The development of residual stresses in Ti6Al4V ... - Marc A. Meyers

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Residual stresses and cracking due to differential CTE rea- sons are well-known ..... [21] X.C. Zhang, B.S. Xu, H.D. Wang, Y. Jiang, Y.X. Wu, Compos. Sci. Tech.
Materials Science and Engineering A 473 (2008) 49–57

The development of residual stresses in Ti6Al4V-Al3Ti metal-intermetallic laminate (MIL) composites Tiezheng Li a,b , Eugene Al Olevsky b , Marc Andr´e Meyers a,∗ a

Department of Mechanical and Aerospace Engineering, Materials Science and Engineering Program, University of California, San Diego, La Jolla, CA 92093-0411, USA b Department of Mechanical Engineering, San Diego State University, San Diego, CA 92182-1323, USA Received 11 January 2007; received in revised form 12 March 2007; accepted 15 March 2007

Abstract Residual stresses in the metal (Ti6Al4V)-intermetallic (Al3 Ti) laminate composite are generated when cooled from the processing temperature (∼700 ◦ C) to ambient temperature, because of the difference in thermal expansion coefficients between Ti6Al4V and Al3 Ti. Two stress release mechanisms, creep and crack propagation, are proposed to explain the development of residual stress during cooling process. Both analytical calculations and finite element simulations are performed. In the analytical modeling, a critical stress criterion is proposed in order to determine the initiation of crack propagation. In the finite element simulation, the J-integral is used as a criterion for crack evolution; it enables the establishment of the distribution of the residual stress as a function of temperature. The results of both analytical modeling and finite element simulation show good agreement with the experimental results obtained through X-ray diffraction. © 2007 Published by Elsevier B.V. Keywords: Residual stress; Creep; Crack propagation; Laminate composites; Finite element simulation; Thermal expansion

1. Introduction Metal-intermetallic composites have become increasingly attractive in recent years because of their low density, high stiffness, and high strength [1,2]. However, these desirable properties are often accompanied by a high level of residual stresses generated during the cooling phase from high temperature fabrication. Since the composite ceramic or intermetallic matrices typically have limited plasticity at temperatures below one-half of their absolute melting temperature, residual stresses generated cannot be relieved by plastic deformation. As a result, the selection of a reinforcement-matrix combination must carefully consider the coefficients of thermal expansion (CTE) of each component to avoid the introduction of damage during cooling in processing and from thermal cycling experienced during service. The stresses are schematically presented in Fig. 1 for the Ti6Al4V-Al3 Ti system, with the intermetallic under tension, an undesirable situation. Therefore, it is essential that existing residual stresses can be correctly anticipated and managed ∗

Corresponding author. Tel.: +1 619 534 4719; fax: +1 619 534 5698. E-mail address: [email protected] (M.A. Meyers).

0921-5093/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.msea.2007.03.069

during heat treatment, processing and use. Under uniform temperature change, stresses induced in a composite depend on the following factors: 1. Reinforcement volume fraction and morphology (i.e., particle, fiber or plate size, shape, orientation distribution, and continuity). 2. Matrix crystallographic texture and porosity. 3. Possible voids or lack of adhesion at matrix-reinforcement interfaces. 4. The development of cracks. Generally speaking, the residual stress is determined primarily by the elastic constants, relative concentration and the thermal expansion coefficients of the matrix and reinforcement materials. While realizing that material microstructural properties have an effect on both strength and thermal strain, depending on the magnitude of the residual stress, the thermal expansion or contraction can be (a) a reversible phenomenon when the materials undergo elastic deformation, or (b) an irreversible process, if the materials experience plastic yielding, microcracking, interface decohesion or creep.

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Fig. 1. Schematics of residual stress in Ti6Al4V-Al3 Ti MIL composites.

