materials
ArticleDetermination of Local Strain Distribution at theLevel of the Constituents of Particle Reinforced
Composite: An Experimental and Numerical Study
Witold Ogierman *and Grzegorz Kokot

Department of Computational Mechanics and Engineering, Faculty of Mechanical Engineering,
Silesian University of Technology, Konarskiego 18a Street, 44-100 Gliwice, Poland; grzegorz.kokot@polsl.pl* Correspondence: witold.ogierman@polsl.pl; Tel.: +48-32-237-2446
Received: 22 July 2020; Accepted: 1 September 2020; Published: 3 September 2020


Abstract: This paper is devoted to numerical and experimental investigation of the strain field at thelevel of the constituents of two-phase particle reinforced composite. The research aims to compare
the strain distributions obtained experimentally with the results obtained by using a computationalmodel based on the concept of the representative volume element. A digital image correlation method
has been used for experimental determination of full-field strain. The numerical investigation was
conducted by the finite element analysis of the representative volume element. Moreover, usage ofthe novel method of assessment of the speckle pattern applicability for the measurement of local
fields by using the digital image correlation method has been proposed. In general, the obtained
experimental and numerical results are in good agreement although some discrepancies between the
results have been noticed and discussed.
Keywords: particle-reinforcement; strain transfer; micromechanics; mechanical properties

1. Introduction
The local strain and stress fields at the level of the constituents of composite materials are
inhomogeneous due to contrast in the mechanical properties of matrix and reinforcement. The strain
and stress fields depend on several factors like the shape, volume fraction and spatial distributionof the reinforcement, and also the interface between the phases. The fundamental strategy incontinuum micromechanics is the analysis of the behavior of the representative volume element(RVE), which statistically represents the real microstructure of the material. The RVE can be generated
in a way to model the real microstructure very closely, for example by basing on the microtomographyscans [1–4]. However, such detailed description of the microstructure may lead to a prohibitive
computation time, particularly in the case of multi-scale computations. Therefore, very often, the RVE
is considered as a model for simplifying the geometrical features of the real microstructure to provide a
reasonable computation time [5,6]. Various numerical methods can be used for the analysis of the RVE,
like, for instance, the boundary element method (BEM) [7–10], or the fast Fourier transform-based
method (FFT) [11–13]. However, the most frequently used is the finite element method (FEM),
which has become a standard approach. The literature presents numerous works that use the FEM
for evaluation of the stress and strain fields at the level of the constituents [14–19]. On the other
hand, there is a limited number of studies devoted to the experimental determination of local fields
that allows for the validation of numerical models. In this case, the digital image correlation (DIC)
can be applied, since it is a non-contact method and has got no inherent length scale. It is widelyused in the determination of the behavior of composite materials at the macroscopic level [20–27],
although several recent studies have reported on the application of the DIC in measuring the local

