Modelling the test methods used to determine material compatibility for hydrogen pressure vessel service

Abstract

This work aims to create finite element models to simulate the three ISO 11114–4 test methods applicable to hydrogen gas cylinders, coupled with calibrated constitutive models, to predict the deformation response of each. Experimental measurements are used to calibrate a monotonic constitutive model and a constitutive model of cyclic deformation. Six finite element solid models are discussed: monotonic tensile test of dog bone-shaped specimens, strain-controlled fatigue test of dog bone-shaped specimens, ISO test Method A, ISO test Method B, and ISO Method C (from ISO 11114–4), and a gas cylinder. Each finite element solid model is paired with the appropriate constitutive model based upon loading conditions. The modeling results are then combined with a new damage parameter in an attempt to compare each of the test methods to the others, as well as to in-service conditions. It is shown that the proposed damage parameter may be used to correlate all test methods considered (except for ISO Method A, a burst-disc test) as well as in-service conditions. The calibrated damage parameter may be coupled with any geometry, loading condition, and boundary condition modeled within a finite element package to predict the onset of critical damage in the material for which the coupled constitutive model is calibrated to. Parametric modelling study results provide estimated cycles to the onset of crack extension for DOT 3AA cylinders having varying sizes of internal thumbnail-shaped cracks. This work provides the baseline for measurements and models in air, with similar work in hydrogen to follow.

1. Introduction

The most common pressure vessels used to transport gaseous hydrogen are made of modified AISI 4130 (25CrMo4) steel. Most relatively small pressure vessels used to transport hydrogen over roadways are designated as DOT 3AA cylinders [1]. The DOT 3AA 4130X steel used for these vessels is relatively inexpensive to manufacture and the strength of the steel can be varied by quench and temper processing. The materials which comprise the DOT 3AA cylinders approved for hydrogen use may, in the future, also require certification by use of ISO-11114–4: Test Methods for Selecting Steels Resistant to Hydrogen Embrittlement of the ISO-11114 Transportable Gas Cylinder- Compatibility of Cylinders and Valve Materials with Gas Contents standard [2]. The ISO-11114–4 standard details three different test methods which may be used to certify material for hydrogen pressure vessel service.

The first method, Method A, is a burst-disk test. Essentially, a nominally 58 mm diameter by 0.75 mm thick membrane-like specimen is machined from material extracted from a sample vessel. The membrane disk is then clamped near its outer circumference and one side is exposed to gas at pressure. One set of tests must be performed in an inert gas and one set of tests in hydrogen. The gas pressure is increased until failure of the membrane disk results. Failure in this case is defined as specimen (or membrane) rupture. The test data resulting from this method may then be used to rank materials relative to one-another based upon the respective materials’ burst pressure ratio of inert gas to hydrogen gas. The data may also be used to certify materials if this ratio meets or exceeds a prescribed value.

The second method, Method B, is a traditional rising-step-load fracture test performed on a compact tension (CT) specimen machined from a sample vessel. Method B is performed in hydrogen only. The resulting fracture toughness in hydrogen, $K_{\text{IH}}$, is then used to rank materials for hydrogen service or certify new materials if the fracture toughness in hydrogen is sufficiently high.

The third method, Method C, is a wedge-opening-loaded (WOL) constant-displacement test utilizing a CT specimen machined from a sample vessel. The WOL test is performed by imposing a displacement which creates a crack tip stress intensity that is one half the critical value for the environment of interest (either inert or hydrogen). If the crack does not extend in the hydrogen environment, a null test results. A null test in hydrogen is viewed as a “pass” in this case.

Methods B and C use CT specimens that have thicknesses more than 85% of the wall thickness of the sample vessel, where a DOT 3AA cylinder for hydrogen use typically has a wall thickness of 6.8 mm or more. Method A uses specimens only 0.75 mm thick. Each of these test methods has benefits. Method A, for example, may be performed with a relatively simple test apparatus. Method B produces an engineering parameter, $K_{\text{IC}}/K_{\text{IH}}$, that can be used in a traditional fracture mechanics analysis. Method C may also be conducted by use of relatively simple equipment. Unfortunately, the three disparate test methods do not produce results that are easily compared to one another. Furthermore, it is unknown at this time how to correlate the results of the three test methods to the other tests.

This work aims to create a methodology to allow direct comparison of the three ISO 11114–4 test methods, as well as enabling comparison between the test methods and in-service conditions. This work employs two constitutive models to capture the materials monotonic and cyclic response. The monotonic constitutive model has been calibrated to experimental data such that it can predict the material response through failure. The cyclic constitutive model is calibrated by use of strain-controlled test data. Though strain-controlled data is used to calibrate the model, the particular constitutive model employed captures material evolution regardless of loading conditions (i.e. captures ratcheting as a function of load control conditions). Finite element solid models were created for the three ISO test methods, A, B, and C; a monotonic tensile test; strain-controlled tests; and a gas cylinder having an internal thumb-nail shaped crack experiencing in-service conditions. Each individual finite element solid model was paired with the appropriate constitutive model based upon loading conditions. The three tests methods A, B, C, the monotonic tensile test, the strain-controlled fatigue tests, and the gas cylinder with in-service conditions were then compared by use of a damage parameter. Future work will develop similar models and cumulative damage parameters in pressurized hydrogen gas.

2. Materials and test methods

The material used for this study was sourced from a single commercially manufactured DOT 3AA cylinder removed from hydrogen service. The manufacturer’s identifying marks indicate that the cylinder was designed for an operating pressure of 41.37 MPa (6000 psi) and was constructed out of AISI 4130 steel. The cylinder had a nominal height of 1295 mm (${51}^{″}$), a nominal inside diameter of 207 mm (${8.147}^{″}$), and a wall thickness of 14.43 mm (${0.568}^{″}$). The manufacturer’s drawing indicates that the cylinder was manufactured by hot billet piercing followed by hot drawing. Table 1 provides the chemical composition of the alloy used in this study. The chemical composition was measured by a commercial laboratory using atomic emission spectrometry. The uncertainty of the chemical analysis can be found in [3].

