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Computational analysis of deformation and fracture in a composite material on the mesoscale level
Computational Materials Science 37 (2006) 110–118 www.elsevier.com/locate/commatsci
Computational analysis of deformation and fracture in a composite material on the mesoscale level R.R. Balokhonov b
a,*
, V.A. Romanova a, S. Schmauder
b
a Institute of Strength Physics and Materials Science, SB, RAS, 634021 Tomsk, Russia Institut fu¨r Materialpru¨fung, Werkstoffkunde und Festigkeitslehre, University of Stuttgart, Germany
Abstract Presented in this paper are the computational results on deformation and fracture in an aluminum-matrix composite. 2D numerical simulations were carried out using the finite-difference method. To describe the mechanical behavior of elasto-plastic matrix and brittle inclusions use was made of different models—elasto-plastic formulation with the strain hardening and a cracking model with a fracture criterion of Huber type, respectively. The fracture criterion takes into account a difference in critical values for different local stress–strain states: tension and compression. The initial structure of the mesovolume, elastic and strength material constants, as well as the strain hardening function were chosen from the experiments. It has been shown that the composite mesovolume exhibits complex mechanical behavior controlled by both shear band formation in the matrix and cracking of inclusions. When applied to simulation of deformation and fracture in a heterogeneous medium, the criterion proposed allows one to describe adequately direction of crack propagation under different types of loading (tension and compression). The computational results have been analyzed in details, taking into account analytical solution for inclusion problem. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Computational mechanics; Composite materials; Crystal plasticity; Fracture; Mesomechanics
1. Introduction From the viewpoint of physical mesomechanics [1–4], development of non-uniform deformation in heterogeneous materials is primarily attributed to stress concentrators of different physical nature. A value of stress concentration in local areas of a material is mainly defined by both geometrical heterogeneity of different scales (e.g. uneven shape of interfaces and particles, surfaces, corners and geometrical features of the specimen) and a difference in mechanical characteristics of constituent materials. The stress concentration phenomena are most pronounced in composite materials (e.g. metal matrix composites, coated and surface-hardened materials, alloys of different sorts, etc.) due to considerable difference in mechanical properties of their components. *
Corresponding author. Tel.: +7 3822 286937; fax: +7 3822 492576. E-mail address: [email protected] (R.R. Balokhonov).
0927-0256/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2005.12.015
Nowadays it is evident that an adequate description of deformation behavior of complex media requires developing hierarchical models taking into account the interrelation between physical processes going on different scales. Along with it, we introduce a term of ‘‘hierarchical simulation’’ which implies the use of a series of simple models and their modifications, with the initial internal structure of a material being taken into account in an explicit form. Each of the models constructed for a certain material is capable to describe its mechanical behavior—elastic, plastic, brittle, viscous, etc. A combination of different simple models when simulating mechanical response of a complex medium has its goal to take into account the interrelation and interference of different physical processes. The majority of classical fracture criteria when applied to homogeneous material is not capable to describe adequately crack propagation under different kinds of loading without additional assumptions. A criterion of maximum tangential stress, widely used in plasticity, is not considered
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to provide reasonable description of brittle fracture at all. Maximum normal stress criterion well describes crack evolution under tension, but is not valid in the case of compression. Indeed, according to the experimental evidence, fracture under compression primarily takes place in planes, where the stress is equal to zero [5]. This means that the fracture criterion of maximum normal stress is never fulfilled under this type of loading. In order to correct this situation, a fracture criterion of maximum strains is introduced in classical continuum mechanics. The simplest one is the Mariott criterion [5] estimating the highest elongation along planes exhibiting zero level of the stress. In this paper, we show by the example of simulation of deformation and fracture in a metal matrix composite that in the special case a certain modification of the fracture criterion of Huber type allows us to correctly describe fracture of heterogeneous media under both tension and compression. We apply the approach to two-dimensionally simulate deformation and fracture of Al + Al2O3 composite. The mesoscopic structure consisting of an aluminum matrix and Al2O3-inclusions is explicitly incorporated in the calculations. To simulate the elasto-plastic response of the matrix use was made of an elasto-plastic model with strain hardening, and the fracture criterion of Huber type is modified to describe crack nucleation and propagation in the Al2O3-particles. The main goal of this work is to model numerically the composite structure response to loading, to investigate deformation and fracture mechanisms in the matrix and inclusions and to give a reasonable conclusion about the interrelation between processes going on at the different scale levels.
the upper dot and comma in subscripts denote time and space derivative, respectively. The mass conservation law takes the form:
2. Mathematical formulation of the problem
u_ 1 ðx; tÞ ¼ C 0 at t P 0; x 2 B1 ; u_ 1 ðx; tÞ ¼ C 0 at t P 0; x 2 B3 ;
ð7Þ
rij ðx; tÞ nj ¼ 0 ði; j ¼ 1; 2Þ at t P 0; x 2 B2 and x 2 B4 ;
ð8Þ
The mechanical problem of deformation of the composite mesovolume is solved by the method of a step-by-step calculation [6,7]. In order to describe the mechanical behavior of the aluminum matrix and ceramic inclusions, an elasto-plastic model with strain hardening and a cracking model using a fracture criterion of Huber type are introduced into the calculations. Mesoscopic deformation of the composite was simulated within the plane strain formulation. Numerical solutions were performed in terms of Lagrangian variables using the finite-difference method [6,7]. A total system of equations includes equations of motion and continuity, expressions for components of the strain rate tensor and constitutive equations. There are the following non-zero components of the strain rate tensor in the case of plane strain conditions: e_ 11 ¼ u_ 1;1 ;
e_ 22 ¼ u_ 2;2 ;
1 e_ 12 ¼ ðu_ 1;2 þ u_ 2;1 Þ; 2
ð1Þ
where u1 and u2 are the components of the displacement vector, e11, e22 and e12 are the strain tensor components,
q_ þ q ðu_ 1;1 þ u_ 2;2 Þ ¼ 0
ð2Þ
and the equations of motion are written as follows: r11;1 þ r21;2 ¼ q€u1 ;
r12;1 þ r22;2 ¼ q€u2 ;
ð3Þ
where r11, r22 and r12 are the stress tensor components and q is the current density. Taking into account both the resolution of the stress tensor into spherical and deviatoric parts rij = Pdij + Sij (dij-Kronecher delta) and the hypothesis of plastic incompressibility e_ pkk ¼ 0, we obtain for non-zero components of the stress deviator the following expressions: 1 1 S_ 11 ¼ 2l e_ e11 e_ ekk ; S_ 22 ¼ 2l e_ e22 e_ ekk ; 3 3 1 ð4Þ S_ 12 ¼ 2l_ee12 ; S_ 33 ¼ 2l e_ kk ¼ ðS_ 11 þ S_ 22 Þ. 3 The pressure is defined by the linear equation P_ ¼ K e_ kk .
ð5Þ
Here K is the bulk modulus and l is the shear modulus. Let us considering a domain D(x, t) (Fig. 1(a)) with the boundary B(x, t) = B1 [ B2 [ B3 [ B4, where x is the radius vector of a continuum point and t is the process time. At t = 0 all parameters of the stress–strain state are equal to zero throughout the area under study: u_ i ðxÞ ¼ 0;
rij ðxÞ ¼ 0;
qðxÞ ¼ qð0Þ ðxÞ.
