Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 2, pp. 463-474, Warsaw 2013

COMPARISON OF TWO MODELS OF NUMERICAL SIMULATION OF

LOW CYCLE FATIGUE OF A PLASTIC MATERIAL WITH

SMALL RIGID INCLUSIONS

Joury M. Temis, Khakim Kh. Azmetov

Central Institute of Aviation Motors, Moscow, Russia

e-mail: tejoum@ciam.ru

William J. Fleming

Northumbria University, School of Computing, Engineering and Information Sciences, Newcastle upon Tyne, U.K.

Theory of Cells and a numerical FEmethod based on strain cyclic plasticity, damagemodel
and technology of died elements were used for a prediction of the fatigue life of a metal
matrix compositematerial. Results of calculationswere comparedwith experimental fatigue
data. It was shown that the predicted fatigue life of MMC using the method of cells was
in close agreement with the experimental results for life outside of low cycle fatigue regime
of 1000 cycles or less. The results obtained from the mathematical simulation procedure
show that the failure occurs in several steps – the process of damage accumulation in the
material and the process of crack growth. The results of prediction of time of the composite
material full fracture are in good agreementwith experimental data. The comparison shows
that both the numerical method and the theory of cells can be used to predict fatigue life
of MMC to within an acceptable degree of accuracy.

Key words: low cycle fatigue, cyclic plasticity, damage accumulationmodel

1. Introduction

The introduction of inclusions into abasemetalwill give greatly enhancedmechanical properties
to the resultant metal matrix composite (MMC). However, these properties will depend on the
properties of both the base matrix and the inclusions used. The inclusion volume ratio and
its geometrical form will, of course, also be important. These parameters and the parameters
of the interface between the matrix and inclusion define both the composite ultimate strength
under monotonic loading and fatigue strength under cyclic loading. The main problem facing
the material designer is to create composite materials or a structure with superior mechanical
properties. To solve this problem, experimental research is mainly employed. This method is
expensive and it does not allow an easy optimisation of the material or component properties.

Amathematical approachwould appear to offer amore economical methodology of selecting
likely candidate composite materials and thus limiting the amount of experimentation necessa-
ry to produce successful materials. In this study, two mathematical approaches are used, one
analytical using the Theory of Cells (Fleming and Dowson, 1999) and the other, a numerical
method using based on a cyclic plasticity and accumulated plasticity strain damagemodels and
a finite element approach for cycling loading (Temis, 1989, 1994; Putchkov et al., 1995). The
methods were used to predict the fatigue life of ametal matrix composite over a range of strain
values, and then the predicted results were compared to experimental results.



464 J.M. Temis et al.

2. Experimental testing

2.1. Material

Thematerials under consideration in this report are an aluminium alloy Al7075 and ametal
matrix composite with an identical material for the matrix plus 12%SiC fibres, in particulate
form. The chemical composition of the composite was 6.2%Zn, 1.5%Cu, 2.3%Mg, 0.2%Cr,
0.3%Fe and the remainder aluminium. The monolithic material had a Young’s modulus of
72GPa, a Poisson’s ratio 0.33, a 0.2% proof stress of 416MPa, a UTS of 565MPa and an
elongation at failure of 14%.

The SiC particles had the following specification. Diameters range from 0.25 to 20 microns
with an average diameter of 3 microns to 4 microns. The average effective aspect ratio was
1.3 whilst the Young’s modulus was 468GPa and the Poisson’s ratio was 0.25. The particulate
strength was assumed to be statistical in nature, 5% of the particulates failing at a stress of
1.6GPa and 90% having failed at an average stress of 3.1GPa.

The MMC’s were produced by spray forming followed by extrusion. Heat treatment of the
material consisted of an initial high temperature solution treatment at 465◦Cheld for 45minutes
followed by a cold-water quench and thematerial aged for 16hr at 135◦C.ThemeasuredYoung’s
modulus was then 84GPa, the 0.2% proof stress 404MPa, the UTS 490MPa and its elongation
at failure was 2%.

2.2. Testing Procedure

The tests were carried out on a PC driven 50kN capacity Dartec servo-hydraulic testing
machine. All tests were conducted under strain control and constant amplitude. The tests were
conducted in air at a frequency of 0.25Hz using fully reversed loading (R =−1.0) with stress
and strain data being recorded for each cycle.

