Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 2, pp. 419-430, Warsaw 2013

TWO-DIMENSIONAL EXPERIMENTAL ELASTIC-PLASTIC STRAIN AND

STRESS ANALYSIS

Barbara Kozłowska

Warsaw University of Technology, Faculty of Mechatronical Engineering, Warsaw, Poland

e-mail: b.kozlowska@mchtr.pw.edu.pl

In the paper, the quantitative analysis of elastic-plastic state of strain and stress in two-
dimensionalmodelswithdifferent stress concentrators is presented.Theexperimental testing
was carried out by the photoelastic coatingmethod on duralumin elements loadedby tensile
stresses. For strain separation, the analytical method of characteristics, taking advantage of
an isochromaticpatternonly,wasapplied.The strainandstress components in elastic-plastic
areas around stress concentrators were calculated using the multi-sectional schematization
of thematerial curve. The effects of investigations have been comparedwith those obtained
from Moiré method and numerical calculation (FEM). The discussion of results has been
presented.

Key words: mechanics of solids, experimental methods, elastic-plastic states

1. Introduction

Exploitation conditions of many constructions often create partial material plastifying. Consi-
dering the safety of the whole structure, the knowledge about the process of deformation after
crossing the yield point is very important.
The simulation of non-linear problems, like themodelling of a structuralmaterial – the cha-

racter of strain hardening or conversion from elastic to plastic state, may causemany difficulties
as well as the modelling of real object itself (shape, loading conditions, etc.).
Thewidely applied nowadays numericalmethods (FEM, e.g) always contain some inaccuracy

and their results should be verified experimentally.
Experimental elastic-plastic states investigations are conducted inmany research centers all

over the world, and a great variety of problems is considered like: estimation of the bearing
capacity of constructional elements and the localization of places, where plastic pseudo-joints
can be created (Livieri and Nicoletto, 2003; Milke et al., 2000; Padmanabhan et al., 2006);
the analysis of residual stress caused by material plastifying (e.g in the process of machining)
(Gurova et al., 1998; Kelleher et al., 2003; Rasty et al., 2007); estimation of residual (plastic)
strain remaining in thematerial after removing loading (Diaz et al., 2004; Haldrup et al., 2008)
and the investigation of large (plastic) deformation (Fontanari et al., 2006; Franck et al., 2007;
Tong, 2004).
Apart from these specific problems of elastic-plastic states investigation, the general case of

quantitative strain and stress analysis in the plastified zones is of great importance. It concerns
any object working in the over-elastic range and makes possible to estimate how the partial
plastifying of the element affect the state of the total structure.
In such cases, experimentalmethods,which allow analyzing real constructions underworking

conditions, seem to be very useful.
One of the experimental methods, which can be applied to elastic-plastic states analysis, is

the photoelastic coatingmethod. Themethod gives information about deformation of thewhole
tested area (not only at several points). It can be used to investigate objects of different sizes



420 B. Kozłowska

and shapes, loaded in different ways (dynamically and statically) also in the over-elastic range
and at various temperatures (−20◦C-50◦C).

2. Photoelastic coating method in elastic-plastic states analysis

The photoelastic coatingmethodmakes use of the effect of optical birefringencewhich occurs in
some transparentmaterials under loading (Zandman et al., 1977). It is based on the assumption
that there is a univocal dependence between the strain occurring on the surface of an analyzed
element and the deformation of the thin layer of the birefringent material integrally bonded to
this surface.When the object is loaded, the surface strains are transmitted to the coating, which
may be observed through a reflection polariscope. In that case, the loaded coating exhibits two
families of fringe patterns:

• isoclinic fringes – lines, along which the directions of the principal strains are constant

αn = const (2.1)

where αn is the angle between the direction of the greater principal strain component and
the accepted reference direction x;

• isochromatic fringes – lines, along which the difference of the principal strains is constant

ε1−ε2 =Nfε (2.2)

where: ε1, ε2 – principal strain components, N – value of the isochromatic order,
fε – strain constant of the photoelastic coating.