ð6Þ
The boundary conditions (Fig. 1(a)) are given in the form:
where nj are components of the normal vector, C0 is a constant velocity which takes a positive or negative value depending on whether tension or compression is applied. 2.1. Plastic flow in matrix To describe plastic deformation in aluminum use was made of the plasticity theory law _ ij e_ pij ¼ kS
ð9Þ
associated with the yield criterion given in the form: Z req ¼ / depeq . ð10Þ Here k is a scalar parameter, e_ pij are the plastic strain rates, req and epeq are the equivalent stress and plastic strain, respectively:
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Fig. 1. Schematic of the mesovolume under study (a) and zoomed fragments of a matrix–inclusion interface (b).
1 n 2 2 2 req ¼ pffiffiffi ðS 11 S 22 Þ þ ðS 22 S 33 Þ þ ðS 33 S 11 Þ 2 o12 ð11Þ þ6 S 212 þ S 223 þ S 231 ; pffiffiffi n 2 ðep11 ep22 Þ2 þ ðep22 ep33 Þ2 þ ðep33 ep11 Þ2 epeq ¼ 3 o12 2 2 2 . ð12Þ þ6 ep12 þ ep23 þ ep31 R p The function /ð deeq Þ of a certain type prescribes strain hardening in the aluminum matrix. Substituting e_ eij ¼ e_ ij e_ pij in Eq. (4) and considering the associated flow rule (9) together with the criterion (10), the constitutive equations can be written as _ 11 ; _S 11 ¼ 2l e_ 11 1 e_ kk 2lkS 3 _ 22 ; _S 22 ¼ 2l e_ 22 1 e_ kk 2lkS 3 _ 33 ¼ ðS_ 11 þ S_ 22 Þ; _S 33 ¼ 2l 1 e_ kk 2lkS 3 _ 12 . S_ 12 ¼ 2l_e12 2lkS ð13Þ 2.2. Fracture in inclusions To take into consideration fracture in alumina inclusions use was made of the energetic criterion of Huber type (I2, P, Cten, Ccom) = 0: C ten ; if P > 0 ) S ij ¼ 0 and P ¼ 0; req ¼ ð14Þ C com ; if P < 0 ) S ij ¼ 0. Here, Cten and Ccom are the constants that characterize the yield strength of Al2O3 under tension and compression, respectively, I2 is the second invariant of the stress tensor. Mathematically the following conditions are associated with any local region of inclusions: if both bulk deformation ekk takes on a positive value and req reaches its critical value of Ccom then all components of the deviatoric stress tensor in this region are accepted to be zero, and in the case
of ekk < 0 and req P Cten, pressure P is equal to zero as well. Thus, if the fracture criterion is fulfilled, the material of inclusions behave themselves similar to an uncompressed liquid, where the material density maintains to be constant and corresponds to the density of Al2O3. Physically, criterion (14) means that for the inclusion material any local region subjected to tension is fractured, if the equivalent stress in this local place reaches its critical value of Cten. For compressed regions the limiting surface in the space of stresses is confined by the value of Ccom. 3. Calculation results Fig. 1 shows a map of the test cut-out. The image corresponds to a real mesostructure experimentally investigated in [8,9]. Mechanical properties of the matrix and inclusions are presented in Table 1 in accordance with the experimental data [10]. A strain hardening function takes the following form in the calculations: /(eeq) = 170 65 exp(eeq/ 0.048) [MPa], where r0 = /(0) = 105 [MPa] is the yield point. This dependence reflects plastic response of the experimental aluminum alloy used as the matrix. Boundary conditions (7) and (8) on the right and left sides of the area under calculation simulate grip displacement speed, while on the top and bottom surfaces they correspond to free surface conditions (Fig. 1(a)). The stress–strain diagrams of the mesovolume under study are presented in Fig. 2 for the cases of tension and compression. The stress hri was calculated as an average value of the equivalent stress over the mesovolume:
Table 1 Mechanical properties [10] Material
K l m [–] q Strength under Strength [GPa] [GPa] [kg/m3] under compression, tension, Cten Ccom
Al2O3 Al(6061)
438 76
141 26
0.35 3990 0.35 2700
260 –
4000 –
R.R. Balokhonov et al. / Computational Materials Science 37 (2006) 110–118 70
< σ >, MPa
113
a
65
f
b
150 60
e c
120
153
55
k 152
d l
90 151
-0.49
60
-0.485
-0.48
129
h
i 126
30
j 123
120
0
g 117 -0.28
-0.4
-0.24
-0.2
0.0
0.2
Ε,%
Fig. 2. Stress–strain curves under tension and compression.
P k k k¼1;N req v hri ¼ P ; k k¼1;N v where N is the amount of grid nodes, vk is the kth local volume. On the abscissa the true strain of the mesovolume is plotted: E = L L0/L0, where L0 and L are the initial and current lengths of the specimen along the X1-axis. Fig. 2 shows that the composition withstands the higher level of load under compression then under tension. The behavior qualitatively well corresponds to the experimental behavior of metal matrix composites. At first glance, the difference between the curves calculated for tension and compression could be explained by a considerable variation in the compressive and tensile strength of inclusions. Since Ccom Cten, fracture in local compressed regions is expected to arise at essentially higher average stress levels then in the places undergoing tensile deformation. However, there is no proportional increasing in the macroscopic current strength relative to the ratio of the volume fractions
of aluminum and alumina. This is a result of two reasons. Firstly, the aluminum matrix deforms plastically that results in a decrease of the average stress level throughout the mesovolume because the stress deviator is limited and the presence of the free surfaces hinders an increase of the pressure. Secondly, the calculations of the mesovolume deformation under different types of loading show that the value of Ccom is never reached. Due to structural heterogeneity and the presence of the interfaces, there are local tensile regions even under compression of the mesovolume. Let us consider this in greater details. Images of stress tensor components, Fig. 3, demonstrate a non-homogeneous distribution of stresses even at the elastic stage of tension. Fig. 3(a) indicates that r11 is more then or equal to 0, i.e. there is no compressive stress in this direction throughout the area. Different situation is observed for the r22- and r12-components which describe material response to deformation perpendicular to the axis of tension and shear of local areas, correspondingly. Note that in the case of a homogeneous medium subjected to
Fig. 3. Distribution of the stress tensor components at the elastic stage of tension.