2.3. Results

The tests were carried out on both the monolithic andmetal matrix material at a constant
strain rate with the stress being allowed to vary (Fig. 1). In both cases, for the low strain rates,
the stresswas fairly constant throughout each individual experiment.However, once thematerial
was strained into the plastic region,work hardening took place and therefore the stress increased
throughout the life of the test (Fleming andDowson, 1999).

Fig. 1. Strain controlled fatigue test on monolithic Al7075 (a) and on Al7075/12%SiC (b)



Comparison of two models of numerical simulation of low cycle fatigue... 465

3. Theory

3.1. Theory of Cells

TheTheory of Cells (TOC) for short fibre and particulatemetalmatrix composites has been
developed byAboudi (1991). In themethod, an elasticmatrix is consideredwhose unidirectional
fibres of short length reinforce it. The fibres are assumed rectangular with dimensions of d1, l1
and h1 and arranged in thematrix as shown inFig. 2a. In a particulatemetalmatrix composite,
h1 will have a similar value to l1.

Given that the arrangement of fibre andmatrix are periodic, only one fibre and its surround
matrix need be analysed to give a representative section of the compositematerial (Fleming and
Dowson, 1999). This area is called a cell and is shown in Fig. 2b.

Fig. 2. (a) Schematic of anMMCwith a periodic array of fibres; (b) a representative cell of the
composite showing the eight sub-cells

As can be seen, each cell comprises eight sub-cells, one of fibre, labelled 1,1,1, and seven of
the matrix material. It is assumed, initially, that both fibres and matrix are perfectly elastic
materials, and the stresses are related to the strains as follows

σ
(α,β,γ) =C(αβγ)Z(αβγ)−Γ(αβγ)∆T (3.1)

where: σα,β,γ is the stress matrix of each sub-cell; Cαβγ – stiffness matrix representing mecha-
nical properties of each sub-cell; Γαβγ –matrix representing thermal properties of each sub-cell;
∆T – temperature difference between the reference stress free temperature and the working
temperature; whilst Z, the strain microvariables, is given by

Z
(αβγ) = [Φ

(αβγ)
1 ,X

(αβγ)
2 ,Ψ

(αβγ)
3 ,X

(αβγ)
1 +Φ

(αβγ)
2 ,Ψ

(αβγ)
1 −Φ

(αβγ)
3 ,Ψ

(αβγ)
2 +X

(αβγ)
3 ] (3.2)

with Φ
(αβγ)
1 ,Φ

(αβγ)
2 ,Φ

(αβγ)
3 ,X

(αβγ)
1 ,X

(αβγ)
2 ,X

(αβγ)
3 ,Ψ

(αβγ)
1 ,Ψ

(αβγ)
2 ,Ψ

(αβγ)
3 – representing strain

microvariables.

Note that when discussing the position of a sub-cell, a superscript notation is used, i.e.,
σ
(1,1,1) refers to the fibre whilst, for instance, σ(1,2,1) is the matrix material to the right of the
fibre, and subscript notation is used for the stress direction, e.g. σ11 for the direct stress in the
x1 direction.

Using a first order approximation, the displacement component at any point in the sub-cell
can be expressed as

u
(αβγ)
i =w

(αβγ)
i +x

(α)
1 Φ

(αβγ)
i +x

(β)
2 +X

(αβγ)
i +x

(γ)
3 Ψ

(αβγ)
i (3.3)



466 J.M. Temis et al.

where wi represents the displacement component of the centre of the sub cell and Φi, Xi, Ψi
characterises the lineardependenceof thedisplacements on the local co-ordinates x

(α)
1 ,x

(β)
2 ,x

(γ)
3 ,

and the displacement can be connected to strain using the expression

ε
(α,β,γ)
ij =

1

2

[

∂ju
(αβγ)
i +∂iu

(αβγ)
j

]

(3.4)

where ∂1 = ∂/∂x
α
1 , ∂2 = ∂∂x

β
2 , ∂3 = ∂/∂x

γ
3.

If the continuity of traction and displacement between appropriate sub-cells is assumed, it is
possible to calculate 26 strain microvariables generated by the theory. Once the stresses in each
sub-cell are calculated, the overall stresses can be found as follows

σij =
1

V

2
∑

α,β,γ=1

V (α,β,γ)σ
(α,β,γ)
ij (3.5)

where V represents the total volume of the cell and V (α,β,γ) the volume of a particular sub-cell.