Obtained from the photoelastic coatingmethod information is not sufficient for determining
all strain components in the general case of two-dimensional state of stress occurring on the
surface of the analyzed object. Besides, the measurement of the isoclinic parameter is labour-
consuming and usually not precise. Therefore, to the principal strain separation one may use
additional information obtained from other experimental methods or analytical (or numerical)
methods.

The method of photoelastic coating can be applied to strain analysis in partly plastified
elements. This possibility results from the linear relation between the photoelastic effect and
strain in the birefringentmaterial in awide range. The characteristics ofmost epoxy resins used
as photoelastic coatings are linear in the range of strains, where the tested material is already
plastified.
One of themethods,which enable the experimental data analysis, is themethod of characte-

ristics (Szczepiński, 1964). It allows determining strain components, in a relatively simple way,
using the isochromatic pattern only.

Themethod of characteristics was originally proposed for the elastic states analysis, because
for strain separation the relations between the strain and stress components (Hooke’s law) as
well as the equations of equilibrium and the strain compatibility condition are used (Kapkowski
et al., 1987).

The application of the method to strain separation in plastified zones required formulating
relations between the strain and stress components (for the two-dimensional state of stress) in
the non-linear part of σ(ε) curve (Kapkowski and Kozłowska, 1993; Kozłowska, 1996).

These relations were derived on the basis of the accepted geometrical model of the material
characteristic, where the experimentally obtained curve is replaced by n line segments (Fig. 1a).
Each of these segments describes a different state of the material and is characterized by a
different modulus of elasticity Ei and the Poisson ratio νi. The points Ki define the change of



Two-dimensional experimental elastic-plastic strain and stress analysis 421

the material state and each time they are matched for the constructional material on the basis
of the real σ(ε) curve. They determine the change of the slope angle of straight segments of
the material characteristic (i=1 refers to the elastic state and i=2, . . . ,n−1 stands for the
successive over-elastic states). These points correspond to the stresses σ(K)i.
Deriving relationships between the strain and stress components in the over-elastic range for

a two-dimensional state of stress using the multi-sectional model requires making assumptions
referring to the course of the deformation process after crossing the yield point (Kapkowski and
Kozłowska, 1993; Kozłowska, 1996). They are as follows:

• The relationship σ(ε) has the same character (the same values of themodulus of elasticity
and thePoisson ratio) for both principal directions as under the uniaxial tension. Itmeans
that successive segments of the characteristics: σ(ε), σ1(ε1), σ2(ε2) are parallel (Fig. 1b).

Where: (K1)i, (K2)i – points analogous to the points (K)i (points of the change of the
slope angle of straight segments of thematerial characteristic – Fig. 1a) for both principal
directions of the two-dimensional state of stress.

• The strain hardening of the material has the isotropic character and the change of state
of the material takes place at the constant ratio σ2/σ1.

Fig. 1. Multi-sectional model of the material characteristic: (a) for uniaxial tension, (b) for
two-dimensional state of stress

The principal stresses characterizing material conversion from the elastic state to the first
over-elastic state and from one over-elastic state to every next one can be described as

(σ1)(K1)i =(k1)iσ(K)i (σ2)(K2)i =(k2)iσ(K)i (2.3)

where: (σ1)(K1)i, (σ2)(K2)i – stresses corresponding to the points of the change of the slope
angle of straight segments of the material characteristic for both principal directions of two-
dimensional state of stress (i = 1, . . . ,n− 1); (k1)i, (k2)i – coefficients characterizing stress
components at the moment of material state change in both principal directions.
The assumptions enable univocal determination of themulti-sectional schematization of the

σ(ε) relationship for the two-dimensional state of stress. If the principal stress components in
the points (K1)i, (K2)i are described by formulas (2.3), the coefficients (k1)i, (k2)i satisfy the
relationships

(k1)1 =(k1)2 = · · ·=(k1)i = · · ·=(k1)n−1 = k1

(k2)1 =(k2)2 = · · ·=(k2)i = · · ·=(k2)n−1 = k2
(2.4)

Defining of multi-sectional model of the material characteristic for two-dimensional state of
stress enables one to formulate relationships between the principal strain and stress components
at any over-elastic stage



