CC BY
21
УДК 339.3, 539.422.22
Strain localization in titanium with a modified surface layer
R.R. Balokhonov, V.A. Romanova, A.V. Panin, M.S. Kazachenok, and S.A. Martynov
Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634055, Russia
Plastic flow localization in commercially pure titanium (VT1-0 according to the Russian classification) with the surface modified by low-energy high current electron beams has been numerically studied. The structure and mechanical properties of the modified surface layer and titanium substrate correspond to the experimentally observed ones and are taken into account explicitly as initial data of a dynamic boundary value problem. The tension of titanium structures with a modified surface layer is simulated by the finite difference method in a plane strain formulation. The dependence of the plastic strain localization parameters on the mechanical properties of structural elements in the titanium substrate has been determined.
Keywords: mechanics of media with structure, numerical simulation, modified surface layers, polycrystalline structure, plastic strain localization
Закономерности локализации деформации в титане с модифицированным поверхностным слоем
P.P. Балохонов, В.А. Романова, А.В. Панин, М.С. Казаченок, С.А. Мартынов
Институт физики прочности и материаловедения СО РАН, Томск, 634055, Россия
Численно исследованы процессы локализации пластического течения в технически чистом титане ВТ1-0, поверхностно модифицированном низкоэнергетическими сильноточными электронными пучками. Структура и механические свойства модифицированного поверхностного слоя и титановой подложки соответствуют экспериментально наблюдаемым и учитываются в расчетах явно в качестве начальных данных краевой динамической задачи. Растяжение структур титана с модифицированным поверхностным слоем моделируется методом конечных разностей в постановке плоской деформации. Установлена зависимость характеристик локализации пластической деформации от механических свойств элементов структуры титановой подложки.
Ключевые слова: механика сред со структурой, численное моделирование, модифицированные поверхностные слои, поликристаллическая структура, локализация пластической деформации
1. Introduction
Titanium is widely used in medicine owing to a combination of various factors. First of all, titanium naturally forms a highly biocompatible dioxide film on its surface both in air and in blood, which promotes a rapid ingrowth of living tissues. Other important advantages of titanium are the corrosion resistance in different atmospheres and corrosive liquids, the ability to influence redox reactions on internal surfaces of tissues, enhanced capabilities for magnetic resonance and computed tomography in comparison with stainless steel and chromium-cobalt alloys, low specific gravity, and no related allergic reactions [1]. These properties of titanium allow it to be successfully used for external prosthetics, bone and joint replacement, dental
implantation, as endosseous maxillofacial and craniofacial implants, as well as in the manufacture of cardiovascular devices and surgical instruments.
Pure coarse-grained titanium has a relatively low strength, which is about 200 MPa. Although the bone strength is lower and can reach 60-80 MPa, the safety margin is always included in engineering calculations. For example, the safety factor for brittle materials in static conditions varies in the range 2-5, for the cables of passenger elevators it is 6, and under dynamic and variable loads in cases when human safety is involved it can reach 15 [2]. There are various well-known ways to improve the mechanical properties of pure titanium. First of all, this is doping [1, 3, 4]. Alpha-beta and beta titanium alloys doped with aluminum,
© Balokhonov R.R., Romanova V.A., Panin A.V., Kazachenok M.S., Martynov S.A., 2017
vanadium, molybdenum, niobium, and tantalum have the strength up to 1000 MPa and higher. The main drawback of doping is the reduction of biocompatibility. Another way to increase the strength of pure titanium is its bulk nano-structuring by various methods [5, 6].
One of the preferred and relatively inexpensive ways to improve the mechanical properties of titanium is surface modification [7-10]. It requires no doping, on the one hand, and no structural transformations of the total material volume, on the other.
Physical mesomechanics considers the surface as an independent planar subsystem that plays a decisive role in the initiation and development of plastic shear [10-12]. The modification of even a thin surface layer whose volume fraction is negligibly small in comparison with the specimen volume prevents the generation of defects in the shear-unstable surface layer and can increase the tensile strength of the material by tens of percent. A negative factor in this method can be the presence of curvilinear interfaces within the modified layer, as well as between the layer and titanium substrate. Stresses concentrate near the interface, leading to plastic flow localization and to a decrease in the strength of the surface-hardened specimen [13-16]. A positive factor in this case is the surface roughness that contributes to the increase of adhesion and biocompatibility. In addition to the structural heterogeneity associated with the presence of mesoscopic interfaces, strain localization may be due to implant geometry [17], nonuniform external loading, or natural instability of the deformation process. For example, a complex stress state in indented specimens without structure, including specimens with a rectilinear "coating -substrate" interface, is caused by the curvilinear shape of the indenter [18], even if it moves uniformly and mono-tonically at a constant rate. The instability of deformation in modeling homogeneous specimens can arise from the form of the plastic potential [19], critical values of the strain hardening coefficient and softening rate [20-22], and dynamic loading conditions in case of using a viscoplastic model of the medium [23].