3.2. Inelastic behaviour of metal matrix composites

A typical Metal Matrix Composites (MMC) comprises a fibre that behaves in an elastic
manner up to the breaking stress and amatrix thatwill show the typical elastoplastic behaviour
of a ductile material. To model a composite, an allowance must be made for the plastic defor-
mation of the ductile matrix. InAboudi’s development of the theory of cells, he used the unified
theory of plasticity developed by Bodner and Pardom. In this theory, plasticity is assumed to
be always present throughout the loading process. Although rigorous, Bodner’s et al. approach
adds a level of complexity to the analysis that may not be necessary. In the present study, this
has been abandoned in favour of a theory of plasticity proposedbyMendelson, inwhich the total
strain experienced by a stressed material is made up of both elastic and plastic strains. Using
this approach, plastic strain can be assumed to approximate to zerowhilst thematerial is within
the elastic region. This approach assumes simple elastic breakdown at the yield stress. For a
biaxial loading system, the failure theory can bemodified to include a shear yield component.

3.3. Randomly reinforced metal matrix composites and fatigue failure

The theory developed so far is for unidirectional short fibre andparticulate compositeswhere
the fibres are aligned in the X1 direction as shown in Fig. 2a. This places severe limitations on
the type of systems possible to analyse. However, a transformation can bemade on the stiffness
matrix of each sub-cell, using amethod first suggested byArridge. It is found that for amaterial
with randomly distributed fibres, the stiffness matrix reduces to three non-zero elements. Using
this modified stiffness matrix in the stress strain matrix, the strain in all three dimensions can
be calculated for each sub-cell in theMMC with randomly distributed fibres.
It can be assumed that fatigue failure will occur when one ormore of the sub-cells reaches a

critical stress level. In a one-dimensional loading system, the fatigue failure stress for thematrix
material can be obtained from an S/N curve. It can be assumed that failure of the composite
will occur for a similar number of cycles when anymatrix sub-cell reaches this cyclic stress level
provided the fibre has not reached its critical level. Therefore, the matrix is assumed to fail by
fatigue similarly to the homogenous material.
The fibres however may behave differently. Silicon carbide fibres are brittle and do not

exhibit fatigue failure. In the case of individual fibres, their critical fatigue stress is assumed to
be identical to their tensile failure stress. However, similar fibres exhibit a wide range of tensile
failure stresses, and this has important implications on the fatigue life of an MMC containing
such brittle fibres



Comparison of two models of numerical simulation of low cycle fatigue... 467

When the cyclic loading is high enough to break fibres, the outcome will be one of two
possible behaviour patterns. First is that only a certain percentage of fibres break due to the
fatigue loading, and the resulting redistribution of stresses ensures no more fibre breakage. In
this event, the compositewill eventually fracturebymatrix fatigue failure.The secondpossibility
is that as the fibres break the stress redistribution ensures more fibres will fail. Fatigue failure
of theMMC is then by fibre failure.

3.4. Model of material cyclic plasticity

Technology of FE low-cycle fatigue simulation is based on the usage of a number of models:
material cyclic behaviour model, concerned with one defect accumulation model and “died”
element model describing LCF crack origin and evolution (Temis, 1989; Temis, 1994; Putchkov
et al., 1995; Fleming and Temis, 2002; Temis et al., 2009). All of them are included in an
original FE system for low-cycle fatigue simulation analysis.The theory of cyclic strain plasticity
accompanied by the damage model based on the concepts of ultimate accumulated plasticity
strain has been successfully employed for the prediction of fatigue lives in highly stressed Ti
components (Putchkov et al., 1995). Models of strain cyclic plasticity and strain-accumulated
damage were employed to analyse behaviour and lifetime prediction of a plastic material with
small rigid inclusions in the area of low cyclic fatigue (Fleming and Temis, 2002).
The branch of cycle strain curve at every halfcycle (Temis, 1989; Temis, 1994; Putchkov et

al., 1995) is defined on a basis of three-parametric model ofmaterial behaviour according to the
accumulated plastic strain χ. This is defined as

χ=

nf
∑

n=0

|∆εp| (3.6)

