ISSN 1029-9599, Physical Mesomechanics, 2017, Vol. 20, No. 3, pp. 324333. © Pleiades Publishing, 324 ROMANOVA et al.Ltd., 2017. Original Russian Text © V.A. Romanova, R.R. Balokhonov, A.V. Panin, E.E. Batukhtina, M.S. Kazachenok, V.S. Shakhijanov, 2017, published in Fizicheskaya Mezomekhanika, 2017, Vol. 20, No. 3, pp. 8190.
Micromechanical Model of Deformation-Induced Surface Roughening in Polycrystalline Materials V. A. Romanova1*, R. R. Balokhonov1, A. V. Panin1,2, E. E. Batukhtina1, M. S. Kazachenok1, and V. S. Shakhijanov3 Institute of Strength Physics and Materials Science, Siberian Branch, Russian Academy of Sciences, Tomsk, 634055 Russia 2 National Research Tomsk Polytechnic University, Tomsk, 634050 Russia 3 National Research Tomsk State University, Tomsk, 634050 Russia * e-mail:
[email protected] 1
Received February 10, 2017
AbstractA micromechanical model has been developed to describe deformation-induced surface roughening in polycrystalline materials. The three-dimensional polycrystalline structure is taken into account in an explicit form with regard to the crystallographic orientation of grains to simulate the micro- and mesoscale deformation processes. Constitutive relations for describing the grain response are derived on the basis of crystal plasticity theory that accounts for the anisotropy of elastic-plastic properties governed by the crystal lattice structure. The micromechanical model is used to numerically study surface roughening in microvolumes of polycrystalline aluminum and titanium under uniaxial tensile deformation. Two characteristic roughness scales are distinguished in the both cases. At the microscale, normal displacements relative to the free surface are caused by the formation of dislocation steps in grains emerging on the surface and by the displacement of neighboring grains relative to each other. Microscale roughness is more pronounced in titanium, which is due to the high level of elastic-plastic anisotropy typical of hcp crystals. The mesoscale roughness includes undulations and cluster structures formed with the involvement of groups of grains. The roughness is quantitatively evaluated using a dimensionless parameter, called the degree of roughness, which reflects the degree of surface shape deviation from a plane. An exponential dependence of the roughness degree on the strain degree is obtained. DOI: 10.1134/S1029959917030080 Keywords: polycrystalline structure, uniaxial tension, surface roughness, numerical simulation, crystal plasticity theory
1. INTRODUCTION Most structural metals and alloys have a polycrystalline structure formed during material production. Although experimental and theoretical studies of polycrystalline materials have a long history, a whole range of issues related to multiscale deformation and fracture was formulated only at the end of the twentieth century. This became possible owing to the emergence of novel experimental techniques and devices that provided fundamentally new information about material behavior in a wide range of scales and phenomena, as well as owing to the rapid development of computer technology and numerical methods. A topical subject in physics, mechanics, and materials science is the study of morphological changes on the
free surface of loaded polycrystalline materials. Experiments conducted for a broad range of materials under various loading conditions revealed a large number of possible surface roughening scenarios during deformation. Generally, surface roughness in the deformed material evolves in the entire hierarchy of scales [1, 2]. The smallest scale is associated with the formation of dislocation steps in grains emerging on the surface. The microscale roughening caused by the extrusion and intrusion of individual grains with respect to the surrounding material is known as the orange peel effect [3, 4]. Mesoscale roughness results from a collective displacement of grain groups [1, 5, 6]. Macroscale roughening occurs at the stage of pronounced plastic deformation, indicating the loss of global stability and bearing capacity of the
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material [7, 8]. Experimental and theoretical studies showed that the main cause of out-of-plane displacements on the surface free from external load is a complex stress-strain state governed by structural inhomogeneity [913]. The deformation-induced surface roughening is typically viewed as an undesirable effect. It deteriorates the surface reflectivity, fatigue and wear resistance, corrosion fatigue strength, adhesion, and weldability. It also negatively affects the physical and mechanical properties of the material due to plastic strain localization. Therefore, a key task for many engineering applications is to develop effective methods of suppressing morphological changes on the surface, at least at certain scales. On the other hand, surface deformation is interrelated with the stress-strain evolution in the material bulk and with the loading history, which can provide a basis for the development of nondestructive testing techniques and methods of predicting the service life and residual life of materials. The mesoscale, where not individual grains but grain conglomerates are involved in roughening, plays a