Residual stresses and cracking due to differential CTE reasons are well-known problems for ceramic composite materials, and numerous studies have been carried out on the residual stress in composites [3–31]. However, very few [3,12,28] include crack propagation. The present study investigates the evolution of residual stress in a Ti6Al4V-Al3 Ti metal-intermetallic laminate (MIL) composite using both analytical and finite element simulation methods. The MIL composites used in this investigation were processed in open air and mechanical properties and damage evolution have been evaluated [32,33]. The processing method has been described in detail [34–40]. In the analytical model, two stress release mechanisms, creep and crack propagation, are introduced. The same mechanisms are also implemented in the finite element simulation to investigate the residual stress evolution as well as distribution.

Fig. 2. Montage of part of the cross-section of as-processed 35% Ti6Al4V MIL composite showing cracks (marked by arrows).

3. Analytical modeling 2. Experimental measurements As processed MIL composite blocks [41–43] of 80 mm × 18 mm × 8 mm were ground from 220 to 4000 grit, before final polishing by 0.05 ␮m Al2 O3 , and were then characterized by optical microscopy. Photomicrographs were assembled in montages (e.g., Fig. 2). It is observed that most cracks in the Al3 Ti layers are perpendicular to the interface. In the 35% MIL composites, 72% of the cracks are perpendicular cracks, thus the discussion in the following sections focuses on the development of perpendicular cracks in the MIL composites during cooling. Li et al. [33] provide a detailed discussion of the crack morphology and distribution in the as processed MIL composites analyzed herein. The X-ray diffraction was preceded by a detailed procedure for specimen surface preparation which started with grinding in 250 and 400 grit SiC. This was followed by polishing using diamond paste (6–10 ␮m) and alumina (1 ␮m). The surface was subsequently etched with a 10% HF for 2–5 s in order to remove any strain in the surface due to sample preparation. The X-ray diffraction was carried out by Harvey [44] using Cu K␣ radiation with a Rigaku Rotaflex rotating anode device. Titanium peaks were chosen. Care was taken not to take a peak which coincided with an Al3 Ti peak. Spectra were collected from 2θ between 20◦ and 135◦ . Additional considerations led to the selection of peaks between 121.5◦ and 124.5◦ .

3.1. Residual stress in MIL composites: elastic analysis When the Ti6Al4V-Al3 Ti MIL composite cools down, Al3 Ti (CTE: 13 × 10−6 ◦ C−1 ) shrinks more than Ti6Al4V (CTE: 9.5 × 10−6 ◦ C−1 ) because of its higher CTE. This mismatch of strain puts the Ti6Al4V layer under compression and Al3 Ti layer under tension. For a composite with an infinite number of layers, the residual stress [45] is: σTi =

ETi · (1 − c) · (αTi − αAl3 Ti )T [1 + (ζ − 1) · c] · (1 − υTi )

(1)

where ζ = ((1 − υAl3 Ti )/(1 − υTi )) · (ETi /EAl3 Ti ), αTi and αAl3 Ti are the CTE of Ti6Al4V and Al3 Ti, respectively and c, the volume fraction of Ti6Al4V, is defined as: c=

dTi · W dTi = dTi · W + dAl3 Ti · W dTi + dAl3 Ti

(2)

dTi and dAl3 Ti are the thickness of the Ti6Al4V and Al3 Ti layers (Fig. 1), respectively, and W is the width of the specimen. The residual stresses as a function of temperature for different volume fractions of Ti6Al4V are plotted in Fig. 3, in comparison with experimental measurements (also shown in Table 1). The experimental values were obtained by K.S. Harvey through Xray measurements. Fig. 3 shows that the residual stress increases as the temperature decreases from the processing temperature of

T. Li et al. / Materials Science and Engineering A 473 (2008) 49–57

Fig. 3. Calculated residual stress as a function of temperature from the difference in CTE for different volume fractions of Ti6Al4V; experimental measurements in lower left-hand corner.

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Fig. 4. Experimental and linearized cooling curve (normal cooling rate) of MIL from processing temperature.

∼1000 K. The slow cooling in Table 1 refers to the slow cooling rate after processing as compared to the normal cooling rate. It is observed that the level of residual stresses decreases with the volume fraction of Ti6Al4V and the calculated residual stresses are much higher than the measured ones.

