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5where Ix, I2 and I are the principal moments of inertia of the lower crack arm, the upper crack arm and the un-cracked beam, respectively. Equation (18) coincides with the formula for the strain energy release rate in a beam with rectangular cross-section [2]. This fact verifies the solution derived in the present study.
It can be concluded that equation (17) gives reliable results for the strain energy release rate when analyzing longitudinal fracture behaviour of linear-elastic layered beam structures.
REFERENCES
1. Alexandrov A.V., Potapov V.D. Fundamentals of the theory of elasticity and plasticity. -M.: Vishaya shkola, 1990.
2. Hutchinson W, Suo Z. Mixed mode cracking in layered materials. Advances in Applied Mechanic, 64: 804-10, 1992.
Acknowledgments: The present study is supported by the Research and Design Centre (CNIP) of the UACEG - Sofia (Contract BN - 189/2016).
ELASTIC-PLASTIC FRACTURE BEHAVIOUR OF
LAYERED BEAMS
Dr. Rizov V. I.
Bulgaria, Sofia
Department of Technical Mechanics, University of Architecture
Civil Engineering and Geodesy
Abstract. Elastic-plastic longitudinal fracture behaviour of layered beams is studied theoretically. For this purpose, a constitutive model with linear hardening is applied. It is assumed that the material has the same properties in tension and compression. Fracture is analyzed by using the J-integral approach. Closed form analytical solutions of the J-integral are derived at characteristic magnitudes of the external load by applying the conventional beam theory. The analysis developed is verified by performing comparisons with results published previously in the scientific literature.
Keywords: Elastic-plastic fracture, Layered materials, Linear hardening, Analytical solution
The use of layered materials has increased significantly in the recent decades. The strong focus on load-bearing applications leads to a particular attention towards longitudinal fracture. The present paper deals with an elastic-plastic analysis of longitudinal fracture in layered beams (the paper is motivated by the fact that the publications in this field usually consider linear-elastic fracture behavior).
A stress-strain curve with linear hardening (Fig. 1) is used to describe the mechanical response of the layered beam configuration (Fig. 2) in the present fracture analysis. There is a longitudinal crack of length, a, located in the beam mid-plane. The beam has a rectangular cross-section of width, b, and height, 2h. The loading consists of one vertical force, F, applied at the beam free end.
The ./-integral is used to analyze the fracture behaviour [1]. The integration contour, r, is chosen to coincide with the beam contour in order to facilitate the solution of the J-integral as illustrated in Fig. 2.
The /-integral is written as [1]
du dx
dv
+" dx
ds,
(1)
where r is a contour of integration going around the crack from one crack face to the other in the counter clockwise direction, w0 is the strain energy density, a is the angle between the outwards
u0 cos« -
r
normal vector to the contour of integration and the crack direction, px and are the components of
the stress vector, u and v are the components of the displacement vector with respect to the crack tip coordinate system, xy, and ds is a differential element along the contour r.
Fig. 1. Stress-strain curve with linear hardening
Fig. 2. Geometry and loading of the beam
It is evident that the J-integral value is non-zero only in segments A and B of the integration contour (Fig. 2). Thus, the ./-integral solution can be written as
J — J a + JB ,
(2)
where JA and J are the /-integral values in segments A and B, respectively. In segment A, ds = —dy and cos a = —1 (the angle a between the normal to the contour and the x -axis in segment A is 180°). Besides, p^ = 0 .
If the external load magnitude is low, the beam deforms in linear-elastic stage, i.e. the Hook's law is valid. In this case, the normal stress, px, in segment A is obtained using the equation:
ma ii
Px —
12 FL bh
(3)
bh3
where L =-is the principal moment of inertia of the lower crack arm cross-section.
