Determination of mixed-mode fracture characteristics due dynamic opening and in-plane shear cases/Misraus irimo charakteristiku nustatymas dinaminio atplesimo ir slyties atvejais.
Sadreika, P. ; Ziliukas, A.
1. Introduction
In fracture mechanics the crack propagation problem is mainly
discussed for individual cases of deformation, i.e. opening and
affecting with shear [1, 2]. However, in practice often mixed cases are
encountered when the crack is affected by opening and shear stresses.
Most of the fracture mechanics of crack instability theories are based
on the idea of Griffith [2]. For pure-mode cases, it has been commonly
accepted that fracture will occur when the corresponding stress
intensity factor reaches its critical value. Stress state ahead of a
crack is often of the mixed type where both [K.sup.d.sub.I] and
[K.sup.d.sub.II] are present. Practical engineering cracked structures
are subjected to mixed mode loading, thus in general [K.sub.I] and
[K.sub.II] are both nonzero, yet we usually measure only mode I fracture
toughness [K.sub.IC]. In this cases we have the so-called cumulative
effect of modes I and II. The mode I describes the opening and normal
stress effect and the mode II - the shear cases and shear stress effect
[1-5]. The determination of a fracture initiation criterion for an
existing crack in mode I and mode II would require a relationship
between [K.sub.I], [K.sub.II] and [K.sub.IC] of the form:
F ([K.sub.I], [K.sub.II], [K.sub.IC]) = 0 (1)
and would be analogous to the between the two principal stress and
yield stress (Fig. 1):
[F.sub.Y] ([[sigma].sub.1], [[sigma].sub.2], [[sigma].sub.Y]) = 0.
(2)
[FIGURE 1 OMITTED]
These tests become more important when dynamic effects arise
because this far, dynamic fracture patterns are least examined in the
fracture mechanics. The known works [6, 7] consider not only the
characteristics of fracture, i.e. dynamic stress intensity factors
[K.sup.d.sub.I] and [K.sup.d.sub.II], but also focus on the crack path,
i.e. when the crack propagation angle depends on both loads: opening and
shear. Sih at al [8] proposed a mixed-mode criterion of fracture, which
states, that the combination of mode I and mode II stress intensity
factors present will cause crack initiation upon reaching some critical
value [K.sub.IC]. This critical intensity of the local stress field is a
material constant and not depends on geometry of specimen. Also Sih at
al [9] discussed the dynamic counterpart of Griffith crack
configuration.
The dynamic crack propagation behaviour has attracted extensive
attention during the past decades [10]. There are a number of
experiments, theoretical models and simulations constructed and
performed to understand the phenomena of dynamic fractures. Zhang at al
investigated dynamic crack growth and branching of running crack under
mixed-mode loading.
In order to predict the fracture loadings of cracked materials
under the general mixed-mode state Chang at al [11] proposed general
fracture criterion based on the concept of maximum potential energy
release rate.
However, the obtained dependences show little representation of
mechanical properties of materials, which determine the fracture
process. Therefore the work is focused on the parameters of the material
strength and fracture that allow easier assessment of the crack growth
specimens depending on the complex stress condition in the tip of the
crack. This would allow describing critical dangerous levels also in the
general load case.
Unlike the static case, solution to the dynamic problem is more
difficult to obtain. The effects of dynamic loading on the distribution
of stresses around a crack - like imperfection have not received
sufficient attention. The elasto-dynamic problems are fundamental
interest in fracture mechanics.
2. Testing procedures
Loading the specimen with an inclined crack, i.e. pulling and
affected by shear according to mode I and mode II (Fig. 2).
In the work [1] from maximum circumferential tensile stress theory
for mixed modes is writing:
[K.sup.d.sub.I]/[K.sup.d.sub.IC] [cos.sup.3] [[theta].sub.0]/2 -
3/2 [K.sup.d.sub.II]/[K.sup.d.sub.IC] cos [[theta].sub.0]/2 sin
[[theta].sub.0] = 1, (3)
where: [[theta].sub.0] is crack angle from the crack position
perpendicular to the load; [K.sup.d.sub.I], [K.sup.d.sub.II] are dynamic
stress intensity factors in the opening (mode I) and in-plane shear
(mode II) cases.