The procedure used in incorporating creep into the residual stress is given below. The cooling is divided into discrete intervals. As the temperature decreases from Ti to Ti−1 , the initial residual stress, σ i i , is obtained from Eq. (1). The residual stress is released during this period of time through the creep effect, and the reduction in residual stress is obtained from:

3.2. Residual stress release mechanism I: creep

σi = ETi · ε˙ s i · (ti − ti−1 )

As seen in Table 1, the experimentally measured residual stresses for slow cooling rate are lower than in the normal cooling rate. This suggests that some time-dependent stress relaxation mechanism is active in the cooling process. The experimental cooling curve, shown in Fig. 4, was linearized to facilitate the calculation. Steady-state creep was described by the power-law equation:   AbμD σ n ε˙ s = (3) kT μ

The residual stress (including creep influence) is the reduction of the change of stress from the initial residual stress, σ i i :

where ε˙ s is the steady-state strain rate, μ the shear modulus, D the diffusion coefficient, b the Burgers vector, T the temperature, k Boltzmann’s constant, and σ is the stress. A and n are materialdependent parameters. Table 2 shows the creep parameters used for Ti6Al4V [46] and Al3 Ti (adapted from Ref. [47]). Table 1 Comparison of calculated residual stresses at ambient temperature by various stress release mechanisms with experimental measurements (marked in bold) Volume fraction of Ti6Al4V (%) Measured, high cooling rate (MPa) Measured, slow cooling rate (MPa) Calculated from difference in CTE (MPa) Calculated, with creep in Ti6Al4V only (MPa) Calculated, with creep in both Ti6Al4V and Al3 Ti (MPa) Calculated, with creep and crack propagation in Al3 Ti (MPa) FE simulation (MPa)

14 65.01

20 32.38

35 8.29

30.79

25.24

10.59

345

327

282

280

260

208

257

240

195

86.16

69.33

46.63

89.53

77.86

60.73

(4)

σr i = σi i − σi

(5)

The calculation is continued as the temperature decreases to room temperature. Residual stresses including the influence of creep in Ti6Al4V, and in both Ti6Al4V and Al3 Ti, are compared with the experimental measurements for different volume fractions of Ti6Al4V, in Fig. 5. The inclusion of creep effectively reduces the level of residual stress, especially at higher temperatures. However, when the temperature decreases below 740 K, the residual stresses begin to build up again, consistent with the fact that creep is not effective at lower temperatures. As expected, creep in Ti6Al4V is more pronounced than in Al3 Ti. 3.3. Residual stress release mechanism II: crack propagation Previous research [32,33] indicates that a significant concentration of cracks is present in as processed MIL composites. They occur mostly in Al3 Ti layers and perpendicular to the interface. This was attributed to the development of residual stresses during the cooling process. A detailed statistical analysis of these cracks is given by Li et al. [33]. As the brittle Al3 Ti layers Table 2 Creep parameters for Ti6Al4V and Al3 Ti

n Ab (nm) μ (GPa)

Ti6Al4V

Al3 Ti

4.55 8.82 × 10−5 43.8

3 4.89 × 10−5 92.3

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Fig. 5. Calculated residual stresses including creep in Ti6Al4V, and in both Ti6Al4V and Al3 Ti; the experimental measurements for different volume fractions of Ti6Al4V are in lower left-hand corner.

undergo increasing tensile stresses, microdefects begin to be activated and cracks develop. These microcracks have a negative effect on the material properties and help to release the level of residual stresses. It was observed that these cracks do not penetrate into the Ti6Al4V layers, and therefore their length is geometrically connected to the thickness of the Al3 Ti layers. Two models, a simple spring model and the Salganik model [48], are used to quantify the effect the crack propagation on the development of residual stress. Fig. 6 shows the schematics of the spring and Salganik models. In the spring model, the initial crack is assumed to be perpendicular to the interface, and its length equals the thickness of the Al3 Ti layer (2a = dAl3 Ti ). The elliptical region (with major axis 4a) denotes the approximate volume of material in which the stored elastic strain energy is released; this unloaded region is assumed to have an elastic modulus equal to zero. This is a reasonable assumption and the same as the one used by Griffith e [49]. The effective Young’s modulus of Al3 Ti, EAl , depends 3 Ti on the separation between two cracks, s, and is a function of the normalized distance, s/a. The relationship between the effective Young’s modulus of Al3 Ti and the theoretical Young’s modulus of Al3 Ti can be expressed, to a first approximation, by consider-

