Construction of siloxane structures with P-Tolyl substituents at the silicon atom

Article abstract of DOI:10.1016/j.jorganchem.2020.121497

In this work, general approaches for preparing p-tolylsilanes as promising precursors for the synthesis of functionalized organosilicon compounds are discussed. Various synthetic techniques, including new and specially developed ones, were used to obtaine

Full text of DOI:10.1016/j.jorganchem.2020.121497

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Procedia Structural Integrity 2 (2016) 744–752
www.elsevier.com/locate/procedia
21st European Conference on Fracture, ECF21, 20-24 June 2016, Catania, Italy
Nonlinear stress intensity factors in fracture mechanics and their
applications
V. Shlyannikova
aKazan Scientific Center of Russian Academy of Sciences, Lobachevsky Street, 2/31, Kazan 420111, Russia
Abstract
A new parameters for characterization of the crack growth resistance for a power law hardening materials and structures
under normal and elevated temperature are introduced in the form of plastic and creep stress intensity factors. A numerical
method is proposed for estimating the governing parameter of the elastic-plastic and creep crack tip fields in the form of an In-
integral along the through-the-thickness straight and curved crack fronts. Equations are also proposed for calculating the In-factor
at both the deepest point along the crack front and the crack tip on the surface. Static and fatigue crack growth and in-plane and
out-of-plane constraint effects are studied through experiments and computations for the material different properties. It is further
demonstrated that for moderately large-scale yielding conditions or plastic deformations, the fracture process can be controlled
by the single parameter KP based on the elastic-plastic numerical solutions and, therefore, is reflected in the influence of the
cracked body geometry and loading conditions. Keeping in mind the purpose of practical application of the introduced nonlinear
stress intensity factors, an engineering approach to the prediction of residual lifetime of cracked steam turbine rotors components
which operate at maintenance under cyclic loading conditions is proposed. Using the nonlinear stress intensity factors
approximate estimations of carrying capacity are presented for the different stress-strain state of steam turbine disks at the
operation. As result it is stated that a one-parameter approach based on the plastic and creep stress intensity factors are more
convenient for practical use in assessing the fracture resistance under monotonic and cyclic loading of materials and structural
elements with respect to the two parameters fracture theories.
Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(
http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Scientific Committee of ECF21.
Keywords: Plastic and creep stress intensity factors; in-plane and out-of-plane constraint; structural integrity assessment.
1. Introduction
The engineering application of the fracture mechanics of solids to real cracked structures requires an appropriate
parameter to quantify the crack tip constraint. Moreover, practical structural components have finite thicknesses, and
Copyright © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license
(
http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer review under responsibility of the Scientific Committee of ECF21.
10.1016/j.prostr.2016.06.096

V Shlyannikov / Procedia Structural Integrity 2 (2016) 744–752
745
the stress-strain state changes between plane stress and plane strain. From a practical point of view, the most useful
approach for assessing the fracture resistance of materials, components and structures would involve one common
parameter, which, unlike the two parameter models and the higher order term solutions, would preserve the one-
term representation. Nevertheless, the basic parameters of the model must be modified such that they are able to take
into account both the in-plane and out-of-plane constraint effects.
In this paper, the new fracture resistance parameters of materials and structures in the form of the plastic and
creep stress intensity factors is employed to generalization and solution actual fracture mechanics problems. It is
further demonstrated that the nonlinear stress intensity factors accounting for the in-plane and out-of-plane
constraint effects can be used to characterize the static and cyclic fracture resistance as well as structural integrity as
a unified single parameter for a variety of cracked body configurations and loading conditions.
2. Plastic stress intensity factor
The plastic stress intensity factor Kp in pure Mode I can be expressed directly in terms of the corresponding
elastic stress intensity factor as follows:

n1

1/


1/


n1
K12
Y2(a/w)

2

(a/w)
1
(1)




Kp  

;K1  Y (a/w)
1
FEM
0In

,n,(a/w)


0

InFEM

,n,(a/w)












where
is elastic SIF normalized by a characteristic size of cracked body,
w
and n are the hardening