422 B. Kozłowska

(ε1)m =
1

Em
[(σ1)m−νm(σ2)m]+

m
∑

i=2

σ(K)i−1

[ 1

Ei−1
(k1−νi−1k2)−

1

Ei
(k1−νik2)

]

(ε2)m =
1

Em
[(σ2)m−νm(σ1)m]+

m
∑

i=2

σ(K)i−1

[ 1

Ei−1
(k2−νi−1k1)−

1

Ei
(k2−νik1)

]

(2.5)

where: (ε1)m, (ε2)m – principal strain components at the m-th stage of over-elastic state of the
material; (σ1)m, (σ2)m – principal stress components at the m-th stage of over-elastic state of
the material; νm – Poisson’s ratio at the m-th stage of over-elastic state of the material; Em –
modulus of elasticity at the m-th stage of over-elastic state of thematerial; νi,Ei –modulus of
elasticity andPoisson’s ratio corresponding to succeeding segments of thematerial characteristic
(i=2. . . . ,m); k1, k2 – plastification coefficients in both principal directions.
The plastification coefficients k1, k2 characterize the state of stress in every point of the

element atmoment of appearing first over-elastic strains. From these assumptions, for themulti-
sectionalmodel of thematerial characteristic it results that in these points theremustbe fulfilled
the yield criterion, because the material is not elastic any longer. Taking into account a more
general Huber-Mises yield criterion and accepting the stress σ(K)1 (Fig. 1a) for the yield point,
we have

k21 −k1k2+k
2
2 =1 (2.6)

where relationship (2.3) for i=1was used.
Introducing a more general coefficient (plastification coefficient of the material)

kpl =
k1−k2
2

(2.7)

the coefficients k1 and k2 can be described as

k1 =
√

1−3k2
pl
+kpl k2 =

√

1−3k2
pl
−kpl (2.8)

To analyze the results obtained by the photoelastic coatingmethod, it is more convenient to
use the sum and the difference of the principal strain and stress components

(ε1−ε2)m =(1+νm)
[ 1

Em
(σ1−σ2)m+2kpl(wR)m

]

(ε1+ε2)m =(1−νm)
[ 1

Em
(σ1+σ2)m+2

√

1−3k2
pl
(wS)m

]

(2.9)

where the following designations were introduced

(wR)m =
1

1+νm

m
∑

i=2

σ(K)i−1

(1+νi−1
Ei−1

−

1+νi
Ei

)

(wS)m =
1

1−νm

m
∑

i=2

σ(K)i−1

(1−νi−1
Ei−1

−

1−νi
Ei

)

(2.10)

The equations of equilibrium for the two-dimensional state of stress described in theprincipal
system can be presented in the form

∂s

∂x
+
∂r

∂x
cos2αn+

∂r

∂y
sin2αn+2r

(∂αn
∂y
cos2αn−

∂αn
∂x
sin2αn

)

=0

∂s

∂y
+
∂r

∂x
sin2αn−

∂r

∂y
cos2αn+2r

(∂αn
∂y
sin2αn+

∂αn
∂x
cos2αn

)

=0

(2.11)

where designations: s=(σ1+σ2)/2 and r=(σ1−σ2)/2 were introduced.



Two-dimensional experimental elastic-plastic strain and stress analysis 423

Using the strain compatibility condition for the two-dimensional state of stress (Timoshenko
and Goodier, 1962), relationships (2.9) and dependence (2.2) for the photoelastic coating, equ-
ations (2.11) can be presented in a form which is more convenient for the analysis using data
obtained from the photoelastic coating method. The difference of principal strain (described by
the values of isochromatic order N) is the known value, while the searched values are: the sum
of principal strain and the angle between the direction of the greater principal strain component
and the accepted reference direction x

1

1−νm

∂

∂x

( e

fε

)

−2N sin2αn
∂αn
∂x
+2N cosαn

∂αn
∂y

=(wS)m
∂

∂x

(

2
√

1−3k2
pl

)

−

(∂N

∂x
cos2αn+

∂N

∂y
sin2αn

)

1

1−νm

∂

∂y

( e

fε

)

