Phân tích trạng thái giới hạn của tấm phẳng có vết nứt bằng phương pháp phần tử hữu hạn




MỤC LỤC
CHƯƠNG 1: TỔNG QUAN 1
1.1.  1
  6
1.3.   8
1.4.  9
1.5.   10
CHƯƠNG 2: LÝ THUYẾT CƠ HỌC RẠN NỨT, LÝ THUYẾT ĐÀN HỒI,
LÝ THUYẾT TẤM 11
2.1.  11
2.2.  17
 21
CHƯƠNG 3: PHƯƠNG PHÁP PHẦN TỬ HỮU HẠN 26
3.1.  26
3.2.  giác 8 nút 28
 30
3.4.  trong FEM 33
3.5.  34
3.6.  35
 37
-  38
3.9. PNewmark 40






CHƯƠNG 4: BÀI TOÁN ÁP DỤNG 45
4.1.  46
4.2.  47
4.3.  thép SM490  50
4.5-6Al-4V. 55
4.6. Kh-4147 70
CHƯƠNG 5: KẾT LUẬN VÀ HƯỚNG PHÁT TRIỂN 86
5.1.  86
4.2.  87

TÀI LIỆU THAM KHẢO



Chương 1
GVHD: PGS.TS Nguy1







P       





liên  
  .
 thái mà


 
   


 
- 1933) 

 
       
     


 







Chương 1
GVHD: PGS.TS Nguy2


:
T  phát sinh, 
 
C 


C 



 i
 


 
kim



1

 tốc độ giải phóng năng lượng (The
energy release rate) 




hệ số cường độ ứng
suất (the stress intensity factor).
Trong  


Chương 1
GVHD: PGS.TS Nguy3


tr, Fracture
Mechanics).
V

        rwin, David Broeke Paris

 [4, 32]








Richard W. Hertzberg. Cơ học biến dạng và rạn nứt của vật liệu cơ khí”
[36].
Irwin, G.R.“Động học rạn nứt”, “Sự rạn nứt của vật liệu” [18]





 




-LEFM), nó
:

Hệ số cường độ ứng suất (Stress Intensity Factor-SIF) [25, 36, 32


 
Chương 1
GVHD: PGS.TS Nguy4


 không


.
 


 
P. Cawley, R.D. Adams. Xác định vị trí và khuyết tật trong kết cấu tấm
bằng kết quả đo đạc thực nghiệm tần số tự nhiên” [31];
Nghiên cứu ảnh hưởng của vết nứt đến tần số dao động của kết
cấu dầm và khung” [21

c
Wu XR, Carlsson AJ. “Hàm trọng số và hệ số cường độ ứng suất” [44].
Brennan FP, Teh LS. “Xác định SIF bằng tổ hợp hàm trọng số” [6].
Chen DH, Nisitani H, Mori K. “Tính toán SIF trên tấm bán vô hạn có vết

nứt elip dưới ứng suất kéo” [7].
Chongmin Song*, Zora Vrcelj. “Đánh giá hệ số cường độ và ứng suất T
bằng phương pháp điều kiện biên” 
-
i nghiên

CTOD – crack tip open displacement
(độ mở đầu vết nứt) .

 J integral.
Cá



  Khiêm  
 

Chương 1
GVHD: PGS.TS Nguy5



  
.
 
Xuân Hoàng -  
.
               
khung   
  

 - .

P


h (FEM)   Matlab

P cho .

.

chọn mô hình Phần tử đẳng tham số tứ giác bậc hai (8 nút), Phần tử
đẳng tham số tam giác bậc hai 

.

Phần tử đẳng tham số tứ giác bậc hai (8 nút) và Phần tử tam
giác suy biến điểm 1/4
xác 

[22, 33, 34, 39].

              
 
 
v.v
Chương 1
GVHD: PGS.TS Nguy6


             
 
,  v.v
 


   
          
 v.v 
 v.v


               

 v.v





.
V
  , 
tài :
Phân tích           
.

1.2  .




.

Chương 1
GVHD: PGS.TS Nguy7


Galileo (1564-1642)          phân tích
  
   

niên sau,
Joseph Sauveur (1653-1716) 

 
a   và


:
Robert Hooke (1635-1703)          
F=kX t.
Newton (1642-1727)   

F=ma.
Leibnitz (1646 - 1716)


Brook Taylor (1658-1731) 




 
    -1783)      


Daniel Bernoulli Euler







 
 
Chương 1
GVHD: PGS.TS Nguy8


 nhà 
Sophie Germaine (1776-
.
             

        
1811 [31].

1.3  .



 -1831), Lagrange
(1736-1813) và Poisson (1781-1840).

Lý thuyết tấm mỏng cổ điển của
Kirchhoff,



, 
  Kirchhoff (1824-           

   

1824-1907) và G.T. Peter (1831-

1, 4, 38].






