Nanomekanikte yerel olmayan elastisite teorisi ve çok-ölçekli modellemeye uygulanması
Nonlocal theory of elasticity in nanomechanics and application to multiscale models
Yazar: MERAL TUNA EROĞLU
Danışman: DOÇ. DR. MESUT KIRCA
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Makine Mühendisliği Ana Bilim Dalı / Makine Mühendisliği Bilim Dalı
Derece: Doktora, 2019
Konu: Makine Mühendisliği = Mechanical Engineering ; Mühendislik Bilimleri = Engineering Sciences

Özet:
Teknolojik gelişmeler nano elemanların uygulama alanlarını genişletmekte ve bu tür yapıların davranışlarını açıklamaya duyulan ihtiyacı her geçen gün arttırmaktadır. İncelenen yapının ölçülerinin kendisini oluşturan atom, molekül veya granül arasındaki mesafeyle kıyaslanabilecek kadar az olması veya yapıya uygulanan dış kuvvetlerin frekansının, yapıyı oluşturan atomların titreşim frekansıyla kıyaslanabilecek kadar yüksek olması, vb. gibi durumlarda klasik elastisite teorisi yetersiz kalmakta ve boyut etkilerini göz önünde bulunduran zenginleştirilmiş sürekli ortam modellerine ihtiyaç duyulmaktadır. Kröner, Krumhansl, Kunin, Eringen ve Edelen tarafından geliştirilmiş olan ve bu tez kapsamında dikkate alınan yerel olmayan elastisite teorisi de bunlardan biridir. Teorinin diferansiyel formunun bazı sınır ve yükleme şartlarında tutarsız sonuçlar verebildiği literatürde rapor edildiğinden, bünye denkleminin orijinal integral formu kullanılarak nano-çubuk, nano-kiriş, nano-levha gibi bir ve iki boyutlu yapısal elemanların mekanik etkiler altındaki davranışları incelenmiş olup oluşturulan sonlu eleman kodu ile atomik zincir, atomik dizi ve karbon nanotüp gibi atomik yapıların eşdeğer sürekli ortam modelleri elde edilmiştir. Bunları takiben, yerel olmayan elastisite teorisinin entegre edildiği yeni bir çok-ölçekli model geliştirilerek, kusura sahip atomik yapıların, bütün yapıyı atomik olarak modellemeye ihtiyaç duymaksızın az maliyet ve yüksek doğrulukla incelenmesi sağlanmıştır.

Çalışmanın ilk adımında Laplace dönüşümü ve konvolüsyon teoremlerinden faydalanarak nano-kirişlerde eğilme, burkulma ve serbest titreşim problemleri incelenmiştir. Literatürde ilk defa bünye denkleminin integral formu kullanılarak, farklı sınır ve yükleme koşulları için elastik eğri denklemleri açıkça sunulmuş ve doğal frekanslar ile kritik burkulma yükleri elde edilmiştir. İntegral form ile ankastre mesnetlerde karşılaşılan tutarsızlıkların giderildiği ve boyut etkilerinin yalnızca eğilme rijitliği üzerinde değil, kayma rijitliği üzerinde de etkili olduğu gösterilmiştir. Kiriş yapılar için geliştirilen sonlu eleman kodu, tez çalışması kapsamında tespit edilen çözümler ile karşılaştırılarak yeni bir ayrıklaştırma şekli önerilmiş ve ilk defa oldukça az sayıda eleman kullanılarak doğruluğu yüksek sayısal sonuçların elde edilmesi sağlanmıştır. 

Çalışmanın ikinci adımında bir boyutlu atomik zincir, iki boyutlu atomik dizi ve üç boyutlu karbon nanotüplerin eşdeğer sürekli ortam modelleri tespit edilmeye çalışılmıştır. Bunun için atomik ilişkileri dikkate alan moleküler dinamik simülasyonları kullanılarak farklı boy ve sınır şartları altındaki karbon nanotüplerin serbest titreşim frekansları ve kritik burkulma yükleri tespit edilmiş olup, Diferansiyel Evrim Algoritması'ndan yararlanarak eşdeğer nano-kiriş modelinin elastiklik modülü, küçüklük (ölçek) parametresi, özgül ağırlığı gibi mekanik ve fiziksel parametrelerinin optimum değerleri elde edilmiştir. Literatürden farklı olarak, aranan mekanik parametrelerin değerlerinin, yapının boyutlarından, incelenen problemden ve sınır şartlarından bağımsız olduğu gösterilmiştir. Ardından, bir boyutlu atomik zincir ve iki boyutlu atomik dizilerin eşdeğer nano-çubuk ve nano-levha modelleri, aynı şekil değiştirme alanına sahip atomik ve sürekli ortam modellerinin iç enerjilerinin belirlenen birim hacimde eşit olması gerekliliğinden faydalanılarak elde edilmiştir. Bu sayede mekanik parametrelerin analitik ifadelerinin atomlar arası potansiyeller cinsinden elde edilmesi sağlanmıştır.