+2N cos2αn
∂αn
∂x
+2N sinαn

∂αn
∂y

=(wS)m
∂

∂y

(

2
√

1−3k2
pl

)

−

(∂N

∂x
sin2αn−

∂N

∂y
cos2αn

)

(2.12)

where

e=
ε1+ε2
2

and fε – strain constant of the photoelastic coating; αn – angle between the direction of the
greater principal strain component and the accepted reference direction x.
To simplificate the form of equations (2.12), a new variable N was introduced, whose value

may be determined on the basis of distribution of the isochromatic pattern N(x,y)

N =
N

2(1+νm)
− (wR)m2kpl (2.13)

The obtained system of equations, quasi-linear in relation to derivatives of the unknown
functions: e/fε and αn, is always of a hyperbolic type andhas two families of real characteristics
described by differential equations:
— first family

dy

dx
=tan

(

αn+
π

4

)

1

2(1−νm)
d
( e

fε

)

+Nd(αn)=
1

2
(wS)md

(

2
√

1−3k2
pl

)

+
1

2

(∂N

∂y
dx−

∂N

∂x
dy
)

(2.14)

— second family

dy

dx
=tan

(

αn−
π

4

)

1

2(1−νm)
d
( e

fε

)

−Nd(αn)=
1

2
(wS)md

(

2
√

1−3k2
pl

)

+
1

2

(∂N

∂x
dy−

∂N

∂y
dx
)

(2.15)

These equations allow calculating the values of e/fε and αn by numerical integration over
the characteristic curves with the appropriate boundary conditions (Kapkowski andKozłowska,
1993;Kozłowska, 1996). They supply information necessary for strain separation in the analyzed
elastic-plastic zone and full determination of the state of strain.
The relationships between stress and strain components found for the nonlinear part of

materials characteristic (2.5) may also be used for calculating stress components in plastified
zones of the analyzed elements as well as for strain separation.



424 B. Kozłowska

3. Experimental testing

The investigation of elastic-plastic states by the photoelastic coating method was performed
on two-dimensional models weakened by different stress concentrators (holes) and subjected to
tensile stresses – Fig. 2. This type of elements and loadings occurs frequently in structural com-
ponents (Pastrama et al., 2005;Wung et al., 2001) andbecause of technological or constructional
cut-outs needs special attention.

Fig. 2. Models of constructional elements

The models were made of a duralumin sheet 3mm thick, from which stripes of 100mm in
width and 450mm in length were cut out. The length of the stripes was taken large enough to
compensate potential non-uniformity of tensile stresses distribution applied at their ends.

The characteristic of the material (alloy EN-AW-2024) was determined experimentally on
the basis of a standard static uniaxial tensile test (according to PN-EN 10002-1 for tests at
ambient temperature). It is shown in Fig. 3.

Fig. 3. Material characteristic

Aftermechanical working and special surface preparation (grinding and etching), themodels
were covered on both sides (to avoid bending effect) with the photoelastic coating made of
epoxy resin. The strain constant of the photoelastic coating: fε = 1.114 · 10

−3 1/fringe order,
was determined experimentally. Next, different holes were cut out in the way which allowed
avoiding creation of plastic strains as a result of machining. Shapes of the stress concentrators



Two-dimensional experimental elastic-plastic strain and stress analysis 425

were designed on the basis of engineering practice (single central holes of various shapes and
groups of circular holes of various configurations) – Fig. 2.

The models were loaded at their ends with uniformly distributed tensile stresses (p). As
the measure of the loading intensity, the ‘loading factor’ (s) was accepted. It was calculated
as a ratio of the average tensile stresses at the cross-section weakened by the hole on the axis
of symmetry perpendicular to the stretching direction in relation to the offset yield strength
R0.2 =182MPa (taken from the material characteristic).

The loading of the models was increased step by step within the over-elastic range of the
material. At selected levels of loading, the photographs of the isochromatic pattern were taken
twice: for both dark- and light-field polariscope. The stand for testingmodels is shown in Fig. 4

Fig. 4. The stand for testing models by photoelastic coating method

4. Strain and stress analysis in elastic-plastic zones

The results of quantitative analysis of the elastic-plastic strain and stress in the neighborhood
of different stress concentrators are presented examplarily for themodel IV (with three circular
holes).

