Elastisite teorisi denklemlerinin mikromekaniğe uygulanması
An application of the theory of elasticity in micromechanics
Yazar: OSMAN BULUT
Danışman: PROF. DR. FATMA NECLA KADIOĞLU ; DOÇ. DR. ŞENOL ATAOĞLU
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / İnşaat Mühendisliği Bölümü / İnşaat Mühendisliği Ana Bilim Dalı / Yapı Mühendisliği Bilim Dalı
Derece: Doktora, 2013
Konu: Makine Mühendisliği = Mechanical Engineering ; Polimer Bilim ve Teknolojisi = Polymer Science and Technology ; İnşaat Mühendisliği = Civil Engineering

Özet:
Bu çalışmada, lineer-elastik, homojen ve izotrop malzeme kabulü altında, yarıçapı
mikron mertebesinde olan küresel parçacıklar veya kesit alanının yarıçapı mikron
mertebesinde olan silindirik geometriye sahip lifler içeren kompozitlerin elastik
sabitlerini veren ifadeler elde edilmeye çalışılmıştır. Burada, kompoziti oluşturan
matris malzemesinin elastik özellikleri, buna eklenen katkı maddesininkilere göre
çok düşüktür. Ayrıca katkı malzemesinin hacminin, toplam kompozitin hacmine
oranının düşük olduğu durumlar göz önüne alınmıştır. İlgili kompozitin mikro yapısı
seviyesinde yapılacak değişiklikle mekanik davranışındaki değişim araştırılmıştır. Bu
amaçla, yapılan kabuller çerçevesinde kompozit malzemeden alınan temsili hacim
elemanının değişik gerilme hallerine maruz olması durumunda, üzerinde biriken
toplam şekil değiştirme işleri hesaplanmıştır. İlgilenilen hacim elemanına eş değer,
fiktif hacim elemanı için aynı iş hesapları yapılmıştır. Bu hesaplar yapılırken
başlangıçta, bilinen klasik elastisite teorisi denklemleri kullanılmıştır. Bilindiği gibi
klasik elastisitede dış etkilere maruz bir cisimde iç kuvvet, gerilme tansörü ile
açıklanır. Başka bir deyişle, cismin içerisindeki bir nokta üç doğrultuda ötelenme
yapar ki, bu ötelenmeler yer değiştirme vektörünü tanımlar. Mikropolar elastisitede ise
ötelenmenin yanında mikro dönmeler de tanımlanmaktadır. Ayrıca gerilme tansörünün
yanında gerilme-çifti tansörü de ortaya çıkar. Buna ek olarak, gerilme tansörü klasik
elastitede simetrikken, mikropolar elastisitede simetrik olmayabilir. Bu çalışmada,
buna ait bir uygulama da yapılmıştır. Ayrıca teorik olarak elde edilen sonuçların
karşılaştırılması açısından deneysel olarak yapılan çalışmalar da son kısımda yer
almaktadır.
Küresel parçacıklar içeren bir kompozitin Bulk modülü ifadesinin elde edilebilmesi
için b yarıçaplı küresel cisimde hidrostatik çekme problemi ele alınmıştır. Klasik
elastisite denklemleri kullanılarak, cisimde biriken toplam şekil değiştirme işi
hesaplanmıştır. İkinci olarak, eş merkezli, a ve b yarıçaplı iç içe iki kürenin, aynı
gerilme hali için üzerlerinde biriken toplam şekil değiştirme işleri hesaplanmıştır. Ele
alınan bu son problem, küresel parçacık içeren kompozit cisimden alınan temsili hacim
elemanına karşı gelmektedir. İlk problemde ele alınan cisim ise bu hacim elemanına
eş değer fiktif bir malzeme olarak düşünülmüştür. Hesaplanan işlerin eşitlenmesi ile
kompozit cismin Bulk modülüne karşı gelen efektif Bulk modülünün ifadesi, matrisin
ve parçacığın elastik sabitleri ile kompozitin hacim konsantrasoyonuna bağlı olarak
elde edilmiştir.
