Statistical strength criterion for materials with hexagonal close-packed crystal lattice/Statistinis stiprumo kriterijus medziagoms su sutankinta sesiakampe kristaline gardele.
Bagmutov, V.P. ; Bogdanov, E.P. ; Shkoda, I.A. 等
1. Introduction
At present in the design and operation of critical products from
non-conventional metal materials their strength and plasticity resources
evaluation is actual. In this context, statistical strength and
plasticity criteria that allow taking into account the structure and
peculiarities of intercrystalline interaction for polycrystalline
materials under deformation and fracture have large opportunities from
our point of view. It is not the least of the factors that directly in
them relation between micro- and macrocharacteristics of the processes
analyzed is set via the corresponding averaging methods. This kind of
criteria takes an intermediate place between phenomenological and
physical approaches. Phenomenological criteria connect strength with
various invariants of a tensor of stresses or deformations. The physical
approach allows to estimate features of elastic-plastic deformation
depending on operating mechanisms of deformation (sliding, doubling)
demand enough difficult procedures of calculation for construction of
macroscopically characteristics (a curve of deformation and others)
[1-3]. Statistical criterion considered in given article, possess
simplicity of use in engineering practice and allow to estimate strength
precisely enough.
Development of the statistical approach was made in works of
Weibull, Frenkel, Fisher and Hollomon, Afanasev, Bolotin, Novozhilov and
others [4]. The statistical approach to material strength evaluation
proposed by Volkov S.D. [5] is based on the hypothesis that
microstresses are distributed by the normal law while fracture occurs
when critical probability of exceeding by microstresses responsible for
fracture is reached. It was assumed that incipient microcracks appear in
the planes with orientation close to the basic area corresponding to the
principal macroscopic stress [[sigma].sub.1] while local criterion of
fracture is [[xi].sub.11] [greater than or equal to] [[xi].sub.c]. Here
[[xi].sub.11] is microstress [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE
IN ASCII] where mathematical expectation is <[[xi].sub.11]> =
[[sigma].sub.1] and local strength is [[xi].sub.c] = const. The
hypothesis acceptance that critical probability does not depend on
stress state kind results in the fact that generalized condition of
strength will be expressed as follows:
[z.sub.c] = [[[xi].sub.c] - [[sigma].sub.1]]/[S ([[xi].sub.11])] =
const, (1)
that is prescribed integration limit of density function
corresponding to the critical probability. The exact criterion kind
depends on the function S ([[xi].sub.11]) = [PHI] ([[sigma].sub.1],
[[sigma].sub.2], [[sigma].sub.3]) determining dependence of
root-mean-square deviation on macrostress state. Volkov S.D. supposed
that dispersion:
D([[xi].sub.11]) = [S.sup.2] ([[xi].sub.11]) = KW, (2)
where W is potential energy of deformation. Having determined K X
[z.sub.c] and [[xi].sub.c] by tests with two different kinds of stress
state, he got two-parameter criterion of strength.
Pisarenko G.S. and Lebedev A.A. proposed to use not the whole of
energy but distortion strain energy only having got as well the
two-parameter criterion which for the main part of materials describes
well the strength for limited range of stress states which are managed
to be imitated in laboratory conditions. However, for a number of
materials such as columbium alloys, the Pisarenko-Lebedev criterion
describes experimental results poor even for the state of plane stress
[6].
In the work [6] the statistical approach has been developed due to
consideration of various mechanisms of discontinuance (oriented and
non-oriented fracture) and use of local strength criteria taking into
account the strength anisotropy of grains. Only for materials with low
strength anisotropy the crystals of which do not have any cleavage
planes or have many families of cleavage planes [[xi].sub.11] [greater
than or equal to] [[xi].sub.c] can be used as local criterion of
fracture as they have low strength anisotropy (oriented fracture). To
such materials refractory of 5a and 6a groups of Mendeleev periodic
system (niobium, vanadium, tantalum, chromium, molybdenum) with cubic
crystalline lattice can be related. However, for such materials or
rather like for all polycrystals the hypothesis Eq. (2) is not
acceptable. We can show this for the case of elastic deformation.