The relationship between the stress σ∗n and strain ε
∗

n is represented in the form

σ∗ =















Eχε
∗ for ε∗ ¬ ε∗sχ

Eχε
∗

sχ+ bχdχ
[

f
(

εS +
ε∗−ε∗sχ
bχ

)

−σs
]

for ε∗ >ε∗sχ

ε∗sχ =
aχ
dχ
εs dχ =

Eχ
E

(3.7)

where σs and εs are the initial stress and strain values at 0.02% offset yield; aχ, bχ and dχ are
material-sensitive parameters describing the plastic deformation response of thematerial under
cyclic load; Eχ is a plastic strainpath χ-dependantmodulus such that E=Eχ(χ=0). Inpractice,
due to the Bauschinger effect, aχ may be defined as aχ =σyχ/σs, dχ is defined as stated above
anda transformation coefficient bχ relates thenon-linear portion of the stress–strain curveunder
monotonic loading to that observed under cyclic loading conditions.
This relationship is valid for thematrixmaterial, which under cyclic load has an alternating

elasto-plastic strain whilst the inclusion material remains elastic. Material constants describing
the abovementioned parameters of cyclic stress-strain were defined on the base of experimental
data obtained for the B95Al alloy, whose parameters are equivalent to that of the Al7075 alloy.
The aluminium alloy B95 under cyclic loading shows a decrease in the amplitude of plastic
strain and the hardening effect with an increase in the number of cycles. It should be noted that
Young’s modulus of the Al alloy under cyclic loading decreases in the first cycles and recovers
its value with increasing numbers of cycles. The main changes in the non-linear part of the Al
alloy stress-strain curve take place in the first fewhalf-cycles withmaximumchange taking place
between the first and the third cycle. After the third cycle, the modulus slowly increases until
the 100th cycles, when the change in modulus ceases.



468 J.M. Temis et al.

Tests conducted fordifferent engineeringmaterials showed that for constant-amplitude stress,
constant-amplitude strain and stress random-amplitude loading, the number of half-cycles nf
before failure at alternating-sign plastic deformation is related to the limiting value χmax by
the power law

nf =
(χmax
δ

)γ

(3.8)

here δ is the constant depending on the residual plastic strain value, γ is the parameter that
characterises the material ability “to cure” the cyclic loading damage.
The model based on relationship (3.8) allows simulating the cyclic life exhaust process of

the specimens under all conditions. If the value D = χ(n)/χmax(n) is taken as the measure
of damage, D = 1 defines the amount of half-cycle loading where the alternating-sign plastic
deformation takes place until a fatigue crack occurs. If, for different loading processes, the pa-
rameters defining the material mathematical model at cyclic deformation are equivalent to the
same damage measure D, all functions of χ in (3.7) may be replaced by the functions of the
dimensionless parameter D. The possibility of applying this approach has been studied experi-
mentally (Putchkov and Temis, 1987; Temis and Putchkov, 1992). The values of parameters to
be used in (3.7) were calculated on the base of experimental data from the results of constant
strain controlled low-cycle fatigue studies that were conducted on un-notched specimens.
Once the value of D of a particular finite element reaches unity, the fatigue damage failure

level of thematerial hasbeen reached and D=Dlimit. The elasticmodulus E of the elementwill
be reduced.The effect of this is that the contribution of the “died” element to the overall system
stiffness will be sharply reduced and a redistribution of stress and strain to the neighbourhood
elements will occur.Overall, some elements will experience a decrease in stresswhilst others will
be more highly stressed, and with the increasing number of cycles more “died” elements will
occur causing further stress redistribution. At that it is possible to estimate a number of cycles
to the crack origin from a condition of obtaining of failure rate to critical value Dlimit and to
retrace a crack evolution process that crack path is simulated the “died” elements (Temis et al.,
2009).
The cyclic elastoplastic material model based on the accumulated plastic strain concept is

not the only possible. Another approaches are possible when different models based on strain
energy estimation are applied formaterial behavior description. Review of thesemodels is given
in the paper by Macha (2001). Experimental researches carried out previously (Putchkov and
Temis, 1987; Temis andPutchkov, 1992) were shown that accumulated plastic strain and plastic
strain work under cyclic elastoplastic deformation bound by a linear relationship. Thismakes it
possible to considermodel (3.7) both in terms of the accumulated plastic strain and in terms of
the plastic strain work under cyclic loading.