Aynı tür kompozitin kayma modülünü hesaplamak için bahsedilen tek küresel cismin
ve iç içe iki küresel cismin, basit kayma haline maruz oldukları durumda toplam şekil
değiştirme işleri hesaplanmıştır. Buradan, bir önceki problemde uygulanan yöntemle,
kompozitin kayma modülünün hacim konsantrasyonuna bağlı değişimi belli bir matris
ve partikül malzemesi için elde edilmiştir.
Lif katkılı kompozitin kayma modülü hesabı için ele alınan problem, silindirik cismin
kendi ekseni etrafında basit burulmasıdır. Buradan, silindirik tek cisimde ve iç içe iki
silindirik cisimde biriken toplam iş ifadeleri klasik elastisite denklemleri kullanılarak
elde edilmiş ve aynı yöntemle kompozite ait kayma modülü ifadesi elde edilmiştir.
Son olarak ele alınan problemde, fiktif malzeme olarak düşünülen tek silindirik
cisim, mikropolar olarak düşünülmüş ve simetrik olmayan elastisite teorisi denklemleri
kullanılmıştır. Ele alınan problem yine basit burulmadır. Burada hesaplanan iş, iç içe
iki silindirik cisim için bir önceki problemde hesaplanan işe eşitlenerek, malzemenin
mikropolar katsayıları olan b ve g arasında bir bağıntı elde edilmiştir.
Deneysel olarak yapılan çalışmalarda matris malzemesi olarak polyester seçilmiştir.
Katkı maddesi olarak silindirik lif için Kevlar lifler kullanılmıştır. Deney
numunelerinin hazırlanması için silindirik geometride hazırlanan kalıba öncelikle
sadece polyester dökülerek numuneler elde edilmiştir. Sonra lifler yerleştirilerek
döküm yapılmıştır. Bunlara ait %1 ve %1.5 hacim konsantrasyonunda iki numune
hazırlanmıştır. Ardından bunların basit çekme testleri yapılarak elastisite modülleri ve
Poisson oranları elde edilmiştir. Buradan fiktif malzemeye ait Bulk modülü ve kayma
modülü elde edilmiştir.

Abstract:
In this study, under the assumptions of linear-elastic, homogeneous, and isotropic
materials, the expressions of the elastic constants of the composites including spherical
particles having radius in micron order or thin cylindrical fibers having radius in
micron level, have been obtained. Here, the elastic constants of matrix material are
very lower than those of particles or fibers. Additionally, it is considered that the
ratio of the volume of inclusion to the volume of composite is small. The mechanical
behaviours of associated composites have been observed while changings have been
occuring on micro-structure of it. For this purpose, the strain energies, which belong
to the different states of stress, have been calculated on the representative volume
element of the composite material. Similar strain energies have also been calculated on
effective volume elements which are assumed to be equivalent to the volume elements
mentioned above. During these calculations, firstly, the equations of the classical
theory of elasticity have been used.
In the classical elasticity, as it is known, the transfer of loadings in a body which is
subjected an effect is occured through a surface element dA situated in the interior
of the body by only the stress vector. In other words, the degree of freedom of a
differential volume element in the interior of the body is three which are displacements
along three independent directions. However, according to the micropolar theory,
volume elements have got abilities of rotation. This means, the degree of freedom
is six and the transfer of loading is occured not only by the stress vector but also by the
stress-couple vector. So, the stress and the strain tensors are not symmetric any more.
In this study, a problem has been solved using to this last theory.
As an application to this type of the composite materials, experimental studies have
been added in the last chapter. The results from experiments are compared with those
from theoretical solutions.
To obtain the expression of the Bulk modulus of a composite including spherical
particles, firstly, hydrostatic tension problem for a single spherical body having radius
b has been solved. Here, only one component of the displacement vector u is
different from zero in spherical coordinates which is in the radial direction. So,
the components of the stress and the strain tensors are expressed in terms of this
component. The equilibrium equations are reduced to a single differential equation of
which dependent variable is radial displacement while independent variable is radial
distance from origin. The solution of this equation involves two integration constants
which can be calculated by boundary conditions. Using this displacement expression
the components of the strain and stress tensors are calculated. The strain energy
accumulated in spherical volume can be calculated by these tensors.
