CC BY
38
УДК 539.42, 539.421.2, 539.421
Modeling of thermomechanical fracture of functionally graded materials with respect to multiple cracks interaction
V.E. Petrova1,2 and S. Schmauder1
1 IMWF, University of Stuttgart, Stuttgart, D-70569, Germany 2 Voronezh State University, Voronezh, 394006, Russia
Different aspects of thermomechanical fracture of functionally graded materials (FGMs) are considered. Among them are the crack interaction problems in a functionally graded coating on a homogeneous substrate (FGM/H). The interaction between systems of edge cracks is investigated, as well as, how this mutual interaction influences the fracture process and the formation of crack patterns. The problem is formulated with respect to singular integral equations which are referred to the boundary equation methods. The FGM properties are modeled by exponential functions. The main fracture characteristics are calculated, namely, the stress intensity factors, the angles of deviation of the cracks from their initial propagation direction and the critical stresses when the crack starts to propagate. The last two characteristics are calculated using an appropriate fracture criterion. The problem contains different parameters, such as the geometry (location and orientation of cracks, their lengths, and the width of the FGM layer) and material parameters, i.e. the inhomogeneity parameters of elastic and thermal coefficients of the functionally graded material. The influence of these parameters on the thermomechanical fracture of FGM/H is investigated. As examples the following real material combinations are discussed: TiC/SiC, Al2O3/ MoSi2, MoSi2/SiC, ZrO2/nickel and ZrO2/steel.
Keywords: edge and internal cracking, thermal fracture, functionally graded coating, stress intensity factors, fracture criteria, fracture angle
Моделирование термомеханического разрушения функциональных градиентных материалов при учете взаимодействия систем трещин
В.Е. Петрова1,2, S. Schmauder1
1 Штутгартский университет, Штутгарт, 70569, Германия 2 Воронежский государственный университет, Воронеж, 394006, Россия
В работе рассматриваются различные аспекты термомеханического разрушения функциональных градиентных материалов, в частности задачи о взаимодействии трещин в функциональном градиентном слое на однородном основании. Исследуются взаимодействие между краевыми трещинами, а также влияние этого взаимодействия на дальнейший процесс разрушения и формирование систем трещин. Сформулирована система сингулярных интегральных уравнений в рамках метода граничных элементов. Свойства функциональных градиентных материалов описываются экспоненциальной функцией. Вычисляются основные характеристики разрушения, такие как коэффициенты интенсивности напряжений, углы отклонения трещин от их начального распространения, критические напряжения начала распространения трещины. Для вычисления последних двух характеристик использовался критерий разрушения. Задача учитывает различные параметры: геометрические (координаты центров трещин и углы их ориентации, длины трещин, толщину функционального градиентного слоя) и характеристики материала (параметры неоднородности упругих и термических коэффициентов). Исследовано влияние этих параметров на характеристики разрушения функционального градиентного слоя на однородном основании. В качестве примера рассматриваются следующие комбинации: TiC/SiC, Al2O3/MoSi2, MoSi2/SiC, ZrO2/никель и ZrO2/сталь.
Ключевые слова: краевые и внутренние трещины, термическое разрушение, функциональное градиентное покрытие, коэффициенты интенсивности напряжений, критерий разрушения, угол разрушения
1. Introduction
Functionally graded coatings (FGCs) are applied in thermal, wear and corrosion barriers which are used in different fields, such as, nuclear energy (e.g., nuclear reactor
components), aerospace (e.g., rocket engine components, space plane body), engineering (e.g., turbine blade, engine components), energy conversion (e.g., thermoelectric generator, fuel cell) as well as many other applications [1-3].
© PetrovaV., Schmauder S., 2017
They are subjected to different thermal and mechanical loading and have to resist high temperature, wear and aggressive environments. However, cracks can initiate from initial defects or microcracks may appear during manufacturing or service. Therefore, the study of fracture of FGC structures is important for a better understanding of the fracture resistance of graded coatings.
Thermal fracture of FGCs is significantly affected by a complex crack interaction mechanism, e.g., interacting cracks can enhance or suppress the propagation of each other. The crack patterns strongly depend on the microstructure of the materials and type of loading. Numerous experimental results (e.g., [4-6]) showed that when FGCs are subjected to thermal shock, multiple cracks often occur at the ceramic surface. The studies from laser thermal shock tests [7, 8] indicated that multiple surface precracks with a large density and short lengths reduce the crack driving force at the interface. At the same time very short precracks were most vulnerable to further surface cracking which in turn induced the interface crack. In [8] a functionally graded plate with an array of periodically spaced surface cracks of alternating lengths subjected to thermal shocks was considered. The crack morphology was described by the ratio of the length of the short cracks to that of the long cracks (initial crack length ratio) as well as by the crack spacing. The thermal properties of the FGM plate were arbitrarily graded in the plate thickness direction, but the elastic properties were assumed to be constant.