3.5. FE calculation procedure

Consider cyclic deformation under which the load vector F is applied to component by the
following program

0→Fmax →Fmin →Fmax → . . .

If the strain and stress vectors in body points εk and σk correspond to the end of the k-th
halfcycle of loading, and εk+1 and σk+1 correspond to the end of the (k+1)-th halfcycle, then
for each halfcycle a variational relationship is carried out
∫

Ω

σ
T
q δεdΩ−

∫

Ω

(FΩ)
T
q δudΩ−

∫

S

(Fs)qδudS =0 (3.9)

where q= k, k+1 halfcycle number.



Comparison of two models of numerical simulation of low cycle fatigue... 469

Subtracting from relation (3.9) at q= k+1 the similar one at q= k, one gets that a problem
of the stress-strain condition simulation by change from the loading halfcycle to unloading one
becomes a solution to the next problem
∫

Ω

(∆σ)Tk+1δε dΩ−

∫

Ω

(∆FΩ)
T
k+1δu dΩ−

∫

SF

(∆FS)k+1δu dS=0 (3.10)

Having specified a relation form ∆σ (∆ε), relation (3.10) by thewell-knownmethod (Temis
and Putchkov, 1992) will become a finite element problem

Kk+1∆Uk+1 =∆Fk+1 (3.11)

where Kk+1 is the stiffnessmatrix of the (k+1)-th halfcycle definedby a step-by-step approach;
∆Uk+1 and ∆Fk+1 are vectors of increments of the displacement and loading at halfcycle,
correspondingly.
In that case, for the displacement vector at the (k+1)-th halfcycle, it is rightly

Uk+1 =Uk+ δUk+1 (3.12)

while the strain and stress in the calculated point are related by similar expressions

εk+1 = εk+∆εk+1 σk+1 =σk+∆σk+1 (3.13)

Abstraction of the concept of a unified curve of strain is used. At the (k+1)-th halfcycle, a
deformation process representing the point is to be on a branch of cycle strain curve (3.7) shown
in Fig. 3 by the thick line outgoing from the point σk.

Fig. 3. Strain curve difinition for (k+1)-th halfcycle: (a) strain curve branches subject to loading
direction

3.6. FE analytical model

In general, the stress-strain analysis of both theMMCand the individual cells within it are a
3Dplasticity problem.However as afirst approach (Fleming andTemis, 2002), the inclusions can
be approximated to a spherical body surrounded by a cylinder of thematrix material as shown
in Fig. 4a. The cylinder represents one cell of theMMC. The cylinder dimensions were defined
by the condition that they were inscribed in the parallelepiped with height H and the base
side 2R, and all calculations were conducted for the model using H = 2R, where R= 20mm
with the strain being in the direction of the cylinder axis, and was equal to the strain in the
specimen. The strains in the transverse directions were equal to −νMMCεy, where νMMC is
Poisson’s ratio of the MMC. In the elastic region, when εy < 0.008, a value of νMMC = 0.324



470 J.M. Temis et al.

was used, which was identical to the calculations carried out using the theory of cells. However,
in the plastic region, Poisson’s ratio must be continuously calculated as its value varies with
increasing strain. For the first approach, we shall calculate νMMC on the base of the volume
mixedmodel

νMMC =
νAl(elast)εe+0.5εp

εy
0.88+vSiC0.12 (3.14)

Here εe is the elastic part of thematrixmaterial strain, εp is the plastic part of thematerial
strain, and εy = εe+εp.

Fig. 4. (a) Schematic model of theMMC structure with a periodic array of fibres; (b) finite element
mesh for theMMCwith aspherical fibre

When considering the behaviour of the MMC under monotonic loading, a FE model with
spherical inclusions containing 4583 nodal points each has two DOFs and 9018 axisymmetric
simplex finite elements was used, as shown in Fig. 4b. When cyclic deformation was analysed,
the meshes needed to be refined to decrease approximation errors and to increase the stability
of FE models during various steps of mathematical simulation, whilst analysing an MMC with
spherical inclusions, the number of nodes in the FE model during cyclic plasticity analysis was
above 7000.

On the boundary AD, displacements in the x (radial) direction were fixed whilst displa-
cements on the boundary AB were fixed in the y (cylinders axial) direction. On the bounda-
ry DC, displacements ν = εyR in the y direction were given. Corresponding displacements
u=−νMMCεyR were given in the x direction on the boundary BC.

