Engineering, 2010, 2, 318-321
doi:10.4236/eng.2010.25041 Published Online May 2010 (http://www.SciRP.org/journal/eng)
The Fatigue Fracture Criterion Based on
the Latent Energy Approach
Alexander R. Arutyunyan, Robert A. Arutyunyan
Faculty of Mathematics and Mechanics, Sankt-Petersburg State University, Sankt-Petersburg, Russia
Received November 24, 2009; revised February 8, 2010; accepted February 12, 2010
Abstract
The extensive literature on the fatigue problem, published for more than one hundred years, is reviewed by
the known scientists [1,2]. As it follows from these investigations, the fundamental amount of failures in en-
gineering practice connected with the fatigue fractures of materials and structure elements. The fatigue prob-
lem is complicated one and it is not solved yet. So the theoretical and experimental investigations of this
problem will be continued. In our paper the energy approach to formulate the fatigue strength criterion is
proposed. The criterion is based on the conception of the latent energy [3-7]. This conception was not ap-
plied previously to the fatigue problem. The latent energy is consumed to generate the irreversible deforma-
tion and to damage and fracture of metallic materials. So the fatigue fracture criterion can be formulated us-
ing the results of latent energy measurements in the macro experiments. This is most impotent advantage of
the proposed approach. The logistic function is used to describe the dependence of latent energy from the
value of irreversible deformation. It is assumed that the cyclic strength of metals is defined by the latent en-
ergy, stored in specimen, when it is reached the critical value in accordance with the logistic curve in a satu-
ration zone. This proposal is used to formulate the fatigue strength criterion. The functions and parameters of
received criterion are concretized and comparisons with experimental results for axial cyclic tension for
sheet aluminum alloy specimens are given.
Keywords: Fatigue Fracture Criterion, Energy Approach, Latent Energy, Heat Energy, Logistics Function,
Damage Parameter
1. Introduction
The energy methods are the most fundamental approach
for receiving the deformation and fracture laws for mate-
rials and structures. It is well known that the fracture me-
chanics as a science was originated by a Griffith’s energy
concept. In this paper the results of recent experimental
investigations on latent energy stored in metallic materials
during the plastic deformation are used to formulate the
fatigue fracture criterion for metallic materials.
In the process of plastic deformation the significant
changes in metallic materials due to increasing of the inte-
rior energy are occurred. During the plastic deformation
heat is emitted so the deformation energy is transformed
into the heat. But not the whole work is emitted to heat. A
part of it called the latent energy of deformation [3-7] is
transformed to the potential energy of lattice distortion.
The latent energy in average constitutes 10-15% of full
deformation energy.
The methods of measuring the latent energy are re-
duced to measure, with a sufficient accuracy, the defor-
mation energy and emitted heat. In this case the energy
balance is defined by the first low of thermodynamics,
which can be written as U = E – Q, where U is interior
(latent) energy, E is the deformation energy and Q is the
heat.
The results of experiments on latent energy carried out
in recent years [8-10] show that the relation of the latent
energy or the stored energy is expressed in the form of
logistics function. Our experiments [11] conducted on
cyclic bending of specimens made of construction mate-
rials show that the relations of different physical and
mechanical characteristics from number of cycles has
two precisely expressed regions. Similarly to the logis-
tics function these relations have the precisely expressed
point of inflection. These results are the additional con-
firmation for the introduction of the logistics function
when formulating the fatigue strength criterion. We as-

A. R. ARUTYUNYAN ET AL.319
sume that metals cyclic strength is defined by the value
of latent energy, when it is reached the critical value.
2. Damage Parameter, Based on the Relative
Changes of Latent Energy
Let’s introduce the parameter , which characterizes the
relative changes of latent energy
γ
*
W
γ=W, where W is the
current and W* is the limiting values of latent energy,
0*