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uniaxial loading, these stress components are identically zero throughout the area under consideration. Thus, the complex stress state characterized by non-zero values of r22 and r12-components is mainly attributed to the material heterogeneity and presence of structural interfaces of different scales. Refer to r22 image in Fig. 3(b), in the direction perpendicular to the axis of tension there are regions under compression and those subjected to tension. The inclusions can be under action of positive and either negative shear stress components (areas marked by ‘‘+’’ and ‘‘’’ in Fig. 3(c)), which is caused by different modes of deformation, i.e. the ‘‘+’’ and ‘‘’’-marked inclusions are stretched into ellipse and tend to take a spherical shape, correspondingly. In the case of compression, the stress–strain patterns are equal in values to those under tension but opposite in sign. Quantitative analysis of the stress state in the mesovolume under study has shown that r22 and r12 take maximum values in the vicinity of inclusion boundaries, with max 1 max rmax 22 r12 5 r11 . Taking into account the fact that the critical values Cten and Ccom differ more then by order, the r22- and r12-stresses have essential influence on the deformation and fracture processes in local areas of the composite material. Thus, in the mesovolume under compression, there are local areas undergoing tension, and tensile stresses in these areas can take considerable values. This conclusion is very important for the analysis of plastic deformation in the matrix and dynamics of crack propagation in brittle inclusions at later stages of loading. According to the experimental data, the composite material under study consists of the metal matrix and ceramic inclusions. Let us analyze in more detail the stress distribution in the vicinity of a single inclusion which, for the sake of simplicity, is circle-shaped, Fig. 4. Such a simplification can be accepted since, statistically, inclusions are of a round shape in average. The analytical solution obtained in [11] for a so-called inclusion problem were used, where a cylindrical inclusion of round cross-section is embedded into an infinite area, with the inclusion and matrix materials are distinguished by their mechanical properties. The area with the embedded inclusion and schematic of its loading are presented in Fig. 4. For the case of arbitrary materials, the problem solution in cylindrical coordinates gives the following expressions for stress components in matrix [11]:
D S
C
μ 1, ν 1
μ 2, ν 2 A
S ϕ
B
Fig. 4. Schematic of the inclusion problem.
S 1 þ 2a1 b 2a2 b a2 1þ 2 2a2 þ b 1 r
4ð1 bÞ a2 3ð1 bÞ a4 1 þ cos 2u ; 1 b þ 4a1 b r 1 b þ 4a1 b r S 1 þ 2a1 b 2a2 b a2 1 ¼ 2 2a2 þ b 1 r
3ð1 bÞ a4 1þ cos 2u ; 1 b þ 4a1 b r S 2ð1 bÞ a2 1þ ¼ 2 1 b þ 4a1 b r 3ð1 bÞ a4 sin 2u. 1 b þ 4a1 b r
rrr ¼
ruu
rru
ð15Þ l2 ; l1
Here, a is the inclusion radius, b ¼ ai ¼ for plane strain 1 mi , i = 1, 2; mi is the Poison 1 ð1 þ mi Þ for plane stress ratio, and S is the external stress. For the metal matrix composite investigated in this work, b 5.4 and a1 a2 = 0.65 under plane strain conditions. The stress components on the boundary between matrix and inclusion, where r = a, are rrr = r11 = 1.35S, ruu = r22 = 0.57S, rru = r12 = 0 at u = 0, p and rrr = r22 = 0.11S, ruu = r11 = 0.2S, rru = r12 = 0 at u = p/2, 3p/2. Therefore, all non-zero components of the stress tensor in the direction of tension are positive in points A–D (refer to Fig. 4), with the maximum values in points A and C. These local areas also undergo tensile loading along the X2-axis. In points B and D material is under compression in the direction perpendicular to the axis of tension. The results of the analytical solution well agree qualitatively and quantitatively with the numerical results in the case of inclusions of similar shape (e.g. inclusion 1 in Fig. 3). It is of interest to estimate the stress distribution in the vicinity of inclusions, analyzing limiting cases. If the elastic moduli of the inclusion considerably exceed those of the matrix, i.e. l2 l1, then b ! 1 and Eq. (15) take the form: 2a1 cos 2u ; rrr ¼ S a1 þ 4a1 1 2ða1 1Þ cos 2u ; ruu ¼ S 1 a1 4a1 1 2Sa1 rru ¼ sin 2u. ð16Þ 1 4a1 In this case, at a1 = 0.5 in points A and C rrr = ruu = 1.5S, i.e. material is under tension, and in B and D regions rrr = ruu = 0.5S, i.e. these areas undergo compression along both X1- and X2-axes. The estimations given above hold for the scale level associated with the average size of inclusions. According to the terminology of physical mesomechanics this scale can be defined as a mesoII scale [3]. For the structure under consideration, this characteristic size is about 20 lm (Fig. 1(a)). The stress concentrators appearing in
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the matrix at the elastic stage of loading give rise to the formation of macroscopic shear bands under plastic deformation. The same analysis can also be done for the mesoI- and micro-level, where the objects of investigation are individual inclusions and the scale under study is mainly defined by the uneven geometrical shape of the matrix–inclusion interface and the extension of local fractured areas in the inclusions as well. Under further loading, the stress concentration near the interface irregularities leads to the plastic strain localization in the matrix and crack nucleation and propagation in the inclusions. Areas of the highest stress concentration in the structure under study are presented in Fig. 1 on an enlarged scale. Let us analyze the stress distribution in the vicinity of points A, C and D, using expressions (15). On this scale, the problem is reduced to the consideration of a metal inclusion surrounded by ceramic material. Characteristic size of such an irregularity is 2 lm (Fig. 1(b)). Taking into account the fact that the interface area is of a semi-circle shape, only qualitative estimations will be given in order to analyze the criterion (14) for the mesoI scale. At l1 > l2 for the composite under study b 0.18, then at r = a the stress tensor components under tensile loading take the following values: rrr = r22 = 0.06S, ruu = r11 = 2.2S and rru = r12 0 in D point (u p/2) and rrr = r11 = 0.4S, ruu = r22 = 0.7S and rru = r12 0 in A and C areas (u 0, p). Thus, under external tension the stress tensor components and, hence, req take their maximum values in the D-type areas which undergo tensile deformation. It is evident that first cracks will nucleate in the places 1 and 2 as the condition req = Cten fulfils. In the case of compression under external stress S, the D-areas are under compressive deformation along both X1- and X2-axes, i.e. there is a negative pressure in these places and fracture is defined by the criterion req = Ccom, according to Eq. (14). In points A and C ruu = r22 = 0.7S, i.e. material undergoes tensile deformation in the direction perpendicular to the axis of external compression. In these places the pressure is positive and fracture is controlled by criterion req = Cten. Refer to Table 1, for the material of inclusions Ccom Cten. Because of this, first cracks are expected to nucleate in the vicinity of points A and C, even though the equivalent stress req takes its maximum value in the D-points. We can conclude, therefore, that in the mesovolume under tension and compression, cracks nucleate in different places. It has been shown that the composite mesovolume exhibits a complex mechanical behavior controlled by both shear band formation in the matrix and cracking of inclusions. Evolution of fracture in the inclusions and plastic deformation in the matrix under tension and compression is illustrated in Figs. 5 and 6, correspondingly. There is a marked difference in the crack orientation relative to the axis of loading under tension and compression. The cracks tend to propagate perpendicular to the loading axis in the case of tension and parallel under compression. This frac-
