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Design of β-Titanium microstructures for implant materials
Materials Science & Engineering C 110 (2020) 110715
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Design of β-Titanium microstructures for implant materials Şafak Çallıoğlu , Pınar Acar 1
∗
T
Virginia Tech, Blacksburg, VA 24061, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: β -Titanium alloy Microstructure Design Implant
The present work addresses the design of β-Titanium alloy, TNTZ, microstructure to be used in biomedical applications as implant materials. The TNTZ alloy has recently started to attract interest in the area of biomedical engineering as it can provide elastic modulus values that are comparable to the modulus of the human bone. Such a match between the implant and bone significantly increases the compatibility and functionality of the implant material with the human body. Experimental studies reveal that the modulus of TNTZ varies around 55–60 GPa, whereas the bones typically have modulus around 25–30 GPa. Therefore, to achieve a better match in modulus values and further improve the compatibility of the implant, we present a computational design study. As the properties of materials are significantly affected by the underlying microstructure, we focus on identifying the optimum microstructures. Our goal is to minimize the difference between the elastic modulus values of the microstructure and the bone. To ensure the manufacturability of such an optimum design solution, we analyze the microstructural evolution during deformation processing to obtain the optimum microstructure that can be processed. The outcomes of our analysis demonstrated that the elastic modulus of TNTZ can be as low as 48 GPa.
1. Introduction Estimation and control of material properties are critical for the structural components utilized in medical applications due to certain restrictions for better performance. With the introduction of the Integrated Computational Materials Engineering (ICME) [1], the goal in material design has been set to predict the properties by modeling the material microstructure and processing. The microstructural texture can be controlled during deformation processing, which arises the necessity of designing the processes to achieve desired microstructures that produce targeted properties. Besides material selection, the design of the crystallographic texture orientations in the material microstructure leads to improved properties. Techniques that allow the control of properties of polycrystalline materials involve tailoring of the preferred orientations of various crystals that constitute the material. This was done before with the graphical quantification of propertyperformance relations using the property cross-plots, as presented by Ashby [2]. A more systematic design approach would be combining the processing, structure, and property through multi-scale computational material models with the help of the recent developments in materialsby-design [3]. In the area of composites, the design of structures holding intriguing properties such as negative thermal expansion [4]
and negative Poisson's ratio [5] has existed with techniques that facilitate tailoring of microstructure topology. However, in the case of polycrystalline materials, the design strategy aims to tailor the preferred orientations of distinctive crystals forming the polycrystalline alloy. During forming processes, the formation of texture and variability in property distributions in such materials are driven with mechanisms such as crystallographic slip and lattice rotation. Understanding the interactions between processing, microstructure, and property would enable tailoring of the properties of metallic structures, when they are used as bio-materials, in accordance with the requirements in medical applications, as represented in Fig. 1. The previous work in the field of multi-scale modeling and design includes studies from our group [6-10] that were focused on the optimization of α-Titanium and Galfenol microstructures for improved mechanical properties, including thermal buckling performance and vibration frequencies. In the present work, we use a similar multi-scale modeling approach for β-Titanium alloys to enhance their compatibility in biomedical applications. To the best of authors' knowledge, this is the first time such a computational approach is applied for the design of microstructures for implants. Bio-materials are pervasively being used in the medical area. Applications include cardiovascular tissue repair and regeneration [11,
∗
Corresponding author. E-mail address: [email protected] (P. Acar). 1 Visiting student at Virginia Polytechnic Institute and State University; full-time student at Bilkent University, Ankara, Turkey. https://doi.org/10.1016/j.msec.2020.110715 Received 31 August 2019; Received in revised form 30 January 2020; Accepted 31 January 2020 Available online 06 February 2020 0928-4931/ © 2020 Elsevier B.V. All rights reserved.
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Fig. 1. Example representation of microstructure design for an implant material. Tailoring of properties is possible (such as elastic modulus (in GPa)) for metallic biomaterials by designing the material microstructures.
bone. However, identification of the optimum microstructure requires testing of numerous processing conditions and microstructures which would be overwhelmingly expensive with pure experimental effort. Therefore, we present a computational design study for metallic implants, and there is no previous study on the computational design of TNTZ microstructure for implants to the extent of our knowledge. The presented computational methodology is also applicable for other design problems for biomedical applications that target other material properties (e. g. strength, stiffness) through the optimization of the underlying microstructures. The organization of the paper is as follows. Section 2 discusses the computational methodology to model the material microstructure while the deformation processing model is given in Section 3. Section 4 presents the optimum TNTZ microstructure designs. A summary of the work with potential future extensions is provided in Section 5.
