated at the very beginning of the loading and the entire life of the structure is spent in propagating the crack. The initiation and propagation phase can be distinguished from the length of the crack (which is again not fixed). However based on classical fracture mechanics theories, a distinction is made on the fact that crack propagation theories are based on the presence of a pre-crack in the structure.
Based on the theories explained in Cui (2002), Fatemi and Yang (1998), Suresh (2001), it can be divided into three parts based on the crack length.

Continuum mechanics: an overview

To obtain the global admissibility conditions, it is necessary that a reference problem is introduced in a spatial domain , as shown in fig. 2.1. The region has a boundary ∂ , which is subdivided into ∂ 1 and ∂ 2, such that ∂ = ∂ 1 ∪ ∂ 2. The body is subjected to specified surface force per unit surface area ~Fd on ∂ 2 and to specified displacement ~Ud on ∂ 1. The outward normal vector ~n is defined at any material pointM on the boundary ∂ . The body is also subjected to body force per unit volume ~ fd. The evolution of the structure is considered to be quasi-static within time t ∈ [0, T], with T being the total time.

Concept of plasticity

The most idealistic plastic behaviour is perfect plasticity, which rarely happens for metals in practical scenario. Perfect plasticity or ideal plasticity results in infinite deformations after the elastic limit is reached, or in other words the load carrying capacity is completely lost after the elastic limit is crossed. Strain hardening, however in general, results in a non-linear increase in stress with respect to the strain after yielding. Due to hardening, the material does not loose the load carrying capacity at the onset of yielding. The stress-strain relationship is complicated and can in the most simplified case be defined in terms of hardening modulus which is a function of total or plastic strain, or can be described as power law, e. g. the classical case of Ramberg-Osgood equation (see Ramberg and Osgood, 1943). Perfect plasticity and strain hardening are illustrated in fig. 2.2. However, Ramberg-Osgood type equations do not capture the stress-strain behaviour accurately, especially for cyclic plasticity. For an accurate description of hardening, it is more convenient to use separate internal variables. Classically, hardening can be classified into kinematic and isotropic. For isotropic hardening, the radius of the yield surface (the surface of elastic limit) in the principal stress space increases. In this case the yielding during tension and compression is considered to be the same. This kind of hardening fails to take into account Bauschinger effect, which indicates the loss of isotropy and lower strength during compression, prevalent mostly in metals. Thereby, kinematic hardening is introduced, where the yield surface only translates in the principal stress space without any change in radius. These phenomena are shown in fig. 2.3. For metals, in general, either pure kinematic hardening or mixed hardening model is used. It has to be noted that reverse phenomena called softening also exist, which have been detailed in Lemaitre and Chaboche (1990).

CDM approaches in fatigue

Different methodologies have been been adopted over the years to model as well as to simulate the
fatigue mechanisms using continuum damage mechanics. A few most important approaches are presented here.
One of the most novel approaches used is the formulation of the “two-scale damage model” by Lemaitre (see Lemaitre et al., 1997). This is mainly used for high cycle fatigue. The global behaviour remains elastic, which is obtained through a macroscopic elastic structural calculation. After that the Gauss points with maximum stress concentrations are identified. Eshelby-Kroner scale transition law is then applied to get the micro-strain at chosen Gauss points from the macro-strain. Thereafter, elasto-plastic analysis coupled with damage is performed at the micro-scale. The final outcome of this is the damage variable D, which is considered to be equal in both the scales. An important development in the solution framework of fatigue is the popular “jump cycle” approach (see Lemaitre and Desmorat, 2005). The basic idea here is confined to loads that are periodic or at
least periodic by blocks, where calculations of full blocks of load cycles are skipped. The quantities of interest are calculated for a certain number of initial load cycles. The damage and accumulated plastic strain increments over the last computed cycle are then estimated. From this information, an estimation of the number of cycles that can be skipped is made with an assumption that damage and accumulated plastic strain are linear with respect to the number of cycles. After that the quantities of interest are again calculated for one cycle and an estimation is made for the number of cycles to skip. This continues till Dc is achieved. The “two-scale damage model” is generally used together with the “jump cycle” technique, e.g. in Bhamare et al. (2014), Lemaitre et al. (1997). Another important approach is the time homogenisation technique initially proposed in Guennouni and Aubry (1986) for quasi-static elasto-viscoplastic problems. Later on Puel and Aubry extended this method for dynamics (see Puel and Aubry, 2012) and also for CDM (see Devulder et al., 2010). Fish and his co-workers, also proposed similar techniques for fatigue, using a viscoplasticity like damage model (see Oskay and Fish, 2004b) and also for GTN damage model (see Fish and Oskay, 2005). The main idea is to divide the time into two separate scales: a fast scale (micro-time scale) and a slow scale (macro-time scale). Thereafter, the quantities at each slow time scale are represented as homogenised quantities of the fast time scale. This solves the problem of numerical simulation of all the quantities of interest at the fast time.

