Transcript Slide 1

1
Structure of the lecture:
1. Preface
2. General characteristics of the problem
3. Classical and non-classical approaches
4. Griffith-Irwin concept and linear fracture mechanics
5. Nonlinear fracture mechanics
6. Problems of materials fatigue fracture
7. The influence of environment on corrosion
resistance of materials
8. The influence of hydrogen containing environment
9. Conclusions
2
Classical and non-classical
approaches
Classical approaches
(1)
Fc (I1, I2, I3, C1, C2, C3 …) = 0
I1 = σ1 + σ2 + σ3; I2, I3 are the stress (strain) invariants,
Ci are the constants
(2)
σ1≤σB, σ1 – σB = 0
σ1›σ2›σ3 are the main stresses
σB is the strength of the material under tension
σin  m1σin  m2
1
σi  (2)
(3)
2
1 / 2
σ  σ 
2
1
2
a
 σ 2  σ3   σ3  σ1 
2

b
a – classical
2 1/ 2
b – non-classical
(П - stage = CS- stage)
n1, n2, m1, m2 are the constants which are determined on the basis of the experiment
(Pisarenko-Lebedev formula)
3
Material Failure Model at the Crack Tip
THE BASIC MODELS OF CRACKS
Fig.2.
Fig.1.
a
b
c
4
Example. Classical and non-classical
approaches. Griffith concept.
1. Tension of the plate with the elliptical hole
σ max  σ B , σ B  cons t
σ max  σ y a, o   k , p
kk  1  2 l
b
p  σ B
k
When b  0crack , k  
and p  0
Fig. 1
(1)
(2)
(3)
(4)
(5)
2. Griffith concept
To determine the value of fracture loading for the cracked plate subjected to tension (Fig. 1
when b = 0), Griffith proposed (1924) so-called power method. It is eliminated to
П  П 0  W  p, l   U l 
δП δ
 U l   W  p , l   0
δд δl
Equation (7) gives us Griffith formula:
2 Εγ
p 
; p l  const
2

π 1 m l


γ – the effective surface energy of the body square unit
(6)
(7)
(8)
5
Griffith-Irwin Concept
σ ij 
1
2 πr
K I0 f1ij (θ)  K II0 f2ij (θ)  K III0 f3ij (θ) 0(1),
(1)
(Δl<<l0)
where i, j = x, у, z in the Cartesian coordinates or i, j = r, 
, z in polar (cylindrical) coordinate system, KI0 = KI0(p, l),
KII0 = KII0(p, l), KIII0 = =KIII0(p, l) are stress intensity
factors (SIF), which are the functions of body
configuration, crack dimensions (l) and the value of loading
p; 0(1) is limited value when r0; fkij() are known
functions (k = 1, 2, 3).
KI0* = KI0(p*, l) = KIc
For the case
KI0 ≠ 0, KII0 = 0, KIII0 = 0
(2)
Local coordinates system near
the crack front (line OZ)
and components (I, II, III) of
the vector of crack edges
displacement
6
Plate containing arbitrary oriented crack
Fig.1
σ θ r , θ  
F K I , K II 
 01
r
(1)
K I  p, l   0; K II  p, l   0; K III  p, l   0
criteria. Initial crack propagation occurs in the plane, for which fracture stresses σθ
have the maximum SIF value. Then on the base of σθ-criteria and Irwin concept
we obtain the following criteria equations
7
Generalized Griffith-Irwin Concept
Equations (1)-(3)
are proved
experimentally
and now present
the linear fracture
mechanics of
materials (LFM).
KI(p,l) ≠ 0, KII(p,l) ≠ 0, KIII(p,l) ≠ 0
Tension of the plate with the arbitrary oriented crack
lim 2πr σθθ (α, p , r , θ )  K Ic ,
r 0
 σθθ (α, p, r , θ

lim 2πr 
  0,
r 0
θ
θ θ 


where θ* is the angle of the initial direction of crack growth
cos3
θ
2
θ


K
(
p
,
α
,
l
)