In the present work the problem of thermomechanical fracture of FGCs on a homogeneous substrate (semi-infinite isotropic medium) is investigated in the frame of linear elastic fracture mechanics. It is supposed that the materials are brittle or quasi-brittle.
In our previous works [9-13] the thermal fracture in the vicinity of an interface in a bimaterial compound which consists of an FGM and a homogeneous material (H) was studied (an infinite FGM/H compound). The bimaterial contains an interface crack and internal arbitrary located cracks in the FGM and is subjected to a heat flux [9, 10] and/or a tensile load [11, 12] or a shear load [13]. For a special case when an interface crack is significantly larger in size than internal cracks in the FGM, the asymptotic analytical solution of the problem was obtained as a power series of the small parameter (the small parameter is equal
to the ratio of the size of small internal cracks to the interface crack size). Approximate analytical formulas for the stress intensity factors (SIFs) at the interface crack tips were obtained. In [14] some results for edge cracks in FGM/ homogeneous structures (a semi-infinite medium) were obtained in the frame of the approach used in [15, 16].
2. Model
2.1. Geometry of the problem
Structures consisting of FGCs on top of homogeneous substrates is considered with the presence of pre-existing systems of cracks in FGCs. The crack morphology in FGC systems depends on their composition profile, methods of producing and exploitation (loading) conditions (e.g., maximum-minimum temperature in the cycle of heating-cooling, number of cycles, additional mechanical loads). Different scenarios of propagation of cracks could be examined for typical crack patterns resulting from experiments available in literature [4, 6, 17].
At present the investigations devoted to the crack interactions in FGMs are restricted only to special cases of crack locations and the interaction of arbitrary located cracks has been not well examined, e.g. see [15]. In our previous works [9-13] an approximate approach for systems of arbitrarily located cracks in infinite FGM/H bimaterials was suggested and will be used partially in the present study of fracture of FGCs on a homogeneous substrate.
A functionally graded material on a homogeneous substrate is modeled by the geometry depicted in Fig. 1. The substrate is a homogeneous material 1 with subscribed properties and the upper material (on the top of the surface) is a pure material 2, at the interface the material has the same properties as the substrate, i.e. material 1. Through the thickness of the layer the properties are changed continuously from one material to the other.
The layer contains a system of cracks and we consider some particular cases of the interaction of edge cracks arbitrarily inclined to the surface.
The positions of cracks are determined exactly by the midpoint coordinates z° = x° + iy0 (i2 = -1) and the inclination angles Pn to the surface. The cracks have sizes 2an. The thickness of the FGM layer is h while the substrate is considered as a semi-infinite medium.
Fig. 1. FGC/H with system of cracks (a); edge cracks inclined arbitrarily with an angle P to the surface of the FGC: a — a half-length
ooo n n
of n-th crack, z° = x° + iyn — the crack midpoint coordinate, h — the width of the FGC (b)
The coordinate systems are chosen as follows: the global coordinates (x, y) with the x-axis on the surface and the local coordinates (xn, yn) with centers in the midpoint of the n-th crack (Fig. 1, b).
The relation between global coordinates (x, y) and local coordinates (xn, yn) systems is written as
z = z0 + zne"^, (1)
where z0 is the origin coordinate of the local system in the global system and at the same time it is the midpoint coordinate of the «-th crack.
2.2. Modeling of functionally graded materials and examples of FGMs
It is assumed that thermomechanical properties of an FGC are continuous functions of the thickness coordinate y. As in previous works [9-14] the exponential form ofthis properties is used. Thus, the thermal expansion coefficient of the FGM layer is written as
at (y) = ati exp (e(y + h)), -h < y < 0, (2)
and the Young's modulus is given as
E(y) = Ei exp (ro(y + h)), -h < y < 0, (3)
with nonhomogeneity parameters e and ro. They are also called graded parameters.
The magnitudes of the dimensionless graded parameters eh and roh (h is the thickness of the FGM layer) are obtained from Eqs. (2) and (3) as
eh = ln(at2/ at1), roh = ln( E2/ E1), (4)
at2 = at(y)y=0. ati = at(y)y=-h.