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ture behavior is well known from experiments [5] and additionally confirmed by estimations given above. Let us analyze the stress state in the vicinity of a local fractured area whose specific size is about 250 nm (micro-level, Fig. 1(b)). For the sake of simplicity, let us consider the fracture area in the form of an ideal circle and find the solution (15) at b ! 0. Due to this simplification, the problem is reduced to a classical solution of elasticity theory on estimation of the effect of a round hole on the stress distribution in a plate [12]. Such an idealization can be accepted, because of the local area under study is much less in size in comparison with the mesovolume, and inclusions are treated as elastic. According to (15), at r = a the only component of the stress tensor remains to be non-zero: ruu ¼ Sð1 2 cos 2uÞ. This stress takes its maximum value ruu = 3S in points B and D and a negative magnitude ruu = S corresponding to the compressive response in places A and C. When q S ffiffiffiffiffiffiffi is further increased, the criterion (14) in the form ffi req ¼ r2uu ¼ jruu j ¼ C ten is first fulfilled in points B and D, which results in crack nucleation in these areas. Under further loading, the process repeats so that the crack propagates perpendicular to the axis of loading. Under compression, cracks demonstrate different behavior. There exists a compressive stress of ruu = 3S in points B and D. Even though the equivalent stress in these points reaches its maximum value, the strength of Al2O3 under compression isqorders of magnitude higher than ffiffiffiffiffiffiffiffi under tension, req ¼
r2uu ¼ jruu j ¼ C com C ten . In this
connection, fracture takes place in the vicinity of areas A and C which undergo tensile deformation, ruu = S. Thus, the crack propagates parallel to the direction of external tensile loading. Presented in Figs. 5 and 6 are the patterns of equivalent stresses and plastic strains and velocity vector fields laid over the structure image. Compare with the average curves in Fig. 2, pictures (a–f) and (g–l) correspond to tension and compression, respectively. Fig. 5 shows that under compression the first crack nucleates in the biggest inclusion near point D (see Fig. 1), where the concentration of equivalent stress req takes its maximum value. The initial crack propagates perpendicular to the direction of tension until it reaches the opposite boundary of the inclusion (Fig. 5(a)–(d)). This process is accompanied with unloading of adjacent material due to release waves propagating from newly formed free surfaces, which corresponds to the descending part of the stress–strain curve, confined between points a and d in Fig. 2. The velocity fields in Fig. 5(a)–(c) illustrate the crack opening. The complex stress state forms due to an interaction of elastic waves with interfaces, with the development of vortex movements. Rotation of separate local areas leads to an additional stress concentration (Fig. 5(c)). Under further loading an average level of stresses in the mesovolume increases and stress concentration appears in smaller inclusions, that results in nucleation of
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Fig. 5. Equivalent stress, plastic strain patterns and velocity vector fields for different steps of tension.
new cracks which, in turn, unload surrounding material (Fig. 5(e)–(f)). Let us analyze the images of plastic strains, presented in Fig. 5(d)–(f), where fractured areas are plotted in black color. These pictures indicate that under tension cracks nucleate and propagate at the elastic stage of loading. This is due to the fact that even though an average value of stresses in the matrix is lower than the yield point r0 = 105 MPa, the stress concentration in local places of the inclusions can exceed its critical fracture value Cten = 260 MPa. Weak plastic deformation in the form of localized shear bands is observed near the places of crack nucleation (Fig. 5(d)–(f)). Under compression, Fig. 6, first cracks nucleate in places of highest concentration of tensile stresses (points A and C in Fig. 1). Note, in the places of local compression (points D) the equivalent stress reaches considerably higher values. This is the reason that a fracture process
under compression goes at a higher average level of stresses than under tension (Fig. 2) and is accompanied by intensive plastic deformation in the matrix (Fig. 6(j)–(l)). Cracks propagate parallel to the axis of compression and, in contrast to the case of tension, this process demonstrates an intermittent character. Average velocity of crack propagation through inclusions under compression is lower than that under tension. Intensive plastic deformation in the matrix considerably retards stress increase in the places of stress concentration. As a result, cracking in inclusions goes in a so-called ‘‘switching’’ mode, which is illustrated by velocity fields in Fig. 6(g)–(i). Refer to Fig. 6(g), where nucleation of first crack causes moderate unloading in the mesovolume. At the next stage of loading the crack stops to move, whereas the average stress curve demonstrates a rather prolonged ascending part confined within dots g and h in Fig. 2 due to strain hardening in the matrix. Finally, a new crack nucleates in
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Fig. 6. Equivalent stress, plastic strain patterns and velocity vector fields for different steps of compression.
the other inclusion, Fig. 6(h). Compare part of the curve between points i and k in Fig. 2 and pictures (i)–(k) in Fig. 6, as the second crack stops to grow, the first one continues to propagate again in a jumping manner toward the opposite boundary of the inclusion. Under further loading the third crack nucleates (Fig. 6(l)) and the process repeats. 4. Conclusion In this paper, a fracture criterion of Huber’s type is modified and applied in numerical investigations of deformation and fracture processes in the mesovolume of the composite material consisting of the aluminum matrix and ceramic inclusions. Using an analytical solution for a plate with a round hole, the results of numerical simulation for the mesovolume under tension and compression have been analyzed. A qualitative and quantitative comparison of the stress concentration caused by structural heterogene-
ities of different scales, such as individual inclusions, irregularities of matrix–inclusion interfaces and local areas of fracture, has been carried out. Specific sizes of heterogeneities on the mesoII-, mesoI- and micro-levels are 20–30 lm, 2–3 lm and 250 nm, correspondingly. It has been shown that, from the viewpoint of mechanics, stress concentration has geometrical meaning and identical nature on different scale levels. Summing up the computational results for tension and compression, we can make the following conclusions: 1. Cracks propagate perpendicular to the loading direction under tension and parallel under compression as expected. 2. Due to strain incompatibility near the aluminum-alumina interfaces, local areas of tensile deformation form under compressive loading. And vice versa, under macroscopic tension along the same axis these local regions undergo compressive stresses.
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3. Local fracture areas first appear at the places of most powerful concentration of tensile stresses which give rise to crack propagation under both tension and compression, i.e. all cracks nucleating in the mesovolume are so-called ‘‘tensile cracks’’ regardless of the kind of loading. 4. Under compression cracking of inclusions is accompanied by intensive plastic deformation in the matrix, whereas under tension first cracks originate and begin to grow at the elastic stage of deformation.
Acknowledgments The support of CRDF and Russian Federation Ministry of Science and Education (BRHE016-02) and Russian Science Support Foundation is gratefully acknowledged. References [1] V.E. Panin, V.A. Lihachev, Ju.V. Grinyev, Structural Levels of Deformation of Solids, Nauka, Moscow, 1985, 230 p.