Table 1.

Measured mechanical properties (mean values ± standard deviation) and chemical composition (in mass fraction) of DOT 3AA 4130X steel.

Yield Strength (0.2% offset) [MPa]	Ultimate Tensile Strength [MPa]	Young’s Modulus [GPa]	Elongation to Failure [%]
604 ± 7	780 ± 8	200 ± 14	41.9 ± 1	Longitudinal
567 ± 8	741 ± 11	200 ± 4	38.0 ± 1	Circumferential
Fe	Cr	Mn	C	Si	Mo	S	P
0.97749	0.0103	0.005	0.0032	0.0021	0.0017	0.0001	0.00011
Nominal
Fe	Cr	Mn	C	Si	Mo	S	P
balance	0.008–0.011	0.004–0.006	0.0028–0.0033	0.0015–0.0035	0.0015–0.0025	< 0.0004	< 0.00035

Optical microscopy (Fig. 1) was performed on three orthogonal orientations of the pressure vessel material. The orientations of interest are: longitudinal-circumferential (L-C), longitudinal-radial (through-thickness) (L-R), and circumferential-radial (C-R). Representative micrographs of the microstructure are provided in Fig. 1.

Fig. 1.

Optical micrographs of the DOT 3AA 4130X microstructure in the longitudinal-circumferential (L-C), longitudinal-radial (L-R), and circumferential-radial (C-R) orientations, from left to right.

Note that the grain microstructure does not exhibit a preferential alignment in any of the three orientations shown and may be characterized as homogeneous at this location, although there is a slight amount of banding. The steel microstructure consists of martensite (dark microstructures in images) and ferrite (light microstructures in images). Electron backscatter diffraction analysis determined the grain size to be approximately 3 μm. It is understood that this measurement likely represents the average ferrite grain size and average martensite packet size, and not the prior austenite grain size [4].

Microhardness mapping was performed on the L-R and L-C orientations of the vessel material in order to characterize through-thickness deformation response as a function of microstructural deviations. The experimental procedure and results are provided in [5]. The through-thickness samples were extracted from the mid-height of the vessel. The microhardness data indicates that the material near the outer diameter of the vessel is approximately 20% harder than the material located at the inner diameter. Given the cylinder manufacturing process, this deviation is expected as the quench step is performed with water spray on the outside surface. The tempering step relieves some, but not all of the through-thickness inhomogeneity from the quench. This small difference in hardness, and therefore strength, would not be expected to change the conclusions of the modeling results, which use the assumption of homogeneous material properties. Bear in mind that the outer diameter of the gas cylinder is stronger, and therefore less ductile, than the inner part, but corrosion and crack initiation at the inner surface is the primary issue, particularly in the case for storage of hydrogen gas.

Mechanical tests were performed on specimens extracted from the pressure vessel in both the longitudinal orientation (aligned along the length of the cylinder) and circumferential orientation (aligned along the “hoop” of the cylinder). A single specimen geometry was used for all of the monotonic and strain-controlled tests (Fig. 2). Given the limited vessel diameter and wall thickness, the specimen geometry was selected based upon the circumferential specimen configuration by use of: (a) primary criteria (minimum of 6.35 mm (0.250”) diameter gage section, identical specimens in longitudinal and radial orientations, identical specimens to be used in future gaseous hydrogen testing); (b) secondary criteria (adherence to gage-section geometry proportions recommended in ASTM E606 [6], button head to be sufficiently long as to allow for the use of collet grips during testing); and (c) tertiary criteria (overall specimen length to be maximized). Though care was taken to ensure that the specimen geometry met ASTM E8 and ASTM E606 standards [7,8], it was not possible to extract circumferentially-oriented specimens having an overall length meeting the ASTM E8/E606 requirements. All other specimen dimensions and geometric relationships conform to both ASTM E8 and ASTM E606.

Fig. 2.

Specimen geometry extracted from the DOT 3AA 4130X pressure vessel, dimensions in inches.

Monotonic tensile tests were performed per ASTM E8 on an 89 kN (20 kip) servo hydraulic load frame. The load frame was equipped with an 89 kN (20 kip) load cell, and commercially available closed-loop control software. Strain was measured by use of a commercially available extensometer having a 12.7 mm (0.50″) gage length. Three specimens in each orientation were tested in monotonic tension at a strain rate of 1 × 10⁻³ -/s. Experimental data collected during monotonic testing are provided in Fig. 3, and tabulated mechanical properties for the monotonic tests performed here are provided in Table 1.

Fig. 3.

Monotonic tensile data of DOT 3AA 4130X pressure vessel material extracted from the transverse (or circumferential) orientation and longitudinal orientation of the cylinder. Strain rate of 1× 10⁻³ s⁻¹.

Strain-controlled tests were performed in both the longitudinal and circumferential orientations per ASTM E606. The strain-controlled tests were performed on identical equipment to that used in monotonic testing. The test matrix (Table 2) for the longitudinal orientation was chosen such that a strain-life curve could be produced that was statistically significant per ASTM E739 [9] for “research and development testing of components and specimens.” Specifically, in order to achieve a statistically significant strain-life curve for this purpose, a minimum of four strain ranges with two duplicates must be tested. The magnitudes of the strain ranges chosen for this study were selected based upon the need to collect strain-controlled data at “low” strain ranges (e.g. approaching what would be found in an undamaged in-service cylinder) and “large” strain ranges (e.g. those found in the ISO test Method A) in order to create a robust constitutive model of cyclic plasticity for the pressure vessel material. Three specimens were also tested in strain control in the circumferential orientation (Table 2) to determine if the pressure vessel material exhibited circumferential anisotropy in its accumulated deformation response. Finally, two additional strain-controlled fatigue tests were performed at very low lives ($N_f$ less than 2) for subsequent damage model validation.