Materials 2020, 13, 3889; doi:10.3390/ma13173889

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Materials 2020, 13, 3889

2 of 14strain fields. Berfield et al. [28] used the DIC method for the determination of displacement fields
near isolated silica microspheres embedded in polymer materials. The works of Canal et al. [29] andMehdikhani et al. [30] present the application of the DIC for the analysis of displacement and strain
fields in long fiber-reinforced composites. Mehdikhani et al., in another work [31], performed analysis
for detection and quantification of matrix cracks in carbon fiber composite laminates employing theDIC, and another study related to laminates was conducted by Orell et al. [32]. Koohbor et al. [33]and Anzelotti et al. [34] applied the DIC for analysis of the meso-scale strain localization of a woven
reinforced composite. In the case of the particle-reinforced composites, which are the subject of thepresent work, experimental results presenting deformation behavior at the level of constituents can befound in the work of Hadjem-Hamouche et al. [35]. There are also several interesting papers associatedwith measuring the local strain fields by the DIC related to steels [36–39].
The present paper is devoted to both numerical and experimental determination of the local
strain fields at the level of constituents. For numerical modelling, a standard approach based on
the Finite Element Analysis (FEA) of the RVE containing particles with idealized geometry wasused. The main aim of this paper is to compare the strain field distributions obtained using thesimplified numerical model with the full-field strain measured experimentally by using the DIC.
Another goal of this paper is to develop a novel approach for assessing speckle pattern applicability
for the measurement of local fields at the level of constituents of the composite. One of the mostimportant factors impacting the accuracy of the strain measurements by using the DIC method is thequality of the speckle pattern. In particular, when the analysis is performed on the level of composite
constituents and high strain heterogeneity is expected, unacceptable errors related to the DIC analysis
can be made. Moreover, even if the speckle pattern possesses appropriate parameters, it can lead towrong results when its size is too big in comparison to the characteristic length of the microstructure(for example, the size of the reinforcing particles). In this case, the strain field may be formally correctly
determined; however, the local strain concentration due to reinforcing particles can be unexposed(averaged). Therefore, during this work, an original method for assessing the DIC measurement
accuracy was applied. The conception of the method is based on the generation of images that represent
the artificially deformed specimen with respect to the results of the reference FE simulation related tothe microgeometry behind the speckle pattern. This concept was proposed in the previous work ofauthors [40]; however, it was associated with artificially generated speckle patterns by prescribing arandom grayscale level of pixels. This paper concentrates on a real specked pattern that was used in
DIC analysis at the level of constituents of the composite.
2. Materials and Methods
2.1. Method of Assessment of the Speckle Pattern
A new method of assessment of the DIC measurement accuracy has been proposed to investigate
if the created speckle pattern is appropriate for exposing the local strain field heterogeneity at the levelof the constituents. The idea of the method is based on the generation of the virtually deformed image
referencing to the original image of the speckle pattern. A flowchart illustrating the proposed method
is presented in Figure 1. The input data to the procedure is associated with the image of the speckle
pattern created on the surface of the tested specimen and the FE model based on the microgeometry.
It has to be underlined that purpose of the FE model, within this procedure, is not to approximate thebehavior of a real tested material as close as possible but rather to provide the prediction on how thestrain may be distributed for the tested composite. In general, the microgeometry can be reconstructed
directly from the microscopic image or by using the microtomography. However, for the sake ofsimplicity, it can be created virtually in a way to be statistically equivalent to the real, which should bemuch more convenient in application. Boundary conditions should be applied in a way to model thereal conditions of the experimental, for example when the specimen undergoes compression test theuniaxial compressive strain has to be applied to the FE model. The next step is the generation of theMaterials 2020, 13, 3889

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image-based FE model with respect to the image of the speckle pattern. In this case, the pixels from theMaterials 2020, 13, x FOR PEER REVIEW
3 of 14
image are converted to quadrilateral finite elements. Then the displacements which are the result of
the FEA based on the microgeometry are interpolated to the FE mesh related to the image-based model.
interpolated to the FE mesh related to the image-based model. Afterwards, the FEA of the imageAfterwards, the FEA of the image-based model is performed with the interpolated displacements as thebased model is performed with the interpolated displacements as the boundary condition. The result
boundary condition. The result of the analysis is the artificial image representing the deformed stateof the analysis is the artificial image representing the deformed state which is, together with thewhich is, together with the original image, an input data to the DIC analysis. It has to be underlined
original image, an input data to the DIC analysis. It has to be underlined that the accuracy of the DIC
that the accuracy of the DIC depends not only on the speckle pattern itself but also on the subset
depends not only on the speckle pattern itself but also on the subset size (facet size) which has to besize (facet size) which has to be appropriately adjusted. Finally, the result of the DIC analysis can beappropriately adjusted. Finally, the result of the DIC analysis can be compared with the referencecompared with the reference “accurate” FEM solution. This is done by using the developed original“accurate” FEM solution. This is done by using the developed original procedures enabling the DIC
procedures enabling the DIC results to be transferred directly to the FEM system.
results to be transferred directly to the FEM system.