12], drug delivery systems [13, 14], and skin substitutes [15]. The metallic bio-materials are used for artificial equipment such as artificial hip joints and dental implants [16]. The implant materials in the human body must provide comparable elastic modulus values to the modulus of the bones (20–40 GPa) [16]. This is because any difference between the elastic modulus of the human bone and implant material leads to the stress shielding effect that results in bone absorption [17]. Previous studies were focused on the bio-material selection for matching the elastic modulus of the bone with a material that provides low toxicity and strong corrosion resistance. One group of the metallic bio-materials implemented was the β-phase Titanium alloys, which are perceived as an alternative to the widely used implant materials. The β-Titanium alloys are found to provide low elastic modulus values [18] compared to Co-Cr-Mo alloys [19], α + β-Titanium alloys [20], and stainless steels [21] and low density [22] which is about half of density of stainless steels and Co—Cr alloys. Thus, different Titanium alloys have been developed for biomedical applications such as spinal fusion treatment with meshed plates on the spine for hernia of intervertebral discs [23], fixations of femoral fractures [24] and internal fractures [25], spinal fixation devices [26], and dental implants [27]. Among all Titanium alloys, α + β type alloy, Ti-6Al-4V, has wider use in biomedical applications. However, it contains elements V and Al that may trigger alteration of the kinetics of the enzyme activity associated with the inflammatory response cells [28] and development of Alzheimer's disease [29]. Recently, the β-Titanium alloy, TNTZ, has been found to be a promising structure as it includes the safest alloying elements for human health [18]. Therefore, in this paper, TNTZ is chosen as the material of interest for the implant. Our goal is to match the elastic modulus of the alloy with the modulus of the human bone to improve the material compatibility in the human body by optimizing the microstructural texture. In addition, we perform computational microstructure simulations for different deformation processes to find the optimum TNTZ microstructure, which can actually be manufactured. Many experimental studies [30-34] have been conducted for reducing the elastic modulus of TNTZ to match the modulus of the
2. Microstructure modeling The microstructural behavior is modeled during deformation processing at room temperature (cold working) using the Orientation Distribution Function (ODF) (denoted by A) to capture the crystallographic texture. The ODF represents the local densities of crystals over crystallographic space. The ODF is discretized using a finite element (FE) approach using the Rodrigues parameterization space. The details on the FE scheme and Rodrigues parameterization can also be found in our earlier studies [6-10]. The complete orientation space of a polycrystal is represented with the fundamental region, a small subset, that is obtained after applying the crystal symmetries. Each crystal orientation is described with a unique coordinate, r, the parameterization for the rotation (eg. Euler angles, Rodrigues vector, etc.). The ODF, A(r), represents the volume density of crystals of orientation r. The visualization of the ODFs in the Rodrigues fundamental region for hexagonal closed-packed (HCP) Titanium is shown in Fig. 2, with the locations of the k = 50 independent nodes (the total number of 2
Materials Science & Engineering C 110 (2020) 110715
Ş. Çallıoğlu and P. Acar
Table 2 Optimum Young's modulus values and processing times. Process
E11 (GPa)
E22 (GPa)
t (s)
Tension Compression xy-shear xz-shear yz-shear
48.6969 48.7653 49.2986 49.6134 49.8016
49.7922 48.0061 49.8430 49.6073 49.5210
0.32 0.285 0.0348 0.0078 0.0434
averaged, orientation-dependent material property of the microstructural domain can be obtained as: Nelem Nint
<χ > =
∮R χ (r ) A (r , t ) dv = ∑ ∑ χ (rm) A (rm) wm |Jn | n=1 m=1
1 (1 + rm⋅rm)2 (3)
where the ODF, A ≥ 0, is a function of orientation r, and time t (during processing). The volume-averaged property can be computed with the linear expression given next:
Fig. 2. The visualization of the ODFs in the Rodrigues fundamental region for hexagonal crystal symmetry, where the locations of the 50 independent nodes are shown in red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
< χ > = pT A
where p is the property matrix that contains the material property values in the 50 independent orientations. In this work, the volumeaveraged property expression is used to compute the stiffness of the TNTZ microstructure with upper-bound averaging. Using the stiffness definition, the elastic modulus values are obtained through the compliance matrix. The design is performed for Young's modulus (E11), and the computation procedure is explained next.
Table 1 Improvements with the optimum design solution compared to other studies in the literature.
Tane et al. [17] Niinomi et al. [18] Yilmazer et al. [30] Majumdar et al. [31] Besse et al. [32]
E11 (GPa)
Improvement (%)
35 55 60 57 55
– 34.9 40.3 37.2 34.9
C = pT A
(5)
C −1
(6)
S=
E11 = nodal points is 111). The volume density of any other node in the fundamental region can be determined from the independent nodes using the crystallographic symmetries. The ODF must satisfy a normalization constraint, ∫ Adv = 1, with the integral computed over the fundamental region. The volume normalization constraint on the ODFs can be written as follows: Nelement Nint
∮R A dv = ∑ ∑ A (rm) wm |Jn | n=1
m=1
1 =1 (1 + rm⋅rm)2
1 S (1, 1)
(7)
where C is stiffness matrix, S is the compliance matrix, E11 is Young's modulus value in < 11 > direction. The following values are used for the single crystal orthotropic elastic stiffness tensor coefficients [17]: C11 = 65.1 GPa, C12 = 40.5 GPa, C44 = 32.4 GPa. In the present work, the microstructural texture is visualized in terms of the ODF space as well as the pole figures in different orientations. The pole figure for a particular diffraction plane with a unit normal, h contains the pole density function P(h,yi) measured at locations y1,y2,…,yq on a unit sphere. The value for P(h,yi) at location yi can be obtained from the ODF (A) with a linear equation based on the algorithm presented by Barton et al. [35]:
(1)
where Nelement and Nint represent the number of FEs and integration points, respectively. A(rm) is the volume fraction value at mth integration point with rm parameter, wm is integration weight associated with mth integration point, |Jn| is Jacobian determinant of nth element. Eq. (1) is equivalent to a linear expression in the ODF:
qT A = 1
(4)
k
P (h, yi ) =
∑ Mij Aj j=1
(8)
where Mij is a constant system matrix M that defines the linear relationship between the pole figures and ODFs. The expression can be used for each of the m points in a pole figure. The equations can later be
(2)
where q shows the volume normalization vector. Similarly, the volume-
Fig. 3. The optimum ODF representations. 3
Materials Science & Engineering C 110 (2020) 110715
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Fig. 4. Pole figures for mathematically and processed optimum microstructure.