special role in this context. It serves as a link between the microscale, where plastic deformation is governed by dislocation mechanisms, and the macroscale, where failure is preceded by plastic strain macrolocalization. A weakly pronounced mesoscale roughness is formed already in the initial stage of plastic flow and evolves under subsequent loading up to failure. The mesoscale roughness can be easily detected by scanning and optical microscopy during the entire plastic flow stage, without strong requirements to the processed surface quality. Thus, the study of mesoscale roughness can be of great practical interest for the development of nondestructive testing methods. For efficient suppression of deformation-induced surface roughening under loading and nondestructive testing based on data about surface morphological changes, it is necessary to thoroughly understand the mechanical and physical aspects of this phenomenon. Numerical simulation is a good supplement to experimental methods in this case which allows an explicit investigation of the relationship between deformation processes on the surface and in the bulk of the loaded material. This paper continues the study of surface roughening in loaded materials. Earlier, we studied experimentally and numerically surface deformation in aluminum [11] and titanium alloys [13] and in steel [9, 12] under uniaxial tension. The deformation-induced surface roughening at the micro- and mesoscales was simulated with an explicit account of the material grain structure. The PHYSICAL MESOMECHANICS
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model proposed in this paper also explicitly accounts for the microscale crystallographic anisotropy of grains associated with the crystalline structure of the material. This approach is crucial for describing materials with crystallographic texture, strong elastic-plastic anisotropy, and a limited set of slip systems. The numerical simulation of deformation-induced surface roughening under uniaxial tension is carried out on model specimens of polycrystalline aluminum and titanium. 2. MATHEMATICAL MODEL 2.1. General System of Equations in a Dynamic Formulation The solution of micromechanical problems with an explicit account of the internal structure requires substantial computational resources. On the one hand, the considered microvolume must contain a sufficient number of structural elements to realistically simulate microand mesoscale deformation processes. On the other hand, structural elements and interface regions should be approximated in much detail to ensure acceptable solution accuracy, which necessitates the use of detailed meshes with a large number of elements. It is therefore important for the solution of micromechanical problems to minimize computational requirements without loss of information and solution accuracy. An approach that considerably reduces the requirements for computer memory, disk space, and computational speed implies the solution of quasi-static problems in a dynamic formulation. It allows the transition from implicit to explicit calculations. Explicit methods have significant advantages for solving problems with any type of nonlinearity from the viewpoint of computational capacity. In a static formulation, the equilibrium state is determined by iterations, the number of which can be large depending on the degree of the problem nonlinearity. In this case, explicit methods that do not require the use of iterative schemes have an advantage over implicit methods. A problem can be nonlinear due to various factors, including nonlinear material response (e.g., plastic flow), geometrically nonlinear computational domain, nonlinear boundary conditions, and others. Micromechanical problems with explicitly introduced microstructure are as a rule nonlinear even in the elastic loading region. Thus, we simulated quasi-static loading of polycrystalline structures in a dynamic formulation using an explicit finite element method. The system of solid mechanics equations in a dynamic formulation includes the equations of motion
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continuity equation
ρU i = σij , j ,
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V − Ui ,i = 0, (2) V and relations for the total strain rates 1 ε ij = (Ui , j + U j ,i ), (3) 2 where U i are the velocity vector components, V = ρρ−0 1 is the relative volume, ρ0 and ρ are the initial and current material densities, σij and εij are the components of the stress and total strain tensors; the dot above the symbol denotes the time derivative. The system of Eqs. (1) (3) is closed by constitutive equations defining the relationship between the stress and strain tensors. 2.2. Model of Three-Dimensional Polycrystalline Structure The geometric model of polycrystalline material is constructed with an explicit determination of the coordinate dependence of the physical and mechanical properties. The polycrystalline structure was generated using the step-by-step packing method described in our previous paper [11]. The structure contained 1600 equiaxed grains and was generated on a regular mesh with cubic elements of size 200 × 75 × 200 and a step of 10 µm (Fig. 1). In the numerical implementation, interfaces between different structural elements pass through the mesh nodes; the physical and mechanical properties of the material are specified inside elements. Groups of contacting elements with the same crystallographic orientation form a grain. All grains at the initial time point had the same physical and mechanical characteristics (density, elastic moduli, critical resolved shear stress inducing dislocation glide, etc.) and differed only in the orientation of the crystallographic axes, which remained constant within the grain and changed in passing through the grain boundary. For specifying the )