Fig. 7. The effect of cracking on Young’s modulus by Salganik and Spring models.

ing springs in series, one with Young’s modulus EAl3 Ti and one with E0 = 0, corresponding to the unloaded region. This gives: e EAl 3 Ti

EAl3 Ti

=1−

4 s/a

(6)

Salganik [48] developed a more realistic model. The cracks are also assumed to be perpendicular to the interface, but their length is smaller than the thickness of the Al3 Ti layer. The effect of microcracks on Young’s modulus, E2e , of Al3 Ti, predicted by the Salganik model is E2e =

E2 (1 + ((16 · (10 − 3υ2 ) · (1 − υ22 ))/

(7)

(45 · (2 − υ2 ) · N(a, (s/a)) · a3 ) where E2e is the effective Young’s modulus of Al3 Ti, E2 the Young’s modulus of the uncracked Al3 Ti, υ2 the Poisson’s ratio of Al3 Ti, N(a,(s/a)) the number of cracks per unit volume, a the radius of a mean crack, and s is the distance between two cracks. Fig. 7 shows the effect of cracks on the Young’s modulus with both spring and Salganik models as a function of normalized distance, s/a. It is seen that the spring model is more conservative in predicting the crack influence on the elastic modulus. The Young’s modulus in the spring model decreases to zero when

Fig. 6. Schematic representation of (a) spring model; (b) Salganik model.

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the spacing is less than 4a ((s/a) < 4), by virtue of the assumption that the ellipsis with major axis 4a is unloaded. On the other hand, the Salganik model is more realistic for smaller s/a values. 3.4. Combining creep and crack propagation mechanisms As the specimen is cooled, the Al3 Ti layer is under increasing tensile stress. Due to the brittleness of intermetallics, existing flaws in Al3 Ti will start to propagate when the stress exceeds a critical value. The residual stress, σ r , is a function of crack size, crack density and temperature:  s  σr = σr a, , T (8) a The critical stress, σc∗ , is a function of the effective fracture toughness of Al3 Ti, which is a combination of the intrinsic fracture toughness of Al3 Ti and the extrinsic fracture toughness, determined by the residual stress in Al3 Ti layer, crack size and crack density: σc∗ =

Keffective √ πa

 s  Keffective = Kintrinsic + Kextrinsic a, , σr a

(9) (10)

As the existing cracks start to propagate, the residual stress is released, affecting the effective toughness and the critical stress. The cracks will continue to grow until the residual stress is smaller than the critical stress. As the temperature decreases, the residual stress competes with the critical stress to determine the propagation of the cracks. This process continues until the temperature reaches room temperature or until the crack size reaches the width of the Al3 Ti layer, dAl3 Ti . It is assumed that the initial crack size is 25% of the thickness of the Al3 Ti layer and that the MIL composites are infinitely long. The results obtained by an iterative calculation are shown in Fig. 8. It is observed in Fig. 8(a) that the residual stresses with different concentrations of Ti6Al4V all increase steadily before the critical stresses in Al3 Ti are exceeded, during which time, the creep in Ti6Al4V and Al3 Ti is the dominant mechanism in releasing the residual stresses. After that, crack growth dominates the residual stress release process, as demonstrated by the steady decrease in residual stress in Ti6Al4V. The sharpest decrease in residual stress is observed in 14% MIL, followed by 20 and 35% MIL. This is attributed to the higher residual stress level and larger initial crack size in 14% MIL composites, and implies that the effective fracture toughness in 14% MIL is more negatively affected by crack growth, at the corresponding critical stress. Fig. 8(b) shows the evolution of crack length in the MIL composites during cooling. The cracks remain stable at higher temperature as the residual stress accumulates, and when the critical stress level is reached, the cracks begin to propagate. The larger slope (in magnitude) of crack length evolution in the 14% Ti6Al4V composite corresponds to a higher growth for a certain temperature gradient. The crack length is eventually