The strain separation in the analyzed area was carried out by means of the general field
method of characteristics (equations (2.14) and (2.15)) using the 6-sectional geometrical appro-
ximation for the σ-ε curve (Fig. 5).

Fig. 5. 6-sectional model of the material characteristic

The isochromatic pattern taken fromthedark-fieldpolariscopeat the loading level s=0.928,
which corresponds to tensile stresses p=127MPa, is shown in Fig. 6. Because of the symmetry
of the model and loading, only one quarter of the tested area has been considered.



426 B. Kozłowska

On the basis of the isochromatic pattern obtained from this picture, the strain and stress
components in the nodes of characteristics were determined. Calculations were started from the
non-loaded edge of the central hole (A-B) – (Fig. 6). For determining the isochromatic order in
the design point (on the basis of the sinusoidal distribution of the light intensity level), strain
separation and stress calculation, a special computer programwas used.

Fig. 6. Analyzed area of model IV at the loading level – s=0.928: isochromatic pattern (a), strain εx
distribution (b), strain εy distribution (c)

For better visualization of the determined strain and stress distributions, the values calcula-
ted in the nodes of characteristics (irregular co-ordinates) were recalculated for the nodes of a
rectangular grid and graphically represented in form of a surface.

The distribution of strain values εx and εy in the analyzed area of model IV are shown
in Fig. 6b and Fig. 6c. For better legibility of the diagrams, the values of εx(x,y) are
shown as positive – in reality, they are negative. Extreme values of the calculated strains are:
εx = −0.297% and εy = 0.534%). Figure 7 shows the surfaces representing the stress com-
ponents σx(x,y) and σy(x,y), whose extreme values are: σx = 138MPa and σy = 218MPa.
Because of very small non-dilatational strain and shear stress, the distributions of their values
are not presented.

In Fig. 8, the distribution of reduced stress (according toHuber-Mises hypothesis) is presen-
ted as well as the range of the over-elastic area determined on the basis of this distribution. It
corresponds to the first yield point on the schematization of thematerial characteristic (Fig. 5).



Two-dimensional experimental elastic-plastic strain and stress analysis 427

Fig. 7. Analyzed area of model IV at the loading level – s=0.928: (a) stress σx distribution,
(b) stress σy distribution

Fig. 8. Analyzed area of model IV at the loading level – s=0.928: (a) reduced stress σred distribution,
(b) range of over-elastic area

In many practical cases, we deal with the elements having a geometric and loading plane
of symmetry. The determination of strain and stress components along the axis of symmetry is
simpler in comparison with the solution of the full plane problem. Such analysis often allows
estimating the maximum values of strains and stresses in the whole plastified area.

The distribution of strain and stress components in a partly plastified area along the x axis
of symmetry (perpendicular to the stretching direction) is presented for model II (with slot) at
the loading level s=0.952, which corresponds to tensile stresses p=87MPa.

For the strain separation, the simplified method of characteristics was used, which is pre-
sented in papers by Kapkowski and Kozłowska (1993), Kozłowska (1996). The values of stress
components were calculated on the basis of the same 6-sectional model of the material charac-
teristic.

The isochromatic pattern taken from the dark-field polariscope for model II is shown in
Fig. 9a, and like previously, one quarter of the tested area was considered. The segment A-B
means the part of the models axis of symmetry, where the values of strain and stress were
calculated. The distribution of strain components εx and εy on the segmentA-B is presented in
Fig. 9b. On the same diagram, the strain distribution found using experimental Moiré method
and from numerical (FEM) calculations are shown. These investigations were performed by the
author (Kozłowska, 2008).

As it results from these diagrams, the εx and εy distribution on the x axis of symmetry
obtained from the photoelastic coatingmethod is approximated to this received from theMoiré
method. The differences between both experimental methods are about 6-8%. A little bigger
difference can be foundwhen comparing experimental and FEMdata, but even then it does not
exceed 12%.