[2] V.E. Panin (Ed.), Physical Mesomechanics of Heterogeneous Media and Computer-Aided Design of Materials, Cambridge Int. Science Publishing, 1998, p. 339. [3] V.E. Panin, Physical Mesomechanics 1 (1998) 5–22. [4] V.E. Panin, L.S. Derevyagina, Ye.Ye. Deryugin, A.V. Panin, S.V. Panin, N.A. Antipina, Physical Mesomechanics 6 (2003) 97–106. [5] L.M. Kachanov, Fundamentals of Fracture Mechanics, Nauka, Moscow, 1974, 312 p. [6] P.V. Makarov, S. Schmauder, O.I. Cherepanov, I.Yu. Smolin, V.A. Romanova, R.R. Balokhonov, D.Yu. Saraev, E. Soppa, P. Kizler, G. Fischer, S. Hu, M. Ludwig, Theoretical and Applied Fracture Mechanics 37 (1–3) (2001) 183–244. [7] R.R. Balokhonov, S.V. Panin, V.A. Romanova, S. Schmauder, P.V. Makarov, Theoretical and Applied Fracture Mechanics 41 (1–3) (2004) 9–14. [8] E. Soppa, S. Schmauder, G. Fischer, J. Thesing, R. Ritter, Computation Material Science 16 (1999) 323–332. [9] E. Soppa, S. Schmauder, G. Fischer, J. Brollo, U. Weber, Computation Material Science 28 (2003) 574. [10] A.P. Babichev, N.A. Babushkina, A.M. Bratkovskii et al., in: I.S. Grigorieva, E.Z. Meilihova, M. (Eds.), Energoatomizdat. Physical Values: Reference Book, 1991, 1232 p. ISBN 5-283-04013-5. [11] Mal Ajit K., Singh, Sarva Jit, Deformation of Elastic Solids, Pearson Higher Education, 1990, 534 p. ISBN/EAN: 0132007002. [12] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, Nauka, Moscow, 1979, 560 p.
Computational analysis of deformation and fracture in a composite material on the mesoscale level R.R. Balokhonov b
a,*
, V.A. Romanova a, S. Schmauder
b
a Institute of Strength Physics and Materials Science, SB, RAS, 634021 Tomsk, Russia Institut fu¨r Materialpru¨fung, Werkstoffkunde und Festigkeitslehre, University of Stuttgart, Germany
Abstract Presented in this paper are the computational results on deformation and fracture in an aluminum-matrix composite. 2D numerical simulations were carried out using the finite-difference method. To describe the mechanical behavior of elasto-plastic matrix and brittle inclusions use was made of different models—elasto-plastic formulation with the strain hardening and a cracking model with a fracture criterion of Huber type, respectively. The fracture criterion takes into account a difference in critical values for different local stress–strain states: tension and compression. The initial structure of the mesovolume, elastic and strength material constants, as well as the strain hardening function were chosen from the experiments. It has been shown that the composite mesovolume exhibits complex mechanical behavior controlled by both shear band formation in the matrix and cracking of inclusions. When applied to simulation of deformation and fracture in a heterogeneous medium, the criterion proposed allows one to describe adequately direction of crack propagation under different types of loading (tension and compression). The computational results have been analyzed in details, taking into account analytical solution for inclusion problem. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Computational mechanics; Composite materials; Crystal plasticity; Fracture; Mesomechanics
1. Introduction From the viewpoint of physical mesomechanics [1–4], development of non-uniform deformation in heterogeneous materials is primarily attributed to stress concentrators of different physical nature. A value of stress concentration in local areas of a material is mainly defined by both geometrical heterogeneity of different scales (e.g. uneven shape of interfaces and particles, surfaces, corners and geometrical features of the specimen) and a difference in mechanical characteristics of constituent materials. The stress concentration phenomena are most pronounced in composite materials (e.g. metal matrix composites, coated and surface-hardened materials, alloys of different sorts, etc.) due to considerable difference in mechanical properties of their components. *
Corresponding author. Tel.: +7 3822 286937; fax: +7 3822 492576. E-mail address: [email protected] (R.R. Balokhonov).
0927-0256/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2005.12.015
Nowadays it is evident that an adequate description of deformation behavior of complex media requires developing hierarchical models taking into account the interrelation between physical processes going on different scales. Along with it, we introduce a term of ‘‘hierarchical simulation’’ which implies the use of a series of simple models and their modifications, with the initial internal structure of a material being taken into account in an explicit form. Each of the models constructed for a certain material is capable to describe its mechanical behavior—elastic, plastic, brittle, viscous, etc. A combination of different simple models when simulating mechanical response of a complex medium has its goal to take into account the interrelation and interference of different physical processes. The majority of classical fracture criteria when applied to homogeneous material is not capable to describe adequately crack propagation under different kinds of loading without additional assumptions. A criterion of maximum tangential stress, widely used in plasticity, is not considered
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to provide reasonable description of brittle fracture at all. Maximum normal stress criterion well describes crack evolution under tension, but is not valid in the case of compression. Indeed, according to the experimental evidence, fracture under compression primarily takes place in planes, where the stress is equal to zero [5]. This means that the fracture criterion of maximum normal stress is never fulfilled under this type of loading. In order to correct this situation, a fracture criterion of maximum strains is introduced in classical continuum mechanics. The simplest one is the Mariott criterion [5] estimating the highest elongation along planes exhibiting zero level of the stress. In this paper, we show by the example of simulation of deformation and fracture in a metal matrix composite that in the special case a certain modification of the fracture criterion of Huber type allows us to correctly describe fracture of heterogeneous media under both tension and compression. We apply the approach to two-dimensionally simulate deformation and fracture of Al + Al2O3 composite. The mesoscopic structure consisting of an aluminum matrix and Al2O3-inclusions is explicitly incorporated in the calculations. To simulate the elasto-plastic response of the matrix use was made of an elasto-plastic model with strain hardening, and the fracture criterion of Huber type is modified to describe crack nucleation and propagation in the Al2O3-particles. The main goal of this work is to model numerically the composite structure response to loading, to investigate deformation and fracture mechanisms in the matrix and inclusions and to give a reasonable conclusion about the interrelation between processes going on at the different scale levels.
the upper dot and comma in subscripts denote time and space derivative, respectively. The mass conservation law takes the form:
2. Mathematical formulation of the problem
u_ 1 ðx; tÞ ¼ C 0 at t P 0; x 2 B1 ; u_ 1 ðx; tÞ ¼ C 0 at t P 0; x 2 B3 ;
ð7Þ
rij ðx; tÞ nj ¼ 0 ði; j ¼ 1; 2Þ at t P 0; x 2 B2 and x 2 B4 ;
ð8Þ
The mechanical problem of deformation of the composite mesovolume is solved by the method of a step-by-step calculation [6,7]. In order to describe the mechanical behavior of the aluminum matrix and ceramic inclusions, an elasto-plastic model with strain hardening and a cracking model using a fracture criterion of Huber type are introduced into the calculations. Mesoscopic deformation of the composite was simulated within the plane strain formulation. Numerical solutions were performed in terms of Lagrangian variables using the finite-difference method [6,7]. A total system of equations includes equations of motion and continuity, expressions for components of the strain rate tensor and constitutive equations. There are the following non-zero components of the strain rate tensor in the case of plane strain conditions: e_ 11 ¼ u_ 1;1 ;
e_ 22 ¼ u_ 2;2 ;
1 e_ 12 ¼ ðu_ 1;2 þ u_ 2;1 Þ; 2
ð1Þ
where u1 and u2 are the components of the displacement vector, e11, e22 and e12 are the strain tensor components,
q_ þ q ðu_ 1;1 þ u_ 2;2 Þ ¼ 0
ð2Þ
and the equations of motion are written as follows: r11;1 þ r21;2 ¼ q€u1 ;
r12;1 þ r22;2 ¼ q€u2 ;
ð3Þ
where r11, r22 and r12 are the stress tensor components and q is the current density. Taking into account both the resolution of the stress tensor into spherical and deviatoric parts rij = Pdij + Sij (dij-Kronecher delta) and the hypothesis of plastic incompressibility e_ pkk ¼ 0, we obtain for non-zero components of the stress deviator the following expressions: 1 1 S_ 11 ¼ 2l e_ e11 e_ ekk ; S_ 22 ¼ 2l e_ e22 e_ ekk ; 3 3 1 ð4Þ S_ 12 ¼ 2l_ee12 ; S_ 33 ¼ 2l e_ kk ¼ ðS_ 11 þ S_ 22 Þ. 3 The pressure is defined by the linear equation P_ ¼ K e_ kk .