Table 2.

Test matrix and results for strain control testing of DOT 3AA 4130X pressure vessel steel, where $R$ is load ratio (maximum/minimum) and $N_f$ is number of cycles to failure based upon a 10% decrease in load-carrying capacity.

Orientation	$R$	$\dot{\epsilon }\left[{\text{s}}^{-1}\right]$	$\text{Δ}\epsilon$	Stabilized $\text{Δ}{\epsilon }_p$	Stabilized $\text{Δ}{\epsilon }_e$	$N_f$
Longitudinal	−1	0.008	0.2	–	–	1.25
Longitudinal	−1	0.008	0.3	–	–	1.25
Longitudinal	−1	0.008	0.1000	0.0890	0.0110	49
Longitudinal	−1	0.008	0.0400	0.0325	0.0074	103
Longitudinal	−1	0.008	0.0399	0.0326	0.0072	153
Longitudinal	−1	0.008	0.0196	0.0137	0.0059	1613
Longitudinal	−1	0.008	0.0196	0.0135	0.0061	1381
Longitudinal	−1	0.008	0.0084	0.0033	0.0050	16,000
Circumferential	−1	0.008	0.0399	0.0332	0.0067	81
Circumferential	−1	0.008	0.0186	0.0128	0.0058	1201
Circumferential	−1	0.008	0.0084	0.0033	0.0051	20,121

2.1. Physical testing results for input to constitutive models

Representative hysteresis loops collected at the research programs predetermined “large” strain range, “middle” strain range, and “low” strain range are provided in Figs. 4–6, respectively. The figures also provide the maximum force manifesting in each specimen as a function of cycle data, collected during each test. The data in the figures indicate that, for the strain ranges tested as a function of this program, the DOT 3AA 4130X pressure vessel material exhibited varying amounts of softening as a function of increased cycling.

Fig. 4.

Force-cycle data and hysteresis loops collected during testing ($\text{Δ}\epsilon =0.1$, $R=-1$, $\dot{\epsilon }=0.008$ s⁻¹).

Fig. 6.

Force-cycle data and hysteresis loops collected during testing ($\text{Δ}\epsilon =0.0084$, $R=-1$, $\dot{\epsilon }=0.008$ s⁻¹).

Considerable initial cyclic softening occurred over approximately a quarter of the life of the mid and large strain-range tests, and only approximately one sixteenth of the life of the low strain-range test. After initial cyclic softening, the material response resulting from all three strain ranges stabilized to a linearly decreasing (cyclic softening) trend (relative to the maximum force-cycle data). Low cycle fatigue results presented as maximum force versus cycles are said to be “stabilized” when the data follows a linear, well behaved, and predictable trend regardless of slope. The low strain-range test exhibited a stabilized region with a nearly horizontal slope, while the high strain-range test exhibited marked linear cyclic softening as a function of increased cycles. These results are as expected for material evolution of pressure vessel steels for both low cycle fatigue and high cycle fatigue.

The cycle to failure, $N_f$, was determined in these tests as the cycle in which the maximum load carrying capability of the specimen decreases by 10% from the predicted stabilized (or linear) response. The ASTM standard test method for strain-controlled fatigue testing allows definition of failure relative to any drop in stress as long as the percentage drop is documented [8]. The predicted stabilized response for each test is indicated by an orange line, the 10% reduction in load carrying capacity is indicated by a purple line, and the cycle in which the failure criteria defined above was met is shown as an “X” in Figs. 4–6. Specimen half-life was then determined as $N_f/2$, and is indicated as a circle in Figs. 4–6. Hysteresis loops were extracted and analyzed for the first cycle, half-life cycle, and final cycle for each test. Fig. 7 provides the first cycle and half-life cycle hysteresis loops for representative tests on specimens having their loading orientation aligned with the pressure vessel longitudinal (along the long axis of the vessel) and circumferential (along the “hoop” direction of the vessel) material orientations. The comparison of the first cycle and half life cycle hysteresis loops indicate that the material exhibits isotropic accumulated deformation response, relative to the pressure vessel longitudinal and circumferential orientations. Furthermore, presenting the preliminary strain-life data collected here on a single set of axis (Fig. 8), indicates that the material is homogeneous in its strain-life response.

Fig. 7.

Longitudinal and circumferential oriented strain-controlled hysteresis loops. First cycle, left, and half-life cycle, right. Test parameters are $\text{Δ}\epsilon =0.0084$, $R=-1$, $\dot{\epsilon }=0.008$ s⁻¹.

Fig. 8.

Strain-life data (strain range, $\text{Δ}\epsilon$, versus cycles to failure, $N_f$) for initial fatigue tests performed ($R=-1$, $\dot{\epsilon }=0.008$ s⁻¹).

The strain-controlled data in the longitudinal orientation was fit to the traditional Basquin relationship [10] for the elastic portion of the deformation,

$$\frac{\text{Δ}{\epsilon }_e}{2}=\frac{{\sigma }_f}{E}{\left(2N_f\right)}^b$$ (1)

and the Coffin Manson relationship [11,12] for the inelastic (plastic) portion of the test results

$$\frac{\text{Δ}{\epsilon }_p}{2}={\epsilon }_f^{\prime }{\left(2N_f\right)}^c$$ (2)