3
tg
K
(
p
,
α
,
l
)
 K Ic ,
II 0

 I 0 

2

K I 0  K I20  8K II2 0
θ   2arctg
,
4 K II0
K I0  0, K II0  0,
K I0  p πl sin 2 α, K II0  p πl sin α cos α
Equations (1) and (2) were generalized (O.Ye.Andreikiv et al) for the case
(1)
(2)
(3)
KI0 ≠ 0, KII0 ≠ 0, KIII0 ≠ 0
8
Linear Fracture Mechanics,
Advanced Problems
The Griffith-Irwin concepts
and formulated equations above in previous slide
form the basis of the linear fracture mechanics (LFM).



1. Calculation of the stress intensity factors for limited bodies
with cracks (structural elements) under the influence of force
and temperature factors; preparation of corresponding data
bases and reference books for engineering practice.
2. Determination of crack growth resistance of structural
materials (KIc, KIIc, KIIIc, g) taking into account the structure
of the material and the influence of the environment, in
particular under the action of H2
3. Experimental and theoretical verification of the LFM
statements and the establishment of LFM concepts
applicability in the case of limited elasto-plastic bodies
(structural elements).
9
Linear Fracture Mechanics,
Advanced Problems
The Griffith-Irwin concepts
and formulated equations above in previous slide
form the basis of the linear fracture mechanics (LFM).



1. Calculation of the stress intensity factors for limited bodies
with cracks (structural elements) under the influence of force
and temperature factors; preparation of corresponding data
bases and reference books for engineering practice.
2. Determination of crack growth resistance of structural
materials (KIc, KIIc, KIIIc, g) taking into account the structure
of the material and the influence of the environment, in
particular under the action of H2
3. Experimental and theoretical verification of the LFM
statements and the establishment of LFM concepts
applicability in the case of limited elasto-plastic bodies
(structural elements).
10
Linear Fracture Mechanics,
Advanced Problems
The Griffith-Irwin concepts
and formulated equations above in previous slide
form the basis of the linear fracture mechanics (LFM).



1. Calculation of the stress intensity factors for limited bodies
with cracks (structural elements) under the influence of force
and temperature factors; preparation of corresponding data
bases and reference books for engineering practice.
2. Determination of crack growth resistance of structural
materials (KIc, KIIc, KIIIc, g) taking into account the structure
of the material and the influence of the environment, in
particular under the action of H2
3. Experimental and theoretical verification of the LFM
statements and the establishment of LFM concepts
applicability in the case of limited elasto-plastic bodies
(structural elements).
11
Nonlinear Fracture Mechanics
Typical linear dimensions of П-state zone in materials is great
(comparable) with the linear dimensions of a crack or a considered body.

δ c– model and COD-criteria ( l  l0, figure)
δ p  2v( p* , l0 , σ0 , E )  δc
(1)
*
 d 
2
p*  σ 0 arccos exp  * ,
π
 l0 
d*  πEδc  8σ 0 