Figure 1.
1. Flowchart of the method of the DIC measurement
measurement accuracy.
accuracy.
Figure2.2.
2.2. Experimental
Experimental Test
Test
Experimental
Experimental determination
determination ofof full-field
full-field strainstrain was
was carried
carried out
out byby using
using thethe DICDIC method.
method.
Olympus
SZX16
opticalmicroscope(Olympus
Corp.,
Tokyo,
Japan)(Figure2)
was
Olympus SZX16 optical microscope (Olympus Corp., Tokyo, Japan) (Figure 2) was applied
applied forfor
observationofthespecimen
microstructureand
capturingtheimagesduringthedeformation.
observation of the specimen microstructure and capturing the images during the deformation. A
Amaterial
which
used
in investigation
the investigation
is epoxy
reinforced
with particles.
glass particles.
material
which
waswasusedin theis epoxyresin resin
reinforced
with glass
Cubic
Cubic
specimen
of
the
edge
lengthof
10mm
was
prepared,thesurfaceintendedforthestrainspecimen of the edge length of 10 mm was prepared, the surface intended for the strain measurement
measurement
and thenpattern
the speckle
pattern by
wasusing
createdusing
theairbrushfine point
airbrush
was polished wasand polishedthen the speckle
was created
theby
fine
point
InfinityCR
InfinityCR
plus(Harder&
Steenbeck,
Norderstedt,
Germany).
The
specimenwasplacedin unittheplus (Harder & Steenbeck, Norderstedt, Germany). The specimen was placed in the loading
loading
which iswith
combinedwith the microscope
allows for compressionof the specimens.
which isunit
combined
the microscope
and allows andfor compressionof the specimens.
A schemeA
schemepresenting
the
test
setup
and
the
regionofinterest
withdetailed
view
on
the
specklepatternpresenting the test setup and the region of interest with detailed view on the speckle
patternisispresented
Figure3. Themicroscopic
image
visualize
the reinforcing
particles
embeddedpresented
in inFigure3. Themicroscopic
imagethat thatvisualize
the reinforcing
particles
embeddedin theinthe matrix
the regionof interestis presented
Figure4.4. The
The images
images representing
representing anmatrixplacedplacedin thein regionof interestis presented
ininFigure
anundeformedanddeformed
state
of
the
specimenhave
beencollected
and
thenprocessed
undeformed and deformed state of the specimen have been collected and then processed forfor full-field
full-fieldstrainanalysisby using
DIC method(Istra 4D(Istra
v4.6.1.374
software, Dantec
Dynamics).
strainandanddisplacement
displacement
analysisby using
DIC method
4D v4.6.1.374
software,DantecDynamics). The correlation process for the selected region of interest was conducted by taking thefollowing parameters: facet size—21 pixels, grid spacing—20 pixels. Those parameters were taken as
the optimal according to DIC measurements practice guide [41]. Finally, the obtained DIC results

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Materials 2020, 13, x FOR PEER REVIEW
4 of 14The
correlation process for the selected region of interest was conducted by taking the following
Materials 2020, 13, x FOR
PEER pixels,
REVIEWgrid spacing—20 pixels. Those parameters were taken as the optimal
4 of 14parameters:size—21
Materials 2020,facet
13, x FOR
PEER REVIEW
4 of 14were transferred
to ANSYS
18.2 software for the purpose of convenient comparison with the results
according to DIC measurements practice guide [41]. Finally, the obtained DIC results were transferredof
FEAtransferred
based on the
RVE. 18.2 software for the purpose of convenient comparison with the results
weretoANSYS
ANSYS
towereANSYS
18.2 software
for theconvenientcomparison
withcomparison
the results ofFEAontransferredto18.2purpose
softwareoffor
the purposeof convenientwiththebased
resultsofFEA
FEAbased
basedon
onthe
theRVE.
RVE.
of
the
RVE.