[FIGURE 2 OMITTED]
Evaluating the characteristics of material strength, from minimum
strain energy density criteria [1] can be expressed as:
8[[mu].sub.d]/[kappa] - 1 [[a.sub.11]
[([K.sup.d.sub.I]/[K.sup.d.sub.IC]).sup.2] + 2[a.sub.12]
([K.sup.d.sub.I][K.sup.d.sub.II]/[([K.sup.d.sub.IC]).sup.2]) +
[a.sub.22] [([K.sup.d.sub.II]/[K.sup.d.sub.IC]).sup.2]] = 1, (4)
where [[mu].sub.d] is dynamic sear modulus; coefficients
[a.sub.11], [a.sub.12] and [a.sub.22] are calculated from:
[a.sub.11] = 1/16[[mu].sub.d] (1 + cos [[theta].sub.0])([kappa] -
cos [[theta].sub.0]); (5)
[a.sub.12] = sin [[theta].sub.0]/16[[mu].sub.d] [2[cos
[[theta].sub.0] - ([kappa] - 1); (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (7)
Solving the system of Eqs. (3) and (4), from the Eq. (3) we get:
[K.sup.d.sub.I] = 1 + 2/3[K.sup.d.sub.II] [K.sup.d.sub.IC]cos
[[theta].sub.0]/2 sin[[theta].sub.0]/[cos.sup.3] [[theta].sub.0]/2
[K.sup.d.sub.IC].
Upon inserting the Eq. (8) to Eq. (4), it follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
From the obtained Eq. (9), having [K.sup.d.sub.IC], [[theta].sub.0]
and [[mu].sub.d], [kappa] and [[nu].sub.d] and coefficients [a.sub.11],
[a.sub.12] and [a.sub.21], we can calculate [K.sup.d.sub.II], and from
Eq. (8) also [K.sup.d.sub.I].
Control of the solution can be carried out upon entering the angle
[[theta].sub.0] = 0 and the Eq. (9) then it will be written as:
1 + 2/K - 1[([K.sup.d.sub.II/[K.sup.d.sub.IC]).sup.2] = 1 (10)
from here we get:
2/K - 1([([K.sup.d.sub.II/[K.sup.d.sub.IC]).sup.2] = 0 (11)
i.e. [K.sup.d.sub.II] = 0 and it shows that in this case there is
no shear. We only have the case of opening [K.sup.d.sub.I] =
[K.sup.d.sub.IC], as evidenced by Eq. (8).
In order to evaluate different opening and shear influence on
fracture, the experiments with steel plates under mixed-mode loading
have been carried out. Specimens are made of S355 grade construction
steel, with statitac modulus of elasticity E = 200 GPa, yield stress
[[sigma].sub.y] = 350 MPa, ultimate stress [[sigma].sub.u] = 500 MPa,
Poison's ratio [[nu].sub.d] = 0,30. Size length x width x thickness
of the precracked specimens are respectively 40 x 30 x 3 mm. In center
of the specimen 0.25 mm width and 8 mm length initial crack are made.
This initial crack imitates fatigue crack.
[FIGURE 3 OMITTED]
Testing was done in Kazimieras Vasiliauskas Strength of Materials
lab at the Kaunas University of Technology. During impact testing a
swinging pendulum of 30 kg m potential energy impact tester was used.
Specimen was embedded in swinging pendulum with special fixture and
lifted to starting position (Fig. 4). Released pendulum swings through
and strikes the specimen causing fracture. In the scale of impact tester
the breaking energy [C.sub.V] are shown. Using specimens with different
angles of initial cracks, mixed-mode loading are obtained. The
experimental results under various mixedmode loading conditions are
given in Table 1.
Impact test results (Table 1) are shown in Fig. 5. It should be
mentioned that increasing initial crack angle increases the fracture
energy.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
3. Evaluation of mixed-mode characteristics
Dynamic characteristics of material were obtained from a specimen
without a crack. During the tests, the dynamic shear modulus
[[mu].sub.d] = 1 x [10.sup.5] MPa, and dynamic Poisson's ratio
[[nu].sub.d] = 0.25 were identified. In the case of plane stress [kappa]
= 3 - 4[[nu].sub.d] = 2, and in case of plane strain [kappa] = (3 -
[[nu].sub.d])/(1 + [[nu].sub.d]) = 2.2.
Empirical equation which relates fracture energy CV to fracture
toughness [K.sub.IC] [12]:
[K.sub.IC] = 12.36 [C.sup.2.sub.V], (12)
where stress intensity factor [K.sub.IC] is for static loading.
Empiric formulas having impact strength characteristics can be expressed
in terms of critical stress intensity factor [K.sup.d.sub.IC] [12]:
[K.sup.d.sub.IC] = 1.06 [K.sub.IC]. (13)
In our case the critical stress intensity factor [K.sup.d.sub.IC]
from Eq. (13) is equal to [K.sub.IC] = 47.21 MPa x [m.sup.1/2].