E2 = E(y)y=0, Ei = E(y)|y=-h. (5)
In the local coordinate system (xn, yn) connected with each crack the material parameters a t and E (Eqs. (2) and (3)) take the form
a t( Xn, yn) = atiee(h+y0) eeiX»+e2y», ei = e sin(-Pn), e2 = e cos Pn, E(Xn, yn) = E^0) e*™», ro1 = rosin (-Pn ), ro2 = ro cos Pn and on the crack lines, where yn = 0, we will have
at = at! exp (e(h + y00 - xn sin p„)), (6)
E = Ei exp (ro(h + y° - Xn sin Pn)). (7)
To derive these expressions Eq. (1) was used.
Functionally graded materials, as applied in thermal barrier coatings to protect substrates from high tempera-
tures, should have a low thermal conductivity. At the same time they are desired to have a thermal expansion coefficient close to that of the material for the protected substrate. Some examples of real material combinations, which can be used in the present model, are given in Tables 1 and 2 [18]. In these tables the thermal expansion coefficients and Young's moduli of some ceramic/ceramic and ceramic/ metal FGCs are included as well as the corresponding non-dimensional inhomogeneity coefficients of the FGMs. The thermal conductivity coefficients are also listed (for information about the better combination of FGMs), but the influence of this value on the fracture characteristics is not investigated in the present study.
This exponential model describes the smooth variation of material properties of the FGM in the y-axis direction. For example, if ati is decreased with increasing y-coordi-nate (from y = -h to y = 0), then the inhomogeneity parameter e is negative, and this case can correspond to a ceramic/metal FGM layer on a metal substrate with gradual transition from a metal at y = -h to a ceramic at the upper part of the FGM layer (ac < am).
2.3. Thermal and mechanical loadings
The FGM/H structure is subjected to tensile loadp applied at infinity parallel to the free surface and is cooled by AT from the sintering temperature (Fig. 1, b). In the case of cooling of the FGM/H tensile residual stresses are observed as shown in experimental investigations [6].
If an FGM/H structure is cooled, then residual stresses are arising due to mismatch in the coefficients of thermal expansion [15, 16]. The FGMs inhomogeneity will be accounted via the continuously varying residual stresses, and these stresses are the following [15]:
(y) = [a t( y) -atJATE (y),
°XX(y) =[E(y)lEi - i]a°X> = P, ati and Ei are, respectively, the thermal expansion coefficient and the Young's modulus of a homogeneous material and at the interface, i.e. in the region y < -h; at(y) and E(y) are defined by formulas (2) and (3).
The method of linear superposition is used in the solution of this problem, so that the loads at infinity are reduced to the corresponding loads on the crack faces. Thus, the tensile load is reduced to the load pn on the crack surfaces and written as
Table 1
Material properties of ceramic/metal FGM/H (ZrO2/Ti-6Al-4V)/Ti-6Al-4V
Thermal expansion coefficient (CTE), 10-6 K-1 Inhomogeneity coefficient CTE Young modulus E, GPa Inhomogeneity coefficient of E Thermal conductivity, W-m-1-K-1
ZrO2 a t2 = 10.0 atilati >1 E2 = 200 E2/Ei > i k2 = 2.0
Ti-6Al-4V a t1 = 8.6 ah = 0.15 > 0 Ei = 114 roh = 0.56 > 0 ki = 6.7
(Ti-6Al-4V/ZrO2)/ZrO2 eh = -0.15 < 0 roh = -0.56 < 0
Table 2
Material properties of some FGMs. The Young's moduli of these materials are similar
Material Thermal expansion coefficient, 10-6 K 1 Inhomogeneity coefficient Thermal conductivity, W • m 1 • K
FGM/H (Al2O3/MoSi2)/MoSi2 (ceramic/ceramic)
AA at2 = 5 a tila ti =1 ki = 25
MoSi2 a ti = 5 eh = 0 k = 52
FGM/H (MoSi2/Al2O3)/Al2O3 eh = 0
FGM/H (MoSi2/SiC)/SiC (ceramic/ceramic)
MoSi2 at2 = 5 atilati >1 ki = 52
SiC a t1 =4 eh = 0.22 > 0 k = 60
FGM/H (SiC/MoSi2)/MoSi2 eh = -0.22 < 0
FGM/H (TiC/SiC)/SiC (ceramic/ceramic)
TiC ati = 7 atilati >1 ki = 20
SiC a ti = 4 eh = 0.56 > 0 k1 = 60
FGM/H (SiC/TiC)/TiC eh = -0.56 < 0
FGM/H (ZrO2/Ni)/Ni (ceramic/metal)
ZrO2 a ti = 10 atilati <1 ki = 2
Ni at1 = 18 eh = -0.6 < 0 k = 90
FGM/H (Ni/ZrO2)/ZrO2
eh = 0.6 > 0
FGM/H (ZrO2/steel)/steel (ceramic/metal)
ZrO2 a ti = 10 atilati < 1 ki = 2
Steel at1 = 12 eh = -0.18 < 0 k = 20
FGM/H (steel/ZrO2)/ZrO2 eh = 0.18 > 0
Pn = °n - iTn = P(1 - exp (2ip„))/2, (9)
« = 1,2, ..., N.
