High Throughput Assays for Additively Manufactured Ti-Ni Alloys Based on Compositional Gradients and Spherical Indentation
- 180 Downloads
Recent advances in additive manufacturing (AM) reveal an exciting opportunity to build materials with novel internal structures combined with intricate part geometries that cannot be achieved by traditional manufacturing approaches. The large space of potential material chemistries combined with non-equilibrium microstructures obtained in AM presents a significant challenge for a systematic exploration and optimization of the final properties exhibited by AM parts when using the existing knowledge databases established for conventionally processed materials. In this paper, we demonstrate novel high throughput assays that can be used to prototype a large library of material chemistries (and possibly different process histories) in small quantities, and subsequently apply spherical indentation stress-strain protocols to screen them for their mechanical performance. The potential of these new assays is demonstrated on a class of Ti-Ni alloys, whose Ni composition ranges between 0 and 11%wt.
KeywordsHigh throughput Nanoindentation Ni-Ti Hertzian indentation Additive manufacturing
Metal additive manufacturing (AM) processes today provide a unique pathway for generating novel components with intricate geometry as well as with novel material internal microstructures that could be customized at different sample locations for desired performance . The non-equilibrium microstructures obtained in the AM process, especially, provides additional opportunities to design the material itself to meet the specific demands of the application. Indeed, alloy spaces that are not suitable for conventional processing due to, for example, strong solute partitioning in the liquid phase (e.g., generation of β-fleck in Ti-based alloys  from depressed freezing temperatures ) or the potential precipitation of undesirable phases, may be alternatively explored using additive manufacturing, as the overall undercooling decreases the freezing range and influences the kinetics of both solute segregation in the liquid phase and unwanted solid-state phase transformations. This gives a chance to design new classes of alloys that are not “casting alloys,” but rather are alloys designed to take advantage of the processing attributes associated with additive manufacturing.
When new alloys are designed, it is necessary to develop a high degree of confidence in the performance of the alloy during service. Traditionally, at the most basic level, this is accomplished by developing “design allowables” where the statistical distributions of properties are well understood. Invariably, static properties such as those obtained from tensile coupons (e.g., yield strength, elastic modulus, ultimate strength, % elongation, % reduction in area) form the basics of a “design allowable” data package. However, the space of potential material chemistries combined with non-equilibrium microstructures is enormously large . To obtain the desired data covering the large materials space of interest in AM, it will be necessary to produce and test a large number (i.e., in the hundreds) of standardized bulk tension coupons extracted from material characterized by a good control of processing history and chemistry in the entire sample volume. It thus poses a significant challenge to find the right processing recipes and parameters that fully exploit the unusual characteristics of AM and seems to hinder the potential to design new alloys. Saltzbrenner et al.  have recently developed and demonstrated high-throughput prototyping and tensile testing protocols that are capable of addressing some of the issues described above, including reliable assessment of the variance in the measured properties. These methods continue to employ tensile test specimen, which may not be practical for many AM processes.
One promising strategy to tackle the large space of potential material chemistries is to develop and employ high throughput assays that only require very small material quantities. This is ideally suited for AM processes that can vary material chemistries and process conditions from location to location in the same sample. In essence, this strategy allows one to build a large library of material chemistries using different processing histories, and then perform structure and mechanical characterization at small scales to assemble the desired knowledge databases needed for design of AM parts. Indeed, one can plan and design cost-effective studies that are exclusively aimed at establishing the salient process-structure-property linkages of interest in material systems explored by AM.
A significant current impediment in the implementation of the vision described above comes from the lack of a reliable methodology and protocols for the mechanical characterization of samples in very small volumes. In this regard, recent advances in spherical indentation stress-strain protocols [6, 7, 8, 9, 10, 11, 12, 13] offer many advantages. The conventional practices in indentation testing rely on sharp indenters and analyze the unloading segments after imposing substantial amounts of indentation loads (or depths) to arrive at estimates of modulus and hardness [14, 15, 16]. One of the consequences of adopting these traditional protocols is that they significantly alter the material in the indentation zone (due to the imposed plastic deformation) before extracting the mechanical properties of interest. In contrast, the recently developed spherical indentation stress-strain protocols analyze the very short initial loading segments and recover a meaningful indentation stress-strain curve that reflects the intrinsic elastic-plastic properties of the virgin material in the indentation zone. Since the size of the indentation zone can be systematically varied in the modern instrumented indenters with the use of different indenter radii, this new capability promises to offer a high throughput strategy for multiscale mechanical characterization of AM processed materials. An important inherent advantage of employing indentation techniques is that it allows one to conduct a multitude of tests on a single sample and extract statistically meaningful information on the variability in the location-specific properties exhibited by the produced part in a cost-effective manner.