Today, there is a large amount of literature on the modeling of materials with coatings and modified surface layers. Most of the studies investigate materials without structure and with a rectilinear "coating - substrate" interface. They consider either the indentation process [24, 25] or require the introduction of various defects, e.g., a given set of cracks, as initial data to induce localization under uniaxial deformation [26, 27]. Much less studies take into account the structure of the coated material. Among them are recent works that consider the curvilinear shape of the "coating - substrate" interface [28] and the three-dimensional polycrystalline structure of both the modified surface layer and the substrate [29].
One of the advanced surface modification technologies is electron beam treatment that allows the formation of a
submicrocrystalline hardened layer on the surface of pure titanium. The aim of this paper is to determine and validate numerically and experimentally the strain localization features in titanium with a modified surface layer associated with the presence of curvilinear interfaces.
2. Problem statement and investigation procedure
The dynamic boundary value problem of the mechanical loading of surface-hardened titanium specimens is solved in a plane strain formulation [13-15]. In this case, three components of the strain rate tensor are nonzero:
&11 = e 22 = u2,2' ¿12 = V2(u1,2 + u2,l)> (1)
where u1, u2 are the displacement vector components. The dot and comma denote the derivatives with respect to time and coordinate, respectively. The continuity equation and the law of conservation of momentum are used:
P/P = -(e11 +e22)) ct11,1 + CT21,2 =PU1' CT12,1 + CT22,2 = Pu2' where ct- are the stress tensor components, and p is the density.
By decomposing the stress tensor into the spherical and deviatoric parts ct- = - PS- + S-, the total strain tensor into the elastic and plastic parts = e - + e j, as well as taking into account the hypothesis of plastic incompressibility epk = 0 we come to the following expressions for the deviatoric stress tensor components and pressure:
S11=2^en - 3 & kk-x sn j,
S22 = 2W & 22 -1 & kk -XS22 |,
v 3 J (3)
S33 = 2^-1 ekk -XS33 j = -(Sn + S22), S12 = 2^12 -XS12), P = -K&kk,
where K and ^ are the bulk and shear moduli, X is the scalar factor that is identically zero in the elastic region, and S- is the Kronecker delta; the summation is taken over repeated indices. The plastic flow rule eP = XS- is associated with the yield condition given in the form
CTeq = CTs - (CTs -CT0)eXP(- eeq/ epX (4)
where CTeq and epq are the equivalent stress and plastic strain, ct0 and cts are the yield and ultimate strengths, ep is the characteristic equivalent plastic strain value that determines the strain hardening coefficient.
The model parameters in Eqs. (3) and (4) correspond to titanium and are determined from experimental data. Experiments were conducted on specimens of commercially pure titanium VT1-0 (GOST 19807-91) measuring 10x10x2 mm annealed in vacuum at 1023 K for 1 h, mechanically grinded and polished. The specimens were electron beam irradiated with three pulses of 50 ^s duration using an electron beam generator SOLO. The beam power
Table 1
Microhardness and elastic moduli of the substrate and modified surface layer
H, GPa E, GPa
Substrate 2.6 111
Surface layer 3.3 124
density W was equal to 18-24 J/cm2 at an initial particle energy of 18 keV, the pulse repetition rate was 0.3 s-1. Irradiation was carried out in inert argon atmosphere at a residual pressure of 0.02 Pa. The titanium specimens were tested in uniaxial static tension at room temperature on an Instron machine at a speed of 0.3 mm/min. The hardness and elastic modulus of the modified surface layer were determined on a NanoTest nanoindenter at maximum loads of 2 to 100 mN. The microhardness value was estimated using a microhardness tester PMT-3 under load of 50 g. The surface morphology of the deformed VT1-0 specimens was examined by an atomic force microscope Solver HV and an optical profilometer NewView 6200. The surface roughness after beam irradiation increased more than twice as compared with the polished specimens in the initial state.