In the common case of FGMs, the full load on the n-th crack consists of pn, aTn and aen, where the index n denotes that the functions are written in the local coordinate systems (xn, yn) connected with cracks. If the materials are elastically homogeneous, then Ej = E(y) and consequently aexx = 0 in Eq. (8), see Table 2 for examples of such materials for which the graded parameter of the Young's modulus is ro = 0.
2.4. Main equations
The problem is solved by using the method of singular integral equations. For arbitrary located cracks in a halfplane the system of singular integral equations is written as
[19, 2°], a
ig-^+£ i [ gk (t) Rn, (t, x)+
- an k =1 - a,
n k ^n
+ gk (t)Snk (t, x)]dt = nPn (x),
|x|<an, n = 1,2, ...,N, (10)
and for internal cracks the following condition is fulfilled
J g'n (t)dt = 0.
(11)
It is the condition of displacement continuity at crack tips. An overbar (...) denote the complex conjugate. N is the number of cracks.
The unknown functions in this formulation are the derivatives of displacement jumps on the crack lines
g'n (x) =7
i| d
(K] +i[ vn]).
i(K +1) dx
Here [un ] and [vn ] are shear and vertical displacement jumps, respectively, on the n-th crack line, ^ = E/ 2(1 + v) is the shear modulus, E is Young's modulus, v is Poisson's ratio, k = 3 - 4v for the plane strain state, and k = (3 - v)x x(1 + v)-1 for the plane stress state. The regular kernels contain geometry of the problem and they can be found in [14, 19, 20].
The functions pn in the right side of Eq. (10) are known loadings and they consist of the loads determined by Eqs. (8) and (9).
The stress intensity factors are found as [19, 20]
k±i - ikm = + lim V(an - xn2)/Oigi (x,),
—a
where the upper part of the "±" or " + " signs refers to the right tip and the lower to the left tip of the n-th crack.
2.5. Numerical solution and determination of stress intensity factors
The solution of singular integral equations (Eq. (2)) is obtained by a numerical method which is based on Gauss-Chebyshev quadrature, the description of this method can be found in [19-22].
The equations (10), (11) are rewritten in dimensionless form with the nondimensionless coordinates C = ¿/ak and n = x/an, where 2ak is the length of the k-th crack. The unknown function gn (n) consists of a function un (n) (a bounded continuous function in the segment [-1, 1]) and the weight function l/sjl-n2 , that is
gn (n) = Un (n)A/W. (12)
For edge cracks the function gn (n) possess a singularity less than l/^l + n at the edge point n = -1 and this condition is accounted as [15, 16]
Un (-l) = 0. (13)
In spite of that the exact singularity at the edge points is not taking into account, the numerical results have shown good accuracy [15, 16] when the SIFs at the internal crack tip are calculated. If the stress-strain state in the vicinity of the edge crack tip is examined, then the exact order of singularity at this tip should be taking into account.
Using Gauss's quadrature formulae for the regular and singular integrals the integral equations are reduced to the following system of NxM (N is the number of cracks, M is the number of nodes) algebraic equations
l M N
— XX [uk (Cm )Rnk (Cm > nr ) +
M m=l k=l
+ Uk (Ém )Snk (Ém > )] = nPn far )>
m 2m -1
X (-1) mUn (Ém )tg-m-- n=0,
m=1 4M
(14)
(15)
. N r = 1, 2,
, M - 1,
n = 1, 2,.....
with
t 2m -1 1 -
cm = cos-n, m = 1, 2, ..., M,
m 2M
n = cos—, r = 1, 2, ..., M - 1, r M
M is the total number of discrete points of the unknown functions un (n) within the interval (-1, 1). Applying the conjugate operation to the system (14) and (15) additional NxM equations are obtained, i.e. (2 N)xM equations should be solved, where N is the number of cracks.