ð5Þ
Here K is the bulk modulus and l is the shear modulus. Let us considering a domain D(x, t) (Fig. 1(a)) with the boundary B(x, t) = B1 [ B2 [ B3 [ B4, where x is the radius vector of a continuum point and t is the process time. At t = 0 all parameters of the stress–strain state are equal to zero throughout the area under study: u_ i ðxÞ ¼ 0;
rij ðxÞ ¼ 0;
qðxÞ ¼ qð0Þ ðxÞ.
ð6Þ
The boundary conditions (Fig. 1(a)) are given in the form:
where nj are components of the normal vector, C0 is a constant velocity which takes a positive or negative value depending on whether tension or compression is applied. 2.1. Plastic flow in matrix To describe plastic deformation in aluminum use was made of the plasticity theory law _ ij e_ pij ¼ kS
ð9Þ
associated with the yield criterion given in the form: Z req ¼ / depeq . ð10Þ Here k is a scalar parameter, e_ pij are the plastic strain rates, req and epeq are the equivalent stress and plastic strain, respectively:
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Fig. 1. Schematic of the mesovolume under study (a) and zoomed fragments of a matrix–inclusion interface (b).
1 n 2 2 2 req ¼ pffiffiffi ðS 11 S 22 Þ þ ðS 22 S 33 Þ þ ðS 33 S 11 Þ 2 o12 ð11Þ þ6 S 212 þ S 223 þ S 231 ; pffiffiffi n 2 ðep11 ep22 Þ2 þ ðep22 ep33 Þ2 þ ðep33 ep11 Þ2 epeq ¼ 3 o12 2 2 2 . ð12Þ þ6 ep12 þ ep23 þ ep31 R p The function /ð deeq Þ of a certain type prescribes strain hardening in the aluminum matrix. Substituting e_ eij ¼ e_ ij e_ pij in Eq. (4) and considering the associated flow rule (9) together with the criterion (10), the constitutive equations can be written as _ 11 ; _S 11 ¼ 2l e_ 11 1 e_ kk 2lkS 3 _ 22 ; _S 22 ¼ 2l e_ 22 1 e_ kk 2lkS 3 _ 33 ¼ ðS_ 11 þ S_ 22 Þ; _S 33 ¼ 2l 1 e_ kk 2lkS 3 _ 12 . S_ 12 ¼ 2l_e12 2lkS ð13Þ 2.2. Fracture in inclusions To take into consideration fracture in alumina inclusions use was made of the energetic criterion of Huber type (I2, P, Cten, Ccom) = 0: C ten ; if P > 0 ) S ij ¼ 0 and P ¼ 0; req ¼ ð14Þ C com ; if P < 0 ) S ij ¼ 0. Here, Cten and Ccom are the constants that characterize the yield strength of Al2O3 under tension and compression, respectively, I2 is the second invariant of the stress tensor. Mathematically the following conditions are associated with any local region of inclusions: if both bulk deformation ekk takes on a positive value and req reaches its critical value of Ccom then all components of the deviatoric stress tensor in this region are accepted to be zero, and in the case
of ekk < 0 and req P Cten, pressure P is equal to zero as well. Thus, if the fracture criterion is fulfilled, the material of inclusions behave themselves similar to an uncompressed liquid, where the material density maintains to be constant and corresponds to the density of Al2O3. Physically, criterion (14) means that for the inclusion material any local region subjected to tension is fractured, if the equivalent stress in this local place reaches its critical value of Cten. For compressed regions the limiting surface in the space of stresses is confined by the value of Ccom. 3. Calculation results Fig. 1 shows a map of the test cut-out. The image corresponds to a real mesostructure experimentally investigated in [8,9]. Mechanical properties of the matrix and inclusions are presented in Table 1 in accordance with the experimental data [10]. A strain hardening function takes the following form in the calculations: /(eeq) = 170 65 exp(eeq/ 0.048) [MPa], where r0 = /(0) = 105 [MPa] is the yield point. This dependence reflects plastic response of the experimental aluminum alloy used as the matrix. Boundary conditions (7) and (8) on the right and left sides of the area under calculation simulate grip displacement speed, while on the top and bottom surfaces they correspond to free surface conditions (Fig. 1(a)). The stress–strain diagrams of the mesovolume under study are presented in Fig. 2 for the cases of tension and compression. The stress hri was calculated as an average value of the equivalent stress over the mesovolume:
Table 1 Mechanical properties [10] Material
K l m [–] q Strength under Strength [GPa] [GPa] [kg/m3] under compression, tension, Cten Ccom
Al2O3 Al(6061)
438 76
141 26
0.35 3990 0.35 2700
260 –
4000 –
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< σ >, MPa
113
a
65
f
b
150 60
e c
120
153
55
k 152
d l
90 151
-0.49
60
-0.485
-0.48
129
h
i 126
30
j 123
120
0
g 117 -0.28
-0.4
-0.24
-0.2
0.0
0.2
Ε,%
Fig. 2. Stress–strain curves under tension and compression.
P k k k¼1;N req v hri ¼ P ; k k¼1;N v where N is the amount of grid nodes, vk is the kth local volume. On the abscissa the true strain of the mesovolume is plotted: E = L L0/L0, where L0 and L are the initial and current lengths of the specimen along the X1-axis. Fig. 2 shows that the composition withstands the higher level of load under compression then under tension. The behavior qualitatively well corresponds to the experimental behavior of metal matrix composites. At first glance, the difference between the curves calculated for tension and compression could be explained by a considerable variation in the compressive and tensile strength of inclusions. Since Ccom Cten, fracture in local compressed regions is expected to arise at essentially higher average stress levels then in the places undergoing tensile deformation. However, there is no proportional increasing in the macroscopic current strength relative to the ratio of the volume fractions
of aluminum and alumina. This is a result of two reasons. Firstly, the aluminum matrix deforms plastically that results in a decrease of the average stress level throughout the mesovolume because the stress deviator is limited and the presence of the free surfaces hinders an increase of the pressure. Secondly, the calculations of the mesovolume deformation under different types of loading show that the value of Ccom is never reached. Due to structural heterogeneity and the presence of the interfaces, there are local tensile regions even under compression of the mesovolume. Let us consider this in greater details. Images of stress tensor components, Fig. 3, demonstrate a non-homogeneous distribution of stresses even at the elastic stage of tension. Fig. 3(a) indicates that r11 is more then or equal to 0, i.e. there is no compressive stress in this direction throughout the area. Different situation is observed for the r22- and r12-components which describe material response to deformation perpendicular to the axis of tension and shear of local areas, correspondingly. Note that in the case of a homogeneous medium subjected to
Fig. 3. Distribution of the stress tensor components at the elastic stage of tension.