2. Theoretical
Microstresses in the random point of nonuniform micro-sized medium
with elastic deformation can be found as the sum:
[[xi].sub.11] = [[sigma].sub.1] [[bar.[xi]].sup.(1).sub.11] +
[[sigma].sub.2] [[bar.[xi]].sup.(2).sub.11] + [[sigma].sub.3]
[[bar.[xi]].sup.(3).sub.11] ,
where [[bar.[xi]].sup.(k).sub.11] is microstress caused by single
principal macrostress [[sigma].sub.k], then dispersion D ([[xi].sub.11])
can be found as sum of three dependent random quantities:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)
where [B.sup.k.sub.11] [equivalent to] [B.sup.kk.sub.11] =
d([[bar.[xi]].sup.(k).sub.11]) is dispersions and [B.sup.km.sub.11] =
cov ([[bar.[xi]].sup.(k).sub.11] [[bar.[xi]].sup.(m).sub.11]) =
<[[bar.[xi]].sup.(k).sub.11] [[bar.[xi]].sup.(m).sub.11]> -
<[[bar.[xi]].sup.(k).sub.11]> <[[bar.[xi]].sup.(m).sub.11]>
is covariances (correlation moments) of microstresses caused by single
macrostresses [[bar.[sigma]].sub.k] and [[bar.[sigma]].sub.m]. Here,
angle brackets mean averaging. In the work [6] by the model of
polycrystal based on the deformations homogeneity hypothesis (Voigt
hypothesis) and by the finite element model it is shown that for
quasi-isotropic crystal:
[B.sup.11.sub.11] [not equal to] [B.sup.22.sub.11] =
[B.sup.33.sub.11], [B.sup.12.sub.11] = [B.sup.13.sub.11] [not equal to]
[B.sup.23.sub.11]. (4)
Last equation shows the inconsistency of hypothesis (2) because its
confirmation requires performance of the following conditions for
dispersions [B.sup.11.sub.11] = [B.sup.22.sub.11] = [B.sup.33.sub.11]
and covariances [B.sup.12.sub.11] = [B.sup.13.sub.11] =
[B.sup.23.sub.11]. Moreover [B.sup.ij.sub.11]/[B.sup.ii.sub.11] = v,
where v is Poisson coefficient. It should be noted that
<[[bar.[xi]].sup.(1).sub.11]> = 1,
<[[bar.[xi]].sup.(2).sub.11]> =
<[[bar.[xi]].sup.(3).sub.11]> = 0.
Dispersions [B.sup.k.sub.11] and microstresses covariances
[B.sup.km.sub.11] arising from the elastic anisotropic grains
interaction are defined analytically with use of a hypothesis of strains
uniformity, and also numerically for polycrystal model by means of
finite elements solution for elastic plane problem for cubic crystals
[6-8].
Calculation by a method of final elements of models taking into
account anisotropy of properties of grains is made with the most various
purposes [9-11]. Here the results received at the decision of a volume
problem for materials with close-packed hexagonal space lattice are
discussed. For implementation of this approach the finite element model
of the polycrystal was used in the form of a thin plate consisting of
one layer of grains in the shape of hexagonal prisms (39 hexahedrons and
fragments forming a rectangular plate). Grain thickness was equal to
diameter of the circle around the hexahedron. Each grain was split into
1193 elements in the form of tetrahedron containing four nodes with
three nodal displacements. Anisotropic element type was used for which
the elastic properties [C.sub.i,j] are given by the 6x6 matrix.
Components of the matrix of elastic properties were determined by
conversion of the fourth-rank elastic tensor for different orientations
of crystallographic axes of the grain given by Eulerian angles. An angle
change pitch was [pi]/24 in the range from 0 to [pi]/2. Total number of
considered orientations of grains was 2197. Orientations of grains of
the model were chosen from the resultant aggregation by means of the
random number generator. For each material 5 different combinations of
orientations of 39 grains were considered. For fastening 7 connections
in four points were given. In one angle point of the base [u.sub.1] =
[u.sub.2] = [u.sub.3] = 0, in the most remote from it base point
[u.sub.2] = [u.sub.3] = 0, in two remaining angle points [u.sub.2] = 0.
Such fastening ensures no constraints due to availability of connections
and during extension along any axis in the elastic isotropic model
monoaxial extension occurs in all elements. During calculation of the
polycrystal model complex stress state occurs in the model volume caused
by interaction of anisotropic grains. For each orientation two kinds of
monoaxial extensions were considered along [x.sub.1] axis and along
[x.sub.2] axis caused by single macroscopic stresses on lateral surfaces
[[bar.[sigma]].sub.1] and [[bar.[sigma]].sub.2]. For every kind of
extension six components of microstresses [bar.[[xi].sup.(1).sub.ij]]
and [bar.[[xi].sup.(2).sub.ij]] and deformations caused by single
macrostresses were determined. For evaluation of representativeness of
samplings normal modules of elasticity and Poisson ratios of the model
in [x.sub.1], [x.sub.2] directions were determined. Spread of values of
elastic constants of the model depends on the rate of elastic anisotropy
of crystals. For majority of materials it was in the range [+ or -] 5%.