Figure 2. Equipment used for capturing the images of the tested specimen, and DIC analysis.
Figure 2. Equipment used for capturing the images of the tested specimen, and DIC analysis.
Figure2.2. Equipment
Equipment used
used forfor capturing
capturing the
Figurethe images
images ofof thethe tested
testedspecimen,
specimen,and
andDIC
DICanalysis.
analysis.

Figure 3. Test overview with detailed view on the speckle pattern in the region of interest.
Figure 3. Test overview with detailed view on the speckle pattern in the region of interest.
Figure 3. Test overview with detailed view on the speckle pattern in the region of interest.
Figure 3. Test overview with detailed view on the speckle pattern in the region of interest.

Figure4. The reinforcing particles in the region of interest behind the speckle pattern.
Figure 4. The reinforcing particles in the region of interest behind the speckle pattern.
Figure 4. The reinforcing particles in the region of interest behind the speckle pattern.
Figure 4. The reinforcing particles in the region of interest behind the speckle pattern.

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2.3. Numerical Modeling2.3. Numerical Modeling
The numerical analysis of the particle reinforced composite is performed in standard fashion
The numericalparticle
reinforced
composite isstatistically
performedthein standard
fashionassociatedwith theanalysis
FEA of ofthetheRVE.
A cubic
RVE representing
microstructureofassociated
with
the
FEA
of
the
RVE.
A
cubic
RVE
representing
statisticallythemicrostructureofparticle reinforced composite has been generated by using Digimat 5.1.1 software (MSC Software,particle
reinforced
compositehas
been
generatedby using Digimat
5.1.1 softwarespherical(MSC Software,
NewportBeach, CA,
USA). The
RVE
consistsof approximately
200 nonintersecting
particles
NewportBeach,
CA,
USA).
The
RVE
consistsofapproximately
200
nonintersecting
sphericalof the same size which spatial distribution was determined in a random way (Figure 5a). particles
Usage ofofthesame size
which spatial
was determinedin a random
way(Figure
5a).
Usage
of
the
proposed
simplified
modeldistributionof the microstructure
is justified
by the fact
thatin caseof thetestedthe
proposedmodelof thehave
microstructure
is justified
by the factthat in casewithof thetestedcompositethesimplified
reinforcing
particles
shape varying
from sphericalto ellipsoidal
relatively
compositethereinforcing
particles
have
shape
varyingfromspherical
toellipsoidal
withrelatively
low aspect ratio, the particles are randomly oriented and have smooth surfaces. ANSYS software waslowaspectthe particles
are randomly
oriented
andcreatedhave smoothsurfaces.mesh
ANSYS
softwareappliedforratio,building
the numericalmodel ofthe RVE,
finite element
consistingof
wasapplied elements
for building
thequadratic
numerical
modelof the RVE,
created finite
element
consistingof
tetrahedral
with
shape
functionsis presented
in Figure5b. mesh
Periodic
boundary
tetrahedral
elements
withenforce
quadratic
functionsis presented
in Figure5b. Periodic boundary
conditions [42,43]
which
the shape
uniaxial
compressive
strain havebeen applied.
conditions [42,43] which enforce the uniaxial compressive strain have been applied.

(a)

(b)

FigureFigure5.5.The
Therepresentative
representativevolume
volumeelement
element(RVE):(RVE):(a)(a)geometricalgeometricalmodel;
model;(b)(b)finite
finiteelement
elementmesh.
mesh.

