Journal of Minerals and Materials Characterization and Engineering, 2012, 11, 953-960
Published Online October 2012 (
Finite Element Modeling of Stress Strain Curve and Micro
Stress and Micro Strain Distributions of Titanium Alloys—
A Review
Gangi Setti Srinivasu, Narasimha Rao Raja
Department Mechanical Engineering, National Institute of Technology, Warangal, India
Received July 15, 2012; revised August 20, 2012; accepted August 29, 2012
Most of the alloys like titanium, steel, brass, copper, etc., are used in engineering applications like automobile, aero-
space, marine etc., consist of two or more phases. If a material consists of two or more phases or components it is very
difficult to predict the properties like mechanical and other properties based on simple laws such as rule of mixtures.
Titanium alloys are capable of producing different microstructures when it subjected to heat treatments, so much of
money and time are squandering to study the effect of microstructure on mechanical properties of titanium alloys. This
squandering can be reduced with the help of modeling and optimization techniques. There are many modeling tech-
niques like Finite element method, Mat lab, Mathematical modeling etc. are available. But Finite element method is
widely used for prediction because of capable of producing distributions of stresses and strains at any different loads.
From the literature it is observed that there is a good agreement between the calculated and measured stress strain
curves. This review paper describes the effect of volume fraction and grain size of alpha phase on the stress strain curve
of the titanium alloys. It also can predict the effect of strength ratio on stress strain curve by using FEM. This informa-
tion will be of great use in designing and selecting the titanium alloys for various engineering applications.
Keywords: Titanium Alloys; Finite Element Modeling; Stress-Strain Curve
1. Introduction
Titanium alloys are considered as an important material
because of its excellent combination of properties such as
elevated strength to weight ratio, high fatigue life, tough-
ness and excellent resistance to corrosion. It is heat treat-
able and hot or cold deformed [1,2] and has gained more
and more applications in many fields like aerospace, ma-
rine etc. [3,4]. Titanium alloys are broadly classified into
three types based on the chemical composition of the
alloys and each of these families serves a specific role.
This classification consists of α and near-α alloys, α/β
alloys and β alloys. Low temperature allotrope form of
titanium is α, and the microstructure of α and near-α al-
loys consists predominantly of the α-phase. The α/β al-
loys consist of mostly α phase and they do have more β
phase. β is the high temperature allotropic form of tita-
nium. Mostly β-alloys consist not fully β phase, but in
very general terms, they are capable of retaining 100% β
when quenched from the β-phase field [5-7].
Diffusion and diffusion less transformations taking
place during heat treatment are important factors for de-
termining the functional characteristics of these materials.
These transformations are controlled by means of heat
treatment selection and the chemical composition of the
phases that are present in these alloys, enable advance-
ment in operational properties [8,9]. Mechanical proper-
ties of titanium alloys are important criteria for both in
aerospace as well as industrial applications. Microstruc-
ture of the alloy is one of the important factors control-
ling both the tensile strength and the fatigue strength
[10,11]. The properties of titanium alloys can be varied
over a wide range by heat treatment or thermo-mecha-
nical processing [12-17]. The microstructure of the alloy
can be changed from equiaxial through bi-modal to fully
lamellar. A bi-modal microstructure is reported to have
advantages in terms of yield stress, tensile stress and
ductility and fatigue stress. A fully lamellar structure is
characterized by high fatigue crack propagation resis-
tance and high fracture toughness. The important para-
meters for a lamellar structure with respect to mechanical
properties are the β-grain size, size of the colonies of
α-phase lamellae, thickness of the α-lamellae and the
nature of the inter lamellar interface (β-phase) [18].
1.1. Modelling of Titanium Alloys
Titanium alloys exhibits different morphologies and vo-
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lume fractions of α when it is subjected to heat treat-
ments. The way is to know the effect of these parameters
on the properties of titanium alloys is experimentation,
thus squandering of money, time and material. To reduce
this squandering, modeling techniques are used in order
to get the effect of parameters on properties. Modeling of
material properties can be done by using different tech-
niques such as finite element method (FEM), mathema-
tical modeling, artificial neural networking etc.