The parameters in Eqs. (1) and (2) are as follows: ${\sigma }_f^{\prime }$ is the fatigue strength coefficient of the material, $b$ is the fatigue strength exponent, $E$ is the modulus of elasticity, $\text{Δ}{\epsilon }_e$ is the elastic strain range at half-life, $N_f$, the number of cycles to failure; ${\epsilon }_f^{\prime }$ is the fatigue ductility coefficient, $c$ is the fatigue ductility exponent, and $\text{Δ}{\epsilon }_p$ is the plastic strain range at half-life. The total strain-life curve may then be estimated by the following relationship

$$\frac{\text{Δ}\epsilon }{2}=\frac{{\sigma }_f^{\prime }}{E}{\left(2N_f\right)}^b+{\epsilon }_f^{\prime }{\left(2N_f\right)}^c$$ (3)

where $\text{Δ}\epsilon$ is the total strain range at half-life. Eq. (3) is commonly employed to relate applied strain to resulting life, for a given load ratio. The relationship is typically calibrated to fully reversed test data so that the calibration constants are then representative of material properties at zero mean stress [13]. Several modifications to Eq. (3) have been proposed to account for the effect of non-zero mean stress. A thorough review of the most popular strain-life equations which account for mean stress can be found in [13]. Eq. (3), in its original form, is applied here as all test data was collected by use of fully reversed strain-controlled testing resulting in zero mean stress. The data for the DOT 3AA 4130X pressure vessel specimens from the longitudinal orientation are separated into the total strain-life, the elastic portion of the total strain-life, and inelastic portion of the total strain-life in Fig. 9. The calibrated strain-life constants and uncertainties, provided in Table 3, were calculated for 95% confidence intervals using ASTM E739 [9]. The manner of calculation of the intervals for these tests, stipulated in the ASTM E739 standard, results in uncertainties that are not symmetrical about the calculated values.

Fig. 9.

Strain-life curve (strain amplitude, ${\epsilon }_a$ versus two times cycles to failure, or number of load reversals, $2N_f$) for all tests performed ($R=-1$, $\dot{\epsilon }=0.008$ s⁻¹).

Table 3.

Strain-life and stabilized cyclic stress-strain model parameters.

${\sigma }_{\text{UTS}}$(MPa)	${\sigma }_f^{\prime }$ (MPa)	$b$	${\epsilon }_f^{\prime }$	$c$	$K^{\prime }$ (MPa)	$n^{\prime }$
793	3268	−0.113	0.6592	−0.501	1277	0.18
95% confidence intervals
${\sigma }_f^{\prime }$ (MPa)		$b$		${\epsilon }_f^{\prime }$		$C$
Lower	Upper	Lower	Upper	Lower	Upper	Lower	Upper
1231	11,277	−0.083	−0.399	0.168	1.818	−0.406	−0.744

The half-life hysteresis loops for all longitundinally-oriented strain-controlled tests were then collected in order to determine the cyclic stress-strain curve. The cyclic stress-strain curve (Fig. 10) is fit by use of

$$\epsilon =\frac{\text{Δ}\sigma }{E}+{\left(\frac{\text{Δ}\sigma }{K^{\prime }}\right)}^{1/n^{\prime }}$$ (4)

Where $K^{\prime }$ and $n^{\prime }$ are material- and environment-specific constants (Table 3).

Fig. 10.

Hysteresis loops at half-life for all longitudinal data collected ($\text{Δ}\epsilon =0.0084$, $\text{Δ}\epsilon =0.0196$, $\text{Δ}\epsilon =0.0196$, $\text{Δ}\epsilon =0.0339$, and $\text{Δ}\epsilon =0.040$) coupled with the calibrated cyclic stress–strain curve. All data created at $R=-1$, $\dot{\epsilon }=0.008$ s⁻¹).

3. Modeling

The following provides information on the development of the six finite-element solid models used for this program. The six solid models capture the geometry, loading conditions, and boundary conditions of the monotonic tensile test; the strain-controlled tests; the three ISO 11114–4 test methods A, B, and C; and a representative DOT 3AA gas cylinder having an internal thumb-nail shaped crack experiencing in-service conditions. Each individual finite element solid model was paired with the appropriate constitutive model, based upon loading conditions.

3.1. Monotonic constitutive model

The DOT 3AA 4130X pressure vessel steel exhibits a linear elastic, strain-hardening-inelastic, monotonic deformation response. The onset of non-linear behavior in the transverse orientation is characterized by an inflection in the stress-strain data [7], resulting in a linear-inelastic response between yielding and a strain of approximately $\epsilon =0.010(-)$. Beyond $\epsilon =0.010(-)$, the inelastic response may be described as power-law hardening up to the ultimate tensile strength. The deformation in the longitudinal orientation manifests as linear elastic-strain hardening, with a traditional “round-house” hardening response [7]. The true stress-true strain monotonic deformation response of the DOT 3AA 4130X steel in the transverse (or circumferential) direction was modeled by use of the Deformation Plasticity constitutive relationship in the ABAQUS¹ finite element software package [14]. Conversion from engineering stress and strain to true stress and strain from necking to the point of failure is detailed in [5]. The monotonic test results provided in Fig. 3, and tabulated in Table 1, indicate that the longitudinally-oriented specimens exhibited a 6% larger yield strength and a 5% larger ultimate tensile strength than the circumferentially-oriented specimens. The circumferential orientation was chosen to represent the material response in an isotropic formulation for several reasons. First, the differences in yield strengths and ultimate tensile strengths measured between orientations was slight, as is indicated by the data in Fig. 3. Second, the strain to failure was 9% larger in the circumferential orientation than in the longitudinal orientation. In which case, modelling the circumferential orientation response would not limit the deformation capabilities of the constitutive model. Third, and most importantly, as the circumferential stress is the largest stress manifesting in a pressure vessel, this loading and material orientation will be most critical to failure. The monotonic constitutive model is a generalized Ramberg-Osgood law of the following form

$$E\mathit{\epsilon }=(1+v)\mathit{S}+(1-2v){\sigma }_h\mathit{I}+\frac{3}{2}\alpha {\left(\frac{\sqrt{\frac{3}{2}\mathit{S}:\mathit{S}}}{{\sigma }_0}\right)}^{n-1}\mathit{S}$$ (5)

where $E$ is the modulus of elasticity, $\mathit{\epsilon }$ is the total strain tensor, $\nu$ is the Poisson’s ratio, $\mathit{S}$ is the deviatoric stress tensor defined as

$$\mathit{S}=\mathit{\sigma }-{\sigma }_h$$ (6)

