

Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 2, pp. 489-501, Warsaw 2016
DOI: 10.15632/jtam-pl.54.2.489

EXPERIMENTAL INVESTIGATIONS OF ELASTIC-PLASTIC STRAIN

STATES ON VARIOUS STAGES OF MATERIAL PLASTIFYING

Barbara Kozłowska

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

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

In the paper, the possibility of application of various experimental methods to the analy-
sis of elastic-plastic states at different levels of material plastifying is presented. For tests
carried out on two-dimensional elements with different stress concentrators and loaded by
tensile stresses, three experimental methods have been selected: Moiré method, method of
photoelastic coating and the thermography method. On the basis of the tests results, the
range of the applicability of chosenmethods and their suitability to the elastic-plastic strain
analysis at the development ofmaterial plastifying has been determined. The strain compo-
nents distribution obtained from theMoiré method and themethod of photoelastic coating
has been compared. The possibility of increasing the accuracy of strain determination for
theMoiré method by additional tests at the grid rotated by an angle of 45◦ with respect to
the direction of tensile stress has been shown. The results of the investigations have been
discussed.

Keywords: mechanics of solids, experimental methods, elastic-plastic states

1. Introduction and the experimental method selection

The analysis of elastic-plastic states finds many basic applications in engineering design, espe-
cially nowadays, since the economical trend towards building more lightweight and cheaper
structures accepts partial material plastifying during exploitation.
Experimental studies on a great variety of non-linear problems are conducted inmany rese-

arch centers all over the world. These problems, like non-linear material characteristics, partial
material plastifying, large deformation, material with imperfections, etc, always cause difficul-
ties associatedwith themodelling of constructionalmaterials. In such circumstances, thewidely
nowadays applied numericalmethods as FEMstill need final experimental verification. Especial-
ly the experimental methods which give information about the real object without any model
simplification can be very useful as a good verification tool of theoretical and numerical design.
The experimentalmethodsmost commonly used to researchwork on elastic-plastic problems

are: photoelasticity, especially the method of photoelastic coating (Pacey et al., 2005; Foust et
al., 2011; Lamberson et al., 2012; Diaz et al., 2010),Moiré methods (Min et al., 2006; Livieri and
Nicoletto, 2003; Guo et al., 2006), holographic interferometry (Lin, 2000; Balalov et al., 2007),
electronic speckle pattern interferometrymethod (ESPI) (Diaz et al., 2001; Schajer et al., 2005),
digital image correlation method (DIC) (Vural et al., 2011; Diaz et al., 2004; Tarigopula et al.,
2008), strain gauge technique (Rasty et al., 2007; Olmi, 2010), thermography (Pieczyska et al.,
2006; Connesson et al., 2011).
The selection of the experimental method to study plastic “in-plane” deformation depends

on several elements: the ability and accuracy of the method, the ease of its use in practice, the
character of the obtained results and the possibility of their work out, etc.
The greatest potential taking into account a variety of research techniques and a diversity

of the analyzed problems, create the photoelasticity andMoiré method.


490 B. Kozłowska

To study elastic-plastic states, much more suitable is the method of photoelastic coating
than a traditional photoelasticity which requires the use of optically active materials having
characteristics corresponding to material characteristics of the tested element also in the non-
linear range.

Similarly, among various Moiré techniques – the best applicability to study elastic-plastic
states has the classical geometric Moiré method. The frequently nowadays applied interferen-
tial Moiré method has a very high sensitivity (density of the grid is here of several thousand
lines/mm) and is used to the analysis of small areas. Its additional disadvantage is the necessity
of coherent (laser) light application.

Holographic interferometry, which bases on phenomena occurring during the coherent light
interference, can be used directly for measurement of displacement (or shape) of the structure.
It shows good accuracy (strain measurement with an accuracy of 0.1 · 10−4-1 · 10−4), but has
a high mechanical sensitivity, and the result analysis is labor-consuming. It requires also laser
light.

Making use of coherent light requires also one of the modern experimental methods – elec-
tronic speckle pattern interferometry (ESPI). It has a lot of advantages – non-contact measu-
rement, high sensitivity, resolution and accuracy; it gives surface images of displacement and
strain components. However, ESPImethod has also very serious limitations – high susceptibility
on conditions in which experiment is performed, sensitivity to even slight movements. It is also
not suitable for large deformations and requires the researcher to have high skills and a lot of
experience.