In this paper, we describe our efforts and initial successes in developing high throughput assays for AM processed materials that address sample prototyping as well as characterization of relevant microstructure and properties. Specifically, the additive manufacturing technique known as Laser Engineered Net Shaping (LENS™) will be employed in this study. Ti-Ni alloys, whose Ni composition ranges between 0 and 11%wt, were selected for this study. This system was selected because it is a model eutectoid β-stabilized system, which is traditionally susceptible to problems such as β-fleck and hence is not considered for conventional cast/wrought titanium alloys. Due to the β-fleck, the composition regime is traditionally limited, as most industrially relevant titanium alloys are relatively solute lean in elements that promote β-fleck, such as Fe and Ni. However, if the β-fleck can be avoided, which is the case under AM process conditions, this system may exhibit enhanced strength properties due to the mismatch in the shear moduli and atomic radii of the two elements (these will influence the ease with which dislocations move and the potency for precipitates which will act as barriers for dislocations).
High Throughput Assays
High Throughput Sample Prototyping
High Throughput Mechanical Characterization
As noted earlier, we envision using the recently developed spherical indentation stress-strain protocols for evaluating local mechanical properties. In doing so, we will pay careful attention to quantifying the effects of the macroscale gradients in material chemistry introduced in the samples as well as the inherent microscale heterogeneity in the AM processed samples. While the macroscale chemistry gradients were purposefully introduced in the samples, the microscale heterogeneity is unavoidable and likely to be more severe in the AM processed samples compared to the conventionally processed samples. The indentation methods, because of their very small probe volumes and high throughput nature, are capable of quantifying the variances in the measured mechanical properties at both these scales. The potential benefits of these indentation techniques in extracting reliable process-structure-property linkages have been demonstrated in a few recent studies [7, 25]. Other currently available approaches suitable for mechanical characterization in small material volumes include microscale tension/bending tests [26, 27] and micropillar deformation tests (produced by focused ion beam (FIB) milling) [28, 29]. Both techniques require time-consuming and expensive sample preparation , and thus are not amenable to HT approaches aimed at exploring large design spaces in materials development. Instrumented spherical indentation is indeed the only practical technique of choice for our purpose. In this regard, it should be noted that the recently developed spherical indentation protocols [6, 8] are capable of extracting indentation stress-strain (ISS) curves from very small volumes (indentation zone radii ranging from ~ 50 nm to ~ 500 μm). Significant progress is being made in correlating the properties measured in indentation with the properties measured in conventional testing such as simple compression [6, 7, 31, 32], which will aid tremendously in increasing the interpretability of the indentation results.
where the indentation load and the elastic indentation displacement are denoted by P and he, respectively. The effective modulus and the radius of the indenter-sample system are denoted as Eeff and Reff, respectively. The subscripts i and s correspond to the indenter and the sample, and E and ν denote the Young’s modulus and Poisson’s ratio, respectively. S (=dP/dhe) denotes the elastic stiffness (i.e., the slope of the unloading curve, also known as harmonic stiffness in continuous stiffness measurement (CSM) protocols [14, 35]).
The indentation yield strength, Yind, is defined using a 0.2% offset plastic strain on the indentation stress-strain curve. The indentation work hardening rate, Hind, is computed from the indentation stress-strain curve between 0.5 and 2% offset plastic strains using a linear regression fit [7, 11].
To extract useful insights, one needs to correlate the measured indentation properties with salient descriptors of the microstructures in the indentation zone. If the indentation length scales are well below the typical grain size, relevant microstructure attributes would include the grain orientation and local dislocation density in the indented zone [7, 10, 36, 37]. If the indentation zone cover several grains, the relevant microstructure features might include grain size, shape, and orientation distributions in the indented zones . In the present study on multiphase microstructures, we will focus mainly on the phase volume fraction and an averaged spacing between the intermetallic phase in the microstructure as the main microstructure descriptors. This is because we anticipate these features to exhibit strong correlations with the properties measured in indentation. In order to obtain these microstructure measures, we will employ backscattered electron (BSE) microscopy (in the scanning electron microscope).