Fig. 8. Analytical model for Ti6Al4V-Al3 Ti MIL composites as a function of temperature including both creep and crack propagation: (a) residual stress; (b) crack length.

constrained by the size of the Al3 Ti layers, and the crack stops growing at the interface. 4. Finite element simulation of residual stress distribution The analytical modeling of residual stress evolution discussed above assumes an infinite length of Ti6Al4V and intermetallic layers, and the results reflect the average stress level in Ti6Al4V. In reality, the MIL composites have finite size with boundary conditions, and the distribution of residual stress in Ti6Al4V is uneven. Finite element simulation provides us a powerful tool to simulate not only the magnitude but also the distribution of residual stress during cooling. The commercially available finite element software, ABAQUS, was used to conduct the residual stress simulation during the cooling in 3D formulation. Because of the symmetry, only one eighth of the specimen was simulated and is shown. 4.1. Finite element simulation of residual stress with creep effect A 3-D 17 layer body was created to model the real MIL sample size, in which three faces of the sample were con-

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Fig. 11. FE results of residual stress during cooling, in comparison with analytical modeling and experimental results.

Fig. 9. FE model for MIL composites: initial stage.

stress than that in the analytical modeling in Ti6Al4V layers is observed.

fined in X1 , X2 and X3 global directions as shown by the symmetric boundary conditions. The initial condition assumes an initial temperature of 1023 K, and creep in both Ti6Al4V and Al3 Ti was considered. The meshed model is shown in Fig. 9. Eight-node brick elements are used in the simulation, and the results (for σ 11 ) including creep in both Ti6Al4V and Al3 Ti are shown in Fig. 10. While most Ti6Al4V layers experience relative constant stress levels, there exist areas around the boundaries where the residual stress level is noticeably lower. The residual stress in Ti6Al4V was calculated by averaging the stresses in different Ti6Al4V layers, and the results are compared with the analytical solutions in Fig. 11. FE results predict a similar trend in the residual stress evolution. However, in FE simulation, creep is effective for a longer period of time during cooling and, as a result, a smaller

4.2. Finite element simulation of residual stress with creep and crack propagation

Fig. 10. FE simulation of residual stresses including creep in both phases: magnitude and distribution at room temperature.

Fig. 12. FE simulation at processing temperature (∼1000 K), including crack propagation; existing flaws visible through mesh distortion.

The detailed crack morphology and distribution were measured and are recorded in the two proceeding papers [32,33]. This served as a basis for their introduction into the simulation. The Salganik model was employed to adjust the material properties during crack propagation. The initial crack sizes are assumed to be 25% of the thickness of Al3 Ti layers, and located in the center of the layers. The initial condition is shown in Fig. 12 (notice the existing microcracks in Al3 Ti visible as mesh distortions). The J-integral, which represents the energy required to propagate a crack, was calculated around the crack tips and compared with the critical value to determine whether the cracks would

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further propagate. The J-integral is calculated in ABAQUS as:  J= λ(s)n · H · q dA (11) A

where λ(s) is a virtual crack advance in the plane of a threedimensional fracture; dA is a surface element along a vanishing small tubular surface enclosing the crack tip; n is the outward normal to dA; and q is the local direction of virtual crack extension. H is given by: H = WI − σ ·

∂u ∂x

(12)

where for elasto-viscoplastic material behavior, W is defined as the elastic strain energy density plus the plastic dissipation, thus representing the strain energy in an “equivalent elastic material.” Although, in theory the J-integral is path independent, experience has shown that the first contour calculated in ABAQUS is of significantly lower accuracy. Therefore at least 10 J-integrals around the crack tip were used for each crack. If the J-integral exceeds the critical value, crack length is extended and the elastic modulus is reduced according to the Salganik model; otherwise, the crack remains stable. Since ABAQUS does not have the capability to simulate crack propagation in 3D automatically, the simulation was interrupted periodically, and comparison and the necessary adjustments were made manually. Fig. 13 shows the magnitude and distribution of residual stresses, σ 11 , at room temperature. It shows a dramatic decrease in the stress levels around cracks, in both Ti6Al4V and Al3 Ti layers. It is also observed that from the start of cooling, there exists an elliptical area of stress relaxation in Al3 Ti, which by and large confirms the spring model. This relaxation area even extends into the neighboring Ti6Al4V layers and thus contributes to the release of stress levels in Ti6Al4V. Fig. 14 shows the evolution of residual stress and crack length with creep and crack propagation by FE simulation.

