Strain hardening exponents and strength coefficients for aeroengine isotropic metallic materials – a reverse engineering approach

1 Introduction

Aeroengine material suppliers provide the materials with typical tensile properties, which can be used for the design in the absence of minimum guaranteed (S-Basis), T₉₉ (A-Basis), and T₉₀ (B-Basis) values [1]. The Metallic Materials Properties Development and Standardization (MMPDS) [1] provides the work hardening parameters for typical engine materials in tension and compression for different temperatures. However, this information is inadequate for the structural design, as the strength coefficient is required to be used along with it. As the data collection for material properties with statistically significant numbers takes years, constitutive equations for typical properties help the designer to a great extent in the preliminary design effort. Carrying out the tensile test on the representative component materials for the applicable range of temperature provides the necessary information for processing as per ASTM E-646 [2] to establish the values of the strain hardening exponent and strength coefficient. A concurrent initial design activity sets off as the catalogue data become available. Catalogue information does not provide the material flow constants but presents the elastic modulus, yield, and ultimate strength. The objective of the current work is to establish a Ramberg-Osgood flow rule for isotropic metallic materials of aeroengines for monotonic loading from these three parameters and validate it through ASTM E-646 [2] with the experimental data from the literature.

2 Experiments

A 50-ton MTS universal testing machine was used. The specimens were prepared as per ASTM E-8 [3]. Ti-6Al-4V, stainless steel 526, Inconel 718, and Haynes 188 were tested. The loading rate was 0.5 mm/min. The specimen had a gauge length of 25 mm, gauge width of 12.5 mm, and thickness of 2 mm. The extensometer with a displacement range of 8 mm was fixed on the specimen. The stress-strain data were extracted for ambient temperature. Each test was carried out three times to establish repeatability.

3 Methodology

Aeroengine metallic materials have a smooth transition from elastic to plastic deformation without a sharp yield point [1]. Figure 1 shows the engineering stress-strain diagram of aeroengine metallic isotropic materials from MMPDS [1] illustrating this point. The Ramberg-Osgood flow rule states that, for quasi-static loading at a given temperature and strain rate, the true-stress σ is given as a function of the strength coefficient K, work/strain hardening exponent n, and true plastic strain ε_p as [4]

Figure 1

Engineering stress – strain curves for aeroengine isotropic materials from MMPDS data [1].

It was reported that there could be different strain hardening exponents at different values of strain [5]. However, it is assumed that the true-stress true-strain diagram fairly follows one value of the strain hardening exponent. For the condition of yielding at 0.2% plastic strain, this equation turns out to be

From Equations (1) and (2),

where the value of 1/n is denoted as the Ramberg-Osgood parameter in MMPDS [1], and ε_σ0.2y=0.002

Recasting Equation (3),

At necking, the plastic strain ε_p equals the strain hardening exponent n [6], that is,

Furthermore, engineering ultimate tensile strength S_u is related to true ultimate tensile strength σ_u as [6]

Substituting Equations (5) and (6) into Equation (4),

with

where e is the engineering strain.

The value of the strain hardening exponent n is calculated from Equation (7).

The variation of the strain hardening exponent as a function of the ratio of engineering ultimate tensile strength to true yield stress is shown in Figure 2. For n=0, the ultimate tensile stress is equal to the yield stress of the material. Therefore, after yielding, the stress remains constant for increasing the plastic strain. In other words, the material behaves in an elastic-perfectly plastic manner. For the ratios of engineering ultimate stress to true yield stress 2.0 and 3.0, the strain hardening exponents are 0.194 and 0.279, respectively. Typically, the value of n for aeroengine metallic isotropic materials varies from 0.03 to 0.25.

Figure 2

Variation of strain hardening exponent as a function of the ratio of engineering ultimate tensile strength to true yield stress.

The strength coefficient K is given from Equations (2) and (8) as

where the value of elastic modulus is taken from the published data [1, 7]. The variation of the ratio of the strength coefficient K to 0.2% true proof stress (K/σ_0.2y) as a function of the strain hardening exponent n is shown in Figure 3. For the strain hardening exponents of 0, 0.1116, and 0.1768, the ratios of the strength coefficient to true yield stress are 1, 2, and 3, respectively.

Figure 3

Variation of the ratio of strength co-efficient K to 0.2% true proof stress (K/s_0.2n) as a function strain hardening exponent.

4 Validation

The validation of the methodology that was derived at the preceding (MMPDS) section is done in this section. For Inconel 718, Inconel 706, 9Ni-4Co-0.3C steel, and 2618-T61-Al, the ambient temperature data were taken from Reference [1]. A comparison of the present methodology with ASTM E-646 [2] for the strain hardening exponent and strength coefficient is shown in Figure 4. The deviation of the value of the strain hardening exponent that was estimated through the present methodology from ASTM E-646 is 6.2% for Inconel 718, 0.15% for Inconel 706, 3.8% for 9Ni-4Co-0.3C steel, 2.9% for 2618-T61-Al, and 0% for Ti-6Al-4V. The deviation of the value of the strength coefficient was 2.5% for Inconel 718, 4.3% for Inconel 706, 3.8% for 9Ni-4Co-0.3C steel, 0.9% for 2618-T61-Al, and 1.0% for Ti-6Al-4V.

Figure 4

Comparison of present methodology with ASTM E-646 for: (A) strain hardening exponent and (B) strength co-efficient for aeroengine materials data from MMPDS [1].