The exceptions were cadmium, zinc and graphite (mentioned in ascending
order of degree of anisotropy) for which deviations were two or three
times more.
For averaging and determination of mathematical expectations
<[[bar.[xi]].sup.(k).sub.ij]>, dispersions and covariances,
difference of volumes of finite elements with consideration of FEM grid
topology was taken into account. Here, it was established that
<[[bar.[xi]].sub.ij]> = [[bar.[sigma]].sub.ij] as required by
boundary conditions. Fig. 1 shows microstress fields
[[bar.[xi].sup.(2).sub.22] for titanium polycrystal in the middle plane
of the plate with monoaxial extension [[bar.[sigma]].sub.2].
[FIGURE 1 OMITTED]
Taking into account the quasi-isotropic conditions (4)
determination of average values of dispersions
<[B.sup.ii.sub.ii]>, <[B.sup.kk.sub.ii]> (k [not equal to]
i) was conducted by averaging of ten values of the corresponding
dispersions and for <[B.sup.ik.sub.ii]> (k [not equal to] i) of
twenty values of covariances for each material got by solving five
variants with different orientations of crystallographic axes. It was
not possible to determine covariances of [B.sup.23.sub.11] using only
two kinds of extensions by single macrostresses [[bar.[sigma]].sub.11] =
[[bar.[sigma]].sub.22] = 1 therefore they were found in assumption that
[cov.sub.23] ([[bar.[xi]].sup.(2).sub.11] [[bar.[xi]].sup.(3).sub.11]) =
[cov.sub.12] ([[bar.[xi]].sup.(1).sub.33] [[bar.[xi]].sup.(2).sub.33])
by averaging 10 values [cov.sub.12] ([[bar.[xi]].sup.(1).sub.33]
[[bar.[xi]].sup.(2).sub.33]).
For each material confidence intervals were determined for
mathematical expectation of the corresponding sampling of dispersions
and covariances on the assumption that these values are not correlated
between themselves. For this purpose dispersions of dispersions D
([B.sup.11.sub.11]), D ([B.sup.22.sub.11]) and dispersion of covariances
D ([B.sup.12.sub.11]), D ([B.sup.23.sub.11]) were calculated. Then with
probability of 0.95 the confidence interval b = [square root of
(([Z.sup.2]D ([B.sup.km.sub.11]))/N)] of possible spread of average
values of dispersions and covariances was determined. Here Z is quantile
of normalized normal law of distribution corresponding to the given
probability while N is a number of considered cases (10 for
[B.sup.ii.sub.ii], [B.sup.jj.sub.ii], [B.sup.ij.sub.mm] (i, j = 1,2; m =
3) and 20 for [B.sup.kj.sub.ii].
As it can be seen from the table, the values of P, Q, F parameters
for the studied HCP materials differ greatly. From here it appears the
necessity of more detailed study of effect of these parameters on
strength with complex stress state with application of experimental
data.
As can be seen from Fig. 2, microstresses concentration
[[xi].sub.11] strongly depends on the stress-strain state which is
influenced by the ratios of principal macroscopic stresses
[[sigma].sub.2]/[[sigma].sub.1] and [[sigma].sub.3]/[[sigma].sub.1].
Therefore there are strong reasons for the creation of a strength
criterion considering influence of a type of stress-strain state on
concentration of microstresses. Taking into account Eq. (4), expressing
in formulae (1) [z.sub.c] and [[xi].sub.c] during two test types with
various stress states in the work [6] the statistical criterion of
oriented fracture was got:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [chi] = [[sigma].sub.P]/[[sigma].sub.c], a [[sigma].sub.p],
[[sigma].sub.c] are true fracture stresses at tension and compression, P
= [D [[bar.[xi]].sup.(1).sub.11]]/[D [[bar.[xi]].sup.(2).sub.11]]; Q =
[[cov.sub.12] ([[bar.[xi]].sub.11])]/[[bar.D][[bar.[xi]].sup.(2).sub.11]]; F = [[cov.sub.23] ([[bar.[xi]].sub.11])]/[[bar.D]
([[bar.[xi]].sup.(2).sub.11])]. Taking into account the quasi-isotropic
conditions (4) one can see that F is correlation coefficient of
[[rho].sub.23] ([[bar.[xi]].sup.(2).sub.11]
[[bar.[xi]].sup.(3).sub.11]). It can be shown that Q/[square root of
(F)] is correlation coefficient of [[rho].sub.12]
([[bar.[xi]].sup.(1).sub.11] [[bar.[xi]].sup.(2).sub.11]).
