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5useful 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)
The strain energy release rate was defined as [1]
G = -U (1)
AA
where
AU = Ub - Ua. (2)
is the change of the strain energy (Ub and Ua are the strain energies before and after the crack increase, respectively), when the crack area increases with
AA = b Aa, (3)
where bs is the beam cross-section width at the crack level, Aa is the crack advance (Fig. 1). It should be specified that the beam width, b, is a function of z3. In view of (2) and (3), equation (1) was rewritten as
G =U-Ub. (4)
bs Aa
The strain energy before the crack increase was expressed as
Ub =j]J u0dV , (5)
(V )
where (Fig. 1)
dV = b(z3 )Aadz3. (6)
The strain energy density was found as
1
uo =~os , (7)
where the normal stress, o, was obtained as
M
<* = MZ3, (8)
=JJ z2 dA, (9)
( A)
dA = b( z3 )dz3 (10)
The strain, s, was expressed by the Hooke's law as
a
s = —. (11)
E
Here E is the modulus of elasticity.
In view of (6), equation (5) was re-written as
Ub = Aa J ub(z3 )dz3, (12)
Z3K
where the edge points K and T are shown in Fig. 1.
The strain energy, U , in the lower crack arm after the increase of crack can be obtained again
by (12). However, z , z and z should be replaced with z , z , and z , respectively (the z -axis originates from the lower crack arm cross-section centre and is directed downwards):
Z1T
Uai =Aa J u0ab( zi)dzi . (13)
Z1S
Here the strain energy density, u0 , can be found by (7). For this purpose, M and z3 have to
be replaced with M, and z in equations (8), (9) and (10). Here M, is the bending moment in the lower crack arm cross-section behind the crack front.
Equation (13) was applied also to obtain the strain energy in the upper crack arm, U . For this
purpose, z , z and z were replaced with z , z and z , respectively (the z -axis originates from the upper crack arm cross-section centre and is directed downwards):
z2 S
Ua2 = Aa J Uoab( z2)dz2, (14)
z2 K
where the strain energy density, u , can be found by (7). For this purpose, M and z have
to be replaced with M2 and z2 in equations (8), (9) and (10). Here M2 is the bending moment in the upper crack arm cross-section behind the crack front. It is obvious that
M2 = M -M . (15)
The strain energy after the crack advance was found as
z,
Ua = Ua + Uai =Aa j u0ab{zx )dZl + Aa j b(z2 )dz2 . (16)
Z,,
Finally, (12) and (16) were substituted in equation (4): 1
G = b.
ju0b(z3)dz3 - ju0ab(zi)dzi " juoab(z2)dz2
(17)
Formula (17) enables us to calculate the strain energy release rate by using the bending moments in the cross-sections ahead and behind the crack front. Obviously, (17) can be applied also for beam cross-sections with two axes of symmetry. This fact was used to verify (17). For this purpose, b(z) = b(z2) = b(z3) = bs = b was substituted in (17). After performing the necessary transformations, it was obtained
g=_ML+_ML-.M1, (18)
2bE^ 2bEI2 2bEI
z
z
z
_ z
z
z
where 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