combined with a similar set of equations for n other pole figures with different diffraction normals, h. This is equivalent to a global system of equations P = MA, where P is the pole figure matrix (m × n), M is the constant matrix of the global system ((mn) × (k)) and the ODF is denoted by A using the volume densities of k independent nodes. In order k to account for the normalization constraint, ∑i = 1 qi Ai = 1, the overall system P = MA is adjusted such that Mij = Mij − and Pi = Pi −
Mik . qk
Mik qj qk
Kothari [36]. This framework has also been demonstrated in our previous studies [6-10] and hence it will not be repeated here for brevity. The main highlight is that the texture evolution is driven by the time evolution of the ODF values. The crystals reorient in time and thus the ODFs are updated at every time step by computing the re-orientation velocity, v, as shown in Eq. (9) in terms of the incremental change in the orientation, r:
for j = 1,…,k − 1
v=
1 (ω + (ω⋅r ) r + ω × r ) 2
(9)
In Eq. (9), ω shows the spin vector, which is a function of the deformation gradient of the constitutive model. The process is defined through the macro velocity-gradient (L) [37]. An example definition for tensile (z-axis tension) is given below:
3. Process model 3.1. Texture evolution during deformation processing
0⎤ ⎡− 0.5 0 L = α1 ⎢ 0 − 0.5 0 ⎥ 0 1⎦ ⎣0
We perform simulations for five different deformation processes to find which of these processes is the most likely to produce the desired Young's modulus value. The computational model uses the rate-independent single crystal constitutive model introduced by Anand and
(10)
The readers are referred to as Ref. [7] for more information on 4
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as to 25 GPa (representing Young's modulus of human bone). The optimization is performed using the Sequential Quadratic Programming (SQP) algorithm by inputting the randomly oriented texture sample as the initial guess. The mathematical formulation of the optimization problem is as follows:
process modeling and the ODF update scheme. Using the velocity gradient definition, the simulations are performed for tension, plane strain compression, xy-shear, xz-shear, and yz-shear deformation processes. The goal is to find the optimum time for a fixed strain rate (1 s−1) where any of the processes can produce the closest match for Young's modulus value. The strain rate is determined by checking the evolution of the ODF values in different processes using different strain rates to determine a strain rate value that is providing computationally efficient and convergent results. Next, the ODF snapshots are taken at each time step of the simulations to build a predictive model of the corresponding deformation process using a numerical technique. The computation of Young's modulus value using every available ODF snapshot follows the same volume-averaging approach discussed in Section 2. Since the snapshots correspond to the actual physical data, the numerical technique is expected to provide the same texture response at the times the simulation results are available. Such a solution can be achieved with the use of the Kriging model, as explained in the next section.
min subject
μ [Y:, j] = 0, V [Y:, j, Y:, j] = 1, j = 1, …, q
(12)
(13)
(14)
Z(xk) is assumed to have a zero mean value. The spatial covariance function between Z(xk) and Z(w) can be expressed in following way [38]:
Cov [Z (xk ), Z (x w )] = σz2 R (θ , xk , w )
(18)
Our main goal is to obtain an optimum ‘manufacturable’ microstructure design. However, many sharp texture designs are difficult to be processed, especially with traditional techniques. Therefore, to understand the common textures of deformation processes such as tension, plane strain compression, xy-shear, xz-shear, and yz-shear, we perform process simulations for each case. The initial microstructure is assumed to have randomly oriented texture in all cases. With the ODF snapshots taken at different time steps, a Kriging representation of each deformation process is built to describe the ODF evolution as a function of time. The processing time is optimized to find a microstructure that minimizes Young's modulus value of the alloy. As a result, the following optimum Young's modulus values are obtained for each process: Although the optimum microstructure (mathematically optimum) can provide Young's modulus value of 35.8019 GPa, the process model reveals any deformation processing route can provide a minimum Young's modulus value of 48.0061 GPa. Besides simulating the texture evolution during these processes, various other combinations of sequential processing routes are also checked (for example, tension followed by xy-shear). The texture simulations for sequential processing provide microstructures having higher Young's modulus values than those reported in Table 2 for single processing. This result simply implies that the optimum microstructure design cannot be processed with conventional processing techniques. However, the minimum Young's modulus value obtained with a microstructure that is processed with plane strain compression is still closer to Young's modulus value of the human bone, compared to the previous studies listed in Table 1. The optimum microstructure solutions (mathematical optimum solution and the optimum solution that can be produced by the plane strain compression) are illustrated in Fig. 3 in the Rodrigues orientation space in terms of the ODF values. The pole figures (PFs) of the optimum microstructures (on < 111 > , < 100 > , and < 110 > directions) are shown in Fig. 4. Note that the pole figures in Fig. 4 help us visualize the microstructural texture in 2D views taken from different directions. As represented by Figs. 3 and 4, the mathematically optimum microstructure has a different texture compared to the optimum microstructure that can be achieved with deformation processing. This is because the processed microstructure with conventional techniques
where y ^(xk ) is output of predictor for test point xk, f(xk) is the trend function, which includes a vector of regression functions, and β is the regression parameter. Kriging's predictor is based on the minimization of the mean-square error [39].