Application to Ti-Ni Alloys
High Throughput Processing
After aging, the sample was further sectioned longitudinally (see Fig. 3c), prepared for metallography and indentation tests. The sample preparation involved a specific sequence of steps that proceed through grinding (P240, P1200 SiC paper) and then finished with polishing steps with decreasing abrasive particle size (9, 3, 1 μm diamond suspension), with each step ensuring that the surface deformation introduced is removed in the next step. The importance, particularly in indentation, of removing the damaged layer from mechanical polishing without introducing further mechanical deformation has already been expounded . This is mainly because of the very small indentation volumes involved in the indentation protocols employed in this work. We opted for a final polishing step using solution mixing 5 parts of 0.06 μm colloidal silica suspension with 1 part of hydrogen peroxide. Similar sample preparation protocols were successfully employed in prior nanoindentation work on titanium alloys .
Our focus here is to explore the correlations between chemical composition induced salient microstructure descriptors such as phase volume fractions and spacing of intermetallic phase (same as the length scale of the matrix phase) and their associated mechanical properties measured by indentation (i.e., indentation modulus, indentation yield strength, and indentation hardening rate). For this purpose, five locations were chosen longitudinally for microstructure characterization and indentation tests (see Fig. 3c). Since the longitudinal direction was controlled to have maximum compositional gradient and the horizontal direction should have no compositional gradient, the locations are intentionally picked to maximize the differences in local nickel compositions.
In addition, we are also interested in quantifying and correlating the variabilities in both the microscopy and the indentation measurements. With this goal in mind, multiple measurements were performed at each of the five selected locations. More specifically, they were performed on a 5 × 5 grid with a uniform spacing of 100 μm in both directions between the grid points (see Fig. 3d). The selected spacing was designed to be large enough to avoid interactions between neighboring indented zones (i.e., it is substantially larger than the estimated indentation zone size for each indentation). At the same time, it was designed to be small enough to ensure that the measurements on a single grid represent the microstructure and the property for a single chemistry. In other words, the compositional gradient within each grid is maintained to be very small and negligible for the present study.
Average measurements at each of the five locations studied in the high throughput (HT) sample produced for this study
120.2 ± 9.0
126.5 ± 4.8
120.5 ± 4.6
122.7 ± 4.9
122.3 ± 1.9
1.06 ± 0.09
1.16 ± 0.10
1.15 ± 0.05
1.26 ± 0.06
1.37 ± 0.04
47.0 ± 3.0
53.0 ± 3.7
59.9 ± 4.0
52.5 ± 3.5
70.5 ± 3.5
5.5 ± 0.7
15.98 ± 1.44
23.1 ± 0.9
27.85 ± 1.77
35.8 ± 1.6
4.07 ± 1.02
1.42 ± 0.31
0.73 ± 0.08
0.53 ± 0.04
0.33 ± 0.03
wt% of nickel
High Throughput Mechanical Characterization
Given the type and amount of information aggregated in the study, several idealizations and simplifications had to be made in seeking the structure-property correlations of interest that could guide future efforts. First, it was decided that SEM imaging for microstructure analysis would be done adjacent to the actual indented zones (see Fig. 3). Since it is currently very difficult to document the material structure in the indented zone in a non-destructive but high throughput manner, this approximation is unavoidable as well as reasonable since microstructure heterogeneity would be low at the length scales at which structure is being quantified. Next, it was decided that each microstructure scan would cover a region of 50 μm × 50 μm. This scan size is designed to be somewhat larger than the indented zone size, but not too large that the gradient in the composition would have any significant effect. It was also decided that the microstructure would be characterized using BSE in the SEM, as this is a sufficiently fast technique and provides reliable contrast to allow measurements of phase volume fractions and averaged lengths. The lattice orientations of the grains in the indented zones were not measured in this work. Although the grain orientation is expected to play an important role in the indentation properties of hcp metals [45, 46, 47, 48, 49], these dependencies are most clearly seen in indentations performed with smaller indenter tips where the indented zones are fully embedded in a single grain. In fact, most of the prior work on the use of spherical indentation stress-strain curves was focused on single-phase coarse-grained polycrystalline samples [10, 37].Only recently, these protocols have begun to be applied to multiphase polycrystalline samples . Some anisotropy was observed in the residual indents at the indentation sites with low volume fractions of intermetallic precipitates (see the elongated residual indent shapes at location 1 micrograph in Fig. 6). The anisotropy was significantly lower at indentation sites with higher volume fractions of the intermetallic precipitates, as seen in the more circular residual indent shapes (see location 5 micrograph in Fig. 6). Additionally, the measured values of the indentation moduli and yield strengths (Table 1) at each sample location showed only a modest variance (less than 10% in all cases). This variance is significantly lower compared to the variation observed in the indentation yield strengths with the increase in the volume fraction of the intermetallic precipitates. As a result of these observations, we are fairly confident that the mean values reported in Table 1 are not strongly influenced by the differences in the local microstructure in the multiple indentation volumes probed at each sample location. It also appears that the local mechanical anisotropy in the indentation probe volumes is largely mitigated by the presence of the intermetallic precipitates in our samples.