K1  K1 /
parameters, λ=a/w is the dimensionless crack length, w is specimen width, σ is the nominal stress, and σ0 is the yield
stress, In is governing parameter for 3D-fields of the stresses and strains at the crack tip. Shlyannikov and Tumanov
(2014,a) suggested the procedure for calculating the governing parameter of the elastic–plastic stress–strain fields in
the form of In for the different specimen geometries by means of the elastic–plastic FE-analysis of the near crack-tip
stress-strain fields. In this study, the numerical integral of the crack tip field In changes not only with the strain
hardening exponent n but also with the relative crack length c/w and the relative crack depth a/t.
FEM
FEM  FEM
~
~
du






FEM

n
FEM
dur
n1

e

cos  
uFEM 
r urFEM 
~
~
~
~
~




sin 



 n1
d
d


rr



(2)
InFEM(,n,(c/w),(a/t))
d













1
FEM FEM
FEM
FEM cos.
~
~
~
~


rr ur r u




n1

The subject for the experimental study in this part of the work is carbon steel 34ХН3МА, whose main
mechanical properties are listed in Table 1. The single-edge-notched bend (SENB) and compact (CT) specimens
were used in both experimental studies and numerical analyses. In addition to the standard ASTM thickness-to-
width ratio B/W = 0.5, SENB specimens with B/W = 1.0 were prepared. The relative crack length a/W after inserting
a fatigue precrack varied in the range of 0.3-0.62. The three type of CT specimens with the ratio B/W = 0.1, 0.2, and
0.4 were chosen with the relative crack length a/W change of 0.35-0.645 after precracking. Several sets of
specimens of both configurations, which include the change of the in-plane constraint, the out-of-plane constraint
and both of them, were investigated. It is found that elastic stress intensity factor Kmax varied in the range of 52.15-
68.12 MPam and 59.49-77.46 MPam for SENB and CS, respectively.
Table 1. Main mechanical properties of Steel 34ХН3МА
Properties
Static
Cyclic
'f
n
α
n'
E
0
MPa
b
u
'f

GPa
MPa
%
MPa
MPa
34ХН3МА
790
992
56
1455
196.4
7.49
2.39
1488
0.74
6.67

746
V Shlyannikov / Procedia Structural Integrity 2 (2016) 744–752
Fig. 1. Variation of plastic SIF's as a function of specimen thickness and relative crack length.
To calculate
for the specimen-specified geometry, the expression for the governing In-factor of the elastic-
KP
plastic stress fields in the form of Eqs. (1, 2) as well as the values of the limiting experimental loads Pmax was used.
Note that the KP -curves as a function of the relative crack length and specimen thickness in the mid-plane are
close to each other for the different specimen geometries. Unlike the elastic SIF, the plastic stress intensity factor
KP in the formulation of Eq. (1) is sensitive enough to account for the influence of the specimen geometry
(including of thickness) and the loading conditions. This opinion is supported by the experimental results presented
in Fig. 1, where the values of plastic SIF correspond to the mid-plane of each specimen at z/B = 0.5. In contrast to
the elastic SIF, the distributions of plastic SIF KP as a function of relative crack length and specimen thickness
exhibited a lower scatter average to KP  0.61. Furthermore, this value of KP is independent of the considered
specimen geometry, i.e., it is a material characteristic. Thus, the plastic SIF may treated as a unified parameter for
the characterization of the material fracture resistance property.
3. Creep stress intensity factor
In previous section, a plastic stress intensity factor for fracture toughness and the mixed mode crack growth rate
was introduced, based on the analytical form of the elastic-plastic stress and strain fields in the vicinity of the crack
tip. The study of the present section follows the approach found in Shlyannikov et al. (2014,a) and will allow a more
general expression to be obtained for the first term creep crack tip singularity field, which considers the dependence
of a governing parameter in the form of on both the crack front curvature and the constraint. In particular, in contrast
Inpl

 ij,ui

~ ~
to the elastic-plastic formulation
, in the case of creep, in the equation for the I -integral, the time
n
cr
~
~
derivatives are used for the dimensionless displacement components
. It should be noted that


In  ij,dui dt
cr
contrary to the traditional point of view, the numerical constant In depends only on the creep exponent; this
governing parameter of the creeping crack tip fields is a function of the stress-strain state, the crack front curvature,
the crack length, the creep time, the applied load, the tested specimen thickness and the configuration. According to