$\mathit{\sigma }$ is the stress tensor, ${\sigma }_h$ is the hydrostatic stress, $\mathit{\text{I}}$ is the identity matrix, ${\sigma }_0$ is the initial yield stress, and $\alpha$ and n are the yield offset and stress hardening exponent, respectively. The Ramberg-Osgood implementation is coupled to the von Mises effective stress, $VM$, by use of $VM=\sqrt{\frac{3}{2}\mathit{S}:\mathit{S}}$. Inherent to the use of the von Mises effective stress is the notion that the material exhibits isotropic yielding behavior. As is discussed above, the circumferential orientation was chosen to effectively represent an isotropic yield response in the material due to the nearly isotropic yielding behavior exhibited by the material (Table 1, Fig. 3, and Fig. 7). The finite element software internally calibrated the Deformation Plasticity monotonic constitutive model by fitting to user-supplied true stress-strain data pairs from the circumferential monotonic tensile tests. The user-defined modulus and Poisson’s ratio were set to $E=200$ GPa and $\nu =0.3$, respectively.

3.2. Cyclic constitutive model

To capture the materials cyclic deformation response, the “nonlinear isotropic/kinematic hardening” constitutive model in ABAQUS has been calibrated to the strain controlled data collected during testing. Specifically, separate instances of the cyclic constitutive model were calibrated for large strain range, medium strain range, and low strain range tests in order to capture the full evolution response of the DOT 3AA 4130X pressure vessel material as a function of applied loads. The constitutive model utilizes a flow rule associated with the von Mises yield criterion as:

$${\dot{\mathit{\epsilon }}}^{pl}={\dot{\overline{\mathit{\epsilon }}}}^{pl}\frac{\partial F}{\partial \mathit{\sigma }}$$ (7)

where $F$ is the von Mises yield criterion

$$F=\sqrt{\frac{3}{2}\left(\mathit{S}-{\mathit{\alpha }}_{dev}\right):\left(\mathit{S}-{\alpha }_{dev}\right)}-{\sigma }^0=0$$ (8)

And all other variables are defined as: ${\dot{\epsilon }}^{pl}$ is the plastic strain rate, ${\dot{\overline{\epsilon }}}^{pl}$ is the equivalent plastic strain rate, ${\mathit{\alpha }}_{dev}$ is the deviatoric backstress tensor, and ${\sigma }^0$ is the yield stress. Note that tensorial variables are indicated by bold in this work. Kinematic hardening is modeled by use of

$${\dot{\mathit{\alpha }}}_k=C_k\frac{1}{{\sigma }^0}(\mathit{\sigma }-\mathit{\alpha }){\dot{\overline{\epsilon }}}^{pl}-{\gamma }_k{\mathit{\alpha }}_k{\dot{\overline{\epsilon }}}^{pl}$$ (9)

where $C_k$ are the initial kinematic hardening moduli, ${\gamma }_k$ determines the rate at which the kinematic hardening moduli evolve as a function of plastic deformation, and $\mathit{\alpha }$ is the backstress tensor defined as

$$\mathit{\alpha }=\sum _{k=1}^N{\mathit{\alpha }}_k$$ (10)

While any number of backstresses ($N$) may be used, Chaboche suggests the use of three backstresses [15]. This work used three backstresses for each calibration ($N=3$) as three backstresses were found to be sufficient for the material response modelled here. Note that the backstress evolution term provided in Eq. (9) is comprised of a Ziegler linear hardening law, $C_k\frac{1}{{\sigma }^0}(\mathit{\sigma }-\mathit{\alpha }){\dot{\overline{\epsilon }}}^{pl}$ [16], coupled with a relaxation, or recall term, ${\gamma }_k{\mathit{\alpha }}_k{\dot{\overline{\epsilon }}}^{pl}$. Non-linear isotropic hardening, or the evolution of the yield surface size, is modelled by use of

$${\sigma }^0={\sigma |}_0+Q_{\infty }\left(1-e^{-b{\overline{\epsilon }}^{pl}}\right)$$ (11)

where ${\sigma |}_0$ is the yield stress at zero plastic strain, $Q_{\infty }$ is the maximum change in the yield surface, $b$ describes the rate of change of the yield surface size, and ${\overline{\epsilon }}^{pl}$ is the equivalent plastic strain. The cyclic plasticity response of DOT 3AA 4130X pressure vessel steel was calibrated for four varying levels of cyclic plasticity (${\epsilon }_a=0.05$, ${\epsilon }_a=0.0199$, ${\epsilon }_a=0.0098$, ${\epsilon }_a=0.0042$, modeling constants in Table 4) to ensure that the total potential material evolution could be modelled. The backstress tensor, ${\mathit{\alpha }}_i$, is calibrated within the finite element software package as a function of the specific $C_k$ and ${\gamma }_k$ parameters input for the stabilized cycle of interest. Representative constitutive model predictions are provided with the associated experimental data collected from the first cycle, half-life cycle, and final cycle of a strain-controlled test in Fig. 11.

Table 4.

Calibrated constitutive model constants for DOT 3AA 4130X pressure vessel steel tested in this work. Note that the backstress tensor, ${\mathit{\alpha }}_{\mathit{i}}$, is calibrated internally by the software.

Isotropic Hardening
${\epsilon }_a$	$\sigma \mid \mathit{0}$ [MPa]	$Q_{\infty }$ [MPa]	$b$
0.0500	520	−175	3
0.0199	451	−112	6
0.0098	411	−157	48
0.0042	600	−345	350
Kinematic Hardening
$C_{K_{\_}1}$ [MPa]	${\gamma }_{K_{\_}1}$	$C_{K\text{\_}2}$ [MPa]	${\gamma }_{K\text{\_}2}$	$C_{K\text{\_}3}$ [MPa]	${\gamma }_{K\text{\_}3}$	${\alpha }_i$ [MPa]
32,147	178	3402	6	N/A	N/A	*
79,541	701	13,232	55	N/A	N/A	*
7873	3770	81,557	571	14,488	47	*
371	4398	126,637	373	N/A	N/A

^*Matrices generated internally in ABAQUS.