4. Results

Using the data experimentally obtained on the monolithic material, it was possible to predict
the fatigue life of theMMC using the Theory of Cells and FE simulation.

Thenumerical investigation of the compositewas carried out in strain-controlled tests for the
following strain amplitudes of symmetric cycles, ∆εay =0.5%, 0.6%, 0.7%, 0.8%,0.9%, 1%. As
for static loading simulation, it was assumed in the analysis that the inclusions in the composite
were spherical. At each calculation point, the current values of χ and damage measure D were
defined.
Figure 8 represents the accumulated plastic strain distribution after the fifth deforming half-

cycles with an elongation amplitude of 1%. I denotes the concentration zone of accumulated
plastic strain on the interface between the fiber and matrix. F denotes the concentration zone
in the direction of load action.



Comparison of two models of numerical simulation of low cycle fatigue... 471

The calculation process is stoppedwhen D=1 and an S/N diagramwas plotted. TheTOC
prediction, FE calculation up to D=1 and experimental results have been plotted as an S/N
curve, and this is shown on Fig. 6.

Fig. 5. Distribution of the accumulated Fig. 6. Constant strain against cycles to failure.

plastic strain in the composite after Experimental results and numerically predicted

5 half-cycles with the elongation life by TOC and FE calculation

amplitude of 1% ofMMCAl7075+12%SiC

Cthe clculation results with using the “died” elements technique are shown in Fig. 7.
LCF crack initiation is occurred at the interface zone point (Fig. 7a), then the crack grows
from the interface, initiates at the frontal zone (Fig. 7b), the interface and frontal cracks merge
(Fig. 7c), the crack initiates at the middle zone (Fig. 7d), interface-frontal and middle cracks
merge (Fig. 7e). At last the crack exits to the outside of the specimen and full fracture of the
material is occurred (Fig. 7f). This FE calculation up to full fracture with TOCprediction and
experimental results have been plotted as an S/N curve, and this is shown in Fig. 8.

Fig. 7. LCF crack growth: (a) crack initiation at the interface zone point; (b) interface crack growth and
crack initiation at the frontal zone; (c) interface and frontal crackmerging; (d) crack initiation at the
middle zone; (e) interface-frontal andmiddle cracksmerging; (f) exit of the crack to the outside and full

fracture of the material



472 J.M. Temis et al.

Fig. 8. Fatigue diagram up to full fracture

5. Discussion

InFig. 8,whichgives results for thenumerical investigation during fatigue loading it can easilybe
seen that two zones (I and F) appearwhere the accumulated plastic strain is at themaximum.
The first zone I, is on the interface between the inclusion and matrix. The second zone F is
formed at some distance from the inclusion in the direction of deformation, on the cylinder axis.
This would indicate that the first fatigue crack forms on the inclusion matrix interface, but it
is soon followed by the second crack that forms in the frontal zone after a certain number of
loading and unloading half-cycles. Obviously, these cracks precede the overall MMC’s failure.
As long as some region of the composite is at high enough stress, this process will occur at all
other levels of strain amplitude.

In the zones marked I and F in Fig. 5, the process of alternating-sign deformation occurs
in different ways. Despite the fact that the overall loading process occurs at a constant stra-
in amplitude, the zones I and F see a different stress and strain range. It should be noted
that the process of cyclic elasto-plastic deformation brings an essential non-homogeneity in the
distribution of plastic strain in the region close to the inclusion matrix interface. Due to the
non-uniformity of hardening of aluminium in the region near the interface, a zonewithwave-like
distribution of plastic strain arises. However, with the number of half-cycles increasing, such
a wave-like pattern is smoothed out. On the basic assumption that the number of half-cycles
before the appearance of a low-cycle fatigue crack is definedby the condition D=1, the number
of cycles for crack initiation were calculated. It must be remembered that this finite element
analysis only allows prediction of the number of cycles to crack initiation. For this type of fatigue
loading, it would be expected that the number of cycles from the first crack until failure lies
around 50-80% of the specimen cyclic lifetime. It should be expected therefore that, for a given
strain level, theoretical predictions would under-estimate the life of the material compared to
results obtained experimentally, and it can be seen fromFig. 6 that this is the case.