Son aşamada, klasik olmayan teorilerin dahi yetersiz kaldığı ve doğrudan atomlar, moleküller, granüller arasındaki ilişkilerin göz önüne alınmasını gerektiren problemlerin (çatlak ilerlemesi, dislokasyon hareketi, vb.) az maliyet ve yüksek doğrulukla çözülmesini sağlayan çok-ölçekli yaklaşıma, yerel olmayan elastisite teorisi entegre edilerek yeni bir model geliştirilmiştir. Geliştirilen enerji temelli (energy-based), eş zamanlı (concurrent) ve parçalı (partitioned-domain) tipteki çok-ölçekli modelde; şekil değiştirme gradyeninin nispeten düşük olduğu bölge, yerel olmayan elastisite kuramına dayanan sonlu eleman modeli ile, atomlar arası etkileşimlerin takip edilmesi gereken bölge ise klasik atomik teorilerden faydalanılarak modellenmiş, bu sayede bir ve iki boyutlu atomik yapılar çeşitli mekanik yükler altında incelenmiştir. Yerel olmayan elastisite teorisinin atomik teoride olduğu gibi uzun mesafe etkileşimleri hesaba katması sayesinde, sürekli ortam ve atomik bölgelerin arayüzünde karşılaşılan çarpılmaların, klasik elastisite teorisi kullanılan modele göre, büyük oranda azaltılabildiği gösterilmiştir.

Abstract:
Recent technological developments drastically enlarged the application field of nanoscopic elements, and led to many studies that focus on clarifying the behaviour of such structures. Although different approaches including experimental measurements and atomistic modelling techniques are available, it is favourable resorting enhanced continuum theories that balance accuracy with computational efficiency. The first attempts to address the non-local effects by introducing an internal scale parameter dates back to work of Cosserat brothers in the beginning of 20th century, which then motivates many researchers on developing such non-classical theories. Among them Eringen's nonlocal theory of elasticity, which was presented by Kröner, Krumhansl, Kunin and further improved by Edelen and Eringen, becomes the main object of this study. 

The nonlocal theory of elasticity is concerned with the physics of materials bodies whose behaviour at a material point is influenced by the state of entire body. It exhibits a convolution type constitutive relation where stress at each point is related to the strain of entire domain through an attenuation type kernel function. Since the highly popular differential form of the theory has been reported to present insubstantial results for some loading and boundary conditions, herein the original integral form is employed to study nano structures such as bars, beams and plates under mechanical loads. The established finite element formulation is then employed to detect the equivalent continuum models of one-dimensional atomic chains, two-dimensional atomic arrays and three-dimensional carbon nanotubes. With this way, it is aimed to integrate the nonlocal theory of elasticity into an energy-based, partitioned domain, concurrent multiscale model, which enables investigating a nanoscopic structure with atomic defect without requiring a fully atomistic model.  

As the first step, the closed-form analytical solutions of original integral form of Eringen model is derived for static bending of Euler-bernoulli and Timoshenko beams considering four different loading and boundary conditions. This is achieved by transforming Fredholm integral type field equations into Volterra integral equations of second kind and using Laplace transform and convolution theorem. In contrast to differential form, for all investigated cases, the stiffness of structures with size effects decreases with increasing nonlocality. Moreover, an additional term including the nonlocal parameter appears in the solution of Timosheko beam model pointing out that the nonlocal effects act on beams through not only bending, but also shear deformation.

Since field equations are of integrodifferential type, existence and uniqueness of their solution are not guaranteed. To investigate the existence of exact solutions, the analytical expressions of deflections are put into the corresponding field equations and resulting bending moments are compared with the ones obtained from balance equations. Indeed, except for cantilevered beam with distributed load, an error, which spreads with increasing nonlocality, is encountered around the boundaries of the domain. However, it is later demonstrated that although some of the analytical expressions of the deflections are not exact, the errors are negligible for physically meaningful ratios of nonlocal parameter and beam length. Following the bending analysis, critical buckling loads and free vibration frequencies are obtained by solving the first-order governing differential equation system with using transfer matrix and Laplace transformation rule.

After detecting solutions of homogeneous nanobeams, the finite element formulation for bending, linear buckling and free vibration analysis are developed by minimum total potential energy principle. Different from existing literature, a non-uniform mesh distribution is proposed for the corresponding analytical expressions of the beam deflections. With this aid, the discontinuous nature of rotation angle, which is encountered at boundaries of the beam, is aimed to be captured. To demonstrate the versatility of the non-local finite element method with the proposed mesh configuration many numerical examples are solved, and compared with the exact solutions obtained previously. It is found out that, with the suggested mesh distribution, the number of elements can be decreased dramatically without sacrificing the accuracy, hence, leads to a considerable reduction in the computational cost.