Several different approaches were explored in this study to quantify the microstructure information. First, BSE micrographs were segmented by Otsu’s method [50, 51] which separates two classes of pixels in images, in accordance with the two-phase nature of this sample. Then, the segmented binary images were used to calculate volume fraction of the intermetallic phase. The averaged value of the volume fraction estimated at each of the five locations is reported as Vint in Table 1. Chord length distributions (CLD)  from the segmented images were also collected. CLD effectively captures the shape and size distribution of phases in any given microstructure. It is closely related to the lineal-path function , and is likely relevant to the prediction of the effective plastic properties [52, 54]. For the CLD computation performed in this study, horizontal chords that are fully in the α-Ti matrix with ends abutting a phase boundary are identified and analyzed. The averaged value from each CLD was reported as the feature length scale for the matrix phase, L in Table 1. The direction of chords for CLD computation was chosen arbitrarily because the feature length scale would be measured and calculated from multiple images of multiple grains. Table 1 summarizes the measured values of the volume fractions of intermetallic phase and the averaged feature length of the titanium matrix, along with the corresponding indentation properties measured in this study.
Pearson product-moment coefficient between different structure and property measures obtained on the HT sample
Quantitative correlations of the local mechanical properties measured in indentation to the bulk mechanical properties typically measured in standardized testing (e.g., simple compression) are still under development . The main issues in this correlation are indeed related to the differences in the probe volumes and the inherent anisotropy of material response at the probed length scales. At this time, these correlations are most mature at the two ends of the spectrum of applications: (i) the indented volume is very large compared to the representative volume element of the material microstructure, and allows idealization of the material response in the indentation zone as a homogeneous isotropic medium [13, 57] and (ii) the indented volume is a single phase crystalline region (e.g., a region within a grain in a single-phase polycrystalline sample) [7, 13, 32, 58]. Clearly, the indentations in the sample studied here do not fall into either of these idealized conditions.
As mentioned earlier, the microscale features of interest in interpreting the indentation measurements are the spatial distributions of grain orientations and precipitates in the indentation volumes. In most of the indentations conducted here, the indentation zones are significantly smaller than the grain sizes (200 ~ 300 μm), but are larger than the precipitate sizes. As noted earlier, the presence of the precipitates has mitigated some of the grain orientation-induced anisotropy associated with the indentation measurements reported here. Consequently, our indentations in multiple grains at the same location in the sample have not revealed a strong variation, especially when compared to prior work on α-Ti . We therefore believe that the averaged values from the multiple indentations presented in Table 1 could serve as good surrogate measures of the bulk properties that might be measured in standardized testing. However, it should be noted that the averaged indentation values do not capture any Hall-Petch effects  from the grain boundaries (note that these were not present in any of the indentations reported in this study). If grain size effects are significant (compared to the effects of the precipitate size and volume fraction) then one needs to conduct larger indentations that sample a few grain boundaries in the probed volumes.
Because of its dependence on interatomic bonding , the elastic modulus is expected to be impacted by the alloy composition, distribution of grain orientations [7, 61], and the phase volume fraction [62, 63]. Using the shear and bulk modulus values estimated previously by Toprek et al. , one might expect the elastic modulus of Ti2Ni to be approximately 149 GPa. This value is significantly higher than the value of 105 GPa reported in literature for CP Ti . The indentation moduli measured in this study did not show significant variations between the different locations on the HT sample, and fell in the range of 120–126 GPa. The corresponding values of Young’s moduli (see Eq. (9)) would fall in the range of 109–115 GPa. This range of values is quite reasonable, keeping in mind the values of the Young’s moduli mentioned earlier for the constituent phases. If one were to use the rule of mixture, based on the volume fraction of the intermetallic phase, the predicted values of Young’s moduli for the five locations would be in the range of 107–121 GPa. It is therefore clear that the moduli measured by indentation protocols described in this work are quite reasonable and reliable. The very small location-to-location variation of the measured Young’s modulus also explain the low values of the correlation coefficients between the indentation moduli and both the intermetallic volume fraction as well as the averaged matrix chord length (see Table 2).