The microhardness values measured on the specimen surface before and after electron beam irradiation, and the longitudinal elastic moduli are presented in Table 1. The experimental results show that the surface layer material is by 25% harder and by 10% stiffer than the titanium substrate material. This ratio of the mechanical properties was used to select the average values of the model parameters for the surface layer material, such as the elastic moduli, yield strength, and ultimate strength in constitutive equations (3), (4). The values of these parameters for the titanium substrate material correspond to the experimental stress-strain curve of initial annealed specimens subjected to tension prior to electron beam irradiation. Hardening functions (4) that reflect the average mechanical response of the substrate and surface layer materials are depicted in Fig. 1, a.
The physical statement of the problem is the following. Electron beam irradiation causes melting and then rapid
crystallization of a thin surface layer in VT1-0 specimens. After cooling, titanium transforms back to the a-phase. The rapid crystallization is accompanied by the formation of randomly oriented martensitic lamellas and fine equiaxial grains (Fig. 1, b), with no texture being observed. These are actually the same a-titanium single crystals, but having a complex shape and being an order of magnitude smaller. Owing to the rapid cooling induced by low-energy high current electron beams, the "surface layer-titanium substrate" interface is well defined and characterized by the difference in the mechanical properties between the layer and substrate materials. The polycrystalline structure of the near-surface region that includes the modified layer and near-surface grains of the titanium substrate is schematically illustrated in Fig. 2. This region demonstrates zones of the modified layer which qualitatively differ in that they are located either simultaneously above several grains or above the body of a single grain of the titanium substrate. Corresponding mesovolumes, which explicitly account for the experimentally observed crystalline structure of the modified layer, were chosen as two main zones for calculations (Figs. 2(I) and 2(II)). For a comparative analysis, we performed calculations for a volume with a rectilinear "modified layer-titanium substrate" interface without taking into account the crystalline structure (Fig. 2(III)), and for a homogeneous grain of the titanium substrate (Fig. 2(IV)).
Grain crystallites in the polycrystalline titanium substrate and microcrystallites of the modified layer have a different orientation with respect to the applied load direction (Fig. 2). The more favorably the grain is oriented, the softer it is, i.e. the lower are the elastic and strength properties. Plastic flow originates earlier in favorably oriented grains. In unfavorably oriented grains, the transition from the elastic to plastic state occurs at a higher average stress level. Therefore, the degree of orientation of grains and crystallites was determined in the simulation by the magnitude of the elastic moduli and yield strengths (Table 2). Three cases with low, high, and average values were considered for the substrate grains. For the surface layer crystallites, the elastic moduli and yield strengths were randomly
Fig. 1. Mechanical properties of the modified surface layer and titanium substrate materials used in calculations compared to experiment (a); experimental microstructure of titanium VT1-0 with a modified surface layer (b)
Titanium single grain
Fig. 2. Physical statement of the problem. Model microstructures of titanium VT1-0 with a modified surface layer (I—III), and homogeneous titanium specimen (IV)
scattered with respect to the average values, which are by 10 and 25% higher than the corresponding average values of the titanium substrate according to microhardness measurements. The range of scatter in the average yield strengths was chosen based on estimates of the critical resolved shear stress, which can differ by a factor of 3 in prismatic and pyramidal slip systems. The extreme values of the average elastic modulus differed by 40%.
The boundary conditions on the left and right surfaces simulate the uniaxial tension of the structures in the X1 direction, while those on the upper and lower surfaces correspond to the free surface and symmetry conditions, respectively (Fig. 2(IV)).
ux = -v for (X1?X2)e B1,
U = v for (X1?X2)e B3, (5)
GjUj = 0 for (Xi,X2)e B2,B4.
Here, v is the constant velocity, positive in tension and negative in compression, and nj is the normal to the corresponding surfaces B2 and B4 .
The system of Eqs. (1)-(5) is solved numerically by the finite difference method. The computational domain is divided by a uniform quadrilateral mesh. The partial derivatives are replaced by difference analogs on this mesh. An explicit second-order accuracy scheme is used. The above described scatter of the mechanical properties is taken into account explicitly as the initial conditions of the boundary-value problem. The cells of the computational domain belonging to different crystallites and colored differently (Fig. 2) possess different properties. The boundary between crystallites passes through the mesh nodes. The conditions of displacement continuity, i.e., ideal mechanical contact, are fulfilled on this boundary.