As the second step, an evolutionary optimization approach is presented to provide unique/unified values of nonlocal parameter and Young's modulus through focusing on sample problems of one-dimensional (1D) atomic chains and armchair single walled carbon nanotubes (SWCNTs). Optimum values are calculated by matching the buckling loads and free vibration frequencies obtained from the atomistic simulations, and Eringen's nonlocal theory based finite element models for beam and bar structures. Indeed, results show that material properties does not depend on length, mode shape and analysis type; but exhibit a slight dependence of boundary conditions, which may be considered negligible for practical applications. In this respect, presented material properties may be used with a sufficient accuracy for all types of mechanical problems, boundary conditions, and aspect ratios of 1D atomic chains and armchair SWCNTs which can be modelled as bar and beams. Moreover, the inadequate character of local theory in case of long range interactions is clearly demonstrated since the presented approach estimates nonlocal parameter values greater than zero. Although CNTs with large aspect ratios are considered to be in accordance with Euler-Bernoulli beam theory, it is concluded that the size effects still exist and should be incorporated with nonlocal theory. Besides, since the ill-posed character of integral-type nonlocal constitutive model does not adversely affect the overall behavior, it is possible to conclude that the utilization of well-posed constitutive models would yield equivalent material properties.

Advancing further, equivalent continuum finite element model for two- dimensional (2D) atomic arrays under plane-stress condition is formulated, based on Eringen's two phase local/nonlocal model. The interaction between the atoms is modelled using translational and rotational linear elastic springs including both nearest and second nearest neighbour relations. Explicit relations between those set of springs and material properties of associated continuum model is looked for by means of equivalency of potential energy stored in atomic bonds and strain energy of continuum. Indeed, the capability of equivalent continuum models are highlighted in terms of both accuracy and computational expense via comparing models under various loading scenarios. Numerical experiments show that even though the total number of degree of freedoms is reduced by 75\%, all continuum models are well capable of providing very accurate solutions, while it becomes dependent on nonlocal material parameters for increasing deformation gradients. Such a behaviour is expected due to analogous nature of nonlocal and atomic models, which may help recovering more accurate solutions.

With the motivation of possibly smoother transition between coarse and fine scales of partitioned-domain multiscale models due to the ability of nonlocal continuum model in incorporating long-range interactions, an energy based, concurrent model is built as the last step. The formulation is derived based on zero temperature statics for which no time-dependent effects exist. For the sake of simplicity and computational expense, the transition between atomistic and continuum models are assumed to be direct with excluding any sort of handshake (overlapping) region. To precisely model the contribution of the atoms near the interface, a padding region is introduced due to the nonlocal nature of atomic bonds. The padding region is a continuum region with imaginary padding atoms, the position of which is traced for proper calculation of energy of atoms in atomistic region. The continuum region is modelled based finite element method. The existence of long range interactions inherently induces the concept of what we are called as padding elements. These elements, which are required for construction of cross stiffness matrices, are located at the atomistic region. Even though their energy is not explicitly included to the model, their contribution to other elements should be considered, similar to what was referred about padding atoms. Consequently, the width of padding area depends on both the range of atomic interactions and the influence zone of an element located adjacent to interface. To reduce the computational expense arising from different Jacobians, and to avoid the complexity in mesh generation near the interface, continuum part is discretized using identical elements having same dimensions. One can notice that, if the length of finite elements in continuum region is taken equal to the length of unit cell in atomistic region, even one-to-one correspondence between atoms and nodes can be achieved. However, it is not very practical since the total number of degree of freedoms remains constant comparing with full atomistic model. That is why a different strategy is followed. First, the padding atoms are constrained to move in accordance with coarse-scale displacement field, as if they are glued to the finite elements in which they coincident. Then, compatibility between atomistic/continuum interface is adopted via the master-slave approach. Lastly, for padding elements, strong compatibility, and one-to-one correspondence is enforced via using elements with dimensions equal to atomic distance. The validity and applicability of proposed multiscale model is examined in the context of 1D atomic-chain and 2D atomic arrays with a flat atomistic/continuum interface, considering static problems only. Although, 1D case may not be of interest in practical applications, it is included for the integrity of the paper, and enables one to understand and track each step of proposed modelling technique, easily. For numerical examples, the lattice structure is composed by equally spaced identical atoms connected via linear elastic springs. It is assumed that, the atoms are positioned at equilibrium configuration for initial step, and each node that located at the interface and/or padding region coincident with an atom. To study the impact of element's dimensions over the accuracy, the length of real elements is altered for a fixed value of atomic distance. In order to minimize the spurious effects at transition, the nonlocality of continuum model is arranged via optimizing the nonlocal material properties in constitutive equation using differential evolution method. During optimization the link between strain energy on the continuum level and the energy stored in atomic bonds are ensured to provide equivalent models. Consequently, the numerical experiments performed on 1D atomic chain, and 2D intact as well as damaged atomic arrays show that, with using nonlocal theory integrated multiscale model, the distortions around the interface of atomic and continuum subregions are suppressed about 60-70\% compared with local theory based multiscale model. This improvement can be attributed to the analogous character of Eringen's model with atomic theories, both incorporate long range interactions.