Fig. 11.

First cycle and half-cycle experimental results coupled with model predictions for $\text{Δ}\epsilon =0.02$ and $R=-1$.

The constitutive model predictions fit the data well by capturing the evolution of the material response as a function of accumulated strain. The cyclic constitutive model has the capability to predict the Bauschinger effect, cyclic hardening or softening with plastic shakedown, ratcheting, and relaxation of mean stress.

3.3. Finite element solid modelling of ISO test methods

The three test methods detailed within ISO 11114–4, which may be used to approve a pressure vessel steel for hydrogen service, were modelled by use of the finite element software package ABAQUS. The three test methods include: Method A- the burst disk test, Method B- the rising load fracture mechanics test, and Method C- the wedge-opening load (or displacement-controlled) fracture mechanics test.

3.4. ISO Method A model

Method A was modeled by use of a reduced model employing axisymmetry. The elements used were CAX4R, an axisymmetric stress, linear, reduced integration quadratic element. This element type is that suggested for models with large displacements, as is the case for Method A [14]. In order to determine appropriate modeling boundary conditions, a study was performed to determine the effect of varying magnitudes of coefficient of friction between the burst disk and test fixture. Specifically, simulations were performed which utilize the built-in “Penalty” friction model within ABAQUS. The Penalty friction model allows for elastic slipping at the surfaces. The Penalty friction model was implemented with finite sliding and no surface smoothing. Separate simulations were run with coefficients of friction of $\text{μ}=0.0$ (unitless), $\text{μ}=0.1$ (unitless), $\text{μ}=0.5$ (unitless), and $\text{μ}=0.8$ (unitless). Results indicate that the change in coefficient of friction had minimal effect on the global and local strain and stress magnitudes and distributions, for the conditions studied here. The maximum difference in von Mises stress magnitude and equivalent plastic strain magnitude for the coefficients of friction studied here were determine to be $\text{Δ}{\sigma }_{VM}=0.2\text{\%}$ and $\text{Δ}{\overline{\epsilon }}^{pl}=1.4\text{\%}$, respectively. The final boundary conditions were simulated by use of: 1) surface contact, “Penalty” friction, coefficient of friction of $\text{μ}=0.0$ (unitless), with finite sliding, and no surface smoothing, 2) zero displacement in the X-direction and Y-direction as well as zero rotation about the Z-axis applied to the back edge of the modeled disc (labeled as “fixed edge” in Fig. 12), 3) zero displacement in the X-direction and zero rotation in the Y- and Z-axis applied to the center of the disk (labeled as X-Symmetry in Fig. 12), and 4) the loading condition was applied as a pressure to the underside of the burst disk (Fig. 12).

Fig. 12.

Loading and boundary conditions used to model Method A. Note that the cross-hatched region represents the clamp on the edge of the disc specimen.

Sufficient mesh refinement was ensured by convergence of both the model-predicted maximum von Mises stress and maximum effective plastic strain. A validation of the loading and boundary conditions of this model can be found in Appendix A. Because the material experiences monotonic loading conditions, the monotonic constitutive model (Deformation Plasticity) calibrated for the DOT 3AA 4130X pressure vessel steel was coupled to the finite element solid model of the burst disk. To model failure, a pressure of 48.5 MPa was applied to the bottom of the burst disk, which is the air pressure required to fail DOT 3AA 4130X pressure vessel steels in typical test conditions [17]. An image of the predicted failure location within the burst disk (zoomed in at the clamp) is provided in Fig. 13. The figure provides the predicted equivalent plastic strain (PEEQ) resulting from an applied (air) pressure known to cause failure in this material. Note that nearly half of the burst disk cross section experiences strain values greater than the elongation to failure for both the longitudinal and circumferential orientations (reference Table 1).

Fig. 13.

Predicted equivalent plastic strain (PEEQ) resulting from applied air pressure known to cause failure.

Note that the model predicts maximum equivalent plastic strain occurring at the boundary (clamp) of Method A. Given that the majority of the tests performed with Method A fail at the boundary, this is as expected.

3.5. ISO Method B model

The rising load fracture mechanics test, Method B, was modelled by use of half-symmetry along the crack-path plane. The elements used in this study were CPE8 plane strain, quadratic order quadrilateral elements. The compact tension (CT) specimens were modelled with a blunted crack tip having a radius of 5 μm. The crack tip radius was chosen based upon the need to have the most realistic model of the crack possible, and the requirement that the model results be mesh size independent. Ultimately, a 5 μm radius was chosen as studies indicated that models with this radius produced results that most closely matched the plastic zone size estimates calculated from synchrotron X-ray measurements [18]. Mesh refinement (Fig. 14) was performed until convergence of the ABAQUS-calculated nonlinear stress-intensity J-integral parameter [19], von Mises stress, and equivalent plastic strain was reached. Additionally, mesh quality for all solid models created in this work was ensured by requiring an element aspect ratio of less than 3:1 and an element growth ratio of less than 1.5:1. Average mesh size close to the crack tip was on the order of 1.5 μm. A concentrated force was applied to a reference point in the center of the CT specimen loading holes, as shown in Fig. 14. In order to ensure that the coupled loading condition, boundary condition, and cyclic material calibration were predicting the deformation response at the crack tip correctly, the combined finite element solid model and constitutive material model was used to predict elastic crack-tip strains measured by use of synchrotron x-ray diffraction, shown in Appendix B.

Fig. 14.