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uniaxial loading, these stress components are identically zero throughout the area under consideration. Thus, the complex stress state characterized by non-zero values of r22 and r12-components is mainly attributed to the material heterogeneity and presence of structural interfaces of different scales. Refer to r22 image in Fig. 3(b), in the direction perpendicular to the axis of tension there are regions under compression and those subjected to tension. The inclusions can be under action of positive and either negative shear stress components (areas marked by ‘‘+’’ and ‘‘’’ in Fig. 3(c)), which is caused by different modes of deformation, i.e. the ‘‘+’’ and ‘‘’’-marked inclusions are stretched into ellipse and tend to take a spherical shape, correspondingly. In the case of compression, the stress–strain patterns are equal in values to those under tension but opposite in sign. Quantitative analysis of the stress state in the mesovolume under study has shown that r22 and r12 take maximum values in the vicinity of inclusion boundaries, with max 1 max rmax 22 r12 5 r11 . Taking into account the fact that the critical values Cten and Ccom differ more then by order, the r22- and r12-stresses have essential influence on the deformation and fracture processes in local areas of the composite material. Thus, in the mesovolume under compression, there are local areas undergoing tension, and tensile stresses in these areas can take considerable values. This conclusion is very important for the analysis of plastic deformation in the matrix and dynamics of crack propagation in brittle inclusions at later stages of loading. According to the experimental data, the composite material under study consists of the metal matrix and ceramic inclusions. Let us analyze in more detail the stress distribution in the vicinity of a single inclusion which, for the sake of simplicity, is circle-shaped, Fig. 4. Such a simplification can be accepted since, statistically, inclusions are of a round shape in average. The analytical solution obtained in [11] for a so-called inclusion problem were used, where a cylindrical inclusion of round cross-section is embedded into an infinite area, with the inclusion and matrix materials are distinguished by their mechanical properties. The area with the embedded inclusion and schematic of its loading are presented in Fig. 4. For the case of arbitrary materials, the problem solution in cylindrical coordinates gives the following expressions for stress components in matrix [11]:
D S
C
μ 1, ν 1
μ 2, ν 2 A
S ϕ
B
Fig. 4. Schematic of the inclusion problem.
S 1 þ 2a1 b 2a2 b a2 1þ 2 2a2 þ b 1 r
4ð1 bÞ a2 3ð1 bÞ a4 1 þ cos 2u ; 1 b þ 4a1 b r 1 b þ 4a1 b r S 1 þ 2a1 b 2a2 b a2 1 ¼ 2 2a2 þ b 1 r
3ð1 bÞ a4 1þ cos 2u ; 1 b þ 4a1 b r S 2ð1 bÞ a2 1þ ¼ 2 1 b þ 4a1 b r 3ð1 bÞ a4 sin 2u. 1 b þ 4a1 b r
rrr ¼
ruu
rru
ð15Þ l2 ; l1
Here, a is the inclusion radius, b ¼ ai ¼ for plane strain 1 mi , i = 1, 2; mi is the Poison 1 ð1 þ mi Þ for plane stress ratio, and S is the external stress. For the metal matrix composite investigated in this work, b 5.4 and a1 a2 = 0.65 under plane strain conditions. The stress components on the boundary between matrix and inclusion, where r = a, are rrr = r11 = 1.35S, ruu = r22 = 0.57S, rru = r12 = 0 at u = 0, p and rrr = r22 = 0.11S, ruu = r11 = 0.2S, rru = r12 = 0 at u = p/2, 3p/2. Therefore, all non-zero components of the stress tensor in the direction of tension are positive in points A–D (refer to Fig. 4), with the maximum values in points A and C. These local areas also undergo tensile loading along the X2-axis. In points B and D material is under compression in the direction perpendicular to the axis of tension. The results of the analytical solution well agree qualitatively and quantitatively with the numerical results in the case of inclusions of similar shape (e.g. inclusion 1 in Fig. 3). It is of interest to estimate the stress distribution in the vicinity of inclusions, analyzing limiting cases. If the elastic moduli of the inclusion considerably exceed those of the matrix, i.e. l2 l1, then b ! 1 and Eq. (15) take the form: 2a1 cos 2u ; rrr ¼ S a1 þ 4a1 1 2ða1 1Þ cos 2u ; ruu ¼ S 1 a1 4a1 1 2Sa1 rru ¼ sin 2u. ð16Þ 1 4a1 In this case, at a1 = 0.5 in points A and C rrr = ruu = 1.5S, i.e. material is under tension, and in B and D regions rrr = ruu = 0.5S, i.e. these areas undergo compression along both X1- and X2-axes. The estimations given above hold for the scale level associated with the average size of inclusions. According to the terminology of physical mesomechanics this scale can be defined as a mesoII scale [3]. For the structure under consideration, this characteristic size is about 20 lm (Fig. 1(a)). The stress concentrators appearing in
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the matrix at the elastic stage of loading give rise to the formation of macroscopic shear bands under plastic deformation. The same analysis can also be done for the mesoI- and micro-level, where the objects of investigation are individual inclusions and the scale under study is mainly defined by the uneven geometrical shape of the matrix–inclusion interface and the extension of local fractured areas in the inclusions as well. Under further loading, the stress concentration near the interface irregularities leads to the plastic strain localization in the matrix and crack nucleation and propagation in the inclusions. Areas of the highest stress concentration in the structure under study are presented in Fig. 1 on an enlarged scale. Let us analyze the stress distribution in the vicinity of points A, C and D, using expressions (15). On this scale, the problem is reduced to the consideration of a metal inclusion surrounded by ceramic material. Characteristic size of such an irregularity is 2 lm (Fig. 1(b)). Taking into account the fact that the interface area is of a semi-circle shape, only qualitative estimations will be given in order to analyze the criterion (14) for the mesoI scale. At l1 > l2 for the composite under study b 0.18, then at r = a the stress tensor components under tensile loading take the following values: rrr = r22 = 0.06S, ruu = r11 = 2.2S and rru = r12 0 in D point (u p/2) and rrr = r11 = 0.4S, ruu = r22 = 0.7S and rru = r12 0 in A and C areas (u 0, p). Thus, under external tension the stress tensor components and, hence, req take their maximum values in the D-type areas which undergo tensile deformation. It is evident that first cracks will nucleate in the places 1 and 2 as the condition req = Cten fulfils. In the case of compression under external stress S, the D-areas are under compressive deformation along both X1- and X2-axes, i.e. there is a negative pressure in these places and fracture is defined by the criterion req = Ccom, according to Eq. (14). In points A and C ruu = r22 = 0.7S, i.e. material undergoes tensile deformation in the direction perpendicular to the axis of external compression. In these places the pressure is positive and fracture is controlled by criterion req = Cten. Refer to Table 1, for the material of inclusions Ccom Cten. Because of this, first cracks are expected to nucleate in the vicinity of points A and C, even though the equivalent stress req takes its maximum value in the D-points. We can conclude, therefore, that in the mesovolume under tension and compression, cracks nucleate in different places. It has been shown that the composite mesovolume exhibits a complex mechanical behavior controlled by both shear band formation in the matrix and cracking of inclusions. Evolution of fracture in the inclusions and plastic deformation in the matrix under tension and compression is illustrated in Figs. 5 and 6, correspondingly. There is a marked difference in the crack orientation relative to the axis of loading under tension and compression. The cracks tend to propagate perpendicular to the loading axis in the case of tension and parallel under compression. This frac-