1.2. Artificial Neural Network (ANN) Modelling
Malinov et al. [19] developed a model of artificial neural
network for simulation of time-temperature—transfor-
mation (TTT) diagrams for titanium alloy. This model
predicted the influence of aluminium, vanadium, molyb-
denum and oxygen on transformation kinetics in titanium
alloys. The results are in good agreement with the theory
of phase transformation. Using the model, TTT diagrams
for some commercial alloys were predicted. Some other
authors predicted the tensile properties, correlation be-
tween the processing parameters on the properties of
titanium alloys, etc. by using artificial neural network
modeling [20-23].
1.3. Mathematical Modelling
Masaharu et al. [24] discussed the effects of shape and
volume fraction of a second phase on stress states and
deformation behavior of two-phase materials with the
help of empirical relations. They embedded inhomoge-
neous spheroidal (second phase) inclusions in a matrix.
Analytical expressions to describe the stress states in
elastically and plastically deformed two-phase materials
are obtained with the Eshelby method and the Mori-
Tanaka concept of the “average stress”. Considering that
the second phase is also plastically deformable, the over-
all deformation behavior of the two-phase materials is
discussed with the results obtained by the evaluation of
the stress and strain distributions in the materials. Some
of the authors predicted the stress strain curve with the
help of empirical relations [25-28] and phase transfor-
mations of titanium alloys [29,30].
1.4. Finite Element (FE) Modelling
Finite element method (FEM) is one of the most used
modeling technique in worldwide, different software
packages are available in the market like ANSYS,
NASTRAN based on the FEM. Jindrich Jinoch et al. [31]
calculated the stress strain curve of α-β Ti-8Mn Alloy by
using FEM, there is a small error between the calculated
curve and measured curve but the fit is acceptable. S.
Neti [32,33] modeled the deformation behavior of tita-
nium alloy and the effect of strength ration on the stress
strain curve. The author used NASTRAN software for
the modeling. Sreeramamurthy Ankem et al. [34] calcu-
lated the effect of volume fraction of second phase on the
tensile properties of titanium alloys by using FEM. Some
other authors calculated the mechanical properties of
titanium alloys, fracture surface, porosity effects by using
FEM [35-40].
The aim of this review paper is to give idea on the
importance of finite element modeling and the appli-
cation of Finite element modeling to predict or model the
mechanical properties of titanium alloys. It also explains
how the advancements taking place in the modeling from
past 20 years.
2. Finite Element Modeling (FEM) and
2.1. Modelling of Stress Strain Curve
For the modeling of stress strain curve, a FE model can
be developed in such a way that it should equivalent to
the microstructure and this FE model consist of elements
which are linked with nodes. In 1978 Jindrich Jinoch et
al. [31] calculated the stress strain curve of α-β Ti-8Mn
Alloy by using FEM. The calculated curve lies below the
experimental curve, and this may be due to finer grain
sizes in the Ti-8Mn alloy and the contribution of the
interface phase, which were not considered in modeling.
Figure 1 shows the microstructure of Ti-8Mn alloy at
volume fraction of α is 17%. Figure 2 is the corres-
ponding FE model, after application of different loads the
stress strain curve is calculated and the Figure 3 shows
the comparison between the calculated and measured
stress strain curve.
2.2. Effect of Grain Size of Secondary Phase (α)
on the Stress Strain Curve
In 1982 Sreeramamurthy Ankem et al. [34] calculated the
Figure 1. Microstructure of α-β Ti-8Mn alloy at 17% vo-
lume fraction α.
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Figure 2. FEM Model.
Figure 3. Comparison of stress strain curves between the
measured and calculated one.
effect of particle size, matrix, and volume fraction on the
stress-strain relations of α-β titanium alloys by using
FEM. It was observed that for a given volume fraction,
the calculated stress-strain curve was higher for a finer
particle size than for a coarse particle size within the
range of the strains considered, and this behavior was
seen for all the different volume fraction alloys con-
sidered. For a 50:50 volume percent α-β alloy, the stress-
strain curve with β, the stronger phase, as the matrix was
higher than that with α, the softer phase, as the matrix.
The calculated stress-strain curves for four different vo-
lume percent α alloys were compared with their corre-
sponding measured curves, and good agreement was
found. Figure 4 shows the FEM models of the titanium
alloys at different grain sizes for a particular volume
fraction of α at 16.3% volume percent.
Figure 4. FEM Model (a) Finer mesh; (b) Medium mesh; (c)
Coarser mesh.