*
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crystallographic orientation of grains, in addition to the global coordinate system associated with the specimen geometry (Fig. 1a), we introduced a local coordinate system for each grain associated with the direction of its crystallographic axes (Figs. 1b and 1c). The orientation of the local coordinate system with respect to the global system was specified through randomly generated Euler angles, which points to the absence of crystallographic texture. 2.3. Description of the Elastic-Plastic Response of Grains within the Crystal Plasticity Framework The microscale anisotropy of elastic-plastic properties associated with the crystal lattice structure was taken into account by deriving constitutive relations on the basis of crystal plasticity theory [10, 1417]. The latter considers a polycrystalline aggregate as a set of single crystals with different crystallographic orientations relative to the global coordinate system. This approach is essential for materials with a limited set of slip systems, pronounced texture, and a high degree of elastic-plastic anisotropy at the grain scale. In the general case, the stress and strain tensor components are related by the generalized Hookes law (4) σ ij = Cijkl ε ekl , e
where εij are the elastic strain tensor components, and Cijkl is the elastic modulus tensor. In view of the total strain tensor representation as a sum of elastic εeij and plastic εijp components (5) εij = εeij + εijp , Eq. (4) takes the form σ ij = Cijkl (ε kl − ε pkl ).
The total strain rate tensor in Eq. (6) is unambiguously determined from Eq. (3), yet there is no single approach for determining the plastic strain tensor. Today there are a large number of models for εijp determination which
{111}〈110〉
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are constructed from phenomenological and physical considerations. Some models are reviewed, e.g., in Refs. [16, 17]. In this study, the plastic strain rate tensor ε ijp was determined within the crystal plasticity framework [15], which explicitly accounts for the crystallographic orientation of grains and slip systems with respect to the global coordinate system: ε ijp = ∑ γ (α )Θij(α ), (7) α
where γ (α ) is the plastic shear rate on slip system α, and Θij(α ) is the orientation tensor defining the orientation of slip system α through the components of the slip direction vector si(α ) and normal vector to the slip plane mi( α ) : 1 Θij(α ) = ( si(α ) m(jα ) + s (jα ) mi( α ) ). (8) 2 The plastic shear rate γ (α ) under quasi-static loading at room temperatures can be represented as a function of the resolved shear stress τ( α ) acting on a given slip system, which is defined as τ(α ) = si(α )σij m(jα ). (9) Many authors [1417] express this function in the form of a power law ν
τ( α) (10) γ = γ 0 ( α) sgn τ(α ) , τ* where γ 0 is the reference shear rate that is the same for all slip systems, ν is the coefficient defining the rate sensitivity, and τ*(α ) is the critical resolved shear stress at which the slip system is activated. The values of γ 0 and ν were chosen so that to exclude the rate sensitivity of the material. Test calculations showed that the loading rate exerts no effect on the dynamic problem solution at γ 0 ≈ 0.1ε (ε is the macroscopic strain rate) and ν ~ 10. In the general case, the critical resolved shear stress that initiates dislocation motion in a particular slip system is a function of accumulated plastic strain. Different authors proposed dependencies for determining τ*(α ) of various materials with taking into account the accumulation of dislocations, twinning, dislocation glide on other slip systems, and many other factors [1417]. Such models usually have a large number of approximation coefficients and constants determined from independent experiments. Some phenomena are taken into account automatically in direct modeling with explicit consideration of grain structure and do not require special description using constitutive relations. These phenomena include the stress concentration and nucleation of dislocation defects at grain boundaries, crystallographic rotation of a grain (or its part) during deformation, plastic (α )
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strain localization, and others. As a result, simpler models can be used for τ*(α ) description. In this paper, we propose the following phenomenological dependence for τ*(α ) determination: (11) τ*(α ) = τ(0α ) + k (1 − exp(−γ (Aα ) b )) + k1D−1 2 ,
γ (Aα ) = ∫ | γ | dt , t