min{MSE (y ^(xk ))} = min{E (Y (xk ) − y ^(xk ))2}
(17)
4.2. Optimization of processing route
where X:,j represents jth column vector of input data matrix, X, μ is the mean value and V is the covariance. The Kriging approximation returns output for the new test point xk as the summation of the regression model fTβ and stochastic process Z (xk). The prediction function is given as:
y ^(xk ) = f (xk )T β + Z (xk )
=1
The optimum microstructure design of this problem is found to provide Young's modulus (E11) value of 35.8019 GPa. As a result of the optimization, our model is able to achieve a solution with a Young's modulus value of E11 = 35 GPa using the randomly oriented texture that provides a Young's modulus value of E11 = 49.6 GPa. Therefore, the optimization decreased the E11 value by 29.44%. The optimum solution is compared to the findings in the present literature in Table 1. According to the presented results in Table 1, significant improvement is achieved for Young's modulus value by optimizing the microstructural texture. Moreover, the optimum microstructure design of the present study is a polycrystal, as against the single-crystal solution presented in Ref. [17]. This is important as the production of a polycrystal is usually easier and less costly compared to single crystals. The optimum microstructure design is shown in Fig. 3. Note that the optimum ODF values are visualized in the local finite element mesh in the Rodrigues orientation space in Fig. 2.
The texture evolution during deformation processing is captured by taking several snapshots of the ODFs. Next, the Kriging technique is utilized to predict the evolution of the ODFs in a given process using DACE Toolbox [38]. The Kriging is preferred as the predictive model generated by this method that satisfies the output data points at the corresponding input points. The method is explained next. Given a set of m design sites S = [s1…sm]T, siϵRn, and responses at these sites with Y = [y1…ym]T,yiϵRq, data is normalized to satisfy the following statements [38]. (11)
to
qT A
A≥0
3.2. Process modeling with kriging
μ [S:, j] = 0, V [S:, j, S:, j] = 1, j = 1, …, n
(16)
|E11 − 25|
(15)
σz2
where is process variance, R is correlation function defined by parameter, θ. The correlation function can be defined as a spherical, linear, Gaussian, exponential, or spline function. In our study, the second-order polynomial representation is used for the regression model and a spline function is chosen for the correlation function as these functions are observed to provide the highest accuracy of the fit compared to other options. 4. Optimization of TNTZ microstructure and processing route 4.1. Microstructure optimization The objective of the optimization problem is to determine the best microstructure design producing Young's modulus value that is as close 5
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cannot be as sharp textured as the microstructure found with the mathematical design solution. Instead, the processed microstructure provides a higher intensity texture on the favorable directions in accordance with the deformation processing technique. Additionally, the computation of the volume-averaged stiffness matrix is based on the upper bound averaging technique, which is likely to provide an upper bound (higher value) for the elastic moduli values. Therefore, the actual moduli that can be obtained with the optimized microstructure designs would be even lower and thus closer to the stiffness of the bone.
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5. Conclusion The present work addresses a computational study to design βTitanium alloy, TNTZ, microstructures to be used for biomedical applications. The TNTZ alloy has recently started to attract interest as an implant material since it does not contain toxic elements. In addition, it is compatible with the human body by producing elastic modulus values that are close to the modulus of the bones. However, there is still a mismatch between these modulus values. To further improve the compatibility of the material in the human body, we perform a design optimization by optimizing the microstructural texture to minimize the difference between the modulus values of the TNTZ microstructure and the bone. We have achieved the following conclusions:
• The mathematically optimum microstructure is observed to provide • • •
Young's modulus value as low as 35 GPa. On the other hand, the deformation processing simulations demonstrate that the optimum ‘manufacturable’ microstructure can actually provide Young's modulus value of 48 GPa. Even though Young's modulus value of the optimum processed microstructure is not as low as the mathematically obtained optimum solution, it still provides a significant reduction in the modulus values compared to the previous studies. The presented methodology is applicable to the design of other material properties that can be of importance to biomedical applications by optimizing the material microstructure. The increased variability in the microstructural texture would further improve the compatibility of the alloy. This can be accomplished with the application of other processing techniques such as hot rolling and 3D printing. Therefore, future work may focus on efforts towards the manufacturing of optimum TNTZ microstructures with traditional processing techniques and 3D printing.
CRediT authorship contribution statement Şafak Çallıoğlu: Conceptualization, Methodology, Resources, Writing - review & editing, Supervision, Project administration. Pınar Acar: Methodology, Software, Validation, Investigation, Writing - original draft, Visualization. Declaration of competing interest The authors declare no conflict of interest for the manuscript. Acknowledgements The corresponding author would like to acknowledge the financial support through the Department of Mechanical Engineering at Virginia Tech. References [1] J. Allison, D. Backman, L. Christodoulou, Integrated computational materials engineering: a new paradigm for the global materials profession, JOM 58 (2006) 25–27, https://doi.org/10.1007/s11837-006-0223-5. [2] M.F. Ashby, Materials Selection Mechanical Design, 2nd, Butterworth-Heinemann,
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[31] P. Majumdar, S.B. Singh, M. Chakraborty, Elastic modulus of biomedical titanium alloys by nano-indentation and ultrasonic techniques-a comparative study, Mater. Sci. Eng. A 489 (2008) 419–425, https://doi.org/10.1016/j.msea.2007.12.029. [32] M. Besse, P. Castany, T. Gloriant, Mechanisms of deformation in gum metal TNTZ-O and TNTZ titanium alloys: a comparative study on the oxygen influence, Acta Mater 59 (2011) 5982–5988, https://doi.org/10.1016/j.actamat.2011.06.006. [33] M. Niinomi, Trend and present state of titanium alloys with body centered structure for biomedical applications, Tetsu-To-Hagane/J. Iron Steel Inst. Jpn 96 (2010) 661–670. [34] T. Akahori, M. Niinomi, H. Fukui, M. Ogawa, H. Toda, Improvement in fatigue characteristics of newly developed beta type titanium alloy for biomedical applications by thermo-mechanical treatments, Mater. Sci. Eng. C 25 (2005) 248–254, https://doi.org/10.1016/j.msec.2004.12.007. [35] N. Barton, D. Boyce, P. Dawson, Pole figure inversion using finite elements over
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Rodrigues space, Textures Microstruct. 35 (2002) 113–144, https://doi.org/10. 1080/073033002100000182. L. Anand, M. Kothari, A computational procedure for rate-independent crystal plasticity, J. Mech. Phys. Solids 44 (1996) 525–558, https://doi.org/10.1016/00225096(96)00001-4. V. Sundararaghavan, N. Zabaras, On the synergy between texture classification and deformation process sequence selection for the control of texture-dependent properties, Acta Mater. 53 (2005) 1015–1027, https://doi.org/10.1016/j.actamat.2004. 11.001. S.N. Lophaven, H.B. Nielsen, J. Sndergaard, DACE - A Matlab Kriging Toolbox, Technical University of Denmark, 2002. J.P.C. Kleijnen, W.C.M. van Beers, Customized sequential designs for random simulation experiments: kriging metamodeling and bootstrapping, Eur. J. Oper. Res 186 (2008) 1099–1113, https://doi.org/10.1016/j.ejor.2007.02.035.