The second modern experimental method, the digital image correlation method (DIC) also
allows non-contact measurement and gives surface images of displacement and strain compo-
nents. Compared to the ESPImethod, it has a biggermeasurement range, but lower resolution.
The surface of the element needs special preparation and the results require a lot of calculations.

In contrast to modern experimental methods, one of the oldest commonly known but still
most often used experimental technique is the strain gauge measurement. It enables one to get
strain values with a very high accuracy (∼ 1 · 10−6), but only in several points. That is why
it is usually used in combination with other experimental methods, after determining the most
loaded parts of a structure.

An auxiliary character has also usually the thermography method which is often used to
detectmaterial defects. Its accuracy is difficult to determine anddependsmostly on temperature
resolution of a thermovision camera and external conditions. However, the method allows one
to observe thermal processes taking place in the material, what is a great advantage, especially
concerning the elastic-plastic problems.

Taking into account advantages and disadvantages of various experimental methods and
abilities of their application regarding to the elastic-plastic states analysis, the simplicity of their
use in practice, equipment availability, etc., for further experimental testing, threemethods have
been selected: a method of photoelastic coating, Moiré method and the thermographymethod.

All the threemethods give information about deformation of thewhole tested area (not only
at several points). They can be used to investigate real structure elementsmade of anymaterial,
in working conditions, even under heavy loading causing partial material plastifying. Their
advantage is also excellent visualization of the process of progressive material plastifying. In
particular, themethods of photoelastic coating and thermographyallowdirect observation of the
process of formation and development of plastic zones, changes (expansion) of their boundaries
and the direction of propagation.

The first two methods are optical methods, although each of them gives information about
different physical quantities (method of photoelastic coating – strain,Moiré method – displace-
ment).Theyare also comparable in termsof the level of accuracy (measurementordetermination
of strain with an accuracy ∼ 1 ·10−4). The third method – thermography is based on a quite


Experimental investigations of elastic-plastic strain states... 491

different physical phenomenon (emission of infrared radiation from the surface of the tested
element), which allows, to a certain degree, verification of the results obtained from the first two
methods.

Theadditional advantage of thephotoelastic coatingmethod is thepossibility of direct strain
measurement. The disadvantage is the dependence of the obtained results on the properties of
the tested material (using the analytical method of strain separation).

TheMoiré method has a purely geometrical character (measurement of the displacements is
direct and does not depend on the properties of the tested material), determination of strain,
however, requires differentiation of the obtained displacement values.

The photoelastic coating method and the Moiré method are methods very often used in
worldwide research works, thus their choice seems to be themost evident and proper. Thermo-
graphy, though less precise and giving results more qualitative than quantitative, can provide
a useful and interesting complement to the first two methods, particularly for solving certain
elastic-plastic problems.

2. Experimental testing

The experimental investigation of elastic-plastic states has been performed on two-dimensional
models of structural elements weakened by different stress concentrators (holes) and subjected
to tensile stresses. The elements of this type and loaded in such a way are often used inmodern
structures, particularly as different construction joints. The areas of the elements weakened by
cut-outs are parts of structures which require special and accurate checking (Wung et al., 2001;
Olmi, 2010; Foust et al., 2011). Shapes of stress concentrators have been designed on the basis
of literature data and engineering practice (single central holes of various shapes and groups
of circular holes of various configurations). The objects of discussion presented in the work are
three of the models – Fig. 1. The first two models with one central hole have the same area of
themostweakened cross-section in thex axis of symmetry (the effective cross-section) anddiffer
only in shape of the hole. The thirdmodel is weakened by five circular holes cut symmetrically
not only on the axis of symmetry x.

Fig. 1. Models of constructional elements


492 B. Kozłowska

Themodels have beenmade of a duralumin sheet 3mm thick, fromwhich stripes of 100mm
in width and 450mm in length have been 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 thematerial (aluminumalloy EN-AW-2024) has been determined expe-
rimentally on the basis of a standard static uniaxial tensile test and it is shown in Fig. 2.