where τ(0α ) is the initial critical resolved shear stress on slip system α. The second term of the sum describes isotropic hardening due to accumulation of plastic shear on the slip system α, k and b are the approximation coefficients. The third term of the sum accounts for the Hall Petch strengthening. Here, k1 is the strengthening coefficient, D is the grain diameter determined in numerical implementation as the diameter of a sphere of the same volume. The elastic-plastic behavior of aluminum and titanium grains was simulated by writing Eqs. (4)(11) with regard to the crystal lattice structure of these materials. Aluminum has a face-centered cubic lattice (Fig. 1b). The direction of the xi axes was chosen for the simulation according to the scheme depicted in Fig. 1b. As a result of the cubic crystal symmetry, the elastic modulus matrix of aluminum crystals contains 12 nonzero moduli, three of which are independent: C1111 = 108 GPa, C1122 = 61 GPa, and C 2323 = 28 GPa. Aluminum single crystals have 12 slip systems {111}〈110〉 characterized by the same initial critical resolved shear stress necessary for slip activation, τ(0α) ≈ 10 MPa. The approximation coefficients of Eq. (11) were determined from experimental stress-strain curves for Al1570 alloy with different grain sizes: k ≈ 150 MPa, b ≈ 0.016, k1 ≈ 1.6 MPa m1/2. Titanium has a hexagonal close-packed lattice (Fig. 1c). As a result of the hcp crystal symmetry, the elastic modulus matrix of a titanium single crystal contains 12 nonzero moduli, only five of which are independent: C1111 = 162 MPa, C1122 = 69 MPa, C1133 = 92 MPa, C3333 = 181 MPa, and C2323 = 47 MPa. In view of the geometric features of hcp crystals whose lattice is a hexagonal prism, it is described using four crystallographic axes, of which the c axis is that of the prism and the other three ai axes lie in the base plane at an angle of 120° relative to each other (Fig. 1c). Since the constitutive relations are formulated in the Cartesian coordinate system, we introduced a local rectangular coordinate system oriented with respect to the crystallographic directions as shown in Fig. 1c. Titanium has four slip systems: {1010}〈1210〉 prismatic, {1010}〈1210〉 basal, {10 11}〈1210〉 pyramidal, and {1011}〈 21 13〉 pyramidal. The symmetry of an hcp
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crystal results in several equivalent slip systems of each type: 3 prismatic, 3 basal, 6 pyramidal 〈a〉, and 12 pyramidal 〈c + a〉. An essential feature of titanium is that different slip systems have different critical resolved shear stresses of plastic flow initiation. The literature data for the critical resolved shear stress values on different slip systems are contradictory and differ by an order of magnitude in different papers [1820]. It is known, however, that the critical resolved shear stress on the prismatic slip systems is twice lower than on the basal ones and three times lower than on the pyramidal ones. Therefore, the primary slip systems in titanium are prismatic, where the critical resolved shear stress is the lowest. The secondary slip systems are basal. In polycrystalline titanium, grains with the ñ axis oriented perpendicular to the tensile direction are most prone to slipping. Contrarily, grains with the ñ axis oriented parallel to the loading direction remain plastically undeformed or undergo twinning. In the proposed model, pyramidal slip and twinning were not taken into account. 2.4. Loading Conditions The uniaxial tension of polycrystalline structure is schematically illustrated in Fig. 1a. The symmetry conditions relative to the X 2 axis were specified on the lower surface. The lateral and upper surfaces were assumed to be free from external loading. The boundary conditions governing uniaxial extension along the X 1 axis were
formulated in velocities:
U1 | x1 =0 = −Ut , U1| x1 = L1 = U t ,
(12)
where U t is the velocity amplitude. It is known that the solution of a dynamic problem corresponds to a quasistatic solution under certain conditions. In the general case, these conditions imply the reduction of wave effects by smoothly increasing the load velocity and the use of material models nonsensitive to the strain rate. To satisfy the first condition, we linearly increased the tension velocity to a given value and then kept it constant. The loading parameters at which the dynamic and static solutions coincided with acceptable accuracy were determined in a series of tests where the load velocity amplitude and the time of the load velocity increase up to a given amplitude were varied. 2.5. Numerical Implementation Constitutive equations (4)(11) are formulated with respect to the local crystallographic coordinate system, while Eqs. (1)(3) and the boundary conditions are solved with respect to the global coordinate system (Fig. 1a). The transformation equations for the secondrank tensor components during the coordinate system rotation have the form Tij* = Rki RljTkl , (13) where Tij* and Tij are the tensor components in the new and old coordinate systems, respectively, and Rij are the
(a)
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Fig. 2. Roughness patterns in model aluminum (a, b) and titanium polycrystals (c, d) at ε = 2 (a, c) and 10% (b, d). PHYSICAL MESOMECHANICS