Table 2
Mechanical properties of titanium substrate grains and modified layer crystallites used in calculations
ct0, MPa as, MPa e ¡?, % K, GPa GPa
Crystallites 113-338 388 0.13 73-102 53-74
Grains 90, 180, 270 310 0.13 66, 79, 93 47, 57, 67
Fig. 3. Equivalent plastic strains for the structures shown in Figs. 2(111) and 2(IV). Specimen elongation of 1%. The digits indicate the yield strength in MPa
3. Results of simulation and experimental investigation
The stress and strain localization in titanium with a modified surface layer is analyzed from simple to complex, by sequentially considering the structures shown in Fig. 2. The calculation results on the stress-strain state of a tensile homogeneous specimen and a tensile specimen with a rectilinear interface without structure are obvious, trivial, not novel and not valuable. However, they are necessary for comparative analysis when assessing the effect of structural heterogeneity on the macroscopic response of the specimens and on the stress-strain distribution at the me-soscale.
Let us analyze the influence of the near-surface substrate grain orientation on the deformation of the structures (Fig. 2) for the limiting cases of low (90 MPa) and high (270 MPa) yield strength of the grain body, with an average level of 180 MPa (Table 1).
In the specimen without the surface layer and without structure (Fig. 2(IV)), the equivalent plastic strain distributions are uniform, qualitatively identical for the both cases, and differ quantitatively by the value of the elastic deformation occurring in the specimen with higher yield strength, while the specimen with lower yield strength already deforms plastically (Figs. 3, a and b).
In the specimen with a rectilinear "layer-substrate" interface (Fig. 2(III)), the average flow stress in the modified surface layer is by 25% higher than 180 MPa of the substrate grain, according to microhardness measurements, and
is assumed to be 225 MPa in the calculations. Thus, the surface layer is generally a hardening layer for favorably oriented grains. These grains are more prone to plastic deformation than the hardened layer above them (Fig. 3, c). For grains with high yield strength, on the contrary, plastic flow begins in the surface layer, high stresses arise in the grain, and it deforms less than the layer (Fig. 3, d). Generally, the stress and strain distributions are quasi-homogeneous and plastic flow is not localized, as in the case of a homogeneous specimen without a surface layer. In this sense, the yield strength exerts no qualitative effect on the stress-strain patterns.
A qualitatively different deformation pattern is observed in the specimen if the crystal structure of the modified surface layer is taken into account (Fig. 2(II)). The equivalent plastic strain distributions are highly inhomogeneous due to the curvilinearity of crystallite interfaces in the surface layer and the "layer-grain" interface (Fig. 4). The maximum strains in local regions of the structure are more than 7 times higher than the average level of 1% observed in the specimen without regard to the structure, with the regions that continue to deform elastically at the given stage of deformation which are located in the surface layer.
The stress-strain curves for the above described calculations are shown in Fig. 5. The stress (a) was calculated as the volume averaged equivalent stresses, and the strain e corresponds to the relative elongation of the computational domain in the Xj direction (Fig. 2(IV)):
(a), MPa- ! Elastic-plastic III Jê
- | stage y-
___y___
110- d
90
100- ! // c 90
90- ! J^b 90
p----IV
0.2 0.3 8,°/ O
Fig. 5. Stress-strain curves for the different structures shown in
(a)= X aeq^V X sk,
k=1, N / k=1, N
where N is the number of cells in the computational domain, sk is the area of the kth cell, e = (L-L0)/L0 x100%, where L0 is the initial length of the structure in the X1 direction, and L is the current length.
It is seen from the figure that the stress-strain curve for the homogeneous specimen that simulates the response of a favorably oriented grain without a modified layer (Fig. 5, a, curve IV) passes below the corresponding curves for the structures with regard to the surface layer, because the layer is hardening in this case. An unfavorably oriented homogeneous grain without a layer, on the contrary, withstands the highest stress because the average flow stress of the layer equal to 225 MPa is below the grain yield stress, which is 270 MPa in this case (Fig. 5, b, curve IV).
The stress-strain curves for the specimens with a homogeneous layer have well-defined stages (Fig. 5, curves III). In the first stage, both the substrate and the surface layer are in the elastic state. In the second stage, in one case, the substrate deforms plastically and the layer continues to deform elastically (the elastic-plastic stage is bounded by the dashed lines in Fig. 5, a). In the other case, the grain deforms elastically and the layer undergoes plastic flow
... 0 Wi
(a), MPa - IV \b
;—III—
240- 11 d
270
200- 270
160- / a 270 i
0.2 i 0.3 i 8,°/o
Fig. 2, with the titanium yield strength 90 (a) and 270 MPa (b)
(Fig. 5, b). In the third stage, both the grain and the surface layer deform plastically in the both cases.