[FIGURE 2 OMITTED]
In the work [6, 7] with the use of analytical and numerical models
of the polycrystal P, Q, F parameter values were got for a large number
of metals with cubic crystal lattice. Taking into account that for these
materials the parameter values got by the finite element method
calculation are close to the values P=16/9, Q=-8/9, F=-7/63 got by the
model of the polycrystal with the use of the deformation homogeneity
hypothesis, for materials with the cubic crystal lattice in the work [7]
two-parameter strength criterion was proposed which describes the
strength of columbium alloys well.
However, during the tests of polycrystals with the hexagonal
close-packed space lattice it was established that these parameter
values greatly depend on the test material. This was established both by
the model with the use of the deformations homogeneity hypothesis that
allows getting the analytical solution [8] and by the method of the
finite elements [13]. This is obviously related to the less symmetry of
HCP crystals compared with the cubic ones. Therefore, if there is no
data on P, Q, F values for materials, one should use the five-parameters
criterion in the form Eq. (5) that makes it inconvenient for
application.
Earlier in the work [2] specified in our article, we have
considered process of change of dispersions of micro stresses and
deformations at elastic-plastic deformation. It has been shown that
level of concentration of the micro stresses, arising from interaction
it is elastic the anisotropic grains, characterised in the variation
factor, in a to iron at an input in plastic deformation decreases a
little, and then is stabilised and remains at the further deformation.
The fact of preservation of stability of a picture of
micronon-uniform deformation at the big plastic deformations is
confirmed in a great number of experimental works (Gur'ev A. V.,
Romanov A. H. and others). In them it is shown an invariance of factors
of concentration of the local deformations measured on bases several
times smaller, than the size of grain, in a wide range of plastic
deformations.
It should be noticed that the statistical criterion includes not
value of dispersions, but relative parameters P, Q, F. By working out of
criterion of strength it was supposed that process of plastic
deformation poorly influences these relative parameters. For this reason
it is proposed in a present work to use for the first approximation
parameters P, Q and F found from the numerical finite elements
calculation of a microtresses field for a polycrystal model for an
elastic problem. Results can be specified with use of experimental data
on a fracture of cylindrical specimens with cuts of various sharpness
which will provide various stress-strain states in the fracture zone.
Such approach is often used for an estimation of fracture toughness
[14,15], influences of speed of deformation [16] and destruction
mechanisms [17] and another.
3. Experimental
Test methods were used that can be easily applied in laboratory
conditions namely, tension of cylindrical samples with various sharp
circumferential notches. For analysis of stress state in the stress
concentration zone the well-known Bridgeman and Davidenkov approaches
were used. Comparative analysis of these approaches required building of
deformation curves in the coordinates [S.sub.1] = f ([e.sub.1]). Here
[S.sub.1] = F/A is averaged true stress determined by cross section area
A corresponding to the load F; [e.sub.i] = 2ln([d.sub.0]/[d.sub.i]) is
true deformation in the stress concentration zone determined by initial
[d.sub.0] and current diameter of minimum cross section. For tests 5V
titanium alloy samples were taken: smooth and with notch radius 2.3;
1.5; 0.85; 0.5 mm with ratio of diameters d/D = 0.707. In addition to
this, the test results with six fold resharpening of the sample that
were conducted after the neck started forming and [DELTA][e.sub.i] = 5%
plastic deformation was reached that allowed getting the characteristics
without considerable influence of the neck shape. The technique,
described to work [18] has been used.
For approximation of deformation curve of the smooth sample with
sixfold resharpening the power law hardening equation [s.sub.i] =
[[sigma].sub.y] + K x [e.sup.n.sub.i] was used, where K = 225.16 MPa, n
= 0,21395--hardening parameters; [[sigma].sub.y] = 665 MPa--limit of
elasticity. Approximation error is 1.5%. For justification of the
calculation method for stress intensity [s.sub.i] by [S.sub.1] the
deformation curves for the samples with necks and notches were rebuilt
in the coordinates of stress intensity [s.sub.i] and intensity of
deformations [s.sub.i] = f ([e.sub.i]). For determination of [s.sub.i]
the Bridgeman and Davidenkov solutions were used. Coefficients of
"hardening" [k.sub.[sigma]] taking into account curvature of
the neck causing inhomogeneous stress state in the minimum cross section
area for the smooth sample do not differ greatly.
