Sharif Digital Repository

This project presents an outline of the crack tip stress fields in the presence of a plastic enclave around a growing fatigue crack using RKPM and Drucker-Prager criterion for pressure sensitive materials such as concrete and rock. Two distinguished features of RKPM are the arbitrarily high order smoothness and the interpolation property of the shape function which has kronecker delta property at the associated nodes. This method has proved to handle extreme material deformation and moving discontinuities. For the frictional materials such as rock in geotechnical engineering, a non-associated or associated Drucker- Prager plasticity model is appropriate for modeling its constitutive... 

Traditional continuum theory of elasticity becomes remarkably inaccurate in the vicinity of singularities, and when the size eect is of concern. For example in the study of ultra-small objects and ultra-thin lms, near defects, near point of application of a concentrated load and as such, the classical solutions are not reliable. This work focuses on determination of the elastic elds of an edge dislocation near the grain boundary of two perfectly bonded nano-size crystals. It is proposed to study this problem in the context of surface/interface elasticity, and incorporate the eect of the grain boundary on the elastic elds. In contrast to the surface/interface elasticity theory, traditional... 

In this research, a new Kirchhoff plate model based on the modified couple stress theory has been utilized to derive the corresponding closed-form expression for the buckling load. Moreover, a numerical mesh-less method, Reproducing Kernel Particle Method (RKPM), in combination with Corrected Collocation Method (CCM) has been employed to model the nano-plate and calculate its buckling load in the framework of the modified couple stress theory. To this end, two kinds of nano-plates have been modeled, the square nano-plates with all edges simply supported 1) in the presence of the nano-void and 2) without the nano-void. It should be noted that the analytical and numerical solutions for the... 

In this research, circular and elliptic inhomogeneities are modeled using a numerical meshless method named RKPM. A newly developed accurate and simple method called augmented corrected collocation method, which was previously applied in order to model material discontinuities in the framework of classical continuum theory, is used in combination with the penalty method, considering couple stress elasticity and in micro or nano scales for the first time, where the classical continuum theory has certain shortcomings in predicting the behavior of materials. Simulating nano-composites subjected to anti-plane stresses and comparing the analytical and numerical results show that the augmented... 

A theory of gradient elasticity is used and numerically implemented by a meshless method that is called reproducing kernel particle method (RKPM) to model size effects. Some of the problems are modeled under the consideration of gradient elasticity for the first time and all of them are also modeled with classical elasticity to compare with gradient elasticity. First of all, the RKPM formulation and computing the amount of shape functions and requisite derivatives will be explained with details and a mathematical innovation that will decrease the computational cost seriously proposed for the first time. Several 1D and 2D shape functions with first and second derivatives that are resulted... 

In this thesis, some new models have been solved via RKPM method, which is one of the meshfree methods family. These models have never been solved via meshfree methods and their analytical solutions do not exist. At first, the RKPM shape functions and their first derivative formulation in 1D and 2D have been presented and then by using FORTRAN program, the shape functions and their first derivative have been obtained. To verifying the code some functions have been reproduced. In the next step by using the governing equations and penalty method whose formulation exists in chapter 2; some famous examples in linear elasticity have been solved via RKPM to verify the FORTRAN code. At last; some... 

Recently, there has been a strong interest in the development of a new class of meshfree methods. As an alternative to the finite element method (FEM), mainly due to elimination of high cost mesh generation processes. In addition, the size effect is currently a subject of increasing interest since it is an important parameter in predicting, correctly, the mechanical behavior of materials with microstructure. It was well established that classical linear elastic continua which neglects the higher order terms is not able to describe size effects. In contrast, enhanced continuum theories such as nonlocal or gradient-dependent models do involve an internal length scale. Thorough this length... 

In many structures, crack creation is one of the most significant fracture mechanisms. To predict these fracture mechanisms accurate numerical modeling is necssary. Finite Element Method (FEM) is one of the substantial methods in analysis of numerical fracture problems in recent past decades. But, this method has difficulties in remeshing of elements in each step of calculation in fracture mechanics or large deformation analysis. Therefore, the theory was defined that, without using elements, just with setting of characteristics nodes in geometry of problem, the differential equations can be solved. These methods are called Meshfree or Meshless methods. RKPM is a new meshfree method for... 

The aim of this work is to study a cohesive crack in an elastic solid with meshless method. The procedure uses Reproducing Kernel Particle Method (RKPM) formulation in conjunction with Penalty method for implementing all constraints, including the Essential Boundary Conditions (EBCs) and the constraints related to cohesive crack. Meanwhile subdomain technique is employed to diminish the compiling process to facilitate one. Study of Stress Intensity Factor (SIF) at the tip of the cohesive crack has also been dealt with particular interest  

Meshless Methods using kernel approximation like Reproducing Kernel Particle Method (RKPM) are methods for solving partial differential equations that require only nodal data and a description of the geometry without requiring element connectivity data and mesh producing. An innovative method of nonplanar material partitioning method (NMPM) with implementation of RKPM is employed to calculate the stress intensity factor (SIF) at the tip of an anticrack sited in an isotropic plate under a remote applied loading. Numerical examples in comparison with the exact closed form expressions show that accurate SIF for mode I can be obtained.