The calculated value is [K.sup.d.sub.I] and [K.sup.d.sub.II]
according to the Eqs. (8) and (9) are presented in Table 2.
The obtained results were used to make a graph of changes in the
dynamic stress intensity factors [K.sup.d.sub.I] [K.sup.d.sub.IC] and
[K.sup.d.sub.II] [K.sup.d.sub.IC] depending on [[theta].sub.0] which is
presented in Fig. 6.
As can be seen from the graphs, the effect of stress intensity
factor [K.sup.d.sub.II] increases with increasing initial angle of the
crack. The ratio [K.sup.d.sub.I] [K.sup.d.sub.IC] increases from 0 to
0.48 increasing the crack angle from 0 to 45 and [K.sup.d.sub.II]
[K.sub.IC]increases from 0 to 0.9. It may be seen that [K.sup.d.sub.II]
increases more than Kid .
The intersectin of [K.sup.d.sub.I]/[K.sup.d.sub.IC] and
[K.sup.d.sub.II]/ [K.sup.d.sub.IC] is approximately at 25.8[degrees]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
4. Conclusions
1. In mixed load: for opening and shear strength, the dynamic
fracture with the crack can be calculated. The obtained Eqs. (8) and (9)
are able to calculate dynamic fracture parameters [K.sup.d.sub.I] and
[K.sup.d.sub.II], using fracture characteristic [K.sup.d.sub.IC], shear
modulus and Puason's ratio.
2. The obtained test and calculation results show different effect
of the opening and shear to fracture, and the presented to dependences
allow to assess the effect in qualitative indicators.
3. Test and calculation results can be applied not only to cracks
with a clear path of spreading, but also for bifurcations on the tip of
the main crack, where a mixed opening and shear fracture develops.
http://dx.doi.org/10.5755/j01.mech.20.2.6936
References
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P. Sadreika*, A. Ziliukas**
* Kaunas University of Technology, Kfstucio 27, LT-44025 Kaunas,
Lithuania, E-mail: sadreika@gmail.com ** Kaunas University of
Technology, Kfstucio 27, LT-44025 Kaunas, Lithuania, E-mail:
antanas.ziliukas@ktu.lt
Table 1
Results of the impact testing
No. [[theta] [C.sub.v], kg x m Mode
.sub.0],
[omicron]
1 0 13.6 I mode
2 0 12.8 avr.: 12.9 I mode
3 0 12.2 I mode
4 5 12.8 I, II mode
5 5 13.9 avr.: 13.4 I, II mode
6 5 13.4 I, II mode
7 10 13.9 I, II mode
8 10 14.8 avr.: 14.3 I, II mode
9 10 14.1 I, II mode
10 15 14.3 I, II mode
11 15 14.7 avr.: 14.3 I, II mode
12 15 14.0 I, II mode
13 20 15.7 I, II mode
14 20 16.5 avr.: 16.1 I, II mode
15 20 16.1 I, II mode
16 25 16.5 I, II mode
17 25 17.9 avr.: 17.1 I, II mode
18 25 17.5 I, II mode
19 30 16.4 I, II mode
20 30 17.8 avr.: 17.1 I, II mode
21 30 17.1 I, II mode
22 35 18.2 avr.: 18.9 I, II mode
23 35 19.7 I, II mode
24 35 18.7 I, II mode
25 40 19.5 I, II mode
26 40 20.0 avr.: 19.8 I, II mode
27 40 19.8 I, II mode
28 45 22.1 I, II mode
29 45 21.2 avr.: 21.5 I, II mode
30 45 21.3 I, II mode
Table 2
Results of the impact testing
[[theta]- [K.sup. [K.sup. [K.sup. [K.sup.
.sub.0], d.sub.I], d.sub.II], d.sub.II], d.sub.II],
[sup.- MPa x MPa x [K.sup. [K.sup.
omicron] [m.sup.1/2] [m.sup.1/2] d.sub.IC], d.sub.IC],
0 47.21 0 1 0
5 46.21 8.67 0.99 0.18
10 44.17 13.65 0.95 0.29
15 40.61 19.83 0.86 0.42
20 36.06 25.27 0.76 0.54
25 30.79 29.99 0.65 0.64
30 25.08 33.98 0.53 0.72
35 19.13 37.31 0.41 0.9
40 13.09 40.12 0.28 0.85
45 7.01 42.53 0.15 0.90
