Loading...
| Friend's email | |
| Your name | |
| Your email | |
| enter code | |
This page was sent successfuly
Numerical Modeling of a Smooth Notched Tensile Specimen Via Gradient Elasticity Based RKPM
Alavi, Ali | 2009
2372
Viewed
- Type of Document: M.Sc. Thesis
- Language: English
- Document No: 40222 (53)
- University: Sharif University of Technology, International Campus, Kish Island
- Department: Science and Engineering
- Advisor(s): Mohammadi Shodja, Hosain
- Abstract:
- 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 scale, which enters the model as an additional material parameter, size effects can be modeled properly. On the other hand, in recent years, a class of meshfree or meshless methods, such as smooth particle hydrodynamics, the diffuse element method, the element-free Galerkin method, partition of unity and the reproducing kernel particle method (RKPM) has emerged to demonstrate significant potential for solving moving boundary problems typified by growing cracks. Fundamental to all meshless methods, a structured mesh is not used, since only a scattered set of nodal points is required in the domain of interest. This feature presents significant implications for modeling fracture propagation, because the domain of interest is completely discredited by a set of nodes. In this work, the theory of gradient elasticity is used and numerically implemented by a meshless method to accurately model size effects. The goal of this thesis is to find the elastic fields of different examples like cantilever beam, plate with central square under tension, narrow smooth notched tensile specimen and square with hole under bending moment by the help of gradient elasticity based on reproducing kernel particle method (RKPM).
- Keywords:
- Size Effect ; Reproducing Kernel Particle Method (RKPM) ; Meshless Method ; Finite Element Method ; Element Free Galerkin Method (EFGM) ; Couple Stress ; Gradient Elasticity Theory
Digital Object List-
محتواي پايان نامه - view
Bookmark- Thesis PDF Ali Alavi.pdf
- ali alavi title
- 1.1.Introduction………………………………………………………………………………....…….13
- 1.2. RKPM Formulation……………………………………………………………….……..…....15
- 1.2.2. Examples of Spline Functions ………………………………...……...…………...….....21
- 1.2.3. Dilation Parameter..........………………………………………….………….….…..…..23
- 1.2.4. RKPM Shape Functions………………………………………….………….….…..…..23
- 1.4. Reproducing of Different Functions……………………………………......………….….…...34
- 1.5.2. Reproducing of First Derivative of RKPM Shape Functions………..……………....…..39
- 1.6. RKPM Meshless Method In two Dimensions Formulation…………………..…………..…..42
- 1.6.2. Reproduce Functions in 2D Case……………………………………….....…….………56
- 1.6.3. Reproduce of First Derivate of Functions in 2D Case……………………….………….57
- 1.6.4. Reproduce of Second Derivates of Functions in 2D Case…………………….….……..59
- Chapter II: Gradient Theory
- 2.1. Introduction………………………………………….………………………………………..61
- 2.2. Formulation of Gradient Theory ……………………………………………………………..62
- 2.2.1. Variational Formulation of Motion…………………...……………….....……………….62
- 2.2.2. Constitutive Equations…………………...……………….......………………...................67
- 2.2.3. Explicit Formulation....…………………...………………......………………...................70
- 4.Conclusion…………..…………………………...………..……………………………………….126
- 5.List of References……………………………………...…..…………...………………………….127
- Figure-1-15 Reproducing of Function y=x , No. of nodes=6,window fun. Spline o3, dilation parameter =0.9......................34
- Figure-1-16 Reproducing of Function y=x2 , No. of nodes=6, window fun. Spline order 3, dilation parameter =0.9.............35
- Figure-1-17 Reproducing of Function y=x3-2x210.5x+6 , No. of nodes=7,window fun. Spline 03, dilation parameter =0.9.......................35
- Figure-1-18 Reproducing of Function y=x3-2x2-10.5x+6 , No. of nodes=7, Spline o3 window fun., dilation parameter =1.05...................36
- Figure-1-19 Reproducing of Function y=x3-2x2-10.5x+6 , No. of nodes=7, cubic spline window fun. , dilation parameter =3.5................36
- Figure-1-43 Reproduce of fun. ......................................................................................................56
- Figure-1-44 ...........................................................................................................58
- Figure-1-45 ...........................................................................................................58
- Figure-1-46 Reproduce of fun. ( ..........................................................................................59
- Figure-1-47 Reproduce of fun. ( .........................................................................................60
- end after bending.pdf
- RKPM method formulation
- 1.2. RKPM Formulation
- where
- ,()-.=[,.,,()-,-1..,,()-,-2..,,()-,-3..,…]
- (10)
- and one can also express u(x) by
- ,.= ,-.,. (0)
- (11)
- Substituting equations (9) and (11) into Eq. (8) yields
- ,-.,.=,,0.− ,.,..=0
- (12)
- where
- ,.=,-,−. ,-.(−). ,-.,−.