1 12
Further, the strain energy density is written as:
uo —
2 E
(4)
2
x
where E is the longitudinal modulus of elasticity. By substituting of (3) into (4), we obtain:
uo —
2E
FL
-yi
72 F2 L2 Eb2 h6
■yi
(5)
The partial derivative, ^, is written as:
dx
du
Px
- — _'Sx —--
dx x E
12 FL
Ebh3 yi ■
(6)
Finally, (3), (5) and (6) are substituted into (1). The result is:
f , „ ™ Y
J —J
hi 2
72F2 L2 w x 12 FL
~EbWyi(- )-~bhr
yi
12FL
Eh yi
v J
22
< ~dy i)—
6F2 L'
Eb 2h3
(7)
It is assumed that the external force, F, is uniformly distributed over a small distance, l (this assumption is frequently used when obtaining analytical solutions of the /-integral [2]). Thus, the segment B of the integration contour is defined by L — a — l < x < L — a along the upper edge of the beam. The components of the /-integral in segment B of the integration contour (Fig. 2) are obtained as follows:
F
)y —-bl' (8)
Px — 0 , (9)
u — o. (10)
dv
The partial derivative, —, which is equal to the slope of free end of the beam, is calculated
dx
by using the methods of Mechanics of materials:
dv dx
6F Eb
(L - a)2 a(2L - a)
8h3
h3
(11)
By substitution of (8), (9), (10), and (11) into (1), we obtaine the J-integral value in segment B
L-a-l
J■ —J\- F
L-a
F_ ~bl
6F
Eb
i 2 \ (L - a) a(2L - a)
8h3
h3
J,
>(-dx) —
6F2
Eb2
(L - a)2 a(2L - a)
8h3 h3
The /-integral solution is obtained by substitution of (7) and (12) into (2). The result is:
21F2 (L — a )2
(12)
J —
6F2 L2 6F2
Eb 2h3 Eb2
(L - a)2 a(2L - a)
8h3
h3
4Eb 2h3
(13)
2
1
i
It is known that at linear-elastic behavior of the beam, the ./-integral value is equal to the strain energy release rate. This fact is used here to verify the solution (13). Indeed, (13) coincides with the formula for the strain energy release rate reported in [3].
If the fracture toughness is high, plastic strains will arise prior to the onset of crack growth. It is obvious that plastic strains will arise first in cross-section A of the lower crack arm (Fig. 2). The solution of the /-integral for the case when the stresses in the remotest edges of cross-section A attain the yield stress limit, f, can be obtained by formula (13). For this purpose, the force, F, should be
expressed as a function of f using the formula
f = —, (14)
Jy W
bh2
where W =- is the section modulus of the lower crack arm cross-section. Having in mind
1 6
that MA = FL , from (14) we obtain
fvbh2
F = —-. (15)
6L
Finally, (15) is substituted into (13). Thus, the /-integral solution takes the form
f2
j= ' . (16)
7f2h(L - a)2
48EL
Further increase of the external load leads to formation of two plastic zones in the vicinity of cross-section A (Fig. 2). The length of the plastic zone, lh, is obtained by using the fact that the
magnitude of the moment at distance lh is equal to the moment causing stress in the remotest edges of cross-section which is equal to f . Hence
h =
fybh
1 -iz_
6LF
L. (17)
Due to the fact that plastic zones are formed, the /-integral in cross-section A can be resolved into two components
JA=JA,el + JA,pl, (18)
where J and J are the values in the elastic and plastic zones, respectively (Fig. 3). Expression (1) is used to obtain JAel. The normal stress is expressed by formula (3). The partial derivative is written as
du 1
— = =--y , (19)
ax p
1
where — is the curvature of the lower crack arm in cross-section A in elastic-plastic stage of P
deformation. Further, the strain energy density is expressed as
u0 = - px Zx =
6FL 1
bh3 p
yi •
(20)
Finally, the solution of J A l is obtained as
z
JA,el = j
6FL 1
~bhF P
yi
(-1)"
12FL
bh3
-yi
-pp yi Jf(- Jyi)=Ü pp hel
(21)
where h is the height of the elastic zone (Fig. 3).
Fig. 3. Zones of hardeing and diagrams of stresses and strains in cross-section A
of the lower crack arm
We will use the following equation in order to find the curvature [4]:
nr MPL M = ——
2 E P +
E p
1 - £ '
E
( \ 2
3 - V
p J
(22)
where Ex is the modulus of elasticity in regime of hardening (Fig. 1), p is obtained by the
formula [4]
Py =
Eh
2//
MpL=fy
bh2
6
(23)
(24)
Equation (22) is obtained by considering the equilibrium of the elementary forces in a rectangular cross-section in elastic-plastic stage of deformation [4]. We solve (22) with respect to the curvature, 1 / p, using the method for solution of cubic equation [5]. The result is
( Ö1 + 02 " P
P Py V 3 ,
(25)
where
1
h
el
h
el
2
2
a—3
Ö2 — 3
(P
V 3,
n -1
4n
■ +
4Q
rP
V
n -1 4n
-4Q ,
Q—2P3 r n-i Un -1 ^
27 V 4n J
4n
P —
3MPL (i - n)- 2M
n) 2MPLn
n —
El E
It should be mentioned that at El = E, from (25) - (30) we obtain
1-M
p~ EI '
(26)
(27)
(28)
(29)
(30)
(31)
which is the curvature of the lower crack arm in linear-elastic stage of deformation. The height of the elastic zone is determined using the following formula from Mechanics of materials [4]
Ki— h
p_ Pv
By substitution of (25) and (32) in (21), we obtain
FL i
J A,el
2bh3 p
hi.