3.3.Results
Resultsand
andDiscussion
Discussion
3.1. Testing of the Speckle Pattern
3.1. Testing of the Speckle Pattern
The assessment of the DIC measurement accuracy was performed on the fragment of the speckle
The assessment of the DIC measurement accuracy was performed on the fragment of the speckle
pattern presented in Figure 6 by using the procedure described in Section 2.1. Figure 7a presents
pattern presented in Figure 6 by using the procedure described in Section 2.1. Figure 7a presents aa simplified two-dimensional (plane strain) FE model representing a fragment of the composite’s
simplified two-dimensional (plane strain) FE model representing a fragment of the composite’s
microstructure associated with the tested fragment of the speckle pattern. Periodic boundary conditions
microstructure associated with the tested fragment of the speckle pattern. Periodic boundary
enforcing the uniaxial compressive strain have been applied. Linear-elastic material parameters wereconditions enforcing the uniaxial compressive strain have been applied. Linear-elastic material
considered for both phases where: Em = 3 GPa, υm = 0.36 and Ep = 70 GPa, υp = 0.22 are the Young’s
parameters were considered for both phases where: Em = 3 GPa, υm = 0.36 and Ep = 70 GPa, υp = 0.22
moduli and Poisson ratios for epoxy matrix and glass particles, respectively. The FEA was performed
are the Young’s moduli and Poisson ratios for epoxy matrix and glass particles, respectively. The FEA
by using ANSYS software. Figure 7b shows a visualization of the result of the DIC analysis based on
was performed by using ANSYS software. Figure 7b shows a visualization of the result of the DIC
the artificially deformed image of the speckle pattern in the Istra 4D software. Here, the distribution
analysis based on the artificially deformed image of the speckle pattern in the Istra 4D software. Here,of strain εyy is visualized simultaneously with the accounted speckle pattern. Results of the DIC
the distribution of strain εyy is visualized simultaneously with the accounted speckle pattern. Results
analysis are presented in detail in the further figures where obtained displacement (Figure 8) andof the DIC analysis are presented in detail in the further figures where obtained displacement (Figure
strain distributions (Figures 9 and 10) are compiled with the reference FE solution. In the case of8) and strain distributions (Figures 9 and 10) are compiled with the reference FE solution. In the casevisualization of the strain fields obtained by using FEM, a color scale was adjusted to the results of theof visualization of the strain fields obtained by using FEM, a color scale was adjusted to the results ofMaterials 2020, 13, x FOR PEER REVIEW

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Materials
2020,
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the DIC
to 13,
providea reliable comparison between the results. Moreover, the strain distributions
Materials 2020, 13, x FOR PEER REVIEW
6 of 14relatedto theDIC solution
andthe reference
FE solution
are represented
histograms(Figure 11).
the
DIC toprovidea reliable
comparison
betweenthe results.
Moreover,inthe
strain distributions
relatedtheprovide
DIC
solutionand thereferenceFEthesolution
represented
in histograms
11). to
DICDIC
toto
providea reliable
comparison
betweenresults.
Moreover,the strain
distributions
relatedthetoa reliable
comparison
betweenthe areresults.
Moreover,thestrain (Figure
distributions
the DICtosolution
the reference
solution
represented
in histograms(Figure 11).
relatedthe DICandsolutionand the FE
referenceFEaresolutionare represented
in histograms(Figure 11).

Figure 6. Fragment of the speckle pattern that was tested.
Figure 6. Fragment of the speckle pattern that was tested.
Figure 6.
6. Fragment
Fragmentof
ofthe
thespeckle
specklepatternpattern thatthat was
was tested.
tested.
Figure(a)(b)(b) the speckle pattern: (a)Figure 7. Model of (a)the fragment of the microstructure that is expected behindFEmodel,(b)
overviewontheresultoftheDIC
analysis
in
Istra
4D
software(εyyspeckle
distribution).
(a)(b)theFigure7.7.Model
Modelfragmentof themicrostructureis expectedbehindthespecklepattern:
Figureofofthethefragmentof themicrostructure
thatthatis expectedbehindpattern:(a)(a)model,FE model,(b)ofoverview
on result
theof DICthe DIC
analysis
Istra
4Dbehindsoftware(εspeckle
FE(b) overview
on theof microstructure
theanalysisin Istra4D software(εtheyy distribution).
yy distribution).
Figure7. Modelthe fragmentofresult
the
that
isin
expectedpattern: (a)FE model, (b) overview on the result of the DIC analysis in Istra 4D software (εyy distribution).