Detail of loading and boundary conditions employed to model Method B: (A) Analytical reference point located at the X. The reference point is tied to the inner surface of the CT loading hole. Zero displacement allowed in X-direction. (B) Y-symmetry applied between arrows. (C) Concentrated force applied at the analytical reference point in the Y-direction. The lower-left figure provides the mesh refinement at the crack tip.

To simulate the ISO test Method B, a traction was applied to the coupled model described above to ensure that the critical crack tip stress intensity factor/fracture toughness value ($K_{IC}=125$ MPa-m^1/2) for DOT 3AA 4130X pressure vessel material tested in air [20] was achieved (Fig. 15).

Fig. 15.

Predicted equivalent plastic strain (PEEQ) for test Method B loaded to the critical stress intensity factor ($K_{IC}=125$ MPa-m^1/2), zoomed in at the crack tip for DOT 3AA 4130X pressure vessel steel. Results provided in figure are of deformed crack tip after loading.

3.6. ISO Method C model

The compact tension model used to predict the ISO test Method B was then modified to enable the application of displacements such that the ISO test Method C could be modelled. The element type was identical to those used in modeling ISO Method B. The magnitude of the displacement applied when modelling Method C in air was set to the value that produced a stress intensity factor equal to one-half of the $K_{IC}$ in air ($K=62.5$ MPa-m^1/2), as prescribed in the ISO standard. The predicted equivalent plastic strain (PEEQ) resulting from the modelling of ISO Method C is also provided in Fig. 16.

Fig. 16.

Predicted equivalent plastic strain results for the model loaded to $K=62.5$ MPa-m^1/2. Results provided are of deformed crack tip after loading. Boundary and loading conditions consist of (reference Fig. 14 above): (A) Analytical reference point located at the X. The reference point is tied to the inner surface of the CT loading hole. Zero displacement allowed in X-direction. (B) Y-symmetry applied between arrows. (C) Displacement applied at the analytical reference point in the Y-direction.

3.7. Monotonic tensile test model

A finite element solid model of the dog bone test specimen detailed in Fig. 1 was created to predict the deformation resulting from a circumferentially-oriented monotonic tensile test for comparison to ISO Methods A, B, and C. The elements used were the CAX4R, an axisymmetric stress, linear, reduced integration quadratic element. These elements fit the geometry of the tensile specimen and also work well for the magnitude of plastic strains seen in a tensile test. The model of the tensile specimen was coupled with the monotonic constitutive model calibrated to the DOT 3AA 4130X pressure vessel steel. Mesh convergence was ensured per the methods described above. In order to estimate the deformation response at the onset of void initiation (necking), a displacement condition was applied to the finite element model to produce 10% elongation (average elongation at necking determined here for the circumferential orientation). Necking of the monotonic tensile test was determined to be the most appropriate “failure criterion” to use here as the results are to be compared to the onset of fracture in a rising-load fracture mechanics test, and crack initiation (as defined by a 10% reduction in load carrying capabilities) in strain-controlled tests. The entire volume of the gage section of the monotonic cylindrical specimen, shown in Fig. 2, was predicted to have experienced inelastic deformation at necking.

3.8. Cyclic strain-controlled test model

The finite element solid model used to predict monotonic tensile test results was also used to predict the material response to strain-controlled cyclic loading. The finite element solid model of the test specimen employed CAX4R, an axisymmetric stress, linear, reduced integration quadratic element, as well as the cyclic deformation constitutive model calibrated to the strain-controlled test data collected as part of this work. The combined solid model-cyclic constitutive model was then used to predict the deformation resulting from each of the strain-controlled loading conditions tested as part of this work, and shown in Fig. 9, by application of a cyclic strain boundary condition.

3.9. Damage parameter modelling

One of the primary goals of this work is to determine a single relationship which relates: (1) the critical failure “damage” accumulated in the ISO test Methods A, B, and C; to (2) the critical damage accumulated at failure in typical strain-controlled tests; to (3) the critical damage to failure experienced during in-service conditions. The term “critical damage accumulation” is used in this work as being synonymous with the onset of crack extension for a given loading cycle. As defined, the critical damage accumulation does not speak to the extent of crack extension that may occur during a loading cycle. Rather, critical damage accumulation is defined here as the combined damage resulting from historical deformation accumulation that ultimately leads to crack extension. Additionally, failure is defined as the onset of crack extension in this work.

The Fatemi-Socie (F-S) fatigue indicator parameter (FIP) [21] will be employed here as a basis for a proposed cumulative damage parameter to correlate the damage resulting in failure for all of the fatigue, fracture, and monotonic tests performed here. The F-S FIP was initially proposed to account for the critical accumulation of shear strain and normal stress on a critical plane resulting in failure. The Fatemi-Socie fatigue indicator parameter, $FI{P}_{FS}$, is given by

$$FI{P}_{FS}=\left({\gamma }_{\mathit{max}}\ast \left[1+H\left(\frac{{\sigma }_{\mathit{\text{norm\_max}}}}{{\sigma }_y}\right)\right]\right)$$ (12)

where $FI{P}_{FS}$ is a material-specific and environmental-specific constant for a given fatigue life, ${\gamma }_{\mathit{max}}$ is the maximum shear strain amplitude on a plane of interest, ${\sigma }_{\mathit{\text{norm\_max}}}$ is the maximum normal stress on the plane of interest, ${\sigma }_y$ is the yield stress, and $H$ is a material-specific and environment-specific constant. The constant $H$ is typically calibrated by collapsing uniaxial-only and torsional-only FIP values, $FI{P}_{FS}$, to a single trend, and typically takes a value between $0.5\le H\le 1$. The $FI{P}_{FS}$ was proposed to predict failure occurring on a critical plane as a function of “shear failure mode” [21]. Given that the monotonic, fatigue, and fracture failures manifesting in the material tested here are ductile in nature, and therefore result from shear-driven dislocation motion and accumulation, the $FI{P}_{FS}$ will be used as a basis for the new cumulative damage parameter proposed here. As in the F-S FIP, the proposed parameter,