Furthermore, the solution of J is derived by using the following expression:
(32)
(33)
JA,pl — 2 J \ U0 C0S a -
du
Px
dx
>(-ds),
(34)
where the factor 2 takes into account the fact that there are two symmetric plastic zones in the crack arm cross-section (Fig. 3).
The normal stresses in the plastic zones are obtained as (Fig. 1)
Px — fy + Ei (s-£y ).
(35)
The strain energy density in the plastic zones of the crack arm cross-section is equal to the area enclosed by the stress-strain curve:
U0 — \ fy£y + \ [2fy + Ei(s- £y - ^y ).
(36)
3
3
2
h
el
By substituting s = — y in (36), we obtain P
f2 u0 +
0 2E
f , Ei (yi fy I Jy +~---F
U E J
2
yi fy
^
P E
(37)
J
We substitute (19), (35), and (37) into (34). The result is
JA„=§(h-h„(i-§)+^-hä). (38)
It should be specified that the bending moment in cross-section A (Fig. 2), M=FL, is substituted in (29) when calculating J and J .
By substitution of (33) and (38) in (18), we obtain
JA =-h3 +fi(h -h,)fl -(h3 -h3). (39)
A 2bh3 p el 2E el\ E J 24p elJ ( )
The components of the /-integral in segment B of the integration contour (Fig. 2) are obtained as
dv dx
angle of the free end of the beam, (pB , is obtained by using a formula from Mechanics of materials [4]
follows. The normal stress, py , is expressed by (8). The partial derivative, —, which is equal to the slope
dv
1 r 1
- =-(b =-zi mi-dx . (40)
dx i=11 pt
In view of the fact that plastic zones are formed in the vicinity of cross-section A (Fig. 2), formula (40) is written as
dv \ 1 a 1 L 1
— = -(=-| Mx—dx -I M—dx -I Mx—dx, (41)
dx o P t P1 a P 2
where 1 / p is obtained by (25),
M = 1,
1 _M ( x )
P1 EI1 , M (x)
p2 EI
Here, M(x) = F(L - x) is the bending moment generated by the external load, I =
(42)
(43)
3
bh 12
and I = —— are the principal moments of inertia of the lower crack arm and the un-cracked portion
of the beam, respectively (Fig. 2). In (41), the length of the plastic zone, lh, (Fig. 2) is obtained by formula (17).
The curvature of the lower crack arm in the zone of elastic-plastic deformation, 1/p, that participates in the first integral in (41) is obtained by formulae (25) - (30) at
M(x) = F(L - x), (44)
where 0 < x < lh.
The ./-integral solution in segment B of the integration contour is obtained by substitution of (8), (9), (10), and (41) into (1). The result is
L-a-l
F F
Jb,p, — J [-(-f)(-VB )](-dx) — - - (pB, (45)
bl
L-a
where (pB is calculated by (41). By substitution of (39) and (45) in (2), we obtain
J = -F^ - hi + i- (h - hel (1 - E1 ] + ■(h3 - hi) - F (B . (46)
2bh p el 2E el\ E) 24p el' b B
It should be mentioned that at Ex = E, the model with linear hardening (Fig. 1) transforms into the linear-elastic model. Indeed, by substitution of Ex = E in (46), we obtain the linear-elastic solution (13). This fact is a verification of the analysis developed in the present study.
The closed form analytical solutions derived here are very useful for parametric investigations, since the simple formulae obtained capture the essential of the elastic-plastic fracture behaviour of layered beams. The results of the present study help to obtain a better understanding of failure mechanisms in layered beams.
LITERATURE
1. D. Broek, Elementary engineering fracture mechanics, Springer (1986).
2. B. F. Sorensen and T. K. Jacobsen, Crack growth in composites Applicability of R-curves and bridging laws, Plastics, Rubber and Composites, 29, 119-133 (2000).
3. X.-J. Gong and M. Benzeggagh, Mixed Mode Interlaminar Fracture Toughness of Unidirectional Glass/Epoxy Composite Materials, Fatigue and Fracture - Fifth Volume, ASTM STP 1230, R. H. Martin, Ed., American Society for Testing and Materials, Philadelphia, 100-123 (1995).
4. N. N. Malinin, Applied theory of plasticity and creep. M., Mashinostroenie (1968).
5. G. Korn and T. Korn, Mathematical handbook for scientists and engineers, Nauka, M. (1970).
Acknowledgments: The present study is supported by the Research and Design Centre (CNIP) of the UACEG - Sofia (Contract BN - 189/2016).