πp
l  l  l0  l0  sec
 1,
2σ 0 

∆ℓ*=d*, when δp = δc ,
*
δ c– material characteristics !
(2)
P*/σ0
(3)
1,25
Fig.1
2 (Griffith)
0,75
Fig.2 →
0,25
1
0
2,5
5,0
7,5
l0 d* 12
EXPERIMENTAL DETERMINATION OF STRUCTURAL
MATERIALS DEFORMATIONAL CRACK RESISTANCE (δc)
(Investigations Of Ya.L.Ivanytsky, L.I.Muravsky et. al. )
Use of the correlation method of the specimen surface speckle images in the crack tip vicinity
Fig. 1. Distribution of deformations ey(x) near the
crack tip at different measuring bases (b): (1 – b =
1,28 mm; 2 – 2,56 mm; 3 – 3,08 mm) for loaded
(1, 2, 3) and unloaded (1΄, 2΄, 3) D16AT alloy cracked
specimen
d p
  π p  
 l0 sec    1
  2 σ0  
(1)
Fig. 2. Dependence of the plastic zone on the crack
continuation on p*/σ0 loading value for Д16АТ alloy
(firm line corresponds the values in (2), dots –
experimental data for different length of the initial crack
~*
δ1c  8σ0d p πE  ,
d p  l0 
(2)
13
SOME GENERALIZATIONS OF c- MODEL
A diagram of stepwise interaction
between the model cut edges
i (l0 , p)  A[2 2   2 ( 2   2  2(  i )   2  i2 ) 
4ln 1     0,5( i 2)2 (l , l0 , li )  0,5i2(l , l0 , li )] ,
(1)
l  x  (l  x )(l   )
(l , x,  )  ln
,
l  x  (l  x )(l   )
2
2
2
2
2
2
2
2
2
2
A  20 cl0 ,   d / l0 , i  di / l0 ,
p
20

l  l  d, l  l  d ;
0
i
0
i
1
1
[ 2   2  2  2i   2  i2  arcsin

i
1 
 arcsin
1  i
1  i
]  arccos
.
1 
1 
(l0 , p* )  *c


c
value is approximately determined by
*c  8l0 c0 ln
l0
l0  d*
,
c1
d
in the formula, that arises from
(3)
 c -model
*
d  l0

πE с  80 d* /(E ) when *
,
(2)
(4)
14
SOME GENERALIZATIONS OF c- MODEL
Dependence of critical stresses( p*/0), on critical process zone length (d = d*)
for different lengths (i = di/d*) of the material fracture zone near concentrator
2 2d
1 – by Griffith formula :
;
π
l
2 – according to с-model (i = 0);
p /σ 
*
*
0
0
3 – according to formulas (1)-(4) when i = 0,3;
4 – according to formulas (1)-(4) when i = 1.
Points show experimental values for D16АТ aluminium alloy.
15
ВИСНОВКИ – CONCLUSIONS

1. Nielinijowa mechanika pękania materiałów ( d* ~ l0 )
na dany czas jeszcze nie ma (osiągnęła) odpowiedniego zakończenia.

2. Ważnymi i perspektywnymi badaniami w tej dziedzinie są
opracowania fizyko-matematycznych koncepcii i rozrachunkowych
modeli do wyznaczenia granicznej równowagi deformowanych ciał z
pęknięciami ze względem na rozwinięte (wielkie) zony plastyczności
(deformacji) około frontu pęknięcia.

1. Non-linear fracture mechanics of materials ( d ~ l )
* 0
did not reach its appropriate completion yet.

2. Development of physico-mathematical concepts and
calculation models for the determination of limiting equilibrium
of cracked bodies taking into account the developed (big) zones
of plasticity near the crack front are important and perspective
investigations.
16
MATERIALS FATIGUE:
Crack Initiation and Propagation

The problem of materials fatigue is one of the central problems of fracture
mechanics and prediction of structural elements life time (durability).
Great efforts have been spent for solution of this problem since the 19th
century, when this phenomenon was considered for the first time. This
problem was the topic of special plenary report by J. Schijve* at the 14th
European Conference on Fracture (ECF–14) on September 9 2002 in
Cracow. In this problem solution the concepts of fracture mechanics are
very important. They are the following. For fatigue fracture of the material
two periods are determining: the macrocrack initiation period (N1) and its
propagation period (N2). Determination of these periods is the main task
of the science on materials fatigue and durability (life time) of structural
elements. When periods N1 and N2 are known, the total life time (N*) is
determined by formula
N = N1 + N2
* Commentary to the report by J. Schijve “Fatigue of structures and materials in the 20th
century: state of the art”. In this report the detailed list of references on the above problem is
given, however the author did not consider the investigations of the East European scientists.
This was done as a supplement in the Ukrainian translation of the report by the scientific
editor of the “Physicochemical Mechanics of Materials” journal (“Materials Science”). – 2003. № 3. – P. 7-27.
17
MATERIALS FATIGUE:
Crack Initiation and Propagation