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Design of β-Titanium microstructures for implant materials Şafak Çallıoğlu , Pınar Acar 1
∗
T
Virginia Tech, Blacksburg, VA 24061, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: β -Titanium alloy Microstructure Design Implant
The present work addresses the design of β-Titanium alloy, TNTZ, microstructure to be used in biomedical applications as implant materials. The TNTZ alloy has recently started to attract interest in the area of biomedical engineering as it can provide elastic modulus values that are comparable to the modulus of the human bone. Such a match between the implant and bone significantly increases the compatibility and functionality of the implant material with the human body. Experimental studies reveal that the modulus of TNTZ varies around 55–60 GPa, whereas the bones typically have modulus around 25–30 GPa. Therefore, to achieve a better match in modulus values and further improve the compatibility of the implant, we present a computational design study. As the properties of materials are significantly affected by the underlying microstructure, we focus on identifying the optimum microstructures. Our goal is to minimize the difference between the elastic modulus values of the microstructure and the bone. To ensure the manufacturability of such an optimum design solution, we analyze the microstructural evolution during deformation processing to obtain the optimum microstructure that can be processed. The outcomes of our analysis demonstrated that the elastic modulus of TNTZ can be as low as 48 GPa.
1. Introduction Estimation and control of material properties are critical for the structural components utilized in medical applications due to certain restrictions for better performance. With the introduction of the Integrated Computational Materials Engineering (ICME) [1], the goal in material design has been set to predict the properties by modeling the material microstructure and processing. The microstructural texture can be controlled during deformation processing, which arises the necessity of designing the processes to achieve desired microstructures that produce targeted properties. Besides material selection, the design of the crystallographic texture orientations in the material microstructure leads to improved properties. Techniques that allow the control of properties of polycrystalline materials involve tailoring of the preferred orientations of various crystals that constitute the material. This was done before with the graphical quantification of propertyperformance relations using the property cross-plots, as presented by Ashby [2]. A more systematic design approach would be combining the processing, structure, and property through multi-scale computational material models with the help of the recent developments in materialsby-design [3]. In the area of composites, the design of structures holding intriguing properties such as negative thermal expansion [4]
and negative Poisson's ratio [5] has existed with techniques that facilitate tailoring of microstructure topology. However, in the case of polycrystalline materials, the design strategy aims to tailor the preferred orientations of distinctive crystals forming the polycrystalline alloy. During forming processes, the formation of texture and variability in property distributions in such materials are driven with mechanisms such as crystallographic slip and lattice rotation. Understanding the interactions between processing, microstructure, and property would enable tailoring of the properties of metallic structures, when they are used as bio-materials, in accordance with the requirements in medical applications, as represented in Fig. 1. The previous work in the field of multi-scale modeling and design includes studies from our group [6-10] that were focused on the optimization of α-Titanium and Galfenol microstructures for improved mechanical properties, including thermal buckling performance and vibration frequencies. In the present work, we use a similar multi-scale modeling approach for β-Titanium alloys to enhance their compatibility in biomedical applications. To the best of authors' knowledge, this is the first time such a computational approach is applied for the design of microstructures for implants. Bio-materials are pervasively being used in the medical area. Applications include cardiovascular tissue repair and regeneration [11,
∗
Corresponding author. E-mail address: [email protected] (P. Acar). 1 Visiting student at Virginia Polytechnic Institute and State University; full-time student at Bilkent University, Ankara, Turkey. https://doi.org/10.1016/j.msec.2020.110715 Received 31 August 2019; Received in revised form 30 January 2020; Accepted 31 January 2020 Available online 06 February 2020 0928-4931/ © 2020 Elsevier B.V. All rights reserved.