V Shlyannikov / Procedia Structural Integrity 2 (2016) 744–752
747
definition Shlyannikov et al. (2015,a), the following expression for the creep stress intensity factor which can be
applied to the interpretation of the creep crack growth data

n1



1



f
f
V 

P
BInFEM
c




K cr

t



(3)

t

bWL




where P is applied load, Vc is the force-line displacement rate, b - is specimen thickness, f - is geometry dependent

correction factor,
- is its derivative.
 
f
The subject for both the experimental studies and the numerical analyses are compact specimens under tension,
which are most frequently used for characterizing creep crack growth rate. The test materials are power steam
turbine rotor steel R2M and the piping systems of power plants steel 12X1MF operated at the elevated temperature
of 550C. The main mechanical properties of R2M steel and 12X1MF steel at elevated temperature are summarized
in Table 2.
Table 2. Elevated temperature tensile and creep properties
E
(GPa)
α
m
B
n
ys
(MPa)
t
(MPa)
uts
(MPa)
f
(MPa)
f
T=550°С
(MPa)-n hr-1
Steel
R2M
167
150
557
215
660
239
1623
529
280
175
1.35
2.08
0.179
2.117
5.22
5.97
1.4·10-10
5.7·10-15
2.47
5.07
Steel
12X1MF
a)
b)
c)
Fig.2. Creep SIF as a function of creep time (a) and (b,c) crack growth rate diagram
In Fig. 2,a the creep SIF behavior is represented as a function of creep time in the CT mid-plane and the free
surface for the through-the-thickness straight crack front (steel 12X1MF) and curvilinear crack front (steel P2M).
The relationships for the straight-line crack an approximately are the same at plane stress on the free surface of
specimens and plane strain in the mid-plane. In the case of the curvilinear crack front, the values of the creep SIF on
the free surface (z/b = 0.05) and the deepest point of the crack front (z/b = 0.5) differ significantly from each other
for a power law creeping material. Figures 2,b and 2,c represent the crack growth rate behavior of the R2M and
12Cr1MoV steels at 550C as a function of the dimensionless creep stress intensity factor in the form of Eq. (3) for
the hold time of 60s. It is observed that the experimental data fall within a relatively narrow scatter band. In Fig. 2,c,
one of lines is related to the free surface of the compact tension specimen, and the other belongs to the deepest point
of the crack front. Unlike the definition for the elastic SIF and C(t) parameter, the creep stress intensity factor
approach provides the possibility of obtaining the crack growth rate at the deepest point of the CT specimen. The

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V Shlyannikov / Procedia Structural Integrity 2 (2016) 744–752
comparison shows that the crack growth rates at the deepest point and at the surface point of the crack front are
different.
4. Critical distance determination
The main hypotheses of the strain energy density theory are associated with the concept of a characteristic
distance. It has been considered as a fundamental characteristic that setting an interrelation between the processes
occurring on both the micro level and macro level with respect to the material structure. This characteristic distance
is often identified with the fracture damage zone or fracture process zone. A critical distance rc ahead of the crack
tip is assumed to exist when the strain energy density (SED) in an element reaches a certain critical value. In the
present work, the critical value of the SED is measured from a uniaxial test. The dimensionless total SED is
obtained as a sum of elastic and plastic part. The general equation which is used the total SED was introduced by
Shlyannikov et al. (2015,b) for the elastic-plastic constitutive relation of material behaviour in the form of Ramberg-
Osgood equation
2




2









0
 yn
1
2
n
n 1






 S2  S2  4

S1  SP

S3 
 f2 
 fn1









rC
a


(4)
 







2





0
1
2
n
n 1




2 S3 
 f2 
 fn1




 yn










More details to determine the SED functions (i=1,2,3) for cracked body different configurations are given by
Si
Refs. (Shlyannikov et al. (2015,b)). For materials in which secondary creep dominates, deformation behavior is
described by the Norton elastic-nonlinear viscous constitutive relation based on the strain energy rate density
(SERD) creep critical distance can be written as