Fig. 2. Material characteristic

After mechanical working and special surface preparation, the models have been covered
with: a cross-type grid of 20 lines/mm (for the Moiré method) with a 2mm thick photoelastic
coating of strain constant: fε =1.114·10

−3 1/fringe order (for the photoelastic coatingmethod)
andwith a layer of graphite (for the thermographymethod). Next, different holes have been cut
out in the way which allowed avoiding creation of plastic strains as a result of machining.

The models have been loaded at their ends with uniformly distributed tensile stresses p.
As the measure of the loading intensity, the ‘loading factor’ s has been accepted. It has been
calculated as the 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
strengthR0.2 =182MPa (taken from thematerial characteristic).

The loading of the models has been increased step by step within the over-elastic range of
the material. At selected levels of loading, images of the Moiré pattern (for the displacement
u(x,y) and v(x,y)) and isochromatic pattern (for dark- and light-field polariscope) have been
registered. For the thermographymethod, the loading of themodels increased continuously and
the temperature changes on the specimen surface have been recorded by a thermovision camera.

On the basis of the experimental data obtained from the Moiré method and the method of
photoelastic coating, quantitative analysis of theelastic-plastic strainandstress aroundthe stress
concentrators has beenmade (Kozłowska, 2008, 2013). Due to low resolution of thermal images
obtained from the infrared camera, the thermograms have given only qualitative information
about deformation of the elements (Kozłowska, 2012).

3. Determination of the suitability of selected experimental methods to

elastic-plastic analysis in dependence on the material plastifying level

The experiment has been performed within a wide range of the loading – from the moment of
occurring first plastic deformations to the elements failure. But not at every level of loading all
of the selected methods have been equally useful.

As proved byan experiment, photoelastic coatingmethod allows analysis of the plastic strain
starting from the beginning of their occurring in the material – level of about s ≈ 0.5 (tensile
stress p≈ 45MPa) for the models with a single hole or s≈ 0.3 (tensile stress p≈ 44MPa) for
the model with five holes. At that loading level, the Moiré method is not very useful (to low


Experimental investigations of elastic-plastic strain states... 493

number ofMoiré fringes). Exemplary images of isochromatic andMoiré patterns at thediscussed
loading level for the area around the stress concentrator of model III are shown in Fig. 3.

Fig. 3. Model III (s=0.495) – isochromatic pattern: (a) dark-field polariscope, (b) light-field
polariscope;Moiré fringe pattern: (c) u(x,y) surface, (d) v(x,y) surface

For significant plastic deformation, for a loading level over s≈ 1 (tensile stress p≈ 91MPa)
for themodels with a single hole or s≈ 0.8 (tensile stress p≈ 116MPa) for the model with five
holes, the method of photoelastic coating is no longer useful. It is so because the photoelastic
coating can crack (Fig. 4), come off the base (Fig. 5) or (in the best case) isochromatic fringes in
the most plastified areas become quite unreadable (Fig. 6). TheMoiré method, however, at the
same loading level or even higher, enables proper analysis of elastic-plastic states (Figs. 4-6).

Fig. 4. Model II (s=1.136) – isochromatic pattern: (a) dark-field polariscope, (b) light-field
polariscope;Moiré fringe pattern: (c) u(x,y) surface, (d) v(x,y) surface

Fig. 5. Model III (s=1.136) – isochromatic pattern: (a) dark-field polariscope, (b) light-field
polariscope;Moiré fringe pattern: (c) u(x,y) surface, (d) v(x,y) surface

The discussed examples show an approximate range of the application of theMoiré method
and the method of photoelastic coating depending on the level of plastic deformation of the
material. This rangemaybe changed to some extent because the sensitivity of these twomethods
depends on the selection of the proper “measuring element”. Greater possibilities creates here


494 B. Kozłowska

Fig. 6. Model V (s=0.778) – isochromatic pattern: (a) dark-field polariscope, (b) light-field
polariscope;Moiré fringe pattern: (c) u(x,y) surface, (d) v(x,y) surface

theMoirémethodbychangingdensity of thegrids – for thegeometricMoirémethod, thenumber
of lines per millimeter may vary from a few to several tens (most commonly from 20 to 40).