(a)

(b)

Figureofsolution,
Figure8.
8. Distribution
Distribution(a)of thethe total
total normalized
normalizeddisplacement:
displacement: (a)(a) FEFE reference
reference(b)solution,(b)(b)DIC
DICanalysis.
analysis.

(b)
Figure 8. Distribution of(a)the total normalized displacement: (a) FE reference solution,(b) DIC analysis.
Figure 8. Distribution of the total normalized displacement: (a) FE reference solution, (b) DIC analysis.

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77of

(a)

(b)

(a)
(b)
Figure 9. Distribution of the εxx strain: (a) FE reference solution, (b) DIC analysis.
(a)Figure 9. Distributionof the εxx strain: (a) FE reference solution,(b)(b) DIC analysis.
FigureFigure9.9.Distribution
Distributionofofthe
theεεxxxx strain:strain: (a) FE
FE reference
reference solution,
solution, (b)(b) DICDIC analysis.
analysis.

(a)

(b)

(a)
(b)
Figure 10. Distribution of the εyy strain: (a) FE reference solution, (b) DIC analysis.
(a)Figure10.
10.Distribution
Distribution
theεεyystrain:strain: (a)(a) FEFE reference
reference solution,
solution,(b)(b)DIC
DICanalysis.
analysis.
Figureofofthe(b)yy

Figure 10. Distribution of the εyy strain: (a) FE reference solution, (b) DIC analysis.

(a)

(b)

(a)
(b)
Figure11.11.
Histograms
thethestraindistributions
determined
byby
DIC
and
FEFE
reference
FigureHistogramsrepresenting
representing
straindistributions
determinedDIC
andreference(a)(b)
solution:(a)εcomponent,(b)εcomponent.
solution:(a)εxxcomponent,(b)εyycomponent.
xxyyFigure 11. Histograms representing the strain distributions determined by DIC and FE reference
solution:
εxx component,(b) εyy component.
Figure11.(a)Histograms
representing
the strain distributions determined by DIC and FE referenceThe
displacement
fieldsobtainedby thebyDIC
andthe andreference
FE solutionFEaresolution
in good agreement;
The
displacement
fieldsobtainedtheDIC
the reference
are in goodsolution: (a) εxx component, (b) εyy component.
theagreement;
displacement
frontsare
distortedsimilarly
due
to
the
presenceofreinforcing
particles.
Thethe displacement
are distorted
similarly
the presenceof reinforcing
particles.
The displacement
fields fronts
obtainedby the DIC
and due
the toreference
FE solution
are obtainedingoodstraindistributions
show
that thecreatedspecklepatternallowstoexposetheallows
strainfield
heterogeneity
TheTheobtainedstraindistributions
show
that
the
createdspecklepatterntoexposethestrainagreement;thedisplacement
frontsare
distortedsimilarly
due
to
the
presenceofreinforcing
particles.
displacement fields obtained by the DIC and the reference FE solution are in good
and
track
the
positionsoftheparticles
behindthespecklepattern.
Nonetheless,thesmoothingeffectThe obtained
distributions
that the similarly
created speckle
patternallowsto expose thestrainagreement;
thestrain
displacement
frontsshow
are distorted
due to thepresenceof reinforcing
particles.
associated with the DIC, which is unavoidable at the boundary between the phases, is noticeable.
The obtained strain distributions show that the created speckle pattern allows to expose the strain

