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فهرست محتوای دیجیتالی
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مدل سازی عددی نمونه آزمایشی با انحنای ملایم تحت کشش با استفاده از روش RKPM بر مبنای تئوری گرادیان الاستیسیته
علوی، علی Alavi, Ali
- شماره پایان نامه:40222
- کد دانشکده:53
- پديدآور: علوی، علی
- عنوان:مدل سازی عددی نمونه آزمایشی با انحنای ملایم تحت کشش با استفاده از روش RKPM بر مبنای تئوری گرادیان الاستیسیته.
- نام دانشگاه/پژوهشگاه:صنعتی شریف
- سال اخذ مدرك:1388.
- نام دانشکده:مهندسی و علوم
- مقطع:کارشناسی ارشد
- گرایش:مهندسی عمران - سازه
- یادداشت:129ص.: نمودار، کتابنامه؛ چکیده به فارسی و انگلیسی
- یادداشت:متن به انگلیسی
- توصیفگر: اثر اندازه Size Effect
- توصیفگر: روش بازتولید نقطه با هسته Reproducing Kernel Particle Method (RKPM)
- توصیفگر: روش بدون شبکه Meshless Method
- توصیفگر: روش اجزای محدود Finite Element Method
- توصیفگر: روش گالرکین بدون اجزا Element Free Galerkin Method (EFGM)
- توصیفگر: تنش تزویجی Couple Stress
- توصیفگر: نظریه گرادیان کشسانی Gradient Elasticity Theory
- استاد راهنما. محمدی شجاع، حسین
فهرست محتوای دیجیتالی
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محتواي پايان نامه
- مشاهده
فهرست مطالب
- 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
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