2.3. Effect of Strength Ratio and Volume
Fraction of Secondary Phase (α) on the
Stress Strain Curve
In 1991 S. Neti [32,33] has studied the effect of the
strength ratio on the stress-strain behavior of various
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two-phase materials by the finite element method where
the volume per cent of the second phase was varied from
20 to 80 vol.%. The strength ratio of the harder β phase
to the softer α phase was varied from approximately 2 to
5 by keeping the α phase strength (0.2% YS) was cons-
tant at 368 MPa. From his study it was observed that the
flow stress of any given two-phase material did not vary
linearly with the strength ratio. In addition to that, for a
particular strength ratio, the flow stress of the two-phase
material did not vary linearly with volume per cent. For
materials with less than 40 vol.% β phase, the increase in
the strength of the two-phase materials with either
volume per cent of β or strength ratio was very small.
This was attributed to the fact that the softer α phase,
being the matrix, could deform relatively freely without
the phase β undergoing plastic deformation up to a
plastic strain of 0.5%. When the volume per cent of β
was much greater than 50 vol.%, the softer a phase could
not deform freely without the harder β phase undergoing
plastic deformation, resulting in increased flow stresses
with increased strength ratios. Figure 5 shows the dif-
ferent stress-strain curves used for simulation. Figure 6
shows the FE models with different percentages of α and
β. Figure 7 indicates the β stress strain curves at different
ranges in order to get different strength ratios which are
used in modeling.
2.4. Distributions of Stress and Strain
Recently most of the authors used ANSYS software for
the analysis of stress strain curve and its distributions [39,
41,42]. In the year 2008 Zhao Xiqing et al. [39] did work
on the distributions of stresses and strains within the
phases at different loads. By comparing the calculated
stress-strain curve with the measured one, it can be seen
that the fit is acceptable. Thus, the FE model built in this
work is effective. According to the above mentioned
model, the distributions of stress and strain in the α and β
Figure 5. Comparison between the stress strain curves at
different grain sizes of α.
Figure 6. FE Model (a) 20% β and 80% α; (b) 60% β and
40% α; (c) 50% β and 50% α.
phases were simulated. It is observed from the author
work that the stress gradients exist in both α and β phases,
and the distributions of stress are non-homogeneous. The
stress inside the phase is generally higher than the near
interface. Meanwhile, the stress in the α phase is lower
Copyright © 2012 SciRes. JMMCE
Figure 7. Stress-strain curves of α and β used for FEM cal-
culations; note that four different β curves are used to de-
termine the effect of strength difference.
than that in the β phase, whereas the strain in the α phase
is higher than that in the β phase. Figures 10 and 11
shoes the vonmises distributions of stress and strain at
different loads. These figures are showing the variations
of stresses and strains with the application of different
Finite element modeling is used in so many other
fields like heat transfer in furnaces, and buckling of GRP
etc. [43,44]. FEM calculated stress-strain curves for four
different β to α ratio at (a) 80% β and 20% α (b) 20% β
and 80% α are depicted in Figure 8. A variation of 0.2%
flow stress with strength ratio for different volume per
cents of α and β phases are depicted in Figure 9.
3. Summary
Titanium alloys play an important role in the aerospace
industry. To maintain the prominent position in the in-
dustry efforts must be directed towards cost reduction of
titanium structure. So in this cost reduction process, nu-
merical methods play an important role in characteriza-
tion of titanium alloys. The finite element modeling one
of the useful numerical techniques in predicting me-
chanical properties for titanium alloys, because of the
small error between the measured and predicted results.
It is very much useful in case of titanium alloys because
Figure 8. FEM calculated stress-strain curves for four dif-
ferent β to α ratio at (a) 80% β and 20% α (b) 20% β and
80% α.
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Figure 9. Variation of 0.2% flow stress with strength ratio
for different volume per cents of α and β phases.
Figure 10. Distributions of von Mises stress with different
loads: (a) 276 MPa; (b) 690 MPa.
Figure 11. Distributions of von Mises strain with different
loads: (a) 276 MPa; (b) 690 MPa.
it exhibits different microstructures. It is know that tita-
nium is very expensive material, by using these modeling
techniques a lot of amount and time can be saved. This
finite element modeling will be great useful for the de-
signing and selection of titanium alloys for different en-
gineering applications.
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