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8 of 14Analysis of the histograms (Figure 11) leads to the conclusion that the widths of the strain distributions
obtained by using both methods are similar. However, the smoothing effect is also visible—for the FE
solution, the bimodal pattern is significantly noticeable, while the DIC results make the peaks of thehistogram less pronounced. Especially for the strain εxx , the bimodal character of the distribution isalmost smoothed out. In conclusion, the proposed method of assessment of the speckle pattern madeit possible to verify that the created speckle pattern is applicable for measuring the local strain fields
for the expected size and shape of the reinforcing particles. Moreover, it shows a level of smoothing of
the results due to the application of the DIC method accounting for the current speckle pattern.
3.2. Experimental versus Numerical Results
During the experimental test performed by using the DIC, two images were captured: the firstone was associated with the undeformed state, the second one was related to the uniaxial compressive
loading, the value of which was adjusted to not exceed the linear-elastic regime. Although the strain
measurements were performed on the surface of the specimen, they were related to the composite,whose behavior must be modeled by using the three-dimensional representative volume element (RVE)
to achieve consistency of results. Therefore, the FEA of the three-dimensional RVE was carried out
by applying the periodic boundary conditions in a way to enforce the uniaxial compressive strain
that matches the result of the DIC analysis. In other words, an average strain obtained from the
DIC analysis was prescribed as the boundary condition of FEA; the value of the prescribed strain is
ε = −0.004568. The same material parameters of the constituents as those provided in Section 3.1 wereapplied. The strain distributions computed by the FEM are presented in Figure 12; the color scale wasadjusted to the results of the DIC to provide a reliable comparison between the results. Subsequently,
in Figures 13–15, the strain distributions obtained by the experimental test are complied with the resultsof the FEA (a view of one of the three-dimensional RVE’s faces is illustrated). Moreover, the histograms
representing the strain distributions obtained by both methods are presented (Figure 16).
The experimentally and numerically determined strain distributions presented in Figures 13–15
are in good agreement. Nonetheless, some discrepancies between the results are noticeable. The strain
jumps between the phases are visible in the case of the numerical results, while they are smoothed
out in the case of the experimental results. The peak values of the strains are higher in the case of theexperimental results. The histograms presented in Figure 16 show that the widths of the distributions
are similar in both cases. The histograms also illustrate that the distributions determined experimentallyare smoothed out in comparison with the distributions obtained numerically. In the case of normal
strains, a bimodal shape of the distributions, due to a strain partitioning between the phases, can beobserved both for the experimental and numerical results. However, the bimodality is much more
pronounced in the numerical case. The distributions of the shear strains have got almost symmetricunimodal shape. The discrepancies between the experimental and numerical results are slightly larger
than the discrepancies between the results of the DIC analysis (which was conducted based on this same
speckle pattern as the experimental test) and the reference “accurate” numerical solution (Section 3.1).
Therefore, we conclude that the discrepancies between the experimental and numerical results depend
not only on the errors associated with the usage of the DIC method, but also on the simplifications
related to the numerical model (the same size and spherical shape of the reinforcing particles, the perfectinterface between the phases, etc.). Nonetheless, it has to be pointed out that despite the simplifications
the agreement between results is good. Consequently, usage of the representative volume elements
simplifying the microstructural features, which are very often applied in multiscale computations,
is justified and allows to obtain reasonable accuracy.

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(a)

(b)

(c)
(c) FEM: (a) component εxx, (b) component εyy, (c)Figure 12. Strain distributions obtained by using
componentxy.
Figure12.12.εStrain
Straindistributions obtained
obtained byby using FEM: (a) component, (c)Figuredistributions
component εεxxxx, , (b)(b)component
componentεyyεyy,(c)component
componentεεxy.xy.

(a)

(b)

(a)
(b) results (FEM).
Figuredistributions: (a)(a) experimental
experimental results
results (DIC),(DIC), (b)(b) numericalnumerical
Figure13.
13. Strain
Strain εεxxxx distributions:
results (FEM).
Figure 13. Strain εxx distributions: (a) experimental results (DIC), (b) numerical results (FEM).

(a)

(b)

Figure13. Strain εxx distributions: (a) experimental results (DIC), (b) numerical results (FEM). 10 of 14
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Figure14.
14.Strain
Strainεεyyyy distributions:distributions: (a)(a) experimental
experimental results
results (DIC),(DIC), (b)(b) numericalnumericalresults
results(FEM).
(FEM).
Figure(a)(b)

Figuredistributions: (a)(a) experimental
experimental results
results (DIC),(DIC), (b)(b) numericalnumerical results
results (FEM).
Figure 15.
15. Strain
Strain εεxy
xy distributions:

(a)

(b)

10 of 14

(a)

(b)

Figure 15. Strain εxy distributions: (a) experimental results (DIC), (b) numerical results (FEM).