 

Nowadays, the excellent technological applications of composites have attracted the attentions of industry and numerous scientists. They are advantageous for their high tensile modulus, strength, and promising electrical and thermal properties. In applying the approach of lamellar inhomogeneity to real composites, the micro-geometries of the reinforcement must be considered such that they can be approximated as limiting case of an ellipsoid. In vapor grown carbon nanofiber, the fiber may have a diameter of about 150nm and length of 10-20 µm [1]. The modulus of carbon nanofiber is normally in the range of 100-600 GPa and sometimes even higher, whereas the modulus of some polymers is usually... 

An accurate numerical methodology for capturing the field quantities a center lamellar inhomogeneity within the interface of two different phases of a bimaterial rectangular plate subjected to non-uniform tensile stress, in the context of reproducing kernel particle method (RKPM), is of particular interest. For this purpose the innovative numerical technique, so-called extended augmented corrected collocation method is introduced; this technique is an extension of the augmented corrected collocation method used for imposing continuity condition of the displacement at the interface. The robustness of this methodology is shown by solving problems of material discontinuities, namely plates... 

The present study aims at determining the elastic fields of ultra-small flaws and defects. These defects are often introduced undesirably in elastic solids during fabrication and their sizes are normally in the order of couple of nano-meters. In this work, the elastic fields around a circular nano-void subjected to a uniform farfield uniaxial tension, also the elastic fields of a nano-sized mode I crack under remote uniform loading are studied. In this paper the strain gradient theory developed by Mindlin and co-workers in 1960s is employed. According to this theory, the strain energy density assumes the form of a positive-definite function of the strain components and their first gradient.... 

The so-called moving least square reproducing kernel method (MLSRKM), which is a meshfree Galerkin method, has two special properties in comparison with its prior version, say moving least square method (MLSM): (i) stability in preserving completeness conditions to the desired order for a discrete reproducing formula due to implementation of shifted and scaled basis, (ii) use of more accurate quadrature weights in a discrete reproducing form, especially along boundaries. However, because of employing shifted basis in MLSRKM, some of enrichment techniques, specifically in the context of linear elastic fracture mechanics (LEFM) problems, cannot be applied to this method. Therefore, we are... 

Reproducing kernel particle method (RKPM) is a meshfree method for solving various differential equations. RKPM is based on pure mathematics; therefore, it is in the center of attention of many scientists. One major problem in RKPM is satisfying the essential boundary conditions (EBCs) involving the derivative of the field function. This problem is considered herein and its solution is proposed. To this end, two actions should be undertaken. First, the concept of Hermitian interpolation is employed to add the derivative term to the reproducing equation of RKPM and a new meshless method called gradient RKPM (GRKPM) is introduced. Second, the corrected collocation method is modified so... 

In this thesis one of the mesh-free methods called RKPM is employed to solve the differential equations of strain-gradient elasticity. To this end the corresponding weak form is laid down. Subsequently the relevant stiffness-matrix is obtained by discretization of the weak form. To be sure about the accuracy of the relations, the problem of a plate weakened by a hole under uniform far-field tension, for which the exact solution is available in the literature, is solved. The obtained numerical result is in good agreement with the solution of Eshel and Rosenfeld. Afterwards, a plate containing a crack of finite length subjected to uniform far-field tension (mode I) is considered. This problem... 

During recent years, many researches on meshfree methods to solve differential equations and crack problem have been accomplished, and acceptable results have been obtained. One of these methods which is widely used in fracture mechanics specially in problems including crack is RKPM (reproducing kernel particle method). RKPM is one of the modern numerical methods in solving differential equations that has been lately introduced and developed. In this method, the genuine response of the system is replaced with a good approximation of the real response called ‘Reproduced Function’. The formulation of this method obviates the need for discretizing the domain by meshing with elements. In this... 

During the past decade, element free methods have achieved great successes. One of these methods is the so called RKPM which has a suitable structure for use in fracture mechanics problems. Despite all characteristic abilities of element free methods; these methods due to their higher order continuous differentiable approximations fail to model discontinuous material properties of the subjected domains. In this study by improving the collocation method in RKPM treatment of such conditions have been achieved. Also in this study performance of a new meshfree method in fracture mechanics problems has been analyzed. GRKPM is one of these methods which its suitable accuracy and convergence has... 

It is proposed to obtain the mode I plastic zone size and shape at the crack-tip in a work-hardening material using reproducing kernel particle method (RKPM). RKPM is a meshless technology which has proven very useful for solving problems of fracture mechanics. Ramberg-Osgood stress-strain relation is assumed. In this project the crack-tip stress intensity factor (SIF) before and after formation of the plastic zone will be examined. To impose the essential boundary conditions, penalty method is used. To construct the shape functions in the vicinity of the crack and crack-tip, both the diffraction and visibility methods are employed. The effects of different dilation parameters on SIF under...