  , where , are initial and limiting val-
ues of parameter , consequently, .
0
γ* γ
γ01
*
γ<γ
It is well known that the logistics function describes
the different processes which lead to saturation [12-14],
and it is the solution of the following kinetic equation
(
*
dγ=Aγγ-γ
dε)
(1)
where
A
and are constants. In the common case
*
γ
A
and can be considered as functions of stress
, .
*
γ
()σA= A()γγσ
**
=
Solving Equation (1) with the initial conditions0


,
, we will have the following relation, expressed in
the form of logistic function
0
γ=γ
0
11*
*
-A εγ
*
γ
γ=γ
+-e
γ




(2)
The theoretical curve (2) of latent energy accumula-
tion is shown in Figure 1. The results of experiments on
specimens made of aluminum alloy 2024-T351 [10] are
shown in this figure by points. In calculations the fol-
lowing values of constants were used: A = 26, 852,
and . It is seen that the experimental
points of latent energy accumulation are well described
by the logistic function (2).
00,02γ=0,94 *
γ=
Figure 1. Solid line－the theoretical curve (2), points－
experimental results [10].
Introducing the fracture condition in the form
*
γ=a γ

, where a is constant, , from (2) we
can receive the following relation
0,5 1a
0
11
1
*
*
γ a
ε=ln -
Aγ-aγ









(3)
3. The Fatigue Fracture Criterion
It is known [15,16], that in process of cyclic loadings, in
particular in cyclic tension regime, the accumulation of
plastic deformation from cycle to cycle is occurred. The
character of the plastic deformation accumulation is simi-
lar to creep curves. So to describe such curves the power
Norton’s law can be used. This law, expressed through the
number of cycles, can be written in the following form
m
dε=Bσ
dN (4)
We will assume that the stress in one cycle is constant
and is equal to the amplitude of stress

. Using the ini-
tial condition0N

,0


the solution of Equation (4)
can be found as
m
ε=BσN (5)
From the relations (3) and (5) follows the fatigue frac-
ture criterion
11
1
*
m
0*
γ a
N= ln-
-a γABσγ







(6)
Experiments show [17] that the latent energy de-
creases when stress is increases. So this relation can be
expressed by the decreasing power or exponential func-
tions of stress
(1 )-β
*
γ=+cσ (7)
-ασ
*
γ=e (8)
where ,
α
β
, are constants. c
In view (7) and (8) the fracture criterion (6) will be
written as
0
(1)(1) 1
1
β-β
m
+cσa+cσ
N= ln-
-a γABσ







(9)
0
11
1
-ασ
m-ασ
ae
N=ln-
-a γABσ×e








(10)
4. Comparison of the Fatigue Fracture Criterions
with the Experimental Results
The theoretical fatigue curves (9) and (10) were compared

A. R. ARUTYUNYAN ET AL. 320
Figure 2. The theoretical fatigue curves (9)－solid line and (10)－dash line. Cross points－experimental results [15].
with the experimental results received in axial cyclic
tensile tests on sheet specimens made of aluminum alloy
2024-T3 [15]. The mechanical properties of this alloy are
more similar to the consequent properties of aluminum
alloy used in paper [10].
The theoretical fatigue curves (9) and (10) are shown
in Figure 2 by solid and dash lines consequently. The
following values of constants were accepted:
26,852A
0,9a
, , ,
,
4m
 
-1
-15 -4
3 10MPacyclesB


-3 -1
10MPa

=7,5α,

-1
MPac
0,61β=
,
.
The experimental results are marked by cross points. It
is seen that these curves are well describe the experi-
mental points on the whole range of fatigue curves.
5. Conclusions
The damage parameter defined as the relative changes of
latent energy is introduced. It is shown, that this parame-
ter is governed by the differential equation with the solu-
tion in the form of the logistics function, capable to de-
scribe different processes leading to saturation.
The functions and parameters of received criterion are
concretized and comparisons with the experimental re-
sults for axial cyclic tension for sheet aluminum alloy
specimens are given. A good agreement of theoretical
and experimental results is received.
6. Acknowledgements
Financial support of the Russian Foundation for Basic
Research (Grant N 09-01-00513) is gratefully acknowl-
edged.
7
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