Materials 2020, 13, 3889

11 of 14

(a)

(b)

(c)
FigureHistograms
representing
straindistributions
determined
experimentally
Figure16. 16.
Histograms
representing
thethestraindistributions
determined
experimentally
by by
DICDIC
andandnumerically
by
FEM:
(a)
ε
xx
component,(b)εyycomponent,(c)εxy
component.
numerically by FEM: (a) εxx component, (b) εyy component, (c) εxy component.

4. Conclusions
The experimentally and numerically determined strain distributions presented in Figures 13–15

are
good
agreement.
Nonetheless,some discrepancies
betweenthestrain
results
areatnoticeable.
Inin
this
paper,
both numericaland experimental
investigationof thefield
the level ofThestrainjumps
betweenthephasesarevisible
in
the
case
of
the
numerical
results,while
theyofarethe constituents of two-phase particle-reinforced composites were carried out. Moreover, usagesmoothedout infor
thethecaseof the experimental
results. The
values patternof the strains
arestrainhigherin thea novelapproachassessmentof the applicabilityof peakthe speckle
to local
field
case
of
the
experimental
results.
The
histograms
presented
inFigure
16
show
that
the
widthsofanalysis was proposed by the digital image correlation method. The main findings from this work canthe
be summarized as follows:
•

•

•

•

The proposed method for the assessment of the speckle pattern made it possible to verify thatthe speckle pattern prepared on the surface of the specimen was applicable for measuring a local
strain at the level of constituents of composite by using the DIC method. Moreover, it made it
possible to predict how much the strain distributions obtained by the DIC were smoothed out by
performing a comparison with the “accurate” reference solution.
Good agreement between the local strain distributions determined experimentally by using the
DIC and numerically by using FEA based on the RVE model has been noticed. The agreement
between the results was good, despite the simplifications related to the geometrical model of the
microstructure (RVE).
The main difference in the full-field strains obtained experimentally and numerically was related
to the region of the phase boundaries: the strain jumps between the phases are visible in the caseof the numerical result, while they are smoothed out in the case of the experimental results.
The normal elastic strain distributions had a bimodal character due to strain partitioning between
the matrix and reinforcing particles. The bimodality has been revealed by both experimental

Materials 2020, 13, 3889

12 of 14and numerical results although the bimodality is much more pronounced in the numerical case.
The shear strain distributions have got symmetric unimodal shape. The widths of the strain
distributions are similar for both methods.
It has to be noted that the presented experimental results are based on the surface analysis,
which gives limited information associated with the strain state. Moreover, the strain distributions
may be different in the bulk of a solid compared to the values measured at the surface. To obtain
three-dimensional strain fields, including measurement in the bulk, more advanced methods such as
Digital Volume Correlation [44–48] or Raman spectroscopy [49–53] may be used.
Author Contributions: Conceptualization, W.O.; Methodology, W.O. and G.K.; Software, W.O.; Validation, W.O.
and G.K.; Formal Analysis, W.O. and G.K.; Investigation, W.O. and G.K.; Resources, W.O. and G.K.; Data Curation,
W.O. and G.K.; Writing—Original Draft Preparation, W.O.; Writing—Review and Editing, G.K. All authors haveread and agreed to the published version of the manuscript.
Funding: This research was funded by the National Science Centre, Poland, grant number 2016/21/N/ST8/01119,
the paper was founded by the Faculty of Mechanical Engineering, Silesian University of Technology, statutory
subsidy grant number 10/990/BK/20/0162 and 10/040/BK_20/0094.
Conflicts of Interest: The authors declare no conflict of interest.

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