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components of the rotation matrix determined through the Euler angles. In the inverse transition, the tensor components are determined by the inverse transformation equations Tij = Rik R jlTkl* . (14) In the general case, the numerical solution algorithm of the boundary value problem (1)(12) at each time step includes: (i) solution of Eqs. (1)(3) with regard to the boundary conditions relative to the global coordinate system, (ii) determination of the total strain rate tensor components in the crystallographic coordinate system using transformation equations (13), (iii) solution of constitutive equations (4)(11) with respect to the crystallographic coordinate system by the iteration method (1-2 iterations are enough for solution convergence with the accuracy δ = 107), and (iv) determination of the stress tensor components in the global coordinate system using inverse transformation equations (14).
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3. SIMULATION RESULTS Figure 2 illustrates the deformation-induced surface roughness in model aluminum and titanium polycrystals for different degrees of tension. The surfaces are shown on the same scale and the light source is positioned similarly with respect to the surfaces, which allows a qualitative comparison of the obtained images. For a quantitative evaluation, Figure 3 shows the evolution of surface profiles along and across the tensile axis (see the loading scheme in Fig. 1a). Since the calculations for aluminum and titanium were carried out using the same model structure (Fig. 1a), differences in their surface roughening patterns must be due to the different elastic-plastic response of crystallites of these materials. Twelve slip systems in aluminum crystals enable dislocation glide at almost any grain orientation with respect to the load applied. Consequently, all aluminum grains are involved in plastic deformation
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Fig. 3. Surface profiles in aluminum (a, b) and titanium (c, d) taken along (a, c) and across (b, d) the tensile axis at ε = 0 (1), 2 (2), 4 (3), 6 (4), 8 (5) and 10% (6), and microstructure of cross sections along (e) and across the tensile axis (f). PHYSICAL MESOMECHANICS
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Fig. 4. EBSD data (a, b) and optical image (c) of polycrystalline titanium surface before deformation (a) and after uniaxial tension to 5% (b, c).
already at the initial stage of plastic flow. A significantly different pattern is observed in polycrystalline titanium. Its stress-strain state at the microscale is highly inhomogeneous, which is typical of hcp crystals with a limited set of slip systems. Titanium grains unfavorably oriented with respect to the loading axis remain elastic or are involved in plastic deformation at later stages of loading, which agrees with experimental data (Fig. 4c). The deformation accommodation of such grains can be due to their rotation as a whole or due to plastic flow and rotation of neighboring grains. In the general case, a grain can simultaneously undergo dislocation glide and rotation. These mechanisms can be separated experimentally by comparing EBSD analysis data and metallographic images. As an example, we demonstrate experimental data for polycrystalline titanium (Fig. 4). Grain 1 exhibits pronounced dislocation glide, without rotation of its crystal lattice (compare Figs. 4a4c). Neighboring grain 2 rotates as a whole during loading but remains elastically deformed; no slip traces are observed in this grain. All the other grains undergo both dislocation glide and rotation to varying degrees. (a)
The simulation results were analyzed in the same way. As an example, Schmid factor maps for prismatic slip systems are shown in Fig. 5. The local orientation axes within grains before deformation are directed equally at each point and change direction when crossing the grain boundary. During deformation the lattice rotates in individual grains or individual grain regions. This indicates the rotation of a grain or its part as a whole. Comparison of the Schmid factor maps of grains (Fig. 5b) and plastic strain distributions (Fig. 5c) showed that the crystallographic orientation mainly changes along the grain boundaries as well as inside of and near the bands of localized plastic deformation. Plastic flow in grains is generally governed by a combination of factors, among which is the crystallographic orientation. However, no direct correlation was found between the crystallographic orientation of grains with respect to the tensile axis and the degree of plastic deformation in them. According to crystal plasticity theory, slipping at lower stresses occurs in grains most favorably oriented with respect to the applied load. This dependence is quantitatively determined by the Schmid law, and the ability of the grain to slip is determined by the (b)
(c)
Fig. 5. Schmid factor maps for prismatic (10 10)[12 10] slip systems in an undeformed crystal (a) and deformed to ε = 5% (b), and plastic strain distribution (c). PHYSICAL MESOMECHANICS
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(b)
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Fig. 6. Equivalent plastic strain field (a) and deformation-induced surface roughness (b) in polycrystalline aluminum, ε = 23%.