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Proper orthogonal decomposition (POD): a posteriori model reduction technique

The basic idea of POD is to compute certain full-order problems in order to extract relevant information which can then be used to calculate similar problems more efficiently. The first ideas of POD were developed in Karhunen (1946), Kosambi (1943), Lo`eve (1946), Pearson (1901). The usage of POD in the field of mechanics deals with the creation of POD basis, from the snapshots obtained from the solution of the training phase, and then to use this basis to solve the intended problem in a reduced space (see Ryckelynck, 2009). Also known as principal component analysis (PCA), Karhunen Lo`eve expansion (KLE) and singular value decomposition (SVD) (see Chatterjee, 2000), POD provides an optimally ordered set of basis functions in a least square sense for the full-order solution. These basis functions are called “proper orthogonal modes”, “empirical eigenfunctions”, or just “basis vectors”. A reduced order model (ROM) or a surrogate model can then be generated by truncating the optimal basis (see Liang et al., 2002, Pinnau, 2008). A full solution given by ~u (~x, t) where ~x denotes the spatial coordinate and t being the temporal coordinate can be represented as ~u (~x, t) = Xr i=1 Ti (t)~Xi (~x).

Proper generalised decomposition (PGD): a priori model reduction technique

It is desired, ideally to have a technique where a reduced order approximation can be built without relying on any training stage. One would then be able to assess the accuracy of the reduced order approximation, and enrich the reduced order basis if necessary. This leads to the proper generalised decomposition (PGD), which was first introduced as “radial loading approximation” in the context of LATIN method (see Cognard and Ladev`eze, 1993, Ladev`eze, 1989, 1999, and other works of Ladev`eze) based on separated representation of the quantities of interest. For instance, the desired solution field V (~x, t) is obtained in the separated form as (see Chinesta et al., 2014a) V (~x, t) = Xr i=1 Ti (t) Xi (~x) . (3.14).
Here the number of terms r needed for this finite sum decomposition is not known a priori. The functions Ti (t) and Xi (~x) are constructed by successive enrichments. For a particular enrichment stage v +1, the functions {Ti}v i=1 and {Xi}v i=1 being known from the previous steps, Tv+1 and Xv+1 are sought. This is achieved by invoking the weak form of the problem. Considering a case where the problem is defined by L (V ) = f (t, ~x) ,

Table of contents :

List of figures
List of tables v
List of abbreviations
1 Fatigue: an overview 
1.1 History of fatigue
1.2 Different forms of fatigue in the engineering world
1.3 Phases of fatigue life
1.4 Different domains of fatigue
1.5 Different types of load fluctuations
1.6 Existing fatigue approaches
1.6.1 Cumulative fatigue damage theories
1.6.2 Fatigue crack propagation theories
1.7 Cyclic elasto-(visco)plasticity
1.8 Concluding remarks
2 Continuum damage mechanics 
2.1 Continuum mechanics: an overview
2.1.1 Admissibility conditions
2.1.2 Constitutive relations
2.2 General constitutive behaviour
2.2.1 Concept of plasticity
2.2.2 Rate-independent plasticity
2.2.3 Viscoplasticity
2.2.4 Concept of damage
2.2.5 Damage with elasto-(visco)plasticity
2.3 CDM approaches in fatigue
2.4 Concluding remark
3 Reduced order modelling and large time increment method 
3.1 Classical incremental method
3.2 Model reduction techniques
3.2.1 Proper orthogonal decomposition (POD): a posteriori model reduction technique
3.2.2 Proper generalised decomposition (PGD): a priori model reduction technique
3.3 Large time increment (LATIN) method
3.3.1 First principle: separation of difficulties
3.3.2 Second principle: two-step algorithm
3.3.3 Third principle: model reduction method
3.3.4 A note on “normal formulation”
3.3.5 LATIN method in a heuristic nutshell
3.3.6 Newton-Raphson technique in the light of LATIN method
3.4 Concluding remark
4 LATIN-PGD technique for cyclic damage simulation 
4.1 The proposed problem
4.2 Initialisation
4.3 Local stage
4.4 Search direction for global stage
4.5 Internal variables at the global stage
4.6 PGD formulation of the global stage
4.6.1 Separable representation of the quantities of interest
4.6.2 Hybrid method to construct the PGD reduced-order basis
4.7 Relaxation of the solution field and convergence criterion
4.8 Numerical examples
4.8.1 Bar under traction
4.8.2 “L” shaped structure
4.8.3 Plate with a hole
4.9 Concluding remarks
5 Multi-scale temporal discretisation approach 
5.1 Finite element like time interpolation scheme
5.2 Computation of one “nodal cycle”
5.2.1 Initialisation
5.2.2 Local stage
5.2.3 Global stage
5.3 Numerical examples
5.3.1 Verification with mono-scale LATIN method
5.3.2 Influence of the “training stage”
5.3.3 Simulation of large number of cycles
5.3.4 Pre-damaged structure
5.3.5 Variable amplitude loading
5.3.6 Virtual ε-N curves
5.4 Concluding remark
6 Conclusion and future perspective 
A Calculation of the finite element operators 
B Solution technique of the temporal problem 
C Orthonormalisation of the space functions 
D Alternative method of incorporating non-linear elastic state law 
E Extended summary in French 


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