The stress-strain curves of the specimens with regard to the submicrocrystalline structure of the surface layer (Fig. 5, curves II) pass below the corresponding curves for the specimens with a homogeneous layer. The strength decrease is associated with stress concentrations in local material regions near curvilinear interfaces where localized plastic flow begins, which leads to a highly inhomogeneous stress-strain state under further loading (Fig. 4).
However, the main result of our study is not that the plastic strain distributions are inhomogeneous, but that they are qualitatively and quantitatively different from each other for near-surface substrate grains with different yield strengths (Fig. 4). Note that all other conditions, including the random scatter of mechanical properties over crystallites, their geometry and loading conditions, remained the same. The volume of the surface layer material was also constant and corresponded to the material volume of the layer with a rectilinear interface (Fig. 2(III)). Only the yield strength of the substrate grain was changed. It has been found that both the degree of localization and the spatial distribution pattern of localized deformation bands strongly depend on the yield strength of the surface-modified sub. b
Fig. 6. Relief of equivalent plastic strains for the structure shown in Fig. 2(II), for the states shown in Fig. 5, a. The yield strength of titanium grain is 90 MPa
Fig. 7. Relief of equivalent plastic strains for the structure shown in Fig. 2(II), for the states shown in Fig. 5, b. The yield strength of titanium grains is 270 MPa
strate grain (Fig. 4). Plastic flow localization is suppressed in favorably oriented grains (Fig. 4, a), while unfavorably oriented grains exhibit a system of pronounced shear bands (Fig. 4, b). Additionally, the spatial distributions of the bands in the surface layer differ from each other despite the same initial conditions.
A detailed analysis of the simulation results revealed the reason for this. Plastic shearing in favorably (case 1) and unfavorably (case 2) oriented grains with a submicro-crystalline surface layer initiates and occurs by absolutely different scenarios (Figs. 6 and 7).
In case 1, the surface layer is a hardening layer for the grain and contains no crystallites whose yield strength would be lower than the grain yield strength (see Table 2). Plastic flow therefore initiates exclusively in the grain near the curvilinear interface with the surface layer (Fig. 6, a) and propagates into the grain bulk (Fig. 6, b), forming a system of fine shear bands. The number of the bands is associated with a set of quasi-uniformly distributed stress concentrators at the "layer-grain" interface. With further loading, the surface layer material gradually goes to the plastic state (Figs. 6, c and d), and the current pattern of inhomoge-neous deformation of the grain affects strain localization in the layer. The curvilinear "layer-grain" interface thus plays
a primary role in case 1. It determines the resulting localization pattern for the grain with a hardened surface layer.
A completely different deformation scenario takes place when the substrate grain has high strength (Fig. 7). In this case, strain localization zones first appear in the surface layer that contains crystallites with the yield strength lower than the grain yield strength (Fig. 7, a, Table 2). As the load increases, these crystallites are gradually involved in plastic flow (Fig. 7, b) that forms weakened directions along which a system of mesoscopic shear bands is generated. These bands produce additional stress concentrations at the interface with the elastically deformed grain and initiate noncrystallographic shear banding in the grain (Figs. 7, c and d). Thus, the structural heterogeneity associated with the crystalline structure of the surface layer plays a decisive role in case 2, while the factor of the "surface layergrain" interface curvature is secondary and suppressed.
The above described difference in the deformation scenarios results in that the same surface layer regions can be strongly, in case 1, or slightly, in case 2, deformed at the same degree of relative elongation, and vice versa (dashed circles in Fig. 4).
The resulting plastic strain localization pattern for cases 1 and 2 with a relative specimen elongation of 5% is shown
Fig. 8. Relief of equivalent plastic strains for the structure shown in Fig. 2(II), with the yield strength of titanium grain 90 (a) and 270 MPa (b). Specimen elongation of 5%
b
3
12
u
ffi 1
a
0
■ 1 !ir
> . 90 MPa
--270 MPa
#
ft z,. L£ a 1 1. 1 '. 1 1. ■ 11. 1'M 1. I.I..,.