A comparison of the present methodology with ASTM E-646 [2] for the strain hardening exponent and strength coefficient for the test results of Haynes 188 (Haynes International, USA), S-526 steel (Midhani, India), Inconel 718 (Special metals, USA), and Ti-6Al-4V (Timet, UK) is shown in Figure 5. The deviation of the value of the strain hardening exponent that was estimated through the present methodology from ASTM E-646 is 0.4% for Haynes 188, 3.3% for S-526 steel, 3.0% for Inconel 718, and 4.2% for Ti-6Al-4V. The deviation of the value of the strength coefficient that was estimated through the present methodology from ASTM E-646 [2] is 5.1% for Haynes 188, 8.9% for S-526 steel, 0.9% for Inconel 718, and 2.2% for Ti-6Al-4V. Both MMPDS [1] and test data for ambient temperature show that the strain hardening exponent and strength coefficient through the present methodology have a deviation from the ASTM E-646 [2] within 10%. The strain hardening exponent and strength coefficient for the Inconel 718 and Ti-6Al-4V alloys derived from MMPDS [1] and test data show a respectable agreement with each other.

Figure 5

Comparison of present methodology with ASTM E-646 for (A) strain hardening exponent and (B) strength co-efficient for aeroengine materials from test data.

A comparison of the present methodology with ASTM E-646 [2] for the strain hardening exponent and strength coefficient for high temperature for Ti-6Al-4V from MMPDS [1] data is shown in Figure 6. The deviation of the value of the strain hardening exponent that was estimated through the present methodology from ASTM E-646 is 3.6% for 204°C and 1.17% for 371°C. The deviation of the value of the strength coefficient that was estimated through the present methodology from ASTM E-646 [2] is 3.6% for 204°C and 1.17% for 371°C. The deviations for the present methodology from ASTM E-646 [2] for the high-temperature tensile behavior of Ti-6Al-4V are within 5%.

Figure 6

Comparison of present methodology with ASTM E-646 for (A) strain hardening exponent and (B) strength co-efficient for Ti-6Al-4V for high temperature data from MMPDS [1].

A comparison of the MMPDS [1] and reverse-engineered stress-strain curves for Ti-64 for various temperatures is shown in Figure 7. The maximum deviation of the engineering stress for the present methodology from MMPDS [1] is 1.2% for room temperature, 1.1% for 204°C, and 2.1% for 371°C. The present methodology thus demonstrates that, for both ambient and high temperatures, the derived strain hardening exponent and strength coefficients are usable for the design analysis with an acceptable engineering accuracy.

Figure 7

A comparison of MMPDs [1] and reverse engineered engineering stress – strain curves for Ti-64 for various temperatures.

5 Flow parameters for current engine materials

The strain hardening exponents and strength coefficients for Ti-64 [8], Ti-6242 [9], IM685 [10], IMI-834 [11], Waspaloy [1], Inconel 718 [12], Udimet 720 [13], Haynes 188 [14], C-263 [15], L-605 [16], and CM 247 (equiaxed) [17] are derived.

A variation of the strain hardening exponent as a function of temperature for titanium-based alloys is shown in Figure 8A. The strain hardening exponent for the investigated titanium-based alloys increased with temperature, with the exception that, for Ti-6242, there was a slight dip at 426°C.

Figure 8

Variation of flow parameters for current aeroengine materials with temperature; (A) strain hardening exponents for titanium based alloys; (B) strength co-efficients for titanium based alloys; (C) strain hardening exponents nickel and cobalt based alloys; (D) strength co-efficients nickel and cobalt based alloys.

A variation of the strength coefficient as a function of temperature for titanium-based alloys is shown in Figure 8B. Among the investigated titanium alloys, Ti-834 has the highest temperature capability, followed by Ti-685 and Ti-6242. Although the ambient temperature capabilities of Ti-64 are comparable with that of Ti-6242, at temperatures beyond 400°C, Ti-64 has lower temperature capability than Ti-6242.

A variation of the strain hardening exponent as a function of temperature for nickel- and cobalt-based alloys is shown in Figure 8C. There is a dip in the value of the strain hardening exponent for Waspaloy, Inconel 718, Udimet 720, C-263, Haynes 188, and L-605 for temperatures above 600°C. For L-605, the strain hardening exponent starts increasing from 0.2174 at 200°C to its peak of 0.2857 at 500°C and falls to a minimum of 0.1428 at 770°C. The ductility of Waspaloy, Inconel 718, Udimet 720, C-263, Haynes 188, and L-605 comes down from a temperature of 600°C to 700°C. CM 247 LC (equiaxed) exhibited a fairly constant strain hardening exponent up to 760°C.

A variation of the strength coefficient as a function of temperature for nickel- and cobalt-based alloys is shown in Figure 8D. Among the disk materials, Udimet 720 has a superior temperature capability in comparison with Inconel 718 and Waspaloy. Next to the disk materials, the highest strength coefficient is possessed by L-605 up to 600°C. This property in combination with its highest elastic modulus among typical turbine materials [7] makes L-605 suitable for damper material for turbine blades. Among oxidation-resistant static component materials, Haynes 188 has a superior temperature capability in comparison with C-263 alloy.

6 Conclusions

A reverse engineering methodology was established to arrive at the strain hardening exponent and strength coefficient for monotonic tensile loading of aeroengine metallic isotropic ductile materials from elastic modulus, 0.2% proof stress, and ultimate tensile strength. This methodology was validated with the experimental and literature data. The variation of the strain hardening exponent and strength coefficient as a function of temperature were calculated for the current aeroengine materials through the proposed methodology from the catalogue data, which can serve as an input for the initial design effort.