Schmid factor defined through the angles between the normal to the slip plane and the slip direction and applied load vector. However, the forces applied to the grain surface at the microscale differ substantially from the forces applied to the specimen surface. As a result of microstructural inhomogeneity, the stress-strain state of the material is also highly inhomogeneous. Grains in the material bulk are actually under complex loading conditions and the real Schmid factor values differ from the values determined relative to the external force. The plastic deformation of grains is therefore largely governed by the local loading conditions that, in turn, depend on the loading conditions at the microvolume boundaries and deformation behavior of neighboring grains. Despite the differences in the elastic-plastic behavior of aluminum and titanium polycrystals, there are common features of deformation-induced surface roughening in the both materials. The roughness patterns formed during deformation of these materials exhibit two characteristic scales that can be respectively attributed to the micro- and mesoscales (Figs. 2 and 3). Microscale roughening occurs due to the displacement of neighboring grains (or parts of one grain) relative to each other perpendicular to the free surface (Figs. 2a and 2c). Microscale roughening begins at the stage of microplastic deformation, when macroscopic stresses are lower than the macroscopic yield stress, and becomes more pronounced with the increasing strain degree. The most intensive microscale roughening is observed at the initial stage of plastic flow. As a result of extrusion and intrusion of individual grains relative to the surrounding material, the grain boundaries emerging on the surface are clearly visible already in the early stage of plastic flow (Figs. 2a and 2c). Mesoscale roughening in the both materials occurs along with microscale roughening due to displacements PHYSICAL MESOMECHANICS
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of grain groups normal to the surface. The width of mesoscopic surface undulations is several grain diameters, which is clearly seen from the comparison of surface profiles (Figs. 3a3d) to grain structure (Figs. 3e and 3f). Although mesoscopic undulations appear almost simultaneously with microscale roughness, they become visible at a later stage of loading when their height becomes significantly larger compared to microdisplacements. The shape of the mesoscopic undulations is generally preserved throughout the deformation process. Peaks and valleys remain in the same regions, but evolve at different rates. At the stage of pronounced plastic deformation, mesoscopic undulations make the major contribution to deformation-induced surface roughening (Fig. 6b). The plastic strain distributions demonstrate two characteristic localization scales directly related to the deformation-induced roughening (Fig. 6a). Smaller-scale localization regions are associated with plastic deformation along grain boundaries and with the displacements and rotations of individual grains. A larger scale is associated with the formation of localization regions that encompass groups of grains and have a tendency to form across the tensile axis (Fig. 6a). Earlier, we proposed a dimensionless parameter for quantitative roughness evaluation [1113], called the degree of roughness, which is the ratio of the length of the surface profile to the length of its projection on a plane: L Rd = r − 1, (15) L0 where Lr and L0 are the lengths of the surface profile and its projection on a plane. In the three-dimensional case, this parameter is defined as the surface area to the area of the surface projection on a plane. The more the rough surface deviates from a plane, the higher is the
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tion from a plane was proposed for quantitative roughness evaluation. An exponential dependence of the roughness degree on the strain degree was obtained. ACKNOWLEDGMENTS The work was carried out in the framework of RFBR Projects No. 17-08-00643 A (micromechanical model of titanium) and 16-01-00469-a (micromechanical model of aluminum). Fig. 7. Evolution of the degree of roughness in model aluminum and titanium polycrystals.
