**2.1 Stationary loading**

408 Recent Trends in Processing and Degradation of Aluminium Alloys

Under symmetric three-point bending of a beam with the transverse notch, main inelastic strains are concentrated ahead of the notch tip where stress concentration occurs. Change of mechanical characteristics of material in this region under repeated loading conditions causes the increase in residual deflection of the specimen. Thus, the amplitude of the residual deflection can be used as a measure of damage accumulation in the zone of localization of inelastic strains. This provides, from macroscopic phenomena, a possibility for qualitative and quantitative estimating the changes directly exhibited by the material due to processes of fatigue fracture. Moreover, in this case, the notch can be considered as a model of an edge crack with the blunted tip, and the region of localization of inelastic strains

In the second case, consider a plain specimen with one edge notch since two symmetrical notches lead to uncertainty in a choice of the point of crack initiation. Besides, after a crack initiates, the symmetry of a specimen is broken in one of paired notches and its initial

In the performed tests, the effect of various loading conditions, change in the geometry of a specimen, and preliminary plastic strain of material from which the specimens were made on the process of damage accumulation were studied. The attempt was made to determine parameters, which may be extended from the particular cases considered in the tests, to more general loading conditions. A possibility for description of regularities of damage accumulations with the aid of simple analytical functions involving constants just as

The specimens were made from aluminum alloy D16T in the original state, so from preliminary stretched materials with the various degrees of plastic strain. The composition of the D16Т alloy is as follows: Al was as a base metal, the alloying elements were Fe (0.3%),

The experiments were conducted on electromechanical testing machine (the rated capacity load was 100 kN). The loading was repeated three-point bending with unloading and it was given by travelling of a moveable cross-head with a constant velocity. Loading diagrams

The minimum force of a cycle was min *P* ≈ 0 for all the conducted tests. The maximum force of a cycle *P*max was considered in three forms: *i*) *P const* max = (stationary low-cycle loading), *ii*) *P PN* max max ≡ ( ) was increasing step-function of *N* for which the number of cycles at one step was constant (non-stationary low-cycle loading with increasing load), and *iii*) *P PN* max max ≡ ( ) was analogous decreasing step-function. The *P*max value in all tests was chosen to provide plastic material deformation ahead of the notch tip. Tests under stationary cyclic loadings were conducted for different *P*max values in the range from the limit of elasticity to the strength limit of a specimen. Besides, tests with different notch

pre-fracture zone near the notch tip, and therefore, as it was said before, it is considered as a measure of damage increment in this zone. The increment is considered to achieve the limit \* *<sup>w</sup>*value if subsequent deflection of the beam proceeds without increase in *<sup>P</sup>* .

*w wN* = ( ) depends on the cycle number. The

*w* deflection arises mainly due to mechanical properties of material in a

δ δ

**2. Low-cycle symmetric three-point bending of a beam with edge notch** 

determined from experiments, so specific for every material is considered.

Si (0,19%), Mn (0.76%), Cu (4.0%), Mg (1.29%).

depth to beam height ratios were conducted. Increment of the residual deflection

δ

increment of the

were recorded at each loading cycle.

can be considered as a pre-fracture zone ahead of the tip of this crack.

symmetry loses significance.

Fig. 1 demonstrates, as an example, the experimental diagram represented by curves of beam deflection *w* versus applied force *P* in stationary low-cycle testing a specimen made from D16T. The *P* value is sufficiently large in order that fracture to happen after the limited number of cycles (in this case, \* 390 *N* = ). This allows one to visualize all distinctive features of such diagrams obtained also for test materials for different *P*max values. Curve 1 corresponds to single loading of a specimen until fracture occurs; group of curves 2 corresponds to cyclic loading up to the instant when a crack starts to extend for *P P* < max . Each curve of group 2 corresponds to loading branch of one cycle. All the curves of group 2, except the first one, have the initial horizontal section 0 *P* = . The length of this section is equal to the value of residual deflection accumulated at previous cycles. In this figure, δ *w* is the distance between adjacent curves of group 2.

Damage accumulation can be divided into two specific stages. The first stage (Fig. 1, subgroup *А* of curves 2) is a stage of cyclic strengthening at which decrease in δ *w* is observed with increase of *N*. At this stage, δ *w* achieves some minimum that is characteristic for the given value of *P*max after which the δ *w* value becomes constant within the limits of measurement accuracy. The second stage (Fig.1, sub-group *В* of curves 2) is characterized by the increase in the δ *w* value as *N* increases, and this stage is accomplished by growth of a crack when *P P* < max . Accumulation of micro-defects during the first stage is likely to lead to formation of macroscopic defect, which then progresses during the second stage. Therefore, we call the second stage as a stage of development of a macro-defect. The ratio between the number of cycles in sub-groups *А* and *В* and the law of δ *w* variation depend on material characteristics (Karpov, 2009; Kornev et al., 2010).

Fig. 1. Scheme of low-cycle test of specimen loaded in tree-point bending and plots of beam deflection as a function of force for every loading cycle for D16T alloy

Regularity in the residual deflection can be visualized as δ *w N*( ) diagrams. An example of such diagrams is given in Fig. 2. Here pairs of curves are shown with numerals 1–5 for curves, one of the pairs being given by the analytical function and the second one being a saw-like profile. Here the saw-like profiles represent experimental δ *w N*( ) curves for the D16T alloy. The analysis of experimental diagrams shows that curves can be approximated by plots of some power functions. These functions are to have asymptotes corresponding to the limits beyond which the process described by the diagrams can not take place. That is, the inverse power function with some scaling coefficient can be taken as approximating one.