Problem zmęczenia materiałów – jeden zcentralnych (głównych ) problemów
mechaniki pękania i prognozy resursu (długowieczności(wytrzymałości) elementów
konstrukcii. Na rozwiązanie tego problemu zatracono dużo wysiłków poczawszy od
19-go stulecia. Ten problem był przedmiotem specjalnego plenarnego referatu
Dż.Schajwe* na 14-ej Europejskiej konferencji (ECF-14) 9 września 2002 roku w
Krakowie. W rozwiązaniu tego problemu szczególne miejsce zajmują koncepcii
mechaniki pękania materiałów. Oni są następne. Dla pękania zmęczeniowego
materiału wyznaczalnymi są dwa periody – period zarodkowania makroszczeliny (N1)
i period jej rozpowszechniania (N2). Wyznaczenie tych periodów jest głównym
zadaniem nauki o zmęczeniu materiałów i długowieczności (wytrwałości)(resursie)
elementów konstrukcii. Jeśli są wiadome periody N1 і N2 to ogólną długowieczność
można wyznaczyć za nastepną formułą:
N = N1 + N2
* Komentarz do wykładu Dż.Schajwe „Zmęczenie konstrukcii i materiałów w 20-tym stuleciu:
aktualny stan”. W tym referacie autor podaje wielki spis literatury naukowej z tego problemu
lecz nie bierze do uwagi (pod uwagę) osiągnięć uczonych z Europy Wschodniej. To jest
podane jako dodatek od naukowego redaktora tłumaczenia tego artykułu (patrz.czasopismo
„Fizyko-chemiczna mechanika materiałów”- 2003. – N 3. - P.7-27)
18
MATERIALS FATIGUE:
Macrocrack Growth
Ability of the material to resist the fatigue crack initiation and growth
is characterized by its fatigue crack resistance
v  CK
n
I
v  10
7
K
I
K

* m
v  v0 K I  Kth  K f  K I 
m1


vi  A λ 0 λ 0  λi   1 , (i  1,2)
m2
λ1  1  ε  ε c1 ; λ 2  1  K I  K fc1
K  , m, m1 , m2 , n, C , λ 0– material characteristics
Fatigue crack growth resistance diagram ((v-K)-curve).
1 – section close to threshold Kth; 2 – practically rectilinear section;
3 – section of quick crack growth and complete failure when KImax = Kfc
19
MATERIALS FATIGUE:
Fatigue Crack Initiation
N = N1 + N2

N* is material durability; N1 is the period of fatigue
damaging and macrocrack initiation; N2 is the period
of macrocrack growth up to the critical value.
The main task of the
science about the material
fatigue and strength
of structures is the
development of effective
methods for assessment
of fatigue crack initiation
period N1 !
A scheme for calculation of N1 period
of the fatigue macrocrack formation

We have obtained certain results in this direction, in particular
O.P.Ostash (PhMI) proposed in his papers a new concept in the
terms of which it is possible to estimate N1 if the (n–K)-curve
for this material is known (see slide 14).
20
Approximate Model
for Determination of N1
N1 – initiation period
of fatigue macrocrack
of length l ;
l = l – minimum
length of fatigue
macrocrack.
l
l
dl
1
 vl  , N1   v l dl , N 2   v 1 l dl
dN
l0
0
*
ω1 λ1  when l  l*
v l   
ω 2 λ 2  when l  l*
1
k
(1)
(2)
ω2 λ 2  – known function (diagram of material fatigue
macrocrack resistance);
v  K  – diagram;
ω1 λ1  – unknown function.
λ1  1  ε  ε c1 ; λ 2  1  K I  K If 1 ; λi  λi l 
(3)
m
 λ