value of Rd . The dependencies of the surface roughness degree for model aluminum and titanium polycrystals are shown in Fig. 7. In the both cases, these curves increase nonlinearly with the increasing strain degree. The curve for titanium increases more rapidly, which indicates a more pronounced roughness. In the both cases, a higher degree of deformation-induced surface roughness was obtained for profiles taken along the tensile axis. 4. CONCLUSIONS A micromechanical model was developed to describe deformation-induced surface roughening in polycrystalline materials at the micro- and mesoscales. The three-dimensional polycrystalline structure was introduced into the model in an explicit form. This allowed us to describe nonlinear effects of the geometric curvature of grain boundaries on the inhomogeneous stress-strain state in material microvolumes. The model of the elasticplastic response of grains with regard to the anisotropy governed by the crystal lattice structure was constructed within the crystal plasticity framework. Using the developed model, we numerically studied deformation-induced roughening in microvolumes of polycrystalline titanium and aluminum under uniaxial tension. Two characteristic roughness scales were distinguished in the both cases. At the microscale, normal displacements relative to the free surface are due to the formation of dislocation steps in grains emerging on the surface and due to the displacement of neighboring grains relative to each other. A more pronounced microscale roughness is observed in titanium owing to a high degree of elastic-plastic anisotropy in hcp crystals. The mesoscale roughness is represented by undulations and cluster structures that form with the involvement of grain groups. A dimensionless parameter, called the degree of roughness, reflecting the degree of surface shape devia-
REFERENCES 1. Raabe, D., Sachtleber, M., Weiland, H., Scheele, G., and Zhao, Z. Grain-Scale Micromechanics of Polycrystal Surfaces during Plastic Straining, Acta Mater., 2003, vol. 51, pp. 15391560. doi 10.1016/S1359-6454(02) 00557-8 2. Egorushkin, V.E., Panin, V.E., and Panin, A.V., Influence of Multiscale Localized Plastic Flow on Stress-Strain Patterns, Phys. Mesomech., 2015, vol. 18, no. 1, pp. 812. 3. Lee, P.S., Piehler, H.R., Adams, B.L., Jarvis, G., Hampel, H., and Rollett, A.D., Influence of Surface Texture on Orange Peel in Aluminum, J. Mater. Process. Technol., 1998, vol. 8081, pp. 315319. doi http://dx.doi.org/ 10.1016/S0924-0136(98)00189-7 4. Miranda-Medina, M.L., Somkuti, P., Bianchi, D., CihakBayr, U., Bader, D., Jech, M., and Vernes, A., Characterisation of Orange Peel on Highly Polished Steel Surfaces, Surf. Eng., 2015, vol. 31, pp. 519525. doi 10.1179/ 1743294414Y.0000000407 5. Shin, H.J., An, J.K., Park, S.H., and Lee, D.N., The Effect of Texture on Ridging of Ferritic Stainless Steel, Acta Mater., 2003, vol. 51, pp. 46934706. doi 10.1016/ S1359-6454(03)00187-3 6. Wouters, O., Vellinga, W.P., van Tijum, R., and De Hosson, J.T.M., Effects of Crystal Structure and Grain Orientation on the Roughness of Deformed Polycrystalline Metals, Acta Mater., 2006, vol. 54, pp. 28132821. doi 10.1016/j.actamat.2006.02.023 7. Derevyagina, L.S., Panin, V.E., and Gordienko, A.I., SelfOrganization of Plastic Shears in Localized Deformation Macrobands in the Neck of High-Strength Polycrystals, Its Role in Material Fracture under Uniaxial Tension, Phys. Mesomech., 2008, vol. 11, no. 12, pp. 5162. 