As seen in Table 1, the indentation yield strengths increased systematically with the increase in Ni content. Furthermore, these are highly correlated with the microstructure measures (Table 2). As expected, there is a strong positive correlation between the indentation yield strength and the volume fraction of the intermetallic phase. Similarly, there is a strong positive correlation between the indentation yield strength and the averaged chord length raised to a power of −0.5 (following the well-known Hall-Petch relations ).
Several strengthening mechanisms including solid solution strengthening, strengthening due to secondary phase, and grain boundary strengthening are likely to contribute to the observed increase in indentation strength with the increase in Ni content. In terms of solid solution strengthening in the matrix, the presence of Ni in pure Ti is expected to increase yield strength by 35 MPa per wt% addition . The presence of oxygen is also expected to make a significant contribution to increasing the matrix yield strength. For example, it has been reported in literature that the yield strength of titanium can increase from 100 MPa (pure titanium) to 480 MPa (grade 4 CP titanium ), mainly due to increased oxygen content. Furthermore, we should also expect a contribution from phase strengthening due to the intermetallic phase .
The measured indentation yield strengths can be converted to simple compression yield strengths using a factor of 2.0 . Furthermore, fitting the estimated simple compression yield strengths at the five locations to the self-consistent model of Stringfellow-Parks  (this model uses only the volume fraction and yield strengths of the constituent phases), we estimated that the yield strength of α–Ti would be in the range of 400–600 MPa and that of Ti2Ni would be in the range of 1.1–1.2 GPa. These ranges are quite reasonable based on the values reported in literature for grade 4 CP titanium mentioned above and the reported values of 0.9–1.6 GPa  for Ti2Ni.
The hardening values extracted from the indentation stress-strain protocols employed here are much higher than the values measured typically in conventional simple compression tests. This observation has been made consistently in all prior work with the indentation stress-strain protocols [7, 13] and does not yet have a clear explanation. At this time, the increased hardening rates in the indentation stress-strain curves are generally attributed to the fact that the elastic-plastic transition in these tests occurs over a much larger strain range compared to conventional simple compression stress-strain curves . Nevertheless, these values can be used as relative measures to compare the hardening rates at the different location on the HT sample. A 50% increase is observed in the hardening rates from locations 1 to 5. Location 2 shows a small increase (12.7%) over location 1. Locations 2, 3, and 4 seem to show similar hardening rates. A sharp increase is seen in the hardening rate is noted for location 5 (approximately 34%). It should be generally expected that hardening rates would increase with an increase in the interface area per unit volume. For the present study, this is captured indirectly by the averaged chord length of the matrix. Clearly, this variable has a significant effect on the hardening rates exhibited by the sample, especially as the chord lengths decrease to very low values.
A high throughput approach to explore mechanical properties of AM alloys with different material chemistries has been presented and demonstrated on Ti-Ni alloy systems. The LENS™ process has been shown to successfully produce controlled variations in material chemistry and microstructures (Fig. 5) along the length of the manufactured sample. We were able to achieve reasonable variation in terms of phase distribution and volume fraction of the Ni-rich second phase (Ti2Ni) within a 38-mm sample. Similarly, spherical indentation protocols have effectively generated meaningful trends for structure-property relations, as depicted in Fig. 6, Tables 1 and 2. Hence, the combined approach of using AM processes and spherical indentation shows tremendous potential for high throughput alloy development of hierarchical materials for AM processes. These HT assays can be used for exploring large spaces of alloy chemistry and relevant ranges of process parameters in a wide range of alloy systems.
The authors recognize that the specifics of AMS 4999A may not be suitable for all alloys. However, the Ti-xNi system is an active (fast) eutectoid system, and is expected to reach a near equilibrium condition under these commonly used and industrially accepted conditions.