0
  1 ,
ω2 λ 2   A 
 λ 0  λ 2 

(4)
ω1 λ1   ω2 λ1 
A, λ 0 – materials characteristics, l* ~ ?
21
Kinetics of Fatigue Crack Propagation in
the Zone of Two Bodies Contact
(Investigations of O.P.Datsyshyn et al)
Calculational model
B – direction of the contact
loading movement
Scheme
of contacting pairs
Calculation results of the surface crack
propagation path under rolling (pitting formation)
Damage of the subsurfase
contact zone under rolling:
a – the edge crack growth path
(dashed line) under rolling in
lubrication conditions in
dependence of lubricant
pressure intensity (q = rp0) on
crack edges; b –cross-section of
pitting on the bearing surface
This is a new and important direction of the
investigations in the field of materials fracture mechanics
22
Conclusions and Advanced Research
on the Problem of Materials Fatigue
1. Determination of the macrocrack initiation period (N1) in the
cyclically deformed material – the main task of experimental and
theoretical investigations.
2. Construction of the fatigue crack resistance diagrams for
structural materials in v0δр coordinates, that is (v-δр)-diagrams
where v – macrocrack growth rate, δр – opening of crack edges at
the fixed points, - new opportunities for the assessment of
structural elements durability.
3. Investigations of interaction between the propagation rates of
macro- and microcracks in cyclically deformed materials.
4. Development of the effective methods for the evaluation of the
fatigue macrocrack minimum value for a given structure of the material.
5. Investigations of the crack fatigue propagation in the zone of
two bodies cyclic contact (problems of tribology).
23
Wnioski i nowoczesne (nowe) badania
problemu zmęczenia materiałów
1. Wyznaczenie okresu zarodzenia się makroszczeliny (N1) w
cyklicznie-deformowanym materiale – główne zadanie eksperymentalnych i teoretycznych badań.
2. Budowa diagramu zmęczeniowej odporności na pękanie konstrukcyjnych materiałów w koordynatach v0δp ,czyli (v - δp) –diagramów,
gdzie v – szybkość rozwarcia brzegów szczeliny(pęknięcia) w
punktach fiksowanych – nowe możliwości do oceny długowieczności
(wytrzymałości) konstrukcyjnych materiałów.
3. Badania wzajemnego oddziaływania (współdziałania) szybkości
rozpowszechniania się makro- i mikro szczelin w cykliczniedeformowanym materiale.
4. Opracowanie efektywnych metod wyznaczenia minimalnego
znaczenia (wielkości) zmęczeniowej makroszczeliny dla materiału o
pewnej(wyznaczonej) strukturze.
5. Zbadanie rozpowszechniania się zmęczeniowej szczeliny w zonie
cyklicznego kontaktu dwuch ciał (problemy trybologii)
24
Glówne stadium procesu pękania
korozyjnego (korozyjnej rujnacji)
Korozyjne zmęczenie
konstrukcyjnych
materiałów
Nagromadzenie uszkodzeń korozyjnych
Zarodkowanie i wzrost (rozwój) krutkich szczelin
Wzrost długich szczelin do krytycznego rozmiaru
Rujnacja materiału
Wniosek w proces pękania korozyjnego
25
Influence of Initial Testing Conditions
on Corrosion Crack Growth Behaviour
()
()
( )
( )
( )
Influence of initial level
of stress intensity
factor Ki on corrosion
crack growth rate
( )
Stress intensity factor, K
26
Simultaneous Influence
of Stresses and Environment
Peculiarities of Physical-Chemical Situation
Near the Crack Tip (schematically)
pH s  pH t , Es  Et
v  F K I , pH t , Et 
v  vmax when
(1)
 pHt min , Et min
v  K  – curve will be invariant when
 pHt   const , Et   const
The methods and equipment for pH t,
evaluation were developed in PhMI.
(2)
(3)
Et
27
Equipment and Methods
for pH t , Et Measuring
a
b
c
Equipment for evaluation of the material fatigue fracture
characteristics:
a – scheme of location of the gauges-microelectrodes in the specimen
for local electrochemical investigations;
b – scheme of the automatic testing equipment;
c – general view of the equipment.
28
Basic Diagrams
for Pressure Vessel Materials
Fatigue crack growth diagram
(v-K curves) for pressure vessel metal;
1, 2 – according to ASME method;
3, 4 – according to Bamford (generalized
experimental data);
5 – basic curve, plotted in terms of the
proposed concept
29
Influence of hydrogen containing
environment
Problems:
1. Hydrogen transport to the metal
2. Surface interaction and hydrogen
penetration into the metal
3. Hydrogen state and behavior inside the
metal
4. Hydrogen influence on the fracture
microprocesses
5. Hydrogen influence on the crack
growth resistance of metals and welded
joint
Each of these problems is a separate section of the science about the interaction between the
deformed metal and hydrogen. Scientists and engineers from different countries work at the
solution of these problems. Investigations of some aspects of these problems are planned in
the frames of Polish-Ukrainian scientific collaboration. Consideration of specific investigation
of these problems will be the subject of future lectures.
30
Wpływ wodórmieszczących środowisk
Problemy (zadania):
1. Przeniesienie wodoru do metalu
2. ІІ – współdziałanie powierzchni i
przedostawanie
się
(przeniknięcie)
wodoru do metalu ІІІ – stan wodoru i
jego zachowanie wewnątrz metalu
3. Wplyw wodoru na mikroprocesy
rujnacji
4. Wplyw wodoru na odpornosc metali
I spawanych łączni wzrostu szczeliny
Każdy z tych problemów jest oddzielnym rozdziałem nauki o wspłódziałaniu
deformowanego metalu z wodorem. Nad rozwiązaniem tych problemów pracują pracowniki
naukowe i inżynierowie. Badania oddzielnych aspektów takich problemów planujemy
realizowac w ramkach naukowo-technicznej ukrainsko-polskiej wspólpracy. Rozpatrzenie
konkretnych badan z tego problemu będzie przedmiotem następnych wykladów.
31
CONCLUSIONS AND PERSPECTIVES
1. Fundamentally new tools for studying of surface-active and
corrosion-aggressive environments influence on the physicomechanical characteristics of cracked materials are developed.
2. New conditions of a given system “metal-corrosive environment”,
when the value of crack growth rate in the cyclically deformed
metal reaches its maximum, are determined.
3.Methodology for plotting of metals corrosion cracking basic
diagrams, used for the assessment of high-pressure vessels
reliability in service, is worked out.
4. Perspective and important for engineering practice is the plotting
of corrosion cracking basic diagrams for different material classes
and corrosion environments.
5. Perspective and important are the investigations of the
interaction between deformed metals and hydrogen. Here we have
a number of sub-problems, which must be investigated and solved
considering the strength of long-term operation structures.
32
Crack Tip Opening Displacement Estimation
Under Tention of an Infinite Plate
8  0  d*
c 