8. Zuev, L.B., Autowave Model of Plastic Flow, Phys. Mesomech., 2011, vol. 14, no. 56, pp. 275282. doi 10.1016/ j.physme.2011.12.00613 9. Panin, A.V., Romanova, V.A., Balokhonov, R.R., Perevalova, O.B., Sinyakova, E.A., Emelyanova, O.S., Leontieva-Smirnova, M.V., and Karpenko, N.I., Mesoscopic Surface Folding in EK-181 Steel Polycrystals under Uniaxial Tension, Phys. Mesomech., 2012, vol. 15, no. 12, pp. 94103. doi 10.1134/S1029959912010109 PHYSICAL MESOMECHANICS
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No. 3
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MICROMECHANICAL MODEL OF DEFORMATION-INDUCED SURFACE ROUGHENING 10. Guilhem, Y., Basseville, S., Curtit, F., Stephan, J.M., and Cailletaud, G., Numerical Investigations of the Free Surface Effect in Three-Dimensional Polycrystalline Aggregates, Comput. Mater. Sci., 2013, vol. 70, pp. 150162. doi 10.1016/j.commatsci.2012.11.052 11. Romanova, V.A., Balokhonov, R.R., and Schmauder, S., Numerical Study of Mesoscale Surface Roughening in Aluminum Polycrystals under Tension, Mater. Sci. Eng. A, 2013, vol. 564, pp. 255263. doi 10.1016/j.msea.2012. 12.004 12. Romanova, V., Balokhonov, R., and Zinovieva, O., A Micromechanical Analysis of Deformation-Induced Surface Roughening in Surface-Modified Polycrystalline Materials, Meccanica, 2016, vol. 51, pp. 359370. doi 10.1007/ s11012-015-0294-x 13. Romanova, V., Balokhonov, R., Zinovieva, O., and Shakhidjanov, V., Numerical Study of the Surface Hardening Effect on the Deformation-Induced Roughening in Titanium Polycrystals, Comput. Mater. Sci., 2016, vol. 116, pp. 96102. 14. Needleman, A., Computational Mechanics at the Mesoscale, Acta Mater., 2000, vol. 48, pp. 105124. 15. Diard, O., Leclercq, S., Rousselier, G., and Cailletaud, G., Evaluation of Finite Element Based Analysis of 3D Multicrystalline Aggregates Plasticity. Application to Crystal Plasticity Model Identification and the Study of Stress and
PHYSICAL MESOMECHANICS
Vol. 20
No. 3
2017
16.
17.
18.
19.
20.
333
Strain Fields near Grain Boundaries, Int. J. Plast., 2005, vol. 21, pp. 691722. doi 10.1016/j.ijplas.2004.05.017 Roters, F., Eisenlohr, P., Hantcherli, L., Tjahjanto, D.D., Bieler, T.R., and Raabe, D., Overview of Constitutive Laws, Kinematics, Homogenization and Multiscale Methods in Crystal Plasticity Finite-Element Modeling: Theory, Experiments, Applications, Acta Mater., 2010, vol. 58, pp. 11521211. doi 10.1016/j.actamat.2009.10.058 Trusov, P.V. and Shveykin, A.I., Multilevel Crystal Plasticity Models of Single- and Polycrystals, Phys. Mesomech., 2013, vol. 16, pp. 99124. doi 10.1134/s1029959913 020021 Wang, L., Barabash, R.I., Yang, Y., Bieler, T.R., Crimp, M.A., Eisenlohr, P., Liu, W., and Ice, G.E., Experimental Characterization and Crystal Plasticity Modeling of Heterogeneous Deformation in Polycrystalline α-Ti, Metall. Mater. Trans. A, 2007, vol. 42, pp. 626635. Alankar, A., Eisenlohr, P., and Raabe, D., A Dislocation Density-Based Crystal Plasticity Constitutive Model for Prismatic Slip in α-Titanium, Acta Mater., 2011, vol. 59, pp. 70037009. doi 10.1016/j.actamat.2011.07.053 Becker, H. and Pantleon, W., Work-Hardening Stages and Deformation Mechanism Maps during Tensile Deformation of Commercially Pure Titanium, Comput. Mater. Sci., 2013, vol. 76, pp. 5259. doi 10.1016/j.commatsci. 2013.03.028