If internal cracks are considered then instead of Eq. (15) the following equation should be used
M
X un (cm ) =
m=l
After solution of the algebraic system (14) and (15) the functions un (n) are calculated by the interpolation formula:
- m m-i
Un (n) =—X Un ( Ém ) X Tr ( Ém )T (n) "
M m=1 r=0
1M
- TT X un (Ém )-
M m=0
(16)
The functions Tr are Chebyshev polynomials of the first kind.
The stress intensity factors are obtained from the following formulas [15, 16]
k±i - ik±n
= + lim 1 -n2 gn (n),
K+ -iK+n = -^ânUn( +1) =
i— 1 m 2m -1
= Pn^n — X (-1)mUn (Ém^
M m=1 4 M
K\n - iKHn =4a~nUn (-1) =
2m -1 r 4M
"Pnja; ± X (-1)M+m
M m=1
Un (Ém )tg
(17)
(18)
n = 1, 2, ..., N. The functions (16) are used in this calculation.
3. Fracture criteria, fracture angles, critical loads
For general crack problems the stress intensity factors are both nonzero, i.e. mixed-mode conditions hold in the vicinity of cracks. For this mixed-mode case the cracks deviate from their initial propagation direction. For the prediction of the crack growth and direction of this growth a fracture criterion should be applied. Using the maximum circumferential stress criterion, see [23], the direction of the initial crack propagation (Fig. 2) is evaluated as
(Ki-VK2 + 8K2 )/(4Kii)
and the critical stresses can be calculated from the expression
cos3(</ 2) (Ki - 3Kntg(</ 2) ) = KicA/n. (20)
Here KIc is the fracture toughness of the material. The critical stresses are given as
Pcr = Pcr/P0 = PcrAKic/V2^ ) = = 1/[cos3(</2)( -3kntg(</2))]. (21)
< = 2arctg
(19)
Here kI II are nondimensional SIFs
(1) d/a (2)
(3)
r-r-1
1 il 1 I ■ i ! i ■ ! i ■ : i ■ ■ ■ i ! i i ! ■ 1 I I 1 \ ■ ! P n J 1 / ■ 1 1 1 1
• V ■ ■ ■ ■ 1 I 1
4. i>
Fig. 2. Schematic representation of three edge cracks with fracture angles ^
Fig. 3. Fracture angles for two (a) and three (b) equal cracks with different distances between them, P = 90°
km = WK0, K0 = p4la, (22)
and p0 = KIc/sllna is the critical load for a single crack in a material with the fracture toughness KIc.
For the system of cracks the fracture starts from the crack tip where pr is minimal, i.e. min[p.r(k^p0].
4. Results and parametric analysis
The method described in Sects. 2 and 3 can be applied to different systems of cracks. The verification of the method has been done in [14], where the results for some special cases were compared with the results for SIFs for a single inclined edge crack and with SIFs for periodic edge cracks. In [24] some results with respect to edge crack interaction in a homogeneous medium were presented.
In order to investigate the mutual interaction of cracks consider systems of two and three arbitrary inclined edge cracks. The geometry of cracks is considered with respect to experimental data available in the literature, e.g. the experimental results in Refs. [4-6] showed that, when FGCs are subjected to thermal shock, multiple cracks occur at the ceramic surface during cooling-heating processes. The following nondimensional parameters are used in
the calculations: ^a = 4, d/a = 2, 4, 6, a = max ak, a =
k
= 1 mm, ea = -1 and ma = 0. The nondimensional parameter ea = -1 corresponds to smaller values of the thermal expansion coefficient in the upper part of the FGM layer. It should
be mentioned that the results for fracture angles ^ for the cracks in a material with ea = 0 (homogeneous material) and with ea = -1 are nearly the same. The designations for the distances d/a, the fracture angles ^ and crack numbering can be seen in Fig. 2. The loading is described in Sect. 2.