As the second step, two nested spherical bodies whose centroids are positioned at the
same point have been considered. They have radii a and b. The state of stress is
hydrostatic tension again. The expression in terms of unknown constants which has
been obtained in the previous problem for radial displacement component is re-written
for the inner and the outer sphere, seperately. In this new problem, the number of
unknown constants is two times of that in the previous problem. The number of
boundary conditions is equal to the that of unknown. From the solution of these
equations, constants are obtained. The stress and the strain tensors have been written
and the total energy, which is the total of the energies accumulated in both of two
materials, has been calculated.
Here, the second body having two different materials represents the representative
volume element of the composite including spherical particles. The single spherical
body, which has been defined in the first step, has been thought as an effective single
material which is equivalent to this representative volume element of the composite.
The calculated strain energies have been equalized based on the equivalance of two
bodies. So, the expression of the Bulk modulus of associated composite has been
obtained in the terms of the elastic constants of matrix and particles, and the volume
ratio or concentration.
The total strain energy accumulated in two sphere model has been equalized to the
strain energy of a single sphere made from the effective material under simple shear,
to calculate the shear modulus of the composite. For this problem, the displacement
vector has been written as the combination of the three types solutions of spherical
harmonics to solve a single sphere. Here, there are six unknown constants for the
single sphere. The strain and the stress components have been obtained by writing
three boundary conditions related to the stress components on the spherical surface.
Besides the displacement vector must be finite at the origin. Using this condition,
three solutions can be omitted. Number of the equations are nine while number of
the remaining unknown coefficients is three. But if these equations are examined, the
real number of the linear independent equations comes out as three. Solving these
equations, the displacement vector, the strain and the stress tensors, and the strain
energy of the single sphere have been calculated.
For two nested spheres, the solution involves 12 unknown constants. Three of them can
be eliminated from the condition written at the origin. In addition to three conditions
related the surface traction components at the outer spherical body, there are six more
conditions written on the common spherical boundary of two materials. Three of them,
displacement components, are equal on this boundary. Last three equations which
indicates surface traction vectors are also equal on this boundary for both materials. At
first, on spherical boundaries, 27 equations have been obtained. But only nine of them
linearly independent. Obtaining an alpha numeric solution is difficult. To handle this
difficulty, a numerical solution has been made for a specific value of concentration. It
must be emphasized that the shear moduli of two different materials must be known.
But the strain energy has been obtained for composite in the closed form in the terms
of remaining unknown coefficients.
Equating two strain energies obtained above and using numerical solution, a curve has
been plotted representing the effective shear modulus as a function of concentration.
The problem of simple torsion of a cylindrical body with respect to its own axis is
considered for calculation of the shear modulus of a fiber reinforced composite. Here,
the expressions of the total strain energies on the single cylindrical body and two
nested cylindrical bodies have been obtained using the classical theory of elasticity. So,
following the same method as previous problems the expression of the shear modulus
of composite material has been obtained.
As the last problem, the single cylindrical body which was thought as effective material
can be considered as micropolar material. Here, the equations of the theory of
asymmetric elasticity have been used. The problem has been simple torsion again.
In the micropolar theory, six material constants occur when the constitutive equations
are written. In addition to m and l from the classical theory of elasticity, there are
a, b, g, and e constants. A special material whose some constants, a and e, are
zero has been considered in here. In this condition, the stress and the strain tensors
are symmetric again. Because the analogy between them, the problem of the axial
normal force acting on the top and the bottom surfaces of cylindrical body has been
considered instead of the problem of torsion of it. ?It can be understood that the problem
is rotational symmetric. The calculated energy here has been equalized to the energy
of the previous two nested cylindrical bodies. So, the relation between b and g which
are micropolar constants of material has been obtained.
In the experimental studies, the matrix material has been chosen as a type of poliester.
As the inclusion, a type of kevlar has been used for cylindrical fibers. Firstly, only
poliester has been molded. After that, composite samples have been prepared. Two
samples which have the volume ratio of 1% and 1.5% have been prepared. Simple
tension test of these samples have been achieved and moduli of elasticity and Poisson?s
ratios have been obtained for each sample. The experimental results has been plotted
and compared with those from the analytical calculations.