- (13)
- Since u(x) is an arbitrary Nth order monomial, Eq. (12) implies
- ()=,,.-−1. (0)
- (14)
- and
- ,;−.= ,-.,0. ,-−1.,. (−)
- (15)
- (16)
- Eq. (16) can be recast into the following form
- ,-.,.= ,Ω-,,Φ.-..,;−. ,.
- (17)
- (18)
- (19)
- (20)
- (21)
- (22)
- (23)
- (24)
- (25)
- (26)
- (27)
- 1.2.2. Examples of Spline Functions
- 1.2.4. RKPM Shape Functions
- Figure-1-3 Shape function of Gauss window fun., No. of nodes=6, dilation parameter =1.25
- Discrete the reproducing equation, yields:
- ,-.,.=,=1--,,-..,-.(−)Δ,-..
- (28)
- (29)
- (30)
- (31)
- (32)
- (33)
- (34)
- (35)
- (36)
- 1.4. Reproducing of Different Functions
- More details are explained it the graphs
- /
- Figure-1-15 Reproducing of Function y=x , No. of nodes=6,window fun. Spline o3, dilation parameter =0.9
- /
- Figure-1-16 Reproducing of Function y=x2 , No. of nodes=6, window fun. Spline order 3, dilation parameter =0.9
- /
- Figure-1-17 Reproducing of Function y=x3-2x210.5x+6, No. of nodes=7, window fun. Spline 03, dilation parameter =0.9
- /
- Figure-1-18 Reproducing of Function y=x3-2x2-10.5x+6, No. of nodes=7, Spline o3 window fun, dilation parameter =1.05
- /
- Figure-1-19 Reproducing of Function y=x3-2x2-10.5x+6, No. of nodes=7, cubic spline window fun. , dilation parameter =3.5
- (38)
- (39)
- (40)
- (41)
- (42)
- (43)
- (44)
- (45)
- (46)
- (47)
- /
- /
- Figure-1-21 First derivation of shape fun. of cubic spline window fun. , No. of nodes=6, dilation parameter =1.05
- (48)
- where
- =,,-1.,,-2.,,-3..
- =(,-1.,,-2.,,-3.)
- (49)
- (50)
- (51)
- (52)
- (53)
- (54)
- (55)
- (56)
- (57)
- (58)
- (59)
- (60)
- (61)
- (62)
- (63)
- (64)
- /
- 1.6.2. Reproduce Functions in 2D Case:
- ,,.=,-.−+,- . ≤≤,≤≤ (65)
- Figure-1-43 Reproduce of fun. ,,.=,-2.−+,-2 .
- 1.6.3. Reproduce of First Derivate of Functions in 2D Case:
- The first derivative of function (65) is reproduced with respect to x and y respectively in the domain ,[0,5]-2. .The number of particles is 121 and the dilation parameter equals 1 (a=1). The figures of this reproduce function are shown below.
- /
- Figure-1-44 Reprodece of Fun. ,-. (,-2.−+,-2 .)
- /
- Figure-1-45 Reprodece of Fun. ,-. (,-2.−+,-2 .)
- 1.6.4. Reproduce of Second Derivates of Functions in 2D Case:
- The second derivative of function ,-.−,-.+−,-.+,-. is reproduced with respect to x and y respectively in the domain ,[0,5]-2. .The number of particles is 12×12 and the dilation parameter is equal 0.91 (a=0.91). The fig...
- /
- Chapter II
- 2.1. Introduction
- 2.2. Formulation of Gradient Theory
- (1)
- (2)
- (3)
- (4)
- (5)
- (6)
- (7)
- (8)
- (9)
- (10)
- (11)
- (12)
- (13)
- (14)
- (15)
- (16)
- (20)
- Equation (25) can equally be written as
- Spatial Discretization
- 2.2.3. Explicit Formulation
- The starting point is the weak formulation of the equilibrium equation
- Integrating the stress term by parts yields
- To show the effect of hole on the entire body of our numerical model, we can find the Cartesian stresses for five different radiuses (r=1, 2, 3, 4 and 5).
- /
- These results are shown below and compared with their exact solutions:
- /
- Consequently, as it is shown above, the hole shows its effect when 1≤≤2 and as the radius increases, the effects of hole vanishes.
- On the other hand, the ,-. ,-. stresses have significant amounts around the hole and as the radius increases, their values gradually decrease as it is shown below:
- /
- /
- /
- As it was shown, classical theories are not capable of describing size effects because of the lack of material length scale parameter. On the other hand, enhanced elasticity models, such as gradient theory, are able to capture size effect adequa...
- 5. List of References
- ali alavi title
- ali alavi farsi