0
°M00-
ci g
J 80-
o
>
a
w 60-
o <a a
m
a >
40-
4 8 12
Plastic strain degree, %
16
■§ 20 o 04
A
90 MPa 270 MPa
0
4 8 12
Plastic strain degree, %
Fig. 9. Distributions of local volumes with respect to the equivalent plastic strain values for the cases illustrated in Fig. 8. Probability density (a) and probability (b)
in Fig. 8. There is a clear qualitative difference in the strain localization patterns of favorably and unfavorably oriented near-surface grains with the modified layer. To quantitatively estimate the degree of localization, we performed statistical analysis to calculate the material volume fractions undergoing different degrees of plastic deformation. The results of processing the localization patterns from Fig. 8 are illustrated in Fig. 9. The higher is the dispersion of distributions relative to the average value, the higher is the degree of localization. It is interesting that the equivalent plastic strain distribution for an unfavorably oriented grain is characterized by four local maxima. This means that the systems of localized plastic deformation bands from Fig. 8, b have a quasi-hierarchical four-level structure, when qualitatively similar spatial distributions are observed for systems of bands with a different average plastic strain of 3.4, 4.2, 5.8, and 7.2% (Fig. 9, a). Figure 9, b shows the cumulative summation of the material volume fractions with the corresponding equivalent plastic strain values. The steeper is the curve slope, the lower is the degree of localization.
The formation of mesoscopic shear bands was observed experimentally (Fig. 10, a). This confirms the conclusions drawn from the calculation results on strain localization in unfavorably oriented grains. The high-resolution microscopic image of the modified surface layer demonstrates a system of localization bands that initiate shear banding in the underlying grain (Fig. 10, b). These bands are of non-
crystallographic origin and differ from the coarse slip traces observed in titanium grains inside the volume (Fig. 10, a).
The last series of calculations was carried out for the structure presented in Fig. 2(I) characterizing the near-surface region which contains a triple grain junction. Figure 11 shows the calculations for the case when the most favorably oriented grain was located between grains with higher yield strength. First, these calculations confirmed the earlier conclusion that localization in unfavorably oriented grains is more pronounced (Figs. 11, a-c). Second, it has been found that as a result of plastic flow in the central weakest grain the material bends in the specimen center, and stronger surrounding grains begin to rotate in the directions indicated by the arrows (Fig. 10, b). Systems of localization bands initiate from the grain boundaries in the directions not coinciding with the direction of development of conjugate shear bands at an angle of 45°, which were originally formed in the surface layer and near the "layer-substrate" interface. This process is accompanied by bending of localization bands in the regions indicated by the dashed circles in Fig. 11. The generality of this conclusion is confirmed in a series of calculations for various sequences of random scatter of mechanical properties over the surface layer crystallites (Fig. 11, a-c). Our experimental results, as well as independent studies, are in good agreement with the simulation results and confirm the conclusions drawn (Figs. 11, d and e).
Fig. 10. Plastic strain localization in the near-surface titanium grain. Optical image of a lateral specimen face
Fig. 11. Relief of equivalent plastic strains for the structure shown in Fig. 2(I), for different sequences of random scatter of the yield strength in the modified layer (a-c). Bending of slip bands in experiments conducted Ref. [2] (d) and in this study (e)
4. Conclusions
The stress-strain state arising in near-surface grains under uniaxial tension of surface-modified polycrystalline titanium was numerically studied with regard to the submic-rocrystalline structure of the surface layer formed after electron beam treatment. A boundary value problem was solved by the finite-difference method in a two-dimensional plane strain formulation. The studies were performed on models containing several grains of the base metal with explicit account of the submicrocrystalline structure of the surface layer experimentally observed after electron beam treatment. Since the modified surface layer has no pronounced texture, the difference in the orientation of individual lamellar crystallites relative to the applied load direction was simulated by the random scattering of elastic moduli within the range associated with the degree of ani-sotropy of hcp crystals and yield stresses within the critical shear stress range for prismatic, basic, and pyramidal slip systems. It was shown numerically that uniaxial tension leads to a complex stress state at the interface between the substarte grain and submicrocrystalline modified layer, which governs strain localization in the near-surface grains of loaded VT1-0 specimens. Stress concentration in local regions near the interface depends on the ratio between the mechanical characteristics of contacting substarte grains and submicrocrystallites of the modified surface layer.
The main result of this paper is the following. Plastic strain localization is most pronounced in grains with high yield strength which are unfavorably oriented with respect to the loading direction. Plastic flow begins later in such grains, but occurs in the form of a well-defined system of localized shear bands. Localization in favorably oriented grains with low yield strength is, on the contrary, suppressed. Additionally, the degree of localization in shear bands, as well as their number and spatial distribution appear to be different depending on whether the surface-modified grain is favorably or unfavorably oriented. Calculations showed that this is due to the fundamental difference in plastic shear initiation and development in favorably and unfavorably oriented grains.
