DETERMINATION OF YOUNG’S MODULUS VIA EIGENFREQUENCY ANALYSIS
Innovative methods for characterizing elastic properties
In addition to density, strength, fracture mechanical properties and corrosion resistance, the stiffness of the material plays a crucial role in the component design, particularly in case of lightweight construction. In contrast to other material properties, the stiffness of aluminum alloys can hardly be influenced through adjustments in the chemical composition and the production process. Nevertheless, requirements on the Young’s modulus are an important aspect in material specifications, especially for aircraft materials. The fulfillment of these requirements must be proven in qualification procedures. To meet customer demands on the quality of the material, AMAG investigates the factors influencing the material stiffness and additionally develops suitable characterization techniques for the experimental determination of elastic properties. In addition, the stiffness of the material reflects valuable information about the microstructure, which can be useful for material and process development.
Influence of crystallographic texture on anisotropic elastic properties
In contrast to the influence of alloying elements, which contribute directionally independently to the material stiffness, crystallographic textures lead to anisotropic elastic properties.
The elastic properties of aluminum alloys are typically described by Young’s modulus (approx. 70 GPa) and by shear modulus (approx. 26 GPa), assuming isotropic behavior. However, in practice, the elastic properties are direction-dependent, and the description of the fully anisotropic elastic behavior requires the determination of 21 independent tensor components, which requires extensive experimentation. [1]The anisotropy of the elastic properties originates from the anisotropy of the single crystal. For high-purity aluminum, the Young’s modulus shows its minimum of 63.1 GPa along the [100] direction and reaches its maximum of 74.5 GPa along the [111] direction of the face-centered cubic crystal lattice, as shown in (Figure 1) [2].
However, aluminum as a polycrystal consists of differently oriented grains, which results in a combination of different stiffnesses on the macroscopic scale. In the one-dimensional case, this can be modeled using a strongly simplified model of serial and parallel configurations of stiffnesses (Figure 2). This concept can be extended to the three-dimensional case, which allows for predicting the elastic behavior on macroscopic scale if the orientation distribution and the elastic properties of the single crystal are known. In case of a random crystallographic texture, the combination of different stiffnesses, resulting from the orientation distribution, leads to the typical Young’s modulus of approx. 70 GPa, independently from the testing direction.
However, cold rolling generates strong crystallographic textures, as the deformation leads to the alignment of crystals along preferred orientations with respect to the rolling direction (RD), transverse direction (TD), and normal direction (ND).Cold-rolling leads to crystal orientations on the β-fiber, which represents a curve in the Euler space, reaching from the Brass component through the S component to the Copper orientation.Each point in the Euler space corresponds to a specific crystal orientation, which is described by the three Euler angles (Φ1, Φ, Φ2), as shown in Figure 3.
| Cold-rolling textures | Cold-rolling textures | Recrystallization textures | Recrystallization textures | ||
|---|---|---|---|---|---|
| Brass (%) | S (%) | Copper (%) | Cube (%) | InvGross (%) | |
| Hot-rolled | 5 | 7 | 11 | 22 | 15 |
| Cold-rolled | 18 | 16 | 22 | 7 | 8 |
| Soft-annealed | 10 | 8 | 11 | 20 | 13 |
The cold rolling texture components (Brass, S, Copper) exhibit a higher Young’s modulus along the rolling and transverse directions compared to the diagonal direction (DD). During recrystallization the grains are reconstructed, leading to a reorientation of the crystals. In this process, the Cube and InvGoss texture components are preferentially formed, both exhibit a higher Young’s modulus in the diagonal direction than along the rolling and transverse directions (Figure 4). Consequently, cold rolling and recrystallization affect the direction-dependent Young’s modulus.