Figures 3-6 schematically represent the crack patterns due to the interaction of the cracks. Figure 3 shows the influence of the interaction of two and three equal edge cracks with P = 90° on fracture angles ^ of the direction of crack propagation for different distances between cracks. For a single edge crack with P = 90° the fracture angle is equal to ^ = 0°. In the case of two cracks (Fig. 3, a), both cracks change the direction of their propagation. The direction of the angles ^ and is opposite, i.e. the cracks repulse each other. Besides, the less the distance d/a, the larger the fracture angle ^ (as it could be expected). For three edge cracks (Fig. 3, b) the behavior of the outer cracks is similar to the behavior of the two interacting cracks (Fig. 3, a). The middle crack is in the neutral position ^ =0, i.e. the crack will not change the direction of the propagation.
Figure 4 shows the nonsymmetrical cases of the interaction of two and three unequal edge cracks. The crack 1 slightly deviates away from the small crack 2 if the distance is d/a = 2 and will propagate straight for far distances between the cracks (Fig. 4, a). At the same time the
Fig. 5. Fracture angles for two (a) and three (b) equal cracks with different distances between them, P = 60°
influence of the large crack on the propagation direction of the small one is rather strong (Fig. 4, a, left). The small crack will propagate away and a repulsion of these two cracks is observed.
In the case of three non-equal cracks the propagation of the large crack is similar to the case of two cracks (Fig. 4, b). The small cracks have fracture angles with the same sign, and both small cracks deviate away from the large crack. In contrast to the case of the three equal cracks interaction, where the middle crack was in a neutral position (Fig. 3, b), in this case the behavior of the middle crack is similar to the behavior of the outer small crack. The influence of the large crack on the propagation direction of the small middle crack is rather strong and stronger than on the outer small crack (Fig. 4, b).
Figures 5, 6 show the results for systems of edge cracks with inclination angles P = 60°. For a single edge crack with P = 60° the fracture angle is equal to = 31°. The difference in the fracture angles due to their interaction is dependent on the distance between the cracks and the size of these cracks. The influence of this interaction on the fracture angles is more complicated than for the previous case for cracks with P = 90°. For two equal cracks the crack 1 has the fracture angle > in contrast to the crack 2 with < (Fig. 5, a). The difference between and becomes smaller with larger distance between
the cracks. For three cracks we have the following:
> > and < The direction of the crack propagation for all cracks in Figs. 5, 6 is the same, the fracture angles have the same sign.
Table 3 presents the results for fracture angles for two edge cracks with slightly perturbed inclination angle P to the surface, i.e. for P = 85° and 95°. Small changes in the inclination angle P cause strong influences on the fracture angle
The perturbation effect is known for the crack interaction problem, for example, when two collinear mode I cracks are growing towards each other, they do not merge tip to tip, but instead repeal each other [25]. In [25] the stability of crack paths to small perturbations was investigated experimentally for arrays of cracks in heterogeneous plates under tension. The main result of their investigation was the analysis of the geometrical conditions for which cracks are attracted towards another and when they are repelled.
5. Conclusions
A semi-analytical model for fracture analysis of FGCs on a homogeneous substrate subjected to thermomechanical loadings is described. A system of pre-existing cracks in the FGC is studied in detail. This typical crack system is observed in experiments and the details are available in the
Fig. 6. Fracture angles for two (a) and three (b) cracks with different distances between them, a2 = 0.5a1 (a), a2 = a3 = 0.5a1 (b); P = 60°
Table 3
The perturbation effect for two edge cracks
ß 85° 95° 90°
da 2 4 6 2 4 6 2 4 6
1a и 2 -25° -18° -12° -10° -4° 0° -19° -12° -8°
ф2 (ai = a2) 10° 4° 0° 25° 18° 12° 19° 12° 8°
Ф1 (a2 = 0.5ai) -10° -10° -10° 5° 5° 5° -2° -2° -2°
Ф2 (a2 = 0.5ai) 25° 5° 0° 35° 17° 10° 31° 12° 5°
literature [4-6]. The influence of the geometry of the problem, i.e. the crack sizes, the inclination angles and distances between the cracks on the fracture angles of the cracks was studied. It was shown that the directions of crack propagation depend mainly on the geometry of the problem, while the influence of inhomogeneity parameters of the thermal expansion coefficients of FGM is negligible.
The support of the German Research Foundation under the grant Schm 746/139-2 is greatly acknowledged.
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Поступила в редакцию 13.02.2017 г.
Сведения об авторах
Vera E. Petrova, Dr. Sci., Leading Researcher, University of Stuttgart, Germany, Prof., Voronezh State University, Russia, [email protected], [email protected]
Siegfried Schmauder, Prof., University of Stuttgart, Germany, [email protected]