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ture behavior is well known from experiments [5] and additionally confirmed by estimations given above. Let us analyze the stress state in the vicinity of a local fractured area whose specific size is about 250 nm (micro-level, Fig. 1(b)). For the sake of simplicity, let us consider the fracture area in the form of an ideal circle and find the solution (15) at b ! 0. Due to this simplification, the problem is reduced to a classical solution of elasticity theory on estimation of the effect of a round hole on the stress distribution in a plate [12]. Such an idealization can be accepted, because of the local area under study is much less in size in comparison with the mesovolume, and inclusions are treated as elastic. According to (15), at r = a the only component of the stress tensor remains to be non-zero: ruu ¼ Sð1 2 cos 2uÞ. This stress takes its maximum value ruu = 3S in points B and D and a negative magnitude ruu = S corresponding to the compressive response in places A and C. When q S ffiffiffiffiffiffiffi is further increased, the criterion (14) in the form ffi req ¼ r2uu ¼ jruu j ¼ C ten is first fulfilled in points B and D, which results in crack nucleation in these areas. Under further loading, the process repeats so that the crack propagates perpendicular to the axis of loading. Under compression, cracks demonstrate different behavior. There exists a compressive stress of ruu = 3S in points B and D. Even though the equivalent stress in these points reaches its maximum value, the strength of Al2O3 under compression isqorders of magnitude higher than ffiffiffiffiffiffiffiffi under tension, req ¼
r2uu ¼ jruu j ¼ C com C ten . In this
connection, fracture takes place in the vicinity of areas A and C which undergo tensile deformation, ruu = S. Thus, the crack propagates parallel to the direction of external tensile loading. Presented in Figs. 5 and 6 are the patterns of equivalent stresses and plastic strains and velocity vector fields laid over the structure image. Compare with the average curves in Fig. 2, pictures (a–f) and (g–l) correspond to tension and compression, respectively. Fig. 5 shows that under compression the first crack nucleates in the biggest inclusion near point D (see Fig. 1), where the concentration of equivalent stress req takes its maximum value. The initial crack propagates perpendicular to the direction of tension until it reaches the opposite boundary of the inclusion (Fig. 5(a)–(d)). This process is accompanied with unloading of adjacent material due to release waves propagating from newly formed free surfaces, which corresponds to the descending part of the stress–strain curve, confined between points a and d in Fig. 2. The velocity fields in Fig. 5(a)–(c) illustrate the crack opening. The complex stress state forms due to an interaction of elastic waves with interfaces, with the development of vortex movements. Rotation of separate local areas leads to an additional stress concentration (Fig. 5(c)). Under further loading an average level of stresses in the mesovolume increases and stress concentration appears in smaller inclusions, that results in nucleation of
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Fig. 5. Equivalent stress, plastic strain patterns and velocity vector fields for different steps of tension.
new cracks which, in turn, unload surrounding material (Fig. 5(e)–(f)). Let us analyze the images of plastic strains, presented in Fig. 5(d)–(f), where fractured areas are plotted in black color. These pictures indicate that under tension cracks nucleate and propagate at the elastic stage of loading. This is due to the fact that even though an average value of stresses in the matrix is lower than the yield point r0 = 105 MPa, the stress concentration in local places of the inclusions can exceed its critical fracture value Cten = 260 MPa. Weak plastic deformation in the form of localized shear bands is observed near the places of crack nucleation (Fig. 5(d)–(f)). Under compression, Fig. 6, first cracks nucleate in places of highest concentration of tensile stresses (points A and C in Fig. 1). Note, in the places of local compression (points D) the equivalent stress reaches considerably higher values. This is the reason that a fracture process
under compression goes at a higher average level of stresses than under tension (Fig. 2) and is accompanied by intensive plastic deformation in the matrix (Fig. 6(j)–(l)). Cracks propagate parallel to the axis of compression and, in contrast to the case of tension, this process demonstrates an intermittent character. Average velocity of crack propagation through inclusions under compression is lower than that under tension. Intensive plastic deformation in the matrix considerably retards stress increase in the places of stress concentration. As a result, cracking in inclusions goes in a so-called ‘‘switching’’ mode, which is illustrated by velocity fields in Fig. 6(g)–(i). Refer to Fig. 6(g), where nucleation of first crack causes moderate unloading in the mesovolume. At the next stage of loading the crack stops to move, whereas the average stress curve demonstrates a rather prolonged ascending part confined within dots g and h in Fig. 2 due to strain hardening in the matrix. Finally, a new crack nucleates in
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Fig. 6. Equivalent stress, plastic strain patterns and velocity vector fields for different steps of compression.
the other inclusion, Fig. 6(h). Compare part of the curve between points i and k in Fig. 2 and pictures (i)–(k) in Fig. 6, as the second crack stops to grow, the first one continues to propagate again in a jumping manner toward the opposite boundary of the inclusion. Under further loading the third crack nucleates (Fig. 6(l)) and the process repeats. 4. Conclusion In this paper, a fracture criterion of Huber’s type is modified and applied in numerical investigations of deformation and fracture processes in the mesovolume of the composite material consisting of the aluminum matrix and ceramic inclusions. Using an analytical solution for a plate with a round hole, the results of numerical simulation for the mesovolume under tension and compression have been analyzed. A qualitative and quantitative comparison of the stress concentration caused by structural heterogene-
ities of different scales, such as individual inclusions, irregularities of matrix–inclusion interfaces and local areas of fracture, has been carried out. Specific sizes of heterogeneities on the mesoII-, mesoI- and micro-levels are 20–30 lm, 2–3 lm and 250 nm, correspondingly. It has been shown that, from the viewpoint of mechanics, stress concentration has geometrical meaning and identical nature on different scale levels. Summing up the computational results for tension and compression, we can make the following conclusions: 1. Cracks propagate perpendicular to the loading direction under tension and parallel under compression as expected. 2. Due to strain incompatibility near the aluminum-alumina interfaces, local areas of tensile deformation form under compressive loading. And vice versa, under macroscopic tension along the same axis these local regions undergo compressive stresses.
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3. Local fracture areas first appear at the places of most powerful concentration of tensile stresses which give rise to crack propagation under both tension and compression, i.e. all cracks nucleating in the mesovolume are so-called ‘‘tensile cracks’’ regardless of the kind of loading. 4. Under compression cracking of inclusions is accompanied by intensive plastic deformation in the matrix, whereas under tension first cracks originate and begin to grow at the elastic stage of deformation.
Acknowledgments The support of CRDF and Russian Federation Ministry of Science and Education (BRHE016-02) and Russian Science Support Foundation is gratefully acknowledged. References [1] V.E. Panin, V.A. Lihachev, Ju.V. Grinyev, Structural Levels of Deformation of Solids, Nauka, Moscow, 1985, 230 p.
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