Mechanical approaches to determine the Young’s modulus
The experimental determination of Young’s modulus from the stress–strain relationship in the elastic regime during tensile loading is specified in material specifications and described in testing standards (e.g. ASTM E111). However, a variety of factors may influence the results obtained using this characterization method. These include the accuracy of the specimen geometry, the precision of the force and strain measurements, as well as non-linearities in the stress-strain relationship, which may arise from microplasticity, even in the elastic regime. In addition, the test standards allow different procedures for loading modes, data acquisition and evaluation. Therefore, in practice, determining the Young’s modulus uniquely by tensile loading experiments represents a challenge. To nonetheless enable a reliable determination of the Young’s modulus in tensile loading experiments, state-of-the-art testing machines and highly precise measurement equipment for force and strain measurement are necessary, which AMAG has at its disposal at the Center for Material Innovation (CMI). To ensure that the experimental procedure and the data evaluation are well adjusted, comparative measurements through alternative characterization methods are beneficial. A further method for precise determination of elastic properties is the impulse excitation technique (IET). In this characterization method a bar-shaped specimen is excited to oscillation, allowing to determine the specimen’s eigenfrequencies through a Fourier analysis of the emitted sound signal. The eigenfrequencies of a specimen depend on the material’s density, the specimen geometry and the elastic properties. This allows for the determination of the Young’s modulus in the specimen’s longitudinal direction from the flexural eigenfrequency as well as the shear modulus within the specimen plane from the torsional eigenfrequency (Figure 5).
The impulse excitation technique offers the advantage that oscillation leads to very low strains at high strain rates, which almost eliminates plastic effects. In addition, the accuracy of the result mainly depends on the accuracy of the specimen geometry. Consequently, changes of elastic properties on the same specimen can be captured at high precision, as measurement deviations resulting from imperfections of the specimen geometry effectively cancel out.
Extensive comparisons between results from tensile loading tests and the impulse excitation technique showed that, for aluminum alloys, the stress range up to half of the Rp0.2 yield strength is well-suited for evaluation of the linear stress-strain relationship. The Young’s modulus is determined by applying linear regression to multiple unloading curves. For alloys and tempers showing a low yield strength, the uncertainty in the determined Young’s modulus is higher due to the limited measurement range.
Figure 6 represents a comparison between determined Young’s moduli using tensile loading and IET by means of plate material from the alloy EN-AW 7475 T7351 in different measurement directions and thickness positions. It becomes evident that the spread in results obtained via the impulse excitation technique is significantly lower than that observed in tensile loading tests. The absolute values of the Young’s moduli obtained via the impulse excitation technique are higher than those obtained from tensile testing, as plastic effects are excluded.In addition the results show, that the Young’s moduli in rolling direction and transverse directions are higher than along the diagonal direction. This is characteristic for a partially recrystallized microstructure containing a significant volume fraction of cold-rolling texture components, which lead to higher stiffness in the rolling and transverse directions compared to the diagonal direction.
Determination of orthotropic elastic properties using the impulse excitation technique
The symmetries associated with the rolling process in terms of rolling direction, transverse direction and normal direction are preserved in the crystallographic texture and reflected in orthotropic elastic properties. This allows to fully describe the elastic behavior in the rolling plane by only four tensor components. The impulse excitation technique is standardized by the ASTM E 1876 only for homogeneous and isotropic materials and has been extended for the determination of anisotropic elastic properties. [1] A suitable combination of four eigenfrequencies from three differently oriented specimens enables the complete determination of the anisotropic elastic properties within the rolling plane. [3] The direction-dependent Young’s and shear moduli of the mentioned material EN-AW 7475 T7351, determined using the impulse excitation technique for different thickness positions, are presented in Figure 7.