с, mm
(experim.)
dі/d*
0,125
0,120
0,24
0,118
0,131
0,122
0,22
12,6
0,118
0,131
0,125
0,25
0,73
10,6
0,099
0,110
0,108
0,25
18,1
0,72
13,0
0,121
0,135
0,128
0,22
13,0
0,85
11,0
0,103
0,114
0,110
0,26
с , mm
0
0,2
№
l0 ,
mm
d*/l0
d*,
mm
1
12,5
0,96
12,0
0,112
2
12,6
1,0
12,6
3
13,0
0,97
4
14,5
5
6
Distribution of Deformations Near
the Crack Tip on the Basis of
Digital Specle Correlation Method
of the Specimen Surface Image
Base (mm):
1 – 1,28; 2 – 2,56; 3 – 3,84.
33
DETERMINATION OF CRACK RESISTANCE
VALUES K IIc K IIIc
Test investigations of Ya.L.Ivanytsky et. al.
An outline of a speciman for the
determination of the materials crack
resistance (KIIc) [8, 9]: 1, 2 – circular
concentrator; 3, 4 – symmetric cracks; 5, 6
– the places of the specimen grips during
turning or tension
An outline of the specimen for the
determination of the materials crack
resistance (KIIIc) [8, 9]
№
Materials
Thermal
processing
КІС
КІІС
КІІІС
ІС
ІІС
ІІІС
mm
MPa m
1.
40 KhN steel
hardening,tempering under 733 K
64
124
165
0,11
0,41
1,49
2.
40 KhN steel
hardening,tempering under 833 K
74
136
12
0,13
0,48
1,56
3.
30KhGKhA steel
53
148
169
0,14
0,48
1.39
4.
40 KhN steel
hardening,tempering under 833 K
75
139
176
0,13
0,49
1,41
5.
20Kh13 steel
state of delivery
42
–
170
–
–
–
normalizing
34
CRITERION OF LINEAR FRACTURE
MECHANICS UNDER COMPLEX
LOADING CONDITIONS
(КІ ≠ 0, КІІ ≠ 0, КІІІ ≠ 0)
Griffith-Irwin criterion
K I (ρ , l )  K Ic
(1)
Generalized criterion for the case of complex loading (КІ ≠ 0, КІІ ≠ 0, КІІІ ≠ 0) is as follows:
 KI 