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Fig. 1. Example representation of microstructure design for an implant material. Tailoring of properties is possible (such as elastic modulus (in GPa)) for metallic biomaterials by designing the material microstructures.
bone. However, identification of the optimum microstructure requires testing of numerous processing conditions and microstructures which would be overwhelmingly expensive with pure experimental effort. Therefore, we present a computational design study for metallic implants, and there is no previous study on the computational design of TNTZ microstructure for implants to the extent of our knowledge. The presented computational methodology is also applicable for other design problems for biomedical applications that target other material properties (e. g. strength, stiffness) through the optimization of the underlying microstructures. The organization of the paper is as follows. Section 2 discusses the computational methodology to model the material microstructure while the deformation processing model is given in Section 3. Section 4 presents the optimum TNTZ microstructure designs. A summary of the work with potential future extensions is provided in Section 5.
12], drug delivery systems [13, 14], and skin substitutes [15]. The metallic bio-materials are used for artificial equipment such as artificial hip joints and dental implants [16]. The implant materials in the human body must provide comparable elastic modulus values to the modulus of the bones (20–40 GPa) [16]. This is because any difference between the elastic modulus of the human bone and implant material leads to the stress shielding effect that results in bone absorption [17]. Previous studies were focused on the bio-material selection for matching the elastic modulus of the bone with a material that provides low toxicity and strong corrosion resistance. One group of the metallic bio-materials implemented was the β-phase Titanium alloys, which are perceived as an alternative to the widely used implant materials. The β-Titanium alloys are found to provide low elastic modulus values [18] compared to Co-Cr-Mo alloys [19], α + β-Titanium alloys [20], and stainless steels [21] and low density [22] which is about half of density of stainless steels and Co—Cr alloys. Thus, different Titanium alloys have been developed for biomedical applications such as spinal fusion treatment with meshed plates on the spine for hernia of intervertebral discs [23], fixations of femoral fractures [24] and internal fractures [25], spinal fixation devices [26], and dental implants [27]. Among all Titanium alloys, α + β type alloy, Ti-6Al-4V, has wider use in biomedical applications. However, it contains elements V and Al that may trigger alteration of the kinetics of the enzyme activity associated with the inflammatory response cells [28] and development of Alzheimer's disease [29]. Recently, the β-Titanium alloy, TNTZ, has been found to be a promising structure as it includes the safest alloying elements for human health [18]. Therefore, in this paper, TNTZ is chosen as the material of interest for the implant. Our goal is to match the elastic modulus of the alloy with the modulus of the human bone to improve the material compatibility in the human body by optimizing the microstructural texture. In addition, we perform computational microstructure simulations for different deformation processes to find the optimum TNTZ microstructure, which can actually be manufactured. Many experimental studies [30-34] have been conducted for reducing the elastic modulus of TNTZ to match the modulus of the
2. Microstructure modeling The microstructural behavior is modeled during deformation processing at room temperature (cold working) using the Orientation Distribution Function (ODF) (denoted by A) to capture the crystallographic texture. The ODF represents the local densities of crystals over crystallographic space. The ODF is discretized using a finite element (FE) approach using the Rodrigues parameterization space. The details on the FE scheme and Rodrigues parameterization can also be found in our earlier studies [6-10]. The complete orientation space of a polycrystal is represented with the fundamental region, a small subset, that is obtained after applying the crystal symmetries. Each crystal orientation is described with a unique coordinate, r, the parameterization for the rotation (eg. Euler angles, Rodrigues vector, etc.). The ODF, A(r), represents the volume density of crystals of orientation r. The visualization of the ODFs in the Rodrigues fundamental region for hexagonal closed-packed (HCP) Titanium is shown in Fig. 2, with the locations of the k = 50 independent nodes (the total number of 2
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Table 2 Optimum Young's modulus values and processing times. Process
E11 (GPa)
E22 (GPa)
t (s)
Tension Compression xy-shear xz-shear yz-shear
48.6969 48.7653 49.2986 49.6134 49.8016
49.7922 48.0061 49.8430 49.6073 49.5210
0.32 0.285 0.0348 0.0078 0.0434
averaged, orientation-dependent material property of the microstructural domain can be obtained as: Nelem Nint
<χ > =
∮R χ (r ) A (r , t ) dv = ∑ ∑ χ (rm) A (rm) wm |Jn | n=1 m=1
1 (1 + rm⋅rm)2 (3)
where the ODF, A ≥ 0, is a function of orientation r, and time t (during processing). The volume-averaged property can be computed with the linear expression given next:
Fig. 2. The visualization of the ODFs in the Rodrigues fundamental region for hexagonal crystal symmetry, where the locations of the 50 independent nodes are shown in red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
< χ > = pT A
where p is the property matrix that contains the material property values in the 50 independent orientations. In this work, the volumeaveraged property expression is used to compute the stiffness of the TNTZ microstructure with upper-bound averaging. Using the stiffness definition, the elastic modulus values are obtained through the compliance matrix. The design is performed for Young's modulus (E11), and the computation procedure is explained next.
Table 1 Improvements with the optimum design solution compared to other studies in the literature.