The modified surface layer formed above the base metal grains favorably oriented with respect to the applied load direction is generally a hardening layer. Therefore, plastic deformation begins first in the body of the substrate grain at the interface with the modified surface layer and then propagates into the surface layer material with further loading. In this case, the strain localization pattern is completely determined by the curvilinear shape of the "grain-layer" interface. A slight slip occurs from this boundary, and a fine structure of slip bands is observed experimentally.
The grains in which slipping is inhibited due to unfavorable orientation, on the contrary, experience higher
stresses than the surface layer. In this case, plastic shearing begins in the modified layer and forms the initial system of localized shear bands that fully determines the strain localization pattern in the underlying grain under further loading. Thus, the determining factor for unfavorably oriented grains is not the curvilinearity of the "layer-substrate grain" interface, but the submicrocrystalline structure of this layer. The system of noncrystallographic mesoscopic localization bands is observed experimentally in such grains.
It was also numerically shown that localization bands can bend due to the rotation of grains, which was confirmed experimentally.
The work was carried out in the framework of the Fundamental Research Program of SB RAS III.23.1 for 2017-2020.
References
1. Titanium in Medicine: Material Science, Surface Science, Engineering, Biological Responses and Medical Applications / Ed. by M. Donald, P.T. Brunette, P.T. Marcus Textor. - Berlin: Springer, 2001.
2. Belyaev N.M. Resistance of Materials. - Moscow: Nauka, 1965.
3. Okulov I.V., Wendrock H., Volegov A.S., Attar H., Kuhn U., Firrao D. High strength beta titanium alloys: New design approach // Mater. Sci. Eng. A. - 2015. - V. 628. - P. 297-302.
4. Kovalevskaya Z.G., Khimich M.A., Belyakov A.V., Shulepov I.A. Evalu-
ation of physical and mechanical properties of structural components of Ti-Nb alloy // Adv. Mater. Res. - 2014. - V. 1040. - P. 39-42.
5. Figueiredo R.B., Eduardo R., de Barbosa C., Zhao X. Improving the fatigue behavior of dental implants through processing commercial purity titanium by equal-channel angular pressing // Mater. Sci. Eng. A. - 2014. - V. 619. - P. 312-318.
6. Liu J., Khan A.S., Takacs L., Meredith C.S. Mechanical behavior of ultrafine-grained nanocrystalline titanium synthesized by mechanical milling plus consolidation: experiments, modeling and simulation // Int. J. Plasticity. - 2015. - V. 64. - P. 151-163.
7. Medvedev A.E., Ng H.P., Lapovok R., Estrin Y., Lowe T.C., Anu-malasetty V.N. Effect of bulk microstructure of commercially pure titanium on surface characteristics and fatigue properties after surface modification by sandblasting and acid-etching // J. Mech. Behavior Biomed. Mater. - 2016. - V. 57. - P. 55-68.
8. Gao Y Surface modification of TC4 titanium alloy by high current pulsed electron beam (HCPEB) with different pulsed energy densities // J. Alloys Comp. - 2013. - V. 572. - P. 180-185.
9. Panin A.V., Kazachenok M.S., Borodovitsina O.M. Modification of the structure of surface layers of commercial titanium in the process of treatment by low-energy high-current electron beams // Phys. Met. Metallogr. - 2016. - V. 117. - No. 6. - P. 550-561.
10. Panin V.E., PaninA.V., Pochivalov Y.I., Elsukova T.F., ShugurovA.R. Scale invariance of structural transformations in plastically deformed nanostructured solids // Phys. Mesomech. - 2017. - V. 20. - No. 1. -P. 55-68.
11. Panin VE., Panin A.V. Effect of the surface layer in a solid under deformation // Phys. Mesomech. - 2005. - V. 8. - No. 5-6. - P. 7-14.
12. Panin VE., Egorushkin V.E. Basic physical mesomechanics of plastic deformation and fracture of solids as hierarchically organized non-
linear systems // Phys. Mesomech. - 2015. - V. 18. - No. 4. - P. 377390.
13. Balokhonov R.R., Zinovjev A.V, Romanova V.A., Zinovjeva O.S. The Computational micromechanics of materials with porous ceramic coatings // Meccanica. - 2016. - V. 51. - No. 2. - P. 415-428.