Prediction of elastic properties from texture measurements
In addition to mechanical methods, anisotropic elastic properties can also be determined from texture measurements. If the orientation distribution is known, the anisotropic elastic properties on macroscopic scale of the polycrystal can be predicted from the elastic properties of the corresponding single crystal. In contrast to tensile loading, this approach is also independent of plastic effects and does not require a defined specimen geometry. However, the influence of alloying elements is not included, as commonly the single crystal data of high-purity aluminum from literature is assumed. Additionally, the following aspects must be considered for the determination of the orientation distribution: The crystal orientation must be measured at sufficient accuracy to determine the elastic properties as precisely as possible.For this purpose, electron backscatter diffraction (EBSD) is a highly suitable approach. However, it must be ensured that the determined orientation distribution is statistically representative of the investigated specimen volume by capturing a sufficient number of grains to predict the elastic behavior on macroscopic scale. In case of rolled products a gradient over thickness in the volume fractions of texture components can be observed, as rolling induces more deformation close to the surface compared to the center of the sheet or plate. Taking into account these aspects leads to measurement planes perpendicular to the rolling direction and through the complete thickness of the product. Using a state-of-the-art scanning electron microscope in combination with EBSD, which is available at AMAG’s Center for Material Innovation (CMI), such measurements can be performed within a in few minutes.
Process characterization through evolution of elastic properties
The described impact of cold rolling and recrystallization on the anisotropic elastic properties in combination with the impulse excitation technique allows to determine the recrystallization kinetics during annealing. [3] An adapted setup of the impulse excitation technique placed in a furnace enables the measurement of the specimen’s eigenfrequencies at elevated temperatures, allowing to capture the evolution of the Young’s modulus during soft annealing in an in-situ experiment.This characterization method is demonstrated using a sheet of pure aluminum. The strip was initially hot-rolled to a thickness of 10 mm, then cold-rolled to achieve a thickness reduction of 50% and subsequently soft-annealed at 270°C. Table 1 summarizes the evolution of the volume fractions associated with the texture components, as measured via EBSD, through the processing steps. Starting from the hot-rolled strip, cold rolling increases the volume fractions corresponding to cold rolling textures. Consequently, the Young’s moduli in rolling and transverse direction increase, while they decrease along the diagonal direction. During soft annealing, the volume fractions of recrystallization textures increase at the expense of cold-rolling textures. As a result, through recrystallization, the Young’s modulus decreases in the rolling and transverse directions but increases along the diagonal direction.
Figure 8 shows the evolution of the Young’s modulus in the rolling direction during recrystallization. At a temperature of 270°C, the recrystallization kinetics is very slow, resulting in a duration of approximately 70 hours until complete recrystallization. However, the measurement of the specimen’s eigenfrequencies takes only a few seconds, leading to high temporal resolution. The degree of recrystallization is estimated through the relation between the change in Young’s modulus and the total change in the Young’s modulus at complete recrystallization. From this estimation, the recrystallization kinetics can be approximated using empirical models, such as the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model. In contrast to established methods, which are based on measuring the change in strength or hardness at room temperature, the determination of recrystallization kinetics through changes in the Young’s modulus is not affected by recovery by annihilation of lattice defects and can be conducted in-situ at high temporal resolution. Furthermore, it is remarkably more cost-efficient than diffraction-based in-situ methods, such as X-ray diffraction (XRD) and EBSD.
Customer benefits
The eigenfrequency-based approach for determining elastic properties of rolled aluminum products, further developed and refined at AMAG, provides an innovative tool for the precise determination of the direction-dependent Young’s modulus. This in-situ characterization technique enables direct correlations between changes in mechanical properties and microstructural processes during thermal treatments, such as soft annealing. The application of this method provides a deep insight into the material behavior of wrought aluminum alloys, enabling adjustments of process parameters to influence material properties such as stiffness, springback, and formability. This results in increased process reliability, enhanced product quality and shorter development times, clear competitive advantages in demanding industries such as automotive and aerospace.
Sources:
[1] T. Obermayer, C. Krempaszky, E. Werner, Determination of the anisotropic elasticity tensor by mechanical spectroscopy, Continuum Mech. Therm. 34 (2022) 165-184.[2] J. Vallin, M. Mongy, K. Salama, O. Beckmann, Elastic Constants of Aluminum, J. Appl. Phys. 35 (1964) 1825-1826.[3] T. Obermayer, C. Krempaszky, E. Werner, Stiffness based in-situ assessment of static recrystallization kinetics for cold-rolled aluminum alloys, Mater. Des. 254 (2025) 114019.