n1
Kcr
n1
~



(5)
r

e
cr
 0
 f
Most successful correlations of time-dependent crack growth at elevated temperature are with corresponding
fracture mechanics parameters. One of them is the creep stress intensity factor Kcr included in Eq.(5). It should be
noted that both equations 4 and 5 to determine the critical distance for elastic-plastic and creep materials behavior
are explicit functions of the corresponding stress intensity factors in the form of Eqs.(1,3).
By using Eq. (4), the values of the dimensionless critical distance rc /a along the crack front was calculated for
different combinations of crack length, specimen thickness and specimen geometry which mentioned above for
fracture toughness determination. The subject for the numerical and experimental studies is carbon steel 34ХН3МА.
Based on the test and FEA results, it was concluded that the crack-tip constraint had a noticeable contribution to the
relative crack length and specimen thickness effect on the critical distance distribution along the crack front. It is
interesting to note that the distribution of the critical distance rc along the crack front shown in Fig. 3,a,b for the
compact specimen coincides with the simulation of the shape of the crack front that was performed by Cornec et al.
(2003) using cohesive elements. Figure 3,c shows the behavior of the plastic SIF KP as a function of the values of
critical distance rc in the mid-plane for all tested specimens of the 34ХН3МА. Based on these results, it was
assumed that for the greater fracture resistance of the material that was so far away from the crack tip, conditions
appeared for failure initiation according to the critical strain energy density, which also characterizes the material
properties in the fracture process zone. It is easy to see that the critical distance is not constant, as is often supposed

V Shlyannikov / Procedia Structural Integrity 2 (2016) 744–752
749
in the literature, but depends on the loading conditions and the geometry of the tested specimens. Thus, the critical
distance value is affected by in-plane and out-of-plane constraint interaction.
The creep critical distance behavior represented in Figs. 4,a and 4,b can be considered as the particular case of
the process zone distribution at the polar angle equal  = 0 that is the plane ahead of the crack tip. Substituting the
~
creep SIF and the Mises equivalent stress
into Eq. (5), the values of the critical distance were calculated for

e
both steels, taking into account the behavior of the governing parameter of the power-law creep crack-tip stress
fields In for the C(T) specimens. The distributions of the creep critical distance along the curvilinear crack front in
the thickness direction is plotted in Fig. 4,a for the P2M steel, whereas in Fig. 4,b these distributions are shown
along the straight-line crack front in the CT specimen of 12X1MF steel. This parameter is plotted against the
normalized specimen thickness z/b.
a)
b)
c)
Fig. 3. Total and elastic critical distances distributions (a,b) in CT along crack front and (c) variation of plastic SIF’s in SENB and CT.
a)
b)
c)
Fig. 4. Critical distance distributions along crack front in CT and relationship between creep SIF and critical distance.
Figure 4,c represents the critical distance behavior of the R2M and 12X1MF steels at 550C as a function of the
dimensionless creep stress intensity factor in the form of Eq. (3). In general, as expected according to Eq.(5), the
critical distance increases with an increase in the creep stress intensity factor. An overall reasonable correlation
between the rcr /a and
is an indication that dominantly creep crack growth conditions were maintained
K
cr
throughout the testing. It is seen that the numerical data fall within a relatively narrow scatter band, and the
dependence of the critical distance, rcr /a, on follows a near linear trend on a log-log scale. It is clear that rcr /a is
K
cr
not constant, but is the parameter depending on the loading conditions which are described by the creep stress
intensity factor . Moreover, for the steels P2M and 12X1MF different properties takes place general relationship
K
cr
between a dimensionless parameters rcr /a, and
.
K
cr