Tane et al. [17] Niinomi et al. [18] Yilmazer et al. [30] Majumdar et al. [31] Besse et al. [32]
E11 (GPa)
Improvement (%)
35 55 60 57 55
– 34.9 40.3 37.2 34.9
C = pT A
(5)
C −1
(6)
S=
E11 = nodal points is 111). The volume density of any other node in the fundamental region can be determined from the independent nodes using the crystallographic symmetries. The ODF must satisfy a normalization constraint, ∫ Adv = 1, with the integral computed over the fundamental region. The volume normalization constraint on the ODFs can be written as follows: Nelement Nint
∮R A dv = ∑ ∑ A (rm) wm |Jn | n=1
m=1
1 =1 (1 + rm⋅rm)2
1 S (1, 1)
(7)
where C is stiffness matrix, S is the compliance matrix, E11 is Young's modulus value in < 11 > direction. The following values are used for the single crystal orthotropic elastic stiffness tensor coefficients [17]: C11 = 65.1 GPa, C12 = 40.5 GPa, C44 = 32.4 GPa. In the present work, the microstructural texture is visualized in terms of the ODF space as well as the pole figures in different orientations. The pole figure for a particular diffraction plane with a unit normal, h contains the pole density function P(h,yi) measured at locations y1,y2,…,yq on a unit sphere. The value for P(h,yi) at location yi can be obtained from the ODF (A) with a linear equation based on the algorithm presented by Barton et al. [35]:
(1)
where Nelement and Nint represent the number of FEs and integration points, respectively. A(rm) is the volume fraction value at mth integration point with rm parameter, wm is integration weight associated with mth integration point, |Jn| is Jacobian determinant of nth element. Eq. (1) is equivalent to a linear expression in the ODF:
qT A = 1
(4)
k
P (h, yi ) =
∑ Mij Aj j=1
(8)
where Mij is a constant system matrix M that defines the linear relationship between the pole figures and ODFs. The expression can be used for each of the m points in a pole figure. The equations can later be
(2)
where q shows the volume normalization vector. Similarly, the volume-
Fig. 3. The optimum ODF representations. 3
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Fig. 4. Pole figures for mathematically and processed optimum microstructure.
combined with a similar set of equations for n other pole figures with different diffraction normals, h. This is equivalent to a global system of equations P = MA, where P is the pole figure matrix (m × n), M is the constant matrix of the global system ((mn) × (k)) and the ODF is denoted by A using the volume densities of k independent nodes. In order k to account for the normalization constraint, ∑i = 1 qi Ai = 1, the overall system P = MA is adjusted such that Mij = Mij − and Pi = Pi −
Mik . qk
Mik qj qk
Kothari [36]. This framework has also been demonstrated in our previous studies [6-10] and hence it will not be repeated here for brevity. The main highlight is that the texture evolution is driven by the time evolution of the ODF values. The crystals reorient in time and thus the ODFs are updated at every time step by computing the re-orientation velocity, v, as shown in Eq. (9) in terms of the incremental change in the orientation, r:
for j = 1,…,k − 1
v=
1 (ω + (ω⋅r ) r + ω × r ) 2
(9)
In Eq. (9), ω shows the spin vector, which is a function of the deformation gradient of the constitutive model. The process is defined through the macro velocity-gradient (L) [37]. An example definition for tensile (z-axis tension) is given below:
3. Process model 3.1. Texture evolution during deformation processing
0⎤ ⎡− 0.5 0 L = α1 ⎢ 0 − 0.5 0 ⎥ 0 1⎦ ⎣0
We perform simulations for five different deformation processes to find which of these processes is the most likely to produce the desired Young's modulus value. The computational model uses the rate-independent single crystal constitutive model introduced by Anand and
(10)
The readers are referred to as Ref. [7] for more information on 4
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as to 25 GPa (representing Young's modulus of human bone). The optimization is performed using the Sequential Quadratic Programming (SQP) algorithm by inputting the randomly oriented texture sample as the initial guess. The mathematical formulation of the optimization problem is as follows:
process modeling and the ODF update scheme. Using the velocity gradient definition, the simulations are performed for tension, plane strain compression, xy-shear, xz-shear, and yz-shear deformation processes. The goal is to find the optimum time for a fixed strain rate (1 s−1) where any of the processes can produce the closest match for Young's modulus value. The strain rate is determined by checking the evolution of the ODF values in different processes using different strain rates to determine a strain rate value that is providing computationally efficient and convergent results. Next, the ODF snapshots are taken at each time step of the simulations to build a predictive model of the corresponding deformation process using a numerical technique. The computation of Young's modulus value using every available ODF snapshot follows the same volume-averaging approach discussed in Section 2. Since the snapshots correspond to the actual physical data, the numerical technique is expected to provide the same texture response at the times the simulation results are available. Such a solution can be achieved with the use of the Kriging model, as explained in the next section.
min subject
μ [Y:, j] = 0, V [Y:, j, Y:, j] = 1, j = 1, …, q
(12)
(13)
(14)
Z(xk) is assumed to have a zero mean value. The spatial covariance function between Z(xk) and Z(w) can be expressed in following way [38]:
Cov [Z (xk ), Z (x w )] = σz2 R (θ , xk , w )
(18)
Our main goal is to obtain an optimum ‘manufacturable’ microstructure design. However, many sharp texture designs are difficult to be processed, especially with traditional techniques. Therefore, to understand the common textures of deformation processes such as tension, plane strain compression, xy-shear, xz-shear, and yz-shear, we perform process simulations for each case. The initial microstructure is assumed to have randomly oriented texture in all cases. With the ODF snapshots taken at different time steps, a Kriging representation of each deformation process is built to describe the ODF evolution as a function of time. The processing time is optimized to find a microstructure that minimizes Young's modulus value of the alloy. As a result, the following optimum Young's modulus values are obtained for each process: Although the optimum microstructure (mathematically optimum) can provide Young's modulus value of 35.8019 GPa, the process model reveals any deformation processing route can provide a minimum Young's modulus value of 48.0061 GPa. Besides simulating the texture evolution during these processes, various other combinations of sequential processing routes are also checked (for example, tension followed by xy-shear). The texture simulations for sequential processing provide microstructures having higher Young's modulus values than those reported in Table 2 for single processing. This result simply implies that the optimum microstructure design cannot be processed with conventional processing techniques. However, the minimum Young's modulus value obtained with a microstructure that is processed with plane strain compression is still closer to Young's modulus value of the human bone, compared to the previous studies listed in Table 1. The optimum microstructure solutions (mathematical optimum solution and the optimum solution that can be produced by the plane strain compression) are illustrated in Fig. 3 in the Rodrigues orientation space in terms of the ODF values. The pole figures (PFs) of the optimum microstructures (on < 111 > , < 100 > , and < 110 > directions) are shown in Fig. 4. Note that the pole figures in Fig. 4 help us visualize the microstructural texture in 2D views taken from different directions. As represented by Figs. 3 and 4, the mathematically optimum microstructure has a different texture compared to the optimum microstructure that can be achieved with deformation processing. This is because the processed microstructure with conventional techniques
where y ^(xk ) is output of predictor for test point xk, f(xk) is the trend function, which includes a vector of regression functions, and β is the regression parameter. Kriging's predictor is based on the minimization of the mean-square error [39].