The change in sensitivity of themethod of photoelastic coating can be achieved by variation
of thickness of an optically active layer, but in amuch smaller range.Usually, the thickness of the
applied photoelastic coating is 1 to 3mm, although you can find one of 0.25mm.A thicker layer
causes stiffening of the element and introduces toomuch ofmeasurement inaccuracy (averaging
over the thickness). The upper limit of the capabilities of the photoelastic coatingmethod is the
layer cracking or its coming off the base at higher loading levels.

This disadvantage doesnot applies to theMoirémethod,because even if the grid is imprinted
to a photographic film and is affixed to the surface of the element, it forms a flexible thin layer,
very strongly connected with the base. In the case of a grid applied directly to the surface of
the element (e.g. etched), the problem of the grid coming off does not exist at all.

For the grids used in experimental testing (20 lines per millimeter), the range of measured
strain was ∼ 0.2% to 1.5%, while for the photoelastic coating of 2mm thick, the maximum
determined plastic strain was up to∼ 0.9%.

The thermal images obtained from infrared camera did not give sufficiently precise informa-
tion about the temperature distribution on the surface of the tested element. The accuracy of
temperature measurement by an infrared camera, however, depended mainly on its resolution,
and that increases with the technical possibilities.

Even if the thermograms do not allow one to obtain the values of strain components, they
show directly the full development of plastic zones, from the beginning of their creation to the
failure of the element as e.g. in model III (Fig. 7)

Fig. 7. Thermograms (model III) for loading levels: (a) s=0.989, (b) s=1.172, (c) s=1.304
(first cracks), (d) s> 1.355 (element failure)

The method of thermography has also no limitations resulting from the properties of the
layer covering model (graphite), as it is in the case of the photoelastic coating method (the
optically active layer) or even in the Moiré method, when the grid is affixed to the surface of


Experimental investigations of elastic-plastic strain states... 495

the model. Thus, this method can be used, both as a preliminary tool to select an area of the
element to be tested with a more accurate method (the formation of plastic zones) as well as a
way to observe the mechanism of plastic material failure in the range already out of reach for
other experimental methods.

4. The accuracy of the determination of strain components by selected

experimental methods

Although the Moiré method and the method of photoelastic coating have different ranges of
the best suitability for quantitative analysis of the elastic-plastic strain and stress components,
there is a certain range of loading level for which bothmethods can be properly used.
To compare the accuracy of the Moiré method and the method of photoelastic coating, the

loading levels for which both methods give results freely allowing one to determine the strain
components have been chosen. The analysis of model II and III has been carried out at the
loading level s=0.952, which corresponds to tensile stresses p=87MPa. The average stress on
the x axis was then σav = 173MPa (average strain – εav = 0.38%). For model V the loading
level accepted for analysis, was s = 0.687 (tensile stress p = 100MPa), for which the average
stress on the axis of symmetry x – was σav =125MPa (average strain – εav =0.15%).
At the chosen loading levels, a quite significant plastification of thematerial already occurred

around the stress concentrators, on the one hand large enough to enable measurement by the
Moiré method, on the other hand, still allowing using themethod of photoelastic coating.
The analysis of the elastic-plastic strain state for the models with one hole is shown on the

example ofmodel III, forwhich the images of isochromatic andMoiré patterns around the stress
concentrator at the loading level s=0.952 are presented in Fig. 8.

Fig. 8. Isochromatic pattern for model III – s=0.952: (a) dark-field polariscope, (b) light-field
polariscope;Moiré fringe pattern – (c) u(x,y) surface, (d) v(x,y) surface

To compare the results obtained from the Moiré method and the method of photoelastic
coating, the strain components distribution on a horizontal axis of symmetry x perpendicular
to the loading direction (segment AB – Fig. 1) have been assumed. In addition, on the same
diagram it is also shown the distribution of strain εx and εy obtained from numerical (FEM)
calculations (Fig. 9).
Formodel V (with five circular holes), the images of isochromatic andMoiré pattern around

the stress concentrators at the loading level s=0.687 are shown in Fig. 10.
To compare the results obtained from the Moiré method and the method of photoelastic

coating, the strain components distribution in the horizontal axis of symmetry x perpendicular
to the loading direction (segment AB – Fig. 1) has been assumed (Fig. 11). For this model,
FEM calculations have not been performed.
As follows from the presented diagrams, the strain components εx and εy distribution in the

axis of symmetry x (segmentAB) for the chosen loading level obtained from theMoiré method
and themethod of photoelastic coating are approximate. The differences between the calculated