428 B. Kozłowska

Fig. 9. Analyzed area of model II at loading level – s=0.952: isochromatic pattern (a), strain εx
and εy distribution (b), stress σx and σy distribution (c); on diagrams: (1) – photoelastic coating

method, (2) –Moiré method, (3) – FEM calculations

Thedistribution of stress components σx and σy on the segmentA-B is presented inFig. 9c.
On the diagram, the range of the plastic zone on the x axis of symmetry is marked.

5. Conclusions

Proper estimation of the strain and stress components in plastified zones of constructional ele-
ments is very important considering the safety of the whole construction.

In the paper, the author presents quantitative analysis of strains and stresses, which occur
in stretchedmodels of constructional elements weakened by holes of different shapes and groups
of circular holes.

The experimental testingwas carried outby the photoelastic coatingmethod, and for further
analysis pictures of the isochromatic pattern were only used.

For the strains separation and the values of stresses calculation, the 6-sectional model of
material characteristics was accepted. Such a model (considering the number of segments) is
quite sufficient for a proper representation of the experimental σ(ε) curve, what was confirmed
by the smoothness of stress surfaces in the points, where the material is passing into the non-
elastic state.



Two-dimensional experimental elastic-plastic strain and stress analysis 429

The obtained results showed that the applied method allows determination of strain and
stressing distributions with a satisfactory accuracy. Results of the experimental analysis we-
re compared with those obtained from the investigations carried out by the “in-plane” Moiré
methodandFEMcalculations. Thedifferences between the results obtained fromthe two experi-
mentalmethods,whichwerebasedondifferentmeasurements (thephotoelastic coatingmethod–
strainmeasurement, theMoiré method – displacementmeasurement) were about a few percent.
The differences between experimental and FEM results do not exceed 12%.
The range of application of themethod of photoelastic coating dependsmostly on the thick-

ness of the photoelestic layer (usually 1-3mm) and material properties of the tested object.
For the investigated duralumin elements a 2mm thick layer was used, what enabled one to
distinguish isochromatic lines corresponding to the maximum plastic strain of 0.9%. For the
investigation of the elements byMoiré method, the 20 lines/mm grids were used, which allowed
to determine maximum plastic strain of 1.5% (Kozłowska, 2008). Both experimental methods
enable to measure or determine strain components with accuracy of ∼ 0.01%.
Amongdifferent experimentalmethodswhichmaybeapplied to theanalysis of elastic-plastic

strain fields, the method of photoelastic coating is one of the best.
The greatest advantage of this method is the possibility of easy and prompt localization of

the plastic zones which should be carefully controlled.
The advantage of the method is also the direct strain measurement. However, it must be

noted that the obtained results depend on the material properties of the tested element, when
for strain separation themethod of characteristics is used.
Onthebasis of theobtained results, it canbeconcluded that thephotoelastic coatingmethod

can be applied successfully to the strains and stresses analysis in two-dimensional elastic-plastic
problems. It canbeused, therefore, as anexperimental tool for verification of engineeringdesigns.

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Doświadczalna analiza odkształceń i naprężeń w dwuwymiarowych stanach

sprężysto-plastycznych

Streszczenie

W pracy została przedstawiona ilościowa analiza odkształceń i naprężeń w obszarach sprężysto-
plastycznych w płaskich modelach elementów konstrukcyjnych z koncentratorami naprężeń o różnych
kształtach. Modele, wykonane z duraluminium, zostały obciążone równomiernie rozłożonymi na koń-
cach naprężeniami rozciągającymi wywołującymi częściowe uplastycznienie materiału. Badania ekspe-
rymentalne zostały przeprowadzone przy pomocy metody elastooptycznej warstwy powierzchniowej. Do
rozdzielenia odkształceń zastosowanoanalitycznąmetodę charakterystykwykorzystującą do obliczeńwy-
łącznie obraz izochrom. Składowe stanu odkształcenia i naprężeniaw obszarachuplastycznionych zostały
wyznaczone na podstawie wieloodcinkowego modelu krzywej dla materiału elementu. W pracy została
przeprowadzonadyskusja otrzymanychwyników i ich porównanie zwynikami otrzymanymimetodymory
i obliczeń numerycznych (MES).

Manuscript received May 8, 2012; accepted for print July 6, 2012