K
 Ic 
n1
K 
  II 
 K IIc 
n2
K 
  III 
 K IIIc 
n3
1
(2)
where KIc, KIIc, KIIIc, ni (i = 1, 2, 3) – parameters that characterize material near the concentrator,
that is there, were П-states of the material appear. These states are determined experimentally
or on the bases of certain model calculations.
KI, KII, KIII values are calculated for each case in the frames of the crack mathematical theory.
Under the complex loading conditions criterion (2), КІ ≠ 0, КІІ ≠ 0, КІІІ = 0 and mixed mechanism (I + II)
of fracture is realized, is as follows:
 KI 


 K Ic 
ni
n
 K II  i
 = 1
 
 K IIc 
(3)
In the case when КІ = 0, КІІ ≠ 0, КІІІ ≠ 0 and mixed mechanism (II+III) of fracture is realized, we have
 K II 


K
 IIc 
ni
n
 K III  i
 = 1
 
K
 IIIc 
(4)
where ni equals 4 or 2.
35
SOME EXPERIMENTAL RESULTS
A criterion in the case of complex loading has such a form:
 KI

 K Ic
ni

K
   II

 K IIc
ni

K
   III

 K IIIc
ni

  1

(1)
where: ni(1, 2, 3) - material characteristics received from the experiment or on the bases
of theoretical calculations; KIC, KIIC, KIIIC - characteristics received from the experiment
Diagrams of the deformed body limiting-equilibrium state the conditions of mixed (I+II),
(I+III) fracture mechanisms
Curve 1 – according to formula (1)
when KIII = 0 and ni = 4, curve 2
according formula (1) when KIII = 0
and ni = 2
Curves 1 and 2 are plotted according
to formula (1) when ni = 4 and ni =2
correspondingly
Experimental data: - 40KhN steel, hardening in oil under 1123 K, tempering under 833 K;
- 30KhGSA steel, normalizing;
- 40KhN steel, hardening in oil under 1133 K, tempering under 773 K (test results of Ya.L.Ivanytsky);
- 4340 steel (test
results of A.A.Chuzhuk); aluminum alloy 2219 (E87) (test results of A.A.Chuzhuk); - 9KhF steel: - hardening in oil under
1133 K, tempering under 873 K; - tempering under 773 K; - tempering under 673 K (test results of Ya.L.Ivanytsky)
36