XG, SK, PCC, and MM would like gratefully acknowledge support provided by the National Science Foundation awards 1435237 and 1606567 (“DMREF/Collaborative Research: Collaboration to Accelerate the Discovery of New Alloys for Additive Manufacturing”).
- 1.Collins PC, et al (2014) Progress Toward an Integration of Process – Structure – Property – Performance Models for “Three-Dimensional (3-D) Printing” of Titanium Alloys JOM 66(7):1299–1309Google Scholar
- 5.Salzbrenner B et al (2016) Defect Characterization for Material Assurance in Metal Additive Manufacturing (FY15–0664), 2016, Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)Google Scholar
- 11.Pathak S, Kalidindi SR, Mara NA (2015) Investigations of orientation and length scale effects on micromechanical responses in polycrystalline zirconium using spherical nanoindentation. SMM 113:241–245Google Scholar
- 13.Weaver JS, Kalidindi SR (2016) Mechanical characterization of Ti-6Al-4V titanium alloy at multiple length scales using spherical indentation stress-strain measurements. Mater Des 111(5 Dec 2016):463–472Google Scholar
- 33.Johnson KL (2009) Contact mechanics. Proc Inst Mech Eng Part J-J Eng Tribol 223(J3):254–254Google Scholar
- 34.Hertz H (1896) Miscellaneous papers. MacmillanGoogle Scholar
- 38.Fan Z, Liou F (2012) Numerical modeling of the additive manufacturing (AM) processes of titanium alloy. Titanium alloys–towards achieving enhanced properties for diversified applications: p. 3–28Google Scholar
- 39.Baker AH, Collins PC, Williams JC (2017) New Nomenclatures for Heat Treatments of Additively Manufactured Titanium Alloys. JOM 69(7):1211–1227Google Scholar
- 40.Mendoza MY et al (2017) Microstructures and Grain Refinement of Additive-Manufactured Ti-xW Alloys. Metall Mater Trans A 48(7):3594–3605Google Scholar
- 41.Mantri, S., et al., The effect of boron on the grain size and texture in additively manufactured β-Ti alloys. J Mater Sci: 52(10):12455–12466Google Scholar
- 42.Rolchigo MR, Mendoza MY, Samimi P, Brice DA, Martin B, Collins PC et al (2017) Modeling of Ti-W solidification microstructures under additive manufacturing conditions. Metall Mater Trans A 48:3606–3622Google Scholar
- 43.Dutta B, Froes FHS (2016) Additive manufacturing of titanium alloys. Butterworth-Heinemann LimitedGoogle Scholar
- 44.SAE International (2011) Titanium Alloy Direct Deposited Products 6Al - 4V Annealed, SAE Standard AMS4999AGoogle Scholar
- 45.Williams JC, Baggerly RG, Paton NE (2002) Deformation behavior of HCP Ti-Al alloy single crystals. Metall Mater Trans A 33(3):837–850Google Scholar
- 56.Dipen KP, Kalidindi DR (2016) Engineering M Correlation of spherical nanoindentation stress-strain curves to simple compression stress- strain curves for elastic-plastic isotropic materials based on finite element models. Acta Mater 112:295–302Google Scholar
- 57.Weaver JS, Khosravani A, Castillo A, Kalidindi SR (2016) High throughput exploration of process-property linkages in Al-6061 using instrumented spherical microindentation and microstructurally graded samples. Int Mater Manuf Innov 5(1):1–20. https://doi.org/10.1186/s40192-016-0054-3
- 59.Smith WF et al (2006) Fundamentos de la ciencia e ingeniería de materiales: McGraw-HillGoogle Scholar
- 60.Dieter GE, Bacon DJ (1986) Mechanical metallurgy, vol 3. McGraw-hill, New YorkGoogle Scholar
- 62.Voigt W (1910) Lehrbuch der Kristallphysik, Leipzig, BerlinGoogle Scholar
- 63.Reuss A (1929) Berechnung der fließgrenze von mischkristallen auf grund der plastizitätsbedingung für einkristalle. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 9(1):49–58Google Scholar
- 66.Materials Properties Handbook: Titanium Alloys, ed. E.W.C. R. Boyer, and G. Welsch 1993: ASM InternationalGoogle Scholar
- 67.ASTM B348-13 (2013) Standard Specification for Titanium and Titanium Alloy Bars and Billets. ASTM International, West Conshohocken, PA. https://doi.org/10.1520/B0348