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5ARCHITECTURE AND CONSTRUCTION
NON-LINEAR FRACTURE IN LAYERED BEAMS
Dr. Rizov V. I.
Bulgaria, Sofia
Department of Technical Mechanics, University of Architecture,
Civil Engineering and Geodesy
Abstract. Non-linear longitudinal fracture study was performed of layered beam configurations. The mechanical behaviour of beams was modelled by using a non-linear stress-strain relation. The fracture was analyzed theoretically by applying the J-integral approach. Analytical solutions of the J-integral were derived with the help of the conventional beam theory. The analysis developed holds for non-linear elastic material behaviour. However, the analysis can be applied also for elastic-plastic behaviour, if the beam considered undergoes active deformation. The solutions obtained were compared with known results from the literature for verification. The present paper contributes for the understanding of fracture behaviour of layered beam structures exhibiting material non-linearity.
Key words: Fracture, Non-linear behaviour, Layered beams, Analytical solution
One of the basic disadvantages of the layered beam structures is the fact that they are susceptible to initiation and growth of longitudinal cracks between layers [1]. These cracks reduce substantially the stiffness and load-bearing capacity of the beams. Therefore, analyzing the longitudinal fracture behaviour of layered beams is an important task of the contemporary fracture mechanics. Most of the fracture studies of layered materials are concerned with linear-elastic fracture behaviour [1]. However, very often the layered beams exhibit material non-linearity. Therefore, the purpose of the present paper is to perform a theoretical study of longitudinal fracture in layered beam configurations with taking into account the material non-linearity.
The beam under consideration is shown in Fig. 1. The loading consists of one transverse force, F, applied in the mid-span. There is a longitudinal crack of length, a, located in the beam mid-plane. Thus, the crack arms have equal thickness, h. The beam cross-section is a rectangle of width, b, and height, 2h.
Fig. 1. The beam geometry and loading
The non-linear fracture behaviour was studied by using the J-integral approach developed in [2-4]. The J-integral was written as
du dx
dv
+" sx
ds.
(1)
u0cos a -
i
where r is a contour of integration going from the lower crack face to the upper crack face in the counter clockwise direction, w0 is the strain energy density, a is the angle between the outwards
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.
The beam mechanical behaviour was modelled by using a non-linear stress-strain curve that is not symmetric for tension and compression. The stress-strain relations were written as:
^ = Hts"' (2)
and
= Hcs"c (3)
for tension and compression, respectively. In (2) and (3), H and n are material constants in tension, H and n are material constants in compression.
The J-integral was solved by using an integration contour that coincides with the beam cross-sections ahead and behind the crack tip (Fig. 1), i.e. the integration contour has three segments (A , A and B). Thus, the J-integral solution was found by summation:
J = + Ja2 + JB . (4)
Segments A and A coincide with the cross-sections of the lower and the upper crack arm behind the crack tip. Segment B coincides with the beam cross-section ahead of the crack tip (Fig. 1).
First, the J-integral solution was obtained in segment A of the integration contour. The components of the J-integral (refer to formula (1)) were written as
ds = dzy, cos« = -1, py = 0, (5)
where the z -axis originates from the lower crack arm cross-section centre and is directed downwards. The components of the stress vector, px, were determined as
Pxt =-°t=-Ht s"' (6)
and
Pc =-°c =-HcS"c (7)
for tension and compression zone, respectively.
The equations for equilibrium of the elementary forces in the cross-section were used in order to derive formulae for the curvature, k , and the neutral axis coordinate, z , of the lower crack arm:
h
zini 2
N = bdzl + jat bdz1 , (8)
-h z1ni
2
h
zin1 2
M = bzxdzx + bz1dz1, (9)
-h zm1
2
where (refer to Fig. 1)
N = 0, Fa
M =
4
(10) (11)
By substituting of (6) and (7) in (8) and (9) and taking into account that
s = Ki(zi " zin )•
(12)
the equations for equilibrium were transformed as
N = _ HcbK- ( h
n„+1
---z.
2
in,
+
hM:
n +1
h_ v 2 Zini
(13)
M = _ H cbfc\nc
h
. nr +2
---z
n
nc + 2
+ ■
h
_ ~2 _ z
n+
n
n
nc +1
+
+ HtbKint
h
_ z,
nt + 2
■ +
h
_ z,
nt+1
(14)
Equations (13) and (14) can be solved with respect to zlB and kx as an algebraic system by using the MatLab computer program. Then the partial derivative that participates in (1) was written as
du / ^
— = s = KAzi _ z,„ ) •
dx iVi 1ni'
(15)
The strain energy density is equal to the area enclosed by the stress-strain curve. Thus, for tension and compression the strain energy densities were written as
u0t = IH ts
o
J H ontds
(16)
and
U0c = I HcS
s
J H s s"c ds,
(17)
respectively.
The solutions of integrals (16) and (17) were obtained as
uot = Ht
U0c = Hc
,nt + 1
nt + r
snc +1
Hc + 1
(18) (19)
nt + 1
V
2
n.+2
n, + i
z
n
2
2
0
0
In view of (12), formulae (18) and (19) were written as
\k (z - z, )r
Uot = H/'1' , (20)
nt +1
u _ H k (zi - zim f+1
U0c _ Hc -i- . (21)
nc + 1
The J-integral (1) was solved separately in the tension and compression zone of the lower crack arm cross-section and the results obtained were added up:
j t t 1
JA = -
H.n.K?+1 f h T+2 H пк?+1 f h
---z,„ I + c c 1
(nt + Knt + 2)1 2 1n'1 (nc + Knc + 2)
2- z1?11 . (22)
Due to the symmetry, the J-integral solution in segment A2 of the integration contour (Fig. 1) was found by (22).