The TOC analysis was carried out over a larger number of cycles to failure. The numerical
analysis andthe results given inFig. 6 showgoodagreementbetween theTOCprediction andthe
experimental results at the higher number of cycles. At strain levels below 0.007, the agreement
iswellwithin the experimental error of the test.However, once stresses are in theplastic region of
theMMC, the prediction value of fatigue failure starts to deviate from the experimental results.
At these levels, the stress in the particulates is significantly higher than that experienced in any
sub-cell of the matrix. The matrix accommodates the increase in overall loading on the MMC



Comparison of two models of numerical simulation of low cycle fatigue... 473

by yielding, but the particulates are still deforming elastically. As the load increases, some of
theweaker particulates fail and there is a redistribution of the load between the particulate and
matrix. The theory predicts that, for this MMC, the stress level in the individual particulates
falls causing a corresponding increase in the stress on the matrix. So thefailure throughout the
test should have been by matrix fatigue failure. A comparison of both the numerical method
and the Theory of Cells is given in Fig. 6 and as can be seen the TOCoverestimates the fatigue
life of the composite whilst the numerical method using results obtained for the frontal zone
underestimate the fatigue life. Further work is being carried out on both these methods to give
a more accurate estimate of the fatigue life of metal matrix composites.

6. Conclusion

• Using fatigue data obtained from thematrixmaterial and information on the failure stress
of the particulate, it was possible tomake a prediction of the fatigue life of ametal matrix
composite material using both the Theory of Cells and a numerical method.

• The predicted fatigue life of the MMC using the method of cells was in close agreement
with the experimental results for life outside the low cycle fatigue regime of 1000 cycles or
less.

• Both the numerical method and the theory of cells can be used to predict fatigue life of a
metal matrix composite within an acceptable degree of accuracy.

• A procedure for mathematical simulation of the elastoplastic deformation processes in a
compositematerial including themetal basewith short rigid inclusionsunder cyclic loading
is presented. The results obtained show that the failure of the composite material occurs
in several steps. The presented results of prediction of time of the composite material full
fracture under cyclic loading are in well comparison with experimental data.

With the partially support by Russian Fund of Basic Researches (Project 12-01-00109) and by RF

Fundamental Science School Support Grant SS-4140.2008.8.

References

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sevier, Amsterdam, pp. 328

2. BatheE K.-J., 1996,Finite Element Procedures, New Jersey, Prentice Hall, Inc., pp. 1038

3. Fleming W.J., Dowson A.L., 1999, Prediction of the fatigue life of an aluminiummetal matrix
composite using theory of cells, Science and Engineering of Composite Materials, 8, 4, 181-189

4. FlemingW.J.,Temis J.M., 2002,Numerical simulation of cyclic plasticity anddamageof an alu-
miniummetal matrix composite with particulate SiC inclusions, International Journal of Fatigue,
24, 1079-1088

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Porównanie dwóch modeli symulacji numerycznej niskocyklowego zmęczenia materiału

plastycznego z drobnymi i sztywnymi wtrąceniami

Streszczenie

W pracy zastosowano teorię komórkową i numerycznąmetodę elementów skończonych opartą na od-
kształceniowo cyklicznej plastyczności, wprowadzono model zniszczenia oraz uwzględniono technologię
wytłaczania do określenia trwałości zmęczeniowej kompozytówmetalowo-ceramicznych.Wyniki obliczeń
porównano z badaniami doświadczalnymi. Zaobserwowano, że wyniki teoretyczne uzyskane z zastosowa-
niem teorii komórkowej były w zgodzie z eksperymentem dla niskocyklowego obciążenia zmęczeniowego,
tj. dla 1000 cykli i poniżej. Rezultaty otrzymanew drodze symulacji numerycznychmodelumatematycz-
nego wskazały, że zniszczenie zmęczeniowe przebiega w kilku etapach – w wyniku akumulacji uszkodzeń
i wskutek wzrostu szczeliny. Obliczenia czasu do pełnego pęknięcia kompozytu pokryły się z wynika-
mi doświadczalnymi w dobrym stopniu. Po porównaniu efektywności metody numerycznej oraz teorii
komórkowej stwierdzono w podsumowaniu, że obydwie metody mogą być stosowane do wyznaczania
trwałości zmęczeniowejmetalowo-ceramicznychmateriałów kompozytowych z uzyskaniem zadawalającej
dokładności.

Manuscript received June 15, 2011; accepted for print October 15, 2012