Fig. 13. FE simulation at room temperature including crack propagation.

Fig. 14. Evolution of (a) residual stress and (b) crack length by FE simulation and comparison with analytical solutions.

The residual stress obtained by FE simulation (Fig. 14(a)) at room temperature is comparable with the analytical solutions and the experimental measurements. However, the evolution of the residual stress itself is very different. The effect of creep on the residual stress obtained by FE simulation is much more prominent than that in the analytical solutions, as the creep very effectively prevents the residual stress accumulation in the early stage. In the later stage of the cooling process, however, unlike the analytical solution, the FE simulation predicts a sudden eruption and the cracks keep growing in an unstable fashion until reaching the interfaces, while the residual stress experiences a sharp decrease. One explanation is that at that temperature, the Al3 Ti has become sufficiently brittle and has little resistance to crack growth; therefore, FE prediction simulates what happens in reality. The other explanation is that the nature of this FE simulation restricted the minimum size of the crack increment confined by the finite size of the element, which may not capture the changes within that minimum increment. In that sense, the finite element simulation provides a more accurate solution for residual stress evolution and distribution by taking into consideration the finite size of MIL composites. On the other hand, it is less accurate by increasing crack size in a finite fashion. Table 1 shows the comparison of residual stresses obtained by analytical modeling, FE simulation and experimental measurements. The analytical modeling and FE simulation results

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are comparable with the experimental measurements, although the predictions are still higher than the measurements. One reason for the deviation is that the measurement was conducted on small samples, and the results are average values. Therefore, the measured values may only reflect the local average stress level, which, when measured in the boundary area (as observed in Figs. 10 and 13), are lower than the rest of the sample. Another reason is that the residual stress may be released during specimen preparation, such as cutting and polishing, and therefore, the measured values are lower than the real ones existing in the specimens after just being processed. The results were also checked with a previous study [50], in which stress relaxation mechanisms were not taken into consideration. The analytical modeling and FE simulation thus may provide a more accurate residual stress, especially as the finite element simulation provides not only the stress evolution but also the stress distribution. 5. Conclusions The development of residual stresses in a metal (Ti6Al4V)intermetallic (Al3 Ti) laminate composite made by reaction synthesis at ∼1000 K followed by cooling to ambient temperature was investigated. The residual stresses experimentally measured by X-ray diffraction are much lower than the predictions from differences in thermal expansion coefficients alone. The residual stresses in Ti6Al4V-Al3 Ti MIL composites were calculated analytically and by FE simulation, and two stress release mechanisms, creep and crack propagation, were included in the analysis. The following conclusions were drawn: 1. In the analytical modeling, the stress release mechanisms were implemented by an iterative calculation. Creep is the dominant stress release mechanism at high temperature, while below 740 K, when the residual stress reaches the critical value, crack growth becomes the primary mechanism to reduce the stress level. The analytical prediction was successfully compared with the experimental measurements. 2. Finite element simulation was conducted, including both creep and crack propagation. FE simulation confirmed the analytical calculation of the stress level on average, and demonstrated the presence of an uneven stress distribution, especially at the boundaries. FE simulation predicts an unstable crack growth, as the cracks keep growing until reaching the interfaces; the corresponding residual stress experiences a sharp decrease. The FE results were in good agreement with the analytical solutions and the experimental measurements. 3. The difference of stress level between predictions by analytical modeling and FE simulation with experimental measurements has two explanations: (a) experimental measurements were conducted on small samples, which only reflect the local average stress level, and thus may underestimate the real stress level, especially when the measurement was conducted around the boundary; (b) during specimen preparation, the residual stress level can be reduced through processes like cutting and polishing. Thus, the calculated results provide a better assessment of the residual stresses.

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