750
V Shlyannikov / Procedia Structural Integrity 2 (2016) 744–752
5. Application to structural integrity assessment
The methodology describing in the present study is applied to a 100 MW steam turbine rotor. The turbine
disc is loaded, in general, by thermal and mechanical stresses because it is operated at c.a. 550C and 3000 rpm.
Subject for analysis is the disk of 22nd stage turbine rotor with central bore and through-thickness key. Operation
damage in the form of corner crack with length on the free surface a =12 mm and depth along shaft thickness b =
23mm was detected in the slot fillet of key in the disc of 22nd stage. Prior to lifetime predictions, a 3D FE model
of the 22nd stage turbine disc section generated was available and used for the purpose of this study. The turbine
disc and blades, as well as the rivets incorporated in the orignal 3D model. The startup power conditions was
chosen in stress analysis as it represents the most severe combination of temperature and rotor speed in the
operation profile. The material of turbine disc is Steel 34ХН3МА which main mechanical properties are listed in
Table 1.
The commercial finite element code, ANSYS (2012), is used to calculate the strain and stress distributions in the
turbine disk under operation loading conditions. From FE-analysis, it was observed that the peak equivalent von
Mises stresses mainly occur on the surface of the slot fillets of key (web and disc bore). It should be noted that the
stresses on the free surface and in the slot along hub thickness are 1.2 times higher than the yield stress of Steel
34ХН3МА. The special kind of nonlinear calculations accounting for the plastic material properties were
performed to determine plastic stress intensity factors
for the same crack front profiles in turbine disc after
KP
corresponding loading history at operation. Full-field elastic-plastic FEA are performed using ANSYS finite element
(FE) code to determine the stress-strain parameter distributions along of the crack-front for turbine disc of
considered configuration. With reference to an operation damage the cases considered were a disk with semi-
elliptical corner crack emanating from the intersection of the slot fillet and of the flat surfaces of the disk. The
plastic stress intensity factor
in pure Mode I was expressed directly in terms of the corresponding elastic stress
KP
intensity factor. Figure 5,a shows the dependencies of the plastic SIF on the crack sizes for the turbine disc
considered configuration for these two main points, namely, the slot inner surface of key (point a) and the free
surface of hub (point b). The distributions of the plastic SIF's along the crack front in the turbine disk under
operation loading are shown in Fig.5,b. These distributions correspond to the initial crack front (a0=b0=3 mm) and
the final failure front (a=10 mm, b=20 mm) positions.
a)
b)
Fig. 5. Plastic SIF distributions as a function of (a) crack size and (b) crack front angle
Within the current investigation a fatigue life estimation approach (Eq.6) based on the FEA results and
experimental data is applied to predict residual durability of power steam turbine disk with take into account of
operating time
2
2


1 m
 ynS thS
da
th 
(6)
 2 a
.
c
 
 f  f Ec
dN





V Shlyannikov / Procedia Structural Integrity 2 (2016) 744–752
751
The above rate of crack propagation law contains the mechanical properties of the material, E, cyclic properties
, the governing parameters of elastic-plastic stress-strain field I and a length parameter associated with
,


n

n
c
, the In-factor and equivalent
 ,
the fracture process zone size. More details to determine the SED functions
,
S
S
th
~
stress e for cracked body different configurations are given by Shlyannikov et al. (2016,b). The fatigue crack
growth analysis was performed under harmonic loading using the elastic and elastic-plastic SIF's distributions along
different crack front profiles. An initial circumferential edge crack at the highest elastic-plastic stress location was
chosen to be 1.6 mm in the depth and length direction, which is much smaller than the observed crack size at
operation. It is found that the crack growth in the depth direction is much faster than that in the length direction.
Fig. 6. Lifetime prediction based on elastic and elastic-plastic solutions
Figure 6 represents the comparison between the predicted change in crack length a on the free surface of disc
and the crack depth b along the slot of key as a function of fatigue load cycles for both type of solutions. To
compare predictions of the crack growth rate for the elastic stress state of the turbine disk with elastic-plastic stress
state we used the elastic and total strain energy density factor. The elastic-plastic solution on based on the plastic
SIF's show that the crack would grow on the free surface from 1.6 mm to 10 mm and in the depth direction from 1.6
mm to 20 mm in about 1,300 cycles. At the same time the elastic solution using the elastic SIF's gives overestimate
the lifetime in about 1,700 cycles.
As the predicted crack growth rate according to nonlinear fracture mechanics approach is much faster that elastic
modeling, this indicates that the plastic material properties have a significant effect on the damage accumulation and
growth in an critical zone of turbine disc. It should be pointed out that the elastic solution is not accounted for the
stress-strain state redistributions at the plastic zone close to the crack tip. The implications due to this limitation may
give non-conservative predictions of crack growth rate for this case as the actual stress and strain may be higher than
predicted by elastic solution. As the purpose for analyzing the edge crack in the turbine disc considered at the
operation was to estimate the crack growth rate under extreme situation, the residual fatigue life should be
determined based on the elastic-plastic solution.
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