Interrelation Between Failure and

monotonic nonlinear increase of

overloading for D16T alloy

**2.3 The typical ratio** 

*P*max value. Beginning from the fourth step, ∑

Damage Accumulation in the Pre-Fracture Zone Under Low-Cycle Loading 411

value is applied again. Given in Fig. 4 are curves of *w* versus *P* for cyclic loading of D16Т duralumin when decrease in *P*max is step-like. Here, after some initial *P*max value,

δ

decreasing load, the specimen rapidly losses its load capacity. The tests also show that the increase in *P*max after several steps of such loading leads to significant inelastic strain with strengthening for values of the force *P* , which earlier were appropriate to the linear dependence of the deflection on the applied force. This fact evidences that repeated loadings of a material preliminary experienced overloading lead to significant material damage even

Fig. 3. Non-stationary cyclic loading with increasing maximal applied force for D16T alloy

Fig. 4. Non-stationary cyclic loading with decreasing maximal applied force after initial

The typical value for both stationary and non-stationary low-cycle loading is the ratio between limit deflections for single and repeated loading ( \* *w* and \*\* *w* , respectively). The

*w N*( ) is continued even for significant decrease in the

*w N*( ) also increases. As a result, in spite of

δ

this loading is not followed by noticeable increase in the residual strain.

Besides, in the general case, both descending and ascending branches of the experimental curves should be approximated by different curves.

Fig. 2. Approximation of experimental curves of residual deflection increment for D16T alloy

The connection point of these curves corresponds to the minimum δ *w* value. In this case, the form of curve, which approximately describes either descending or ascending branch, is unique for all values of *P*max . That is, the curves of approximating functions for all *P*max values can be superposed by parallel shift. Fig. 2. shows such curves (branches of hyperbolas corresponding to saw-like profiles 1-5). These curves are given by functions of the form *fN N* () ( ) γ λα β <sup>−</sup> = + normalized in such a way that the experimental minimum point δ *w* would be a connection point of two curves approximating both the descending and ascending branches of the saw-like profile (α , β , γ , λ are experimental constants). If approximating functions have been defined, then the derivatives of these functions calculated for integers of the variable *N* can be used as a magnitude characterizing material damage. The stage of cyclic strengthening *A* of duralumin, which differs from steels by essentially larger grain sizes, is limited to several cycles even for low loads (Kornev et al., 2010). As compared with the *B* stage, the stage *A* can be neglected.

#### **2.2 Non-stationary loading**

Under non-stationary loading conditions, the process of cyclic strengthening begins again each time when *P*max increases, and only the last stage explicitly includes both *A* and *B* stages of damage accumulation. However, if instead of change inδ *w* , the analogous change in ∑δ *w N*( ) (summation is performed over numbers of all cycles at one loading step), the stages *A* and *B* are also evident as in the case of the stationary loading. Fig. 3 demonstrates the experimental diagrams represented by curves of *w* versus *P* for non-stationary cyclic loading of alloy D16T.

The non-stationary loading with decreasing *P*max shows that the initial overload of material provokes consequences, which may affect fracture process when loading with the low *P*max

Besides, in the general case, both descending and ascending branches of the experimental

Fig. 2. Approximation of experimental curves of residual deflection increment for D16T

the form of curve, which approximately describes either descending or ascending branch, is unique for all values of *P*max . That is, the curves of approximating functions for all *P*max values can be superposed by parallel shift. Fig. 2. shows such curves (branches of hyperbolas corresponding to saw-like profiles 1-5). These curves are given by functions of

<sup>−</sup> = + normalized in such a way that the experimental minimum

*w* would be a connection point of two curves approximating both the descending

α , β , γ , λ

approximating functions have been defined, then the derivatives of these functions calculated for integers of the variable *N* can be used as a magnitude characterizing material damage. The stage of cyclic strengthening *A* of duralumin, which differs from steels by essentially larger grain sizes, is limited to several cycles even for low loads (Kornev et al.,

Under non-stationary loading conditions, the process of cyclic strengthening begins again each time when *P*max increases, and only the last stage explicitly includes both *A* and *B*

The non-stationary loading with decreasing *P*max shows that the initial overload of material provokes consequences, which may affect fracture process when loading with the low *P*max

 *w N*( ) (summation is performed over numbers of all cycles at one loading step), the stages *A* and *B* are also evident as in the case of the stationary loading. Fig. 3 demonstrates the experimental diagrams represented by curves of *w* versus *P* for non-stationary cyclic

δ

δ

*w* value. In this case,

are experimental constants). If

*w* , the analogous change

The connection point of these curves corresponds to the minimum

2010). As compared with the *B* stage, the stage *A* can be neglected.

stages of damage accumulation. However, if instead of change in

γ

curves should be approximated by different curves.

alloy

point δ

in ∑δ

the form *fN N* () ( )

**2.2 Non-stationary loading** 

loading of alloy D16T.

λα β

and ascending branches of the saw-like profile (

value is applied again. Given in Fig. 4 are curves of *w* versus *P* for cyclic loading of D16Т duralumin when decrease in *P*max is step-like. Here, after some initial *P*max value, monotonic nonlinear increase of δ *w N*( ) is continued even for significant decrease in the *P*max value. Beginning from the fourth step, ∑δ *w N*( ) also increases. As a result, in spite of decreasing load, the specimen rapidly losses its load capacity. The tests also show that the increase in *P*max after several steps of such loading leads to significant inelastic strain with strengthening for values of the force *P* , which earlier were appropriate to the linear dependence of the deflection on the applied force. This fact evidences that repeated loadings of a material preliminary experienced overloading lead to significant material damage even this loading is not followed by noticeable increase in the residual strain.

Fig. 3. Non-stationary cyclic loading with increasing maximal applied force for D16T alloy

Fig. 4. Non-stationary cyclic loading with decreasing maximal applied force after initial overloading for D16T alloy