496 B. Kozłowska

Fig. 9. Strain components distribution in the axis of symmetry x for model III – s=0.952: (1) Moiré
method, (2) method of photoelastic coating, (3) FEM calculations

Fig. 10. Isochromatic pattern for model V – s=0.687: (a) dark-field polariscope, (b) light-field
polariscope;Moiré fringe pattern – (c) u(x,y) surface, (d) v(x,y) surface

Fig. 11. Strain components distribution in the axis of symmetry x for model V – s=0.687: (1) Moiré
method, (2) method of photoelastic coating

values of strain components do not exceed a few percent (6% to 8%).A higher divergence occurs
between the experimental results and numerical calculations, but even there it does not exceed
10% to 12%.

5. The influence of grid configuration on the accuracy of determination of strain

components distribution

The strain components obtained on the basis of the Moiré fringes at the traditionally affixed
grid (in accordance with the axes of symmetry of the model – the direction of tension) can be
determined accurately not in every part of the testedmodel.Where the surfaces of deformation
are not much diversified in the direction of the axis of the coordinate system (directions of
differentiation), the derivatives can be calculated with a certain error.


Experimental investigations of elastic-plastic strain states... 497

As it has been said, the measurement sensitivity of the Moiré method may be increased by
changing density of used grids. It is not always convenient, when the increasing of the accuracy is
neededonly in thepart of the tested element. In suchacase, an improvement of themeasurement
accuracymay be achieved by performing additional tests with grids rotated with respect to the
direction of the basic grid by a certain angle.

Fig. 12. The grid affixed according to the direction of tensile stresses (a), grid rotated by
an angle of 45◦ (b)

In order to verify the possibility of increasing accuracy of determination of the elastic-plastic
strain distribution around stress concentrators, additional tests have been performed at the grid
rotated by an angle of 45◦ with respect to the direction of tensile stress (coordinate system x-y)
– Fig. 12.

An exemplary comparison of the strain components obtained by grids arranged in different
ways is shown for model V with five holes at the loading level s= 0.778, for which, the strain
state has been already determined using a traditionally affixed grid (Kozłowska, 2008).

The images ofMoiré fringes at the grids affixed in different ways are shown in Fig. 13, where
one can see a larger number of Moiré fringes in selected areas at the rotated grid than at the
grid affixed in the direction of tensile stress.

Fig. 13.Moiré pattern for model V (s=0.778) – grid affixed according to the direction of tension:
(a) u(x,y), (b) v(x,y); grid rotated by an angle of 45◦: (c) clockwise, (d) counter-clockwise

The analysis for the rotated grid has been carried out as in previous cases (because of the
double symmetry of themodel and loading) for one-quarter of the tested area (Kozłowska, 2007)
– Fig. 14.

For strain analysis, the coordinate system x-y associated with an element under tension and
a traditionally affixed grid has been rotated by an angle of 45◦ to forma coordinate system x′-y′

associated with a rotated grid (Fig. 12).

On the basis of the displacement obtained fromMoiré fringes at the rotated grid, the strain
components have been determined in the new coordinate system (x′,y′) bymeans of analytical
differentiation.


498 B. Kozłowska

Fig. 14.Moiré pattern for the analyzed area of model V (s=0.778) – grid rotated by an angle of 45◦:
(a) clockwise, (b) counter-clockwise

Then the values of strain εx and εy in the basic coordinate system (x,y) have been calculated
bymakinguseof formulas enablingconverting the strain statedescribed inonecoordinate system
to another (rotated) one (1), where α=45◦, Fig. 15

εx = εx′ cos
2α+εy′ sin

2α+γx′y′ sinαcosα

εy = εx′ sin
2α+εy′ cos

2α−γx′y′ sinαcosα

1

2
γxy =(εx′ −εy′)sinαcosα+

1

2
γx′y′(sin

2α−cos2α)

(5.1)

Fig. 15. Strain state transformation

To compare the obtained results with those from the analysis of Moiré images at the grid
affixed according to the direction of the tensile stresses, the strain components distribution in
the horizontal axis of symmetry x has been assumed (Fig. 16). In the diagram, the correction of
strain components calculation resulting from the larger number ofMoiré fringes (selected areas)
is shown.