min{MSE (y ^(xk ))} = min{E (Y (xk ) − y ^(xk ))2}
(17)
4.2. Optimization of processing route
where X:,j represents jth column vector of input data matrix, X, μ is the mean value and V is the covariance. The Kriging approximation returns output for the new test point xk as the summation of the regression model fTβ and stochastic process Z (xk). The prediction function is given as:
y ^(xk ) = f (xk )T β + Z (xk )
=1
The optimum microstructure design of this problem is found to provide Young's modulus (E11) value of 35.8019 GPa. As a result of the optimization, our model is able to achieve a solution with a Young's modulus value of E11 = 35 GPa using the randomly oriented texture that provides a Young's modulus value of E11 = 49.6 GPa. Therefore, the optimization decreased the E11 value by 29.44%. The optimum solution is compared to the findings in the present literature in Table 1. According to the presented results in Table 1, significant improvement is achieved for Young's modulus value by optimizing the microstructural texture. Moreover, the optimum microstructure design of the present study is a polycrystal, as against the single-crystal solution presented in Ref. [17]. This is important as the production of a polycrystal is usually easier and less costly compared to single crystals. The optimum microstructure design is shown in Fig. 3. Note that the optimum ODF values are visualized in the local finite element mesh in the Rodrigues orientation space in Fig. 2.
The texture evolution during deformation processing is captured by taking several snapshots of the ODFs. Next, the Kriging technique is utilized to predict the evolution of the ODFs in a given process using DACE Toolbox [38]. The Kriging is preferred as the predictive model generated by this method that satisfies the output data points at the corresponding input points. The method is explained next. Given a set of m design sites S = [s1…sm]T, siϵRn, and responses at these sites with Y = [y1…ym]T,yiϵRq, data is normalized to satisfy the following statements [38]. (11)
to
qT A
A≥0
3.2. Process modeling with kriging
μ [S:, j] = 0, V [S:, j, S:, j] = 1, j = 1, …, n
(16)
|E11 − 25|
(15)
σz2
where is process variance, R is correlation function defined by parameter, θ. The correlation function can be defined as a spherical, linear, Gaussian, exponential, or spline function. In our study, the second-order polynomial representation is used for the regression model and a spline function is chosen for the correlation function as these functions are observed to provide the highest accuracy of the fit compared to other options. 4. Optimization of TNTZ microstructure and processing route 4.1. Microstructure optimization The objective of the optimization problem is to determine the best microstructure design producing Young's modulus value that is as close 5
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5. Conclusion The present work addresses a computational study to design βTitanium alloy, TNTZ, microstructures to be used for biomedical applications. The TNTZ alloy has recently started to attract interest as an implant material since it does not contain toxic elements. In addition, it is compatible with the human body by producing elastic modulus values that are close to the modulus of the bones. However, there is still a mismatch between these modulus values. To further improve the compatibility of the material in the human body, we perform a design optimization by optimizing the microstructural texture to minimize the difference between the modulus values of the TNTZ microstructure and the bone. We have achieved the following conclusions:
• The mathematically optimum microstructure is observed to provide • • •
Young's modulus value as low as 35 GPa. On the other hand, the deformation processing simulations demonstrate that the optimum ‘manufacturable’ microstructure can actually provide Young's modulus value of 48 GPa. Even though Young's modulus value of the optimum processed microstructure is not as low as the mathematically obtained optimum solution, it still provides a significant reduction in the modulus values compared to the previous studies. The presented methodology is applicable to the design of other material properties that can be of importance to biomedical applications by optimizing the material microstructure. The increased variability in the microstructural texture would further improve the compatibility of the alloy. This can be accomplished with the application of other processing techniques such as hot rolling and 3D printing. Therefore, future work may focus on efforts towards the manufacturing of optimum TNTZ microstructures with traditional processing techniques and 3D printing.
CRediT authorship contribution statement Şafak Çallıoğlu: Conceptualization, Methodology, Resources, Writing - review & editing, Supervision, Project administration. Pınar Acar: Methodology, Software, Validation, Investigation, Writing - original draft, Visualization. Declaration of competing interest The authors declare no conflict of interest for the manuscript. Acknowledgements The corresponding author would like to acknowledge the financial support through the Department of Mechanical Engineering at Virginia Tech. References [1] J. Allison, D. Backman, L. Christodoulou, Integrated computational materials engineering: a new paradigm for the global materials profession, JOM 58 (2006) 25–27, https://doi.org/10.1007/s11837-006-0223-5. [2] M.F. Ashby, Materials Selection Mechanical Design, 2nd, Butterworth-Heinemann,
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