Wave propagation in functionally graded and layered materials

Fonksiyonel derecelendirilmiş ve çok katmanlı kompozit malzemelerde dalga yayılması

Abstract

Composite materials have been used by the engineers. Thus by using proper combination of different materials one can produce a material with an improved thermal, acoustical, mechanical or electrical properties according to the needs in design. Composite materials can be classified in three groups, which are fibrous composites, laminated composites and particulate composites. Many different approaches have been used for approximating the mechanical behaviour of composite materials. The mechanical behaviour of composite materials under static loads are well documented. On the other hand less is known about mechanical behaviour of composites under dynamic loads. Behaviour of composite materials, which are used in military equipments, under impact loads and modeling of this behaviour is gaining attention. In design and modelling of composite materials two main approaches have been used. One of these is to model the composite material as layered material. In modelling the layered media most of the attention is paid to the defining the effective material properties. Many theoretical and experimental studies have been done on this topic. The use of layered media in designing composite plates has some disadvantages. Sudden change in material properties causes stress concentration in the material which may cause failure. The later is functionally graded materials (FGMs). The material properties are continuous in FGMs. Therefore no stress concentration occurs. This property of FGMs makes it advantageous for many applications. Modelling of both FGMs and layered media are challenging task. Modelling of graded material increases the computational cost whilst modelling the layered media degrades the accuracy due to the sudden change of material properties. Thus degradation of numerical accuracy affects the conservation of energy, momentum and angular momentum. Conservation properties of numerical methods are important for many structural dynamics problems. Therefore in the recent years more attention paid to the research on the conservative numerical methods for time integration and discretization in space. In the field of computational mechanics researchers are focused on the conservation properties and stability of the time integration algorithms. Newmark time integration method is one of the most widely used time integration method in structural dynamics. On behalf of this, Newmark family of algorithms are not energy and angular momentum conserving. Energy preserving algorithms are developed by the researchers. Thus energy preserving schemes are lack of high frequency dissipation, which is necessary for damping high frequency oscillations in the numerical solution. Space-time finite element method is attractive for researchers due to the stability properties. On the other hand time finite element methods are not energy conserving. One approach in discretizing the balance equations is discontinuous Galerkin method. Time Discontinuous Galerkin method (TDG) has begun to be used in elastodynamics. It has small phase error and has small dissipation error at high frequency regime which damps the high frequency oscillations in the numerical solution makes it attractive for wide range of structural problems. The DGM is widely used in the spatial discretization of first-order partial differential equations( PDEs) in fluid mechanics, because of its local and global conservation properties. Since it allows discontinuities at the element interfaces, it is advantageous to use DGM for shock wave propagation problems. More recently DGM for the discretization of second order PDEs has been developed.Another advantage of the DGM is continuously changing material properties can be defined in an element for FGMs and sudden change of the material properties are allowed at the element interfaces for layered materials. n this study mechanical behaviour of layered materials and FGMs is compared under impact loads. Discontinuous Galerkin method will be used for this purpose. Within these context numerical results obtained from the solution of 4, 8 and 16 layered axi-symmetric composite plate is compared with FGM plate. It is observed that as the number of layers increases the effective stress results obtained from layered plate is getting closer to the FGM plate at the centre of the plate. On the other hand it is seen that as the stress wave propagates in the radial direction effective stress values obtained from the layered plate differs from the results obtained form the FGM plate.

Özet

Askeri ekipmanlarda kullanılan kompozit malzemelerin darbe yükleri altındaki davranışı ve bu davranışın modellenmesi giderek önem kazanmaktadır. Kompozit plakların tasarımında iki temel yaklaşım kullanılmaktadır. Bunlardan bir tanesi farklı malzeme özelliklerine sahip malzemelerden oluşan çok katmanlı plaklardır. Diğeri ise özellikleri fonksiyonel olarak derecelendirilmiş malzemelerdir( FDM). FDM’lerde malzeme özelliklerinin sürekli olmasından dolayı malzeme içerisinde gerilme yığılması meydana gelmemektedir. FDM'nin bu özelliği onu bir çok uygulama için uygun kılmaktadır. Malzemelerin darbe yükleri altındaki davranışlarının incelenmesinde özellikle malzemenin dinamik plastik davranışının modellenmesi, temas yüklerinin tespiti ve darbe yükünün etkisi ile meydana gelen şok dalgasının ilerleyişinin doğru olarak modellenmesi silah sistemlerinin tasarımında büyük öneme sahiptir. Burada kullanılan sayısal yöntemin korunum özellikleri (momentum, açısal momentum ve enerji korunumu) önem kazanmaktadır. Süreksiz Galerkin yöntemi, korunum özellikleri ve eleman sınırlarında süreksizliğe izin vermesi nedeni ile dalga yayılması problemlerinde araştırmacılar tarafından tercih edilmektedir. Bu çalışmada çok katmanlı malzemelerin ve FDM’nin ani darbe yükleri altındaki davranışları süreksiz Galerkin yöntemi kullanılarak incelenmiştir. Bu kapsamda ortasından ani darbe yüküne maruz bırakılmış,4, 8 ve 16 katmandan oluşan eksenel simetrik plak probleminden elde edilen sonuçlar FDM plak probleminden elde edilen sonuçlar ile karşılaştırılmıştır. Yapılan çalışma sonucu katman sayısı arttıkça plağın merkezindeki etkin gerilme değerlerinin FDM plakta hesaplanan etkin gerilme değerlerine yaklaştığı tespit edilmiştir. Diğer taraftan çok katmanlı plaklarda radyal yönde ilerledikçe etkin gerilme değerleri FDM plakda hesaplanan değerlerden farklılaştığı görülmüştür.