However, it was established that for small radius of curvature in
circumferential notches Bridgeman correction ensures better conformity
to the single deformation curve in the coordinates [s.sub.i] -
[e.sub.i]. Therefore it is used in further calculations to determine
stress intensity:
[s.sub.i] = [S.sub.1]/[k.sub.[sigma]]; [k.sub.[sigma]] = (1 + 4R/d)
ln (1 + d/4R), (6)
here d, R are diameter and radius of curvature in the neck or in
the notch. For circumferential notches diameters and radii were
determined both before deformation and after fracture.
For determination of P, Q, F parameters the criterion Eq. (5) is
represented in the form convenient for further calculations:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)
here [s.sup.c.sub.i] is stress intensity corresponding to fracture
in such stress state, [[chi].sub.q] = [s.sup.C.sub.p]/[s.sup.C.sub.s],
[s.sup.C.sub.p], [s.sup.C.sub.s] are true fracture stresses at tension
and compression; [n.sub.1] = [[sigma].sub.1]/[s.sub.i]; [n.sub.2] =
[[sigma].sub.2]/[s.sub.i]; [n.sub.3] = [[sigma].sub.3]/[s.sub.i] are
relative parameters characterizing stress state in the fracture point;
[[sigma].sub.1], [[sigma].sub.2], [[sigma].sub.3] design stresses
according to Bridgeman and Davidenkov; [n.sub.1], [n.sub.2], [n.sub.3]
are expressed via rigidity index in stress state N = ([[sigma].sub.1] +
[[sigma].sub.2] + [[sigma].sub.3)]/[s.sub.i] and Lode parameter
[[mu].sub.[sigma]] characterizing the kind of deviator:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
According to Bridgeman for the central part of the cross section in
the neck where fracture starts:
N = l + 3ln{l + d/{4R)), [[mu].sub.[sigma]] = -1.
Experimental and design values of fracture stress [S.sub.1] were
compared. Satisfactory fit was reached. For more precise definition of
parameters variation of P, Q, F, [[chi].sub.r] was made for minimization
of the sum of squared difference of theoretical and experimental values
of fracture stresses for each type of curvature radii:
[n.summation over (m=1)] Re [([k.sub.[sigma]][s.sup.c.sub.i] (P, Q,
F, [[xi].sub.r]) - [S.sup.exp.sub.1]).sup.2.sub.m] = min, (8)
where stress intensity [s.sup.c.sub.i] (P,Q,F,[[chi].sub.r]) is
calculated by the criterion Eq. (7) in which N and [n.sub.1], [n.sub.2]
respectively are determined by the value of radius in the notch measured
after fracture, n--number of sample types; Re()--means that only real
part of the complex number is taken which can be theoretical value of
stresses and deformations in the process of iterations when no imposed
below given restrictions are provided resulted from the study of
deviator and meridian fracture surface cross section. Taking into
account that the second derivative sign
[d.sup.2]F/d[[sigma].sup.2.sub.0] in the meridian cross section will not
change in the whole interval of spherical tensor change [-[infinity] ...
[[sigma].sup.c.sub.0]], where [[sigma].sup.c.sub.0] is strength with
triaxial uniform tension, general solution was got for the restriction
in [[chi].sub.q]. The most strict is the restriction for
[[mu].sub.[sigma]] = -1,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)
that was used in calculations. From the condition of reality of
values [d.sup.2]/F[[sigma].sup.2.sub.0] the restriction F > - [P -
2[Q.sup.2]]/P follows.
The other restriction is the expression P + 4Q + 2F + 2 [greater
than or equal to] 0 that follows from the condition of non-negativity of
dispersions of microstresses in any stress state.
After minimization of the expression (8) with the use of the above
mentioned restrictions it was got: [[chi].sub.r] = 0.45, P=0.59, Q= -
0.49, F= -0.17. The values of P and Q parameters are inside of the
confidence interval got by calculation of FEM, only F value is beyond
the limits got by finite element modeling that is connected with
imperfectness of the polycrystal model (one layer of grains). So there
are grounds to convert the criterion Eq. (5) in three-parameter one if
the calculated P and Q parameters are used.
Accuracy of description of fracture was compared with other
strength criteria that are particular cases of the criterion Eq. (5).