The analysis of strain components for model V (model with five holes) shows that in its
horizontal axis of symmetry x, where the data obtained at the grid affixed traditionally are
relatively inaccurate (low number ofMoiré fringes), the information found from the rotated grid
allows one to increase the accuracy of strain determination and to correct errors resulting from
differentiation of surfaces of deformation.


Experimental investigations of elastic-plastic strain states... 499

Fig. 16. Strain distribution in the axis of symmetry x (model V – s=0.778): (1) grid affixed according
to the direction of tension, (2) grid rotated by an angle of 45◦

6. Conclusions

The review of experimental methods used in mechanics of solids carried out in terms of elastic-
plastic analysis resulted in the selection of three of them for further investigations: the Moiré
method, the method of photoelastic coating and the method of thermography. The choice has
been dictated not only by their measuring capabilities in the over-elastic range of the material
characteristics, but also by the simplicity of their use in practice and the availability of the
equipment.

The main advantage of all three methods is the ability to conduct an experiment on real
structures in working conditions, also under loading causing partial plastifying of the mate-
rial and to obtain information about the strain state of the whole tested object. An additional
advantage of the chosen methods (especially the method of photoelastic coating and the ther-
mography method) is excellent visualization of the process of progressive material plastifying.
The advantage of theMoiré method is also the simplicity of measurements and the work out of
the experimental results.

TheMoiré method and themethod of photoelastic coating enable a relatively easy and quick
quantitative analysis of the strain state around stress concentrators on the basis of experimental
data. Thermographic tests have shown that this method allows rather getting a general view of
the distribution of plastic strain components than their precise quantitative determination.

The conducted tests and detailed analysis of experimental data enabled definition of the
range of applicability of each of the selected methods and determination of their capabilities in
termsof theaccuracyof calculation of strain components atvarious stages ofmaterial plastifying.

The range of application of the method of photoelastic coating (for an 2mm thick optically
active layer used in the testing of duralumin elements) is up to the maximum plastic strain ∼
0.9%.

TheMoirémethodallowes testing of the elements in awider range of thematerial plastifying.
For the used grids of 20 lines permillimeter, plastic strain can bemeasured up to∼ 1.5%, while
determination of the strain less than∼ 0.2% causes difficulties due to the low number of Moiré
fringes. The measurement sensitivity of the Moiré method can be locally increased by affixing
the grids at different angles. Such a possibility has been verified by additional tests performed
at the grid rotated by an angle of 45◦ to the axis of symmetry of themodel. That gave an effect
similar to applying the rosette of strain gauges and showed that the accuracy of the elastic-
plastic strain distribution around stress concentrators could be increased in the areas where the
number of Moiré fringes is low.

The comparison of the strain components distribution in the horizontal axis of symmetry x
(perpendicular to the direction of tension) obtained from theMoiré method and themethod of


500 B. Kozłowska

photoelastic coating for the range of material plastifying, for which both of them are applica-
ble, shows that the differences between results are about a few percent. The comparison with
numerical calculations (FEM) also shows good agreement of the results.
The quantitative analysis of strain and stress components in the whole area around stress

concentrators proves that theMoiré method is a little more useful. Themethod of photoelastic
coating is more labor-consuming due to the necessity of analytical strain separation (solving
of the system of partial differential equations, Kozłowska, 2013) and converting the obtained
results from irregular grid nodes to the rectangular grid.
The thermography method, although not enough accurate for quantitative strain analysis,

gives an opportunity of observingplastic zones developing in elements in the full range of loading
until their complete failure, so it seems to be useful for the study of elastic-plastic states in co-
operationwith other experimentalmethods (e.g., Moiré method and themethod of photoelastic
coating).

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Manuscript received July 31, 2015; accepted for print September 11, 2015