14. Balokhonov R.R., Romanova V.A. The effect of the irregular interface geometry in deformation and fracture of a steel substrate-boride coating composite // Int. J. Plasticity. - 2009. - V. 25. - No. 11. -P. 2225-2248.
15. Balokhonov R.R., Romanova V.A., Schmauder S., Schwab E. Mesos-cale analysis of deformation and fracture in coated materials // Comp. Mater. Sci. - 2012. - V. 64. - P. 306-311.
16. Romanova V.A., Soppa E., Schmauder S,, Balokhonov R.R. Meso mechanical analysis of the elasto-plastic behavior of a 3D composite structure under tension // Comp. Mech. - 2005. - V. 36. - P. 475483.
17. Pakhaliuk V., Poliakov A., Kalinin M., Pashkov Ye., Gadkov P. Modeling and expanding the simulation of wear in the spherical joint with a polymeric component of the total hip prothesis // Facta Univer. Mech. Eng. - 2016. - V. 14. - No. 3. - P. 301-312.
18. Li Q., Popov V.L. Indentation of flat-ended and tapered indenters with polygonal cross-sections // Facta Univer. Mech. Eng. - 2016. -V. 14. - No. 3. - P. 241-249.
19. Becker R., Needleman A. Effect of yield surface curvature on necking and failure in porous plastic solids // J. Appl. Mech. - 1986. -V. 53. - P. 491-499.
20. Chung K., Lee C., Kim H. Forming limit criterion for ductile aniso-tropic sheets as a material property and its deformation path insensi-tivity. Part II: Boundary value problems // Int. J. Plasticity. - 2014. -V. 58. - P. 35-65.
21. Xue L. Localization conditions and diffused necking for damage plastic solids // Eng. Fract. Mech. - 2010. - V. 77. - P. 1275-1297.
22. Mroginski J.L., Etse G. Discontinuous bifurcation analysis of ther-modynamically consistent gradient poroplastic materials // Int. J. Solids Struct. - 2014. - V. 51. - P. 1834-1846.
23. Voyiadjis G.Z., Faghihi D. Localization in stainless steel using microstructural based viscoplastic model // Int. J. Impact Eng. - 2013. -V. 54. - P. 114-129.
24. Marcin K., Rakowski W., Lackner J.M., Major L. Analysis of spherical indentations of coating-substrate systems: Experiments and finite element modeling // Mater. Design. - 2013. - V. 43. - P. 99-111.
25. Skordaris G., Bouzakis K.D., Charalampous B.P. A Dynamic FEM simulation of the nano-impact test on mono- or multi-layered PVD coatings // Surface Coat. Tech. - 2015. - V. 265. - P. 53-61.
26. Fukumasu N.K., Angelo C.M., Ignat M., Souza R.M. Numerical study of tensile tests conducted on systems with elastic-plastic films deposited onto elastic-plastic substrates // Surface Coat. Tech. - 2010. -V. 205. - P. 1415-1419.
27. Zhu W., Yang L., Guo J.W., Zhou Y.C., Lu C. Determination of interfacial adhesion energies of thermal barrier coatings by compression test combined with a cohesive zone finite element model // Int. J. Plasticity. - 2015. - V. 64. - P. 76-87.
28. Nayebpashaee N., Seyedein S.H., Aboutalebi M.R., Sarpoolaky H., Hadavi S.M.M. Finite element simulation of residual stress and failure mechanism in plasma sprayed thermal barrier coatings using actual microstructure as the representative volume // Surface Coat. Tech. - 2016. - V. 291. - P. 103-114.
29. Donegan S.P., Rollett A.D. Simulation of residual stress and elastic energy density in thermal barrier coatings using fast fourier transforms // Acta Mater. - 2015. - V. 96. - P. 212-228.
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Ruslan R. Balokhonov, Dr. Sci. (Phys.-Math.), Leading Researcher, ISPMS SB RAS, rusy@ispms.tsc.ru Varvara A. Romanova, Dr. Sci. (Phys.-Math.), Leading Researcher, ISPMS SB RAS, varvara@ispms.tsc.ru Aleksey V. Panin, Dr. Sci. (Phys.-Math.), Assoc. Prof., Head of Laboratory, ISPMS SB RAS, pav@ispms.tsc.ru Marina S. Kazachenok, Cand. Sci. (Phys.-Math.), Researcher, ISPMS SB RAS, kms@ispms.tsc.ru Sergey A. Martynov, Post-Graduate, ISPMS SB RAS, martinov@ispms.tsc.ru