With the values of the parameters P = 1, Q = F = - 0.5 the criterion Eq.
(5) corresponds to the Pisarenko-Lebedev criterion. With P = 1, Q = F =
-v, where Poisson ratio v equal for titanium to 0.32, the criterion Eq.
(5) corresponds to the Volkov criterion. With these fixed values of P,
Q, F parameters minimization of the expression (8) was performed to find
[chi] parameter for Pisarenko-Lebedev and Volkov criteria. The criteria
[chi] = 0.63 and 1.72 were got, respectively. According to the Volkov
criterion tensile strength is considerably higher than compression
strength that is not proved by the experiment. Fig. 3 shows meridian
cross sections corresponding to [[mu].sub.[sigma]] = -1 for the
Pisarenko-Lebedev criterion (straight line 1), for the Volkov criterion
with [chi] =1.72 (curve 2) and for [chi]=0.42 (curve 3) corresponding to
the maximum value of restriction Eq. (9). The criterion of oriented
fracture Eq. (5) is represented by curve 4 (Fig. 3).
The range of stress states ensured by the samples with
circumferential notches from N = 1 (smooth sample with resharpenings) to
N = 4.3 is marked with arrows. It is seen that within it as well as for
compression spherical tensors the criterion of oriented fracture and the
Pisarenko-Lebedev criterion give approximately the same results.
However, in the area of the most dangerous stress states close to
uniform tension the Pisarenko-Lebedev criterion enhances strength two
point five times more compared with the criterion Eq. (5). So, for this
area of stress states there are grounds to use alternative
three-parameter criterion that allows higher reliability of
calculations.
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
Fig. 4 shows fracture contours in the plane stress state for the
criteria considered. As is obvious, strength differences in the area of
biaxial tension are not great. Maximum difference observed with
[[sigma].sub.1] = [[sigma].sub.2] is less 7%. However, tests of samples
with circumferential notches when the conditions of volumetric stress
state are implemented, give more 30% differences in strength evaluation
by the Volkov criterion and criterion Eq. (5). This points out that for
determination of F parameters one should use tests not in the plane but
in the volumetric stress state that was implemented in the procedure
proposed.
4. Conclusions
For polycrystals with hexagonal crystal lattice the statistical
strength criterion was proposed that is more reliable in forecasting of
strength in any stress state. The procedure of determination of
criterion parameters was developed when two of these parameters are
determined by FEM calculation on the polycrystal model while three other
parameters were determined experimentally on the samples with
circumferential notches.
Acknowledgments
The reported study was supported by RFBR, research project
N[degrees] 14-08-00837, and within a basic part of state task of
Ministry of Education and Science of the Russian Federation, research
project N[degrees] 2014/16.
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Accepted August 20, 2013
Received May 08, 2014
V. P. Bagmutov *, E. P. Bogdanov **, I. A. Shkoda ***
* Volgograd State Technical University, pr. Lenina 28, 400005,
Volgograd, Russia, E-mail: sopromat@vstu.ru
** Volgograd State Agrarian University, pr. University 26, 400002,
Volgograd, Russia, E-mail: bogdanov_ep@list.ru
*** Kamyshin Technological Institute (Branch of VolgGTU), st.
Lenina 6a,403874, Kamyshin, Russia,
E-mail: diststyle@mail.ru
cross ref http://dx.doi.org/10.5755/j01.mech.20.3.4946
Table
Results of calculation of criterion parameters
and confidence intervals b, ordered by ratio of
extreme modulus of crystal elasticity
Parameters Be Co Mg Ti
[E.sub.max]/[E.sub.min] 1.16 1.16 1.17 1.37
P 2.49 1.12 0.92 0.57
[b.sub.p] 0.266 0.219 0.099 0.096
Q -1.03 -0.38 -0.282 -0.48
[b.sub.q] 0.095 0.113 0.065 0.105
F -0.12 -0.024 0.11 0.095
[b.sub.F] 0.028 0.077 0.087 0.065
Parameters Zr Cd Zn Graphite
[E.sub.max]/[E.sub.min] 1.4 2.7 3.5 6.98
P 1.55 3.448 2.70 1.89
[b.sub.p] 0.210 0.449 0.255 0.882
Q -0.386 -0.83 -0.69 -0.43
[b.sub.q] 0.089 0.125 0.101 0.191
F 0.202 0.175 0.033 -0.28
[b.sub.F] 0.039 0.029 0.054 0.12