Chương 1
GVHD: PGS.TS Nguy9



              
khôn

p
               




p


                
Reissner- 
[1, 38].

1.4  (PTHH)
      Finite element method)
  
               o hàm


Phần tử hữu hạn (Finite element) ình
 1960. 



 

O. C Zienkiewicz, R. L. Taylor (1967, 1971, 1977, 1989), G. Strang, G.
Fix (1973), J. N. Reddy (1984, 1993), S. S. Rao (1982, 1989), T. J. T. Hughes
(1979), R. H Gallagher (1975), E. L. Wilson (1971)
Chương 1

GVHD: PGS.TS Nguy10




[12, 25,
27, 29, 40, 45].

1.5 .


P       

Các :






  - FEA
(Finite Element Analysi
 

 -6Al-
-S
               
Intensity Factor    

  

 
Chương 2: 
 Trang 11

C
 

2.1. 

 


xác, l [20, 22, 23].

2.1.1  [50]
Kh  hình thành 
 
 n. 
 
.  là
Hình 2.1a

  
, Hình 2.1b              
  


 


Chương 2: 
 Trang 12


2.1.2. n v
ba ba hình thành  có y
ra:

- I:  h vuông góc, Hình 2.2a.
- II:  , Hình 2.2b.
- III:  , Hình 2.2c.
hình thành
 I 
.
II hình thành 
  và vuông góc
.
III 
  
.
a) Kiểu I
b) Kiểu II
c) Kiểu III
Hình 2.2: Các kiểu hình thành vết nứt
















 

Phá



Phá

Hình 2.1: nh hưởng của vết nứt đến tần số và độ bền
kết ấ
a)
b)
Chương 2: 
 Trang 13

 
, Hình 2.3



Hình 2.3: Vết nứt trong trường hợp ứng suất kéo

   
hình thành  I cho 



























2

3
cos.
2
cos.
2
sin.
2
.
2
3
sin.
2
sin1.
2
cos.
2
.
2
3
sin.
2
sin1.
2
cos.
2
.







r
a
r
a
r
a
xy
y
x
(2.1)
  r.
      n vô    
0r
    
không pHệ số cường độ ứng suất [17, 41]
  
 2


























2
3
cos.
2
cos.
2
sin.
2
2
3
sin.
2
sin1.
2
cos.
2

2
3
sin.
2
sin1.
2
cos.
2









r
K
r
K
r
K
I
xy
I
y
I
x
(2.2)

Ch


d
x

d
y

2a

y


x

r
y
x

xy

Chương 2: 
 Trang 14




































2

sin21.
2
cos.
2
.
2
2
sin21.
2
cos.
2
.
2
2
1
2
1








r
G
K
u
r

G
K
u
y
x
(2.3)








1
3

G : module 
 isson

0

,
r
K
yx


2
1


,
0
xy

(2.4)
hình thành  II III, 
  là:


























2
3
sin.
2
sin1.
2
cos.
2
2
3
cos.
2
cos.
2
sin.
2
2
3
cos.
2
cos2.
2
sin.
2










r
K
r
K
r
K
II
xy
II
y
II
x
(2.5)



































2
sin21.
2
cos.
2
.
2

2
cos21.
2
sin.
2
.
2
2
2








r
G
K
u
r
G
K
u
II
y
II
x
(2.6)















2
sin.
2
.
2
cos.
2
2
sin.
2









r
K
u
r
K
r
K
III
z
III
yz
III
xz
(2.7)

K
I
, K
II
, K
III
n u III III


lúc thì:
Chương 2: 
 Trang 15


 
Ires


2 2 2
(1 )
Ires I II III
K K K K

   
(2.8)

2 2 2 2
(1 )( ) (1 )
Ires I II III
K K K K

    
(2.9)
 
tính:

III
ij
II
ij
I
ij
total

ij


(2.10)

I
= K
IC

 K
IC
          (Critical Stress Intensity
Factor)
2.hình thành  [50]

     
  
.
              
th, t

= 0,

ch
p
y
r
K





*
2
hay
2
2
2
*
22









chch
p
a
K
r



(2.11)



            ; a là
 ;

ch
.
đàn – dẻo, 


*
p
r

y


ch


               
*
2
pp
rr 
[18, 37], Hình 2.4
Chương 2: 
 Trang 16



Hình 2.4: Vùng biến dạng dẻo tại đáy vết nứt Hình 2.5: Độ mở rộng của đáy vết nứt

 
 [11, 18, 37]:

ch
I
p
E
K
ar
E
CTOD


2
*
4
2.
4

(2.12)
    t COD (Crack Opening Displacement),   ,
Hình 2.5:

 
2
2
*
4
xra

E
COD
p


(2.13)
2.1.4 (Stress Intensity Factor) [17, 50]
 h
 . 
:

)2(lim
0
rK
r



(2.14)
Hình 2.3 và  

0

 K
hình 
.  K



aK



(2.15)
Đàn - dẻo
Đàn hồi
*
p
r

y

CTOD
Vùng biến dạng dẻo

ch

*
2
pp
rr 

x
CTOD
COD
a
*
p
ra 
*
p

r

Chương 2: 
 Trang 17


 



 K
C
. , K
C



C
KK 
(2.16)
K
C

 
hình 
 (2.14

 [36, 40]:



 KbaFK )/(
(2.17)


K


)/( baF
(a/b),  a 
b 
)/( baF

. F(a/b)  s tay
TaDa [41].
2.2. I
L

trong sách “Theory of Elasticity” Goodier [39].
Trong 
g 

2.2.1 bài toán 
6 thành
, Hình 2.6 [12, 39].
Chương 2: 
 Trang 18

   
xzyzxyzyx


,,,,,
      o
x, y, z z, x, y.
   
xzyzxyxxx

,,,,,
      
x, y, z  z, x, y.

              
3D  c  thành phân tích
2D [12].
z 
, Oxy, Hình 2.7,

N
0
yzxzz

(2.18)
z  (
0
z

). 
.
a. Quan hệ ứng suất - biến dạng – nhiệt độ
Hình 2.7: Mô hình bài toán ứng suất phẳng
Hình 2.6: Các thành phần ứng suất và biến dạng

Chương 2: 
 Trang 19

 và  có:














































0
0
0
.
/100
0/1/
0//1
xy
y
x
xy

y
x
xy
y
x
G
EE
EE











(2.19)
,

     
0
1
][



(2.20)


 
0

là vector 
][

; E ;  
:
)1(2



E
G
(2.21)
 
 (2.19:



























































0
0
0
2
.
2/)1(00
01
01
1
xy
y
x
xy
y
x

xy
y
x
E













(2.22)
Hay
     
0
][


(2.23)

   
00
][



.

 
0

:

























0
0
0
0
T
T
xy
y
x





(2.24)


,
T

b. Quan hệ biến dạng – chuyển vị
 
y
u
x
v
y
v
x

u
xyyx













; ;
(2.25)

































v
u
xy
y
x
xy
y
x
//
/0
0/




(2.26)
Chương 2: 
 Trang 20

Hay
   
uD][

(2.27)

c. Phương trình cân bằng























0
0
y
yxy
x
xy
x
q
yx
q
yx



(2.28)

yx
qq ,
 

d. Điều kiện biên

8. Thành
 biên chính S

u
      S
t
[13    S
u
ta có
00
, vvuu 
và trên S
t
ta có
00
,
yyxx
tttt 

yx
tt ,
o 
x, y 
0000
,,,
yx
ttvu

2.3. 
       





Khi
1 h 1

100 l 5


Hình 2.8: Biên S của vật thể
Chương 2: 
 Trang 21



Oxyz Oxy trùng v
z vuông góc v (Hình 2.9a ).
(Hình 2.9a,b).

2.3.1. - 
 
 
xyyx

,,
quan
 
xyyx
MMM ,,
[1,9,38]:

2/

2/



h
h
xx
zdzM

;

2/
2/



h
h
yy
zdzM

;



2/
2/
h
h
xyxy

zdzM

(2.28)

yx
QQ ,



2/
2/



h
h
xzx
dzQ

;



2/
2/
h
h
yzy
dzQ


(2.29)
2.3.2. 
Kirchhoff [1, 9, 12]:
- 

- 
hòa.
- 
a. - - 
t, Hình 2.10
Hình 2.9: a) Các thành phần lực và Momen trên tấm; b) Sự phân bố ứng suất
a)
b)
Chương 2: 
 Trang 22


.
y
x
w
u z z
x
w
v z z
y



  



  

(2.30)


,
xy



),( vuww 
z  



yx
w
z
x
u
y
u
y
w
z
y
u
x

w
z
x
u
x





















2
xy
2
2

y
2
2
2
;
;



(2.31)





































xy
y
x
xy
y
x
E











2/)1(00
01
01
1
2
(2.32)
Hay:



































xy
y
x
xy
y
x
k
k
k
E
z
2/)1(00
01
01
1
2








(2.33)

Hình 2.10: Quan hệ giữa các góc xoay của mặt phẳng trung hòa và đạo hàm độ võng
Chương 2: 
 Trang 23

 
 
T
T
xyyx
yx
w
y
w
x
w
kkkk














2
2
2
2
2
2,,,,
(2.34)
E  isson
{k}



b. :

Ngàm:
0 ,0 



n
w
w

(2.35)

0 ,0  Mw
(2.36)

0 ,0  MQ
(2.37)
 n Hình 2.11

2.3.3ý teissner  Mindlin:
, Reissner  Mindlin
[1, 9, 12, 38]  
ngang, hay

0 ,0 
yzxz

(2.38)
  

     , 
; 
ai, , Hình 2.12
Hình 2.11: Đường biên và pháp vector n