The J-integral components in segment B of the integration contour (Fig. 1) were written as
pxt =at = Htsn' , (23)
Pc =°c = Hcs"c, (24)
Py = 0 , (25)
ds = —dz3, cos a = 1, (26)
where the z3 -coordinate originates from the beam cross-section centre and is directed downwards. The strain was written as
^ = K3 (z3 - Z3n3 ), (27)
where k3 and z3 are the curvature and the neutral axis coordinate, respectively. The partial derivative was found as
du _ _
= S = K3 (z3 — z3n
(Z3 - ). (28)
The expressions for the strain energy density (20) and (21) were re-written as
\\k3 (z3 - z3?3 )]nt
17 LV3 (z3 Z3?3 n m
u0t = Ht -■-, (29)
n +1
и -Я \\k3 (z3 Z3?3 )]n ( л
U0c = Hc -*- . (30)
n + 1
c
The curvature and the neutral axis coordinate were determined from the following equilibrium equations of the beam cross-section ahead of the crack tip:
z3nj h
+
N = jacbdz3 + jatbdz3, (31)
-h z3n3
nc+2
зпз n
M — Iacbz^dz?, + bz3dz3,
-h z,„,
(З2)
where (refer to Fig. 1)
N = 0,
M = —. 2
(33)
(34)
After substitution of (2), (3) and (27) in (31) and (32), we derived
N = - H К
n„+1
h - zзnз >
+1 H Ькз
л--t з
- nt + 1
- \h - zзnз у1
M = - H cb К 'c
+h tb к nt
"i- h - Zзnз Y 2 + (- h - z зпз Y 1 z
n + 2
n +1
(П - Zзnз y
+
+1
n + 2
n+1
(З5)
(Зб)
Equations (35) and (36) can be solved with respect to къ and z3 by using the MatLab computer program.
The components of the J-integral were substituted in (1) and the integral was solved separately in the tension and compression zone. The results derived were added up:
r _ нпкП+1
J D -
in + l)i't + 2)
i- h - Zзnз у1
H n к'
cc3
n +1
i'c + l)i'c + 2)
h - Zзnз Y+2.
(З7)
The J-integral final solution was found by substitution of (22) and (37) in (4) and taking into
account that J, — J,
J=-
+
2 HnK+1
in + lX't + 2)
h
---zV
+2
2
ln,
+
2H n к'
c c 1
n +1
i'c + lX'c + 2)
h
--z-
2
ln,
+
н'Кз
nt +1
in + lX't + 2)
i- h - Zзnз )i
H n к'
cc3
nr +1
i'c + l)i'c + 2)
ih - Zзnз Y +2 .
(3S)
It is obvious that at nt = nc = 1 and H = H = E (here E is the modulus of elasticity) the stress-strain relations (2) and (3) transform into the Hooke's law. This fact was used for verification of (38). For this purpose, nt = nc = 1 and H = H = E were substituted in (38). It was obtained
9F2 a2
J =-
16Eb 2h3
(39)
Equation (39) coincides with the formula for the strain energy release rate in the beam considered [1]. This fact is a verification of (38) since, in principle, at linear-elastic behaviour the J-integral value is equal to the strain energy release rate.
It can be summarized that the present study contributes for the development of the non-linear fracture mechanics of layered beams. The analysis performed can be applied for optimization of beam structures with respect to their fracture performance. Also, the analytical solutions derived are very
z
2
nc +2
useful for parametric investigation of longitudinal fracture behaviour with considering the material non-linearity.
LITERATURE
1. W. Hutchinson, Z. Suo, Mixed mode cracking in layered materials. Advances in Applied Mechanic, 64:804-10 (1992).
2. J. R. Rice, A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks, Journal of Applied Mechanics, 35, 379-386 (1968).
3. G. Cherepanov, Brittle materials fracture mechanics, Nauka, M. (1974).
4. D. Broek, Elementary engineering fracture mechanics, Springer (1986).
Acknowledgments: The present study is supported by the Research and Design Centre (CNIP) of the UACEG - Sofia (Contract BN - 189/2016).
INVESTIGATION OF LONGITUDINAL CRACKS IN LAYERED MATERIALS
Dr. Rizov V. I.
Bulgaria, Sofia
Department of Technical Mechanics, University of Architecture,
Civil Engineering and Geodesy
Abstract. An analytical approach was developed for investigation of longitudinal fracture behaviour of layered materials. The methods of linear-elastic fracture mechanics were applied. The study was carried-out in terms of the strain energy release rate. By applying the conventional beam theory, the strain energy release rate was expressed in a function of the bending moments in the cross-sections ahead and behind the crack front, which substantially facilitates the practical application of the solution derived. Comparisons with known solutions were performed for verification. The present paper contributes for the development of fracture mechanics of layered materials and helps to widen the application of these materials in the engineering practise.
Key words: Layered materials, Linear-elastic fracture mechanics, Analytical investigation
The layered materials are rather prone to longitudinal fracture between layers. This fact hinders the application of layered materials in various branches of engineering. Therefore, the aim of the present study is to perform an analysis of the longitudinal fracture in layered materials.
A layered beam portion with the crack front is illustrated in Fig. 1. There is a longitudinal crack located arbitrarily along the beam cross-section height (the lower and the upper crack arm thicknesses are hx and h2, respectively). The beam height is 2h. The beam cross-section is symmetric with respect to the z3 -axis (this axis originates from the beam cross-section centre, C3, and is directed downwards). The bending moment in the cross-section ahead of the crack front is M.
*——*
Fig. 1. Beam portion with the crack front (1 - front position be fore the crack advance, 2 - front
position after the crack advance)