$$CDP=\left(\text{Δ}{\gamma }_{\mathit{max}}^{pl}\left[1+\left(\frac{{\sigma }_H}{{\sigma }_y}\right)\right]\right)V_c$$ (13)

states that for a given life, the critical damage accumulation at failure, $CDP$, is constant. Specifically, the cumulative damage parameter predicts that failure will occur when a critical amount of mechanical damage occurs over a critical volume,$V_c$. The parameter utilizes the plastic shear strain range on the plane experiencing maximum shear strain $\left({\gamma }_{\mathit{max}}^{pl}\right)$, the maximum hydrostatic stress on the plane experiencing maximum shear strain $\left({\sigma }_H\right)$, and the materials’ initial yield strength $\left({\sigma }_y\right)$ to determine cumulative damage. The $CDP$ parameter has been implemented numerically to compute critical damage on a node-by-node basis for all of the finite element solid models discussed above. The critical volume at failure resulting from monotonic loading conditions was determined as that which produced identical $CDP$ values for a uniaxial tensile test modeled to have just reached the ultimate tensile strength of the material and a rising load fracture toughness test modeled to have been loaded to $K_{\text{IC}}$. In this way the critical volume determined for failure resulting from monotonic loading accounts for smooth geometries as well as those having a sharp crack. The critical volume under monotonic loading for the DOT 3AA 4130X steel used was determined to be $V_{\mathit{\text{c\_mono}}}=0.35$ mm³. Given that the size of cyclic plastic zones are thought to be on the order of one-fourth that of monotonic loading [19,22], for all other things being held equal, a first order estimate of the critical volume under fatigue loading was estimated as $V_{c_{-}\mathit{\text{fatigue}}}=V_{c_{\text{\_}}\mathit{\text{mono}}}/4$.

The value of the $CDP$ parameters at failure for each test was plotted versus the cycles to failure for their respective tests and is provided in Fig. 17.

Fig. 17.

Cumulative damage parameter ($CDP$) for each test method versus cycles to failure.

The data in Fig. 17 indicate that the proposed cumulative damage parameter collapses the monotonic tensile test, the strain-controlled fatigue tests, and the ISO test Method B to a reasonable trend, while the ISO test Method A does not collapse to the trend exhibited by the other methods. That is, when using the proposed cumulative damage parameter to predict damage at failure, the ISO test Method A appears to provide outlier results. Note that per the standard, the ISO test Method C was modeled such that it experienced a crack tip stress intensity factor one-half of that which causes crack extension in a single-cycle rising-load test. In which case, test Method C would not be expected to produce a failure in one cycle and therefore cannot be represented as a single data point in Fig. 17. A power-law function was fit to a selection of the calculated $CDP$ values. The power-law formulation is given by

$$CDP=A\cdot N_f^a$$ (14)

where $A$, $a$, are material-specific and environmental-specific constants determined to be $A=0.024$ and $a=-0.396$. Calibration of the constants in Eq. (14) was performed by use of three sets of test results: the monotonic tensile test results, the $\text{Δ}\epsilon =0.1000$ strain-controlled test results, and the $\text{Δ}\epsilon =0.0084$ strain-controlled test results. The remaining test results (those associated with ISO tests Method B and the three remaining strain-controlled tests) were subsequently used for calibration verification. The cumulative damage-to-life prediction model provided in Eq. (14) is plotted on the same axis as the calibration data (represented as X-markers) and the verification data (represented as solid dots) in Fig. 18.

Fig. 18.

Damage model prediction for monotonic test, ISO test Method C, and strain-controlled tests in air.

Equation (14) performs well at collapsing all of the applicable $CDP$ data to a single trend relative to cycles to failure and may be used to correlate the respective damage experienced by each of the test methods for which it applies (all tests studied here except ISO test Method A).

3.10. Gas cylinder model

In-service mechanical conditions were simulated by use of six finite element solid models of gas cylinders having 6.76 mm (${0.266}^{″}$) wall-thickness. This cylinder geometry is typical for use in pressure vessels intended for 20.68 MPa (3000 psi) service. Each solid model was seeded with a single semi-circular thumbnail-shaped crack having one of the following depths (in percent wall thickness): 3.75%, 8%, 10%, 25%, 37.5%, and 50%. The cracks were oriented such that they would be opened by the hoop-stresses manifested in the vessel. Extensive use of symmetry was employed in order to minimize computational time. The pressure vessels were modeled (Fig. 19) as having a fluctuating internal pressure of 0 MPa to 20.68 MPa (3000 psi). The elements used for in-service conditions were the C3D20RH, 20-node quadratic brick utilizing hybrid formulation and reduced integration. This type of element works well with the geometry of the crack and is recommended by ABAQUS for use at stress concentrations in three-dimensional solid models [14]. Mesh refinement was performed until convergence of the ABAQUS calculated J-integral, von Mises effective stress, and effective plastic strain were achieved (Fig. 20). Additionally, mesh quality for all solid models created in this work was ensured by requiring an element aspect ratio of less than 3:1 post deformation, and an element growth ratio of less than 1.5:1. Note that the $CDP$ modelling required considerably more mesh refinement beyond stress and strain convergence in order to ensure that the $CDP$ solutions converged. The cyclic constitutive model was employed to predict the in-service conditions.

Fig. 19.

Model geometry of in-service conditions indicating loading and boundary conditions employed.

Fig. 20.

Predicted equivalent plastic strains (PEEQ) at the crack tip of the in-service model experiencing ${\text{P}}_{\text{min}}=0$ MPa and ${\text{P}}_{\text{max}}=20.68$ MPa.

Note that though the global pressure ratio $\left(R=P_{\min }/P_{\max }\right)$, and in turn the global hoop-stress ratio $\left(R={\sigma }_{\text{h\_min}}/{\sigma }_{\text{h\_max}}\right)$, were equal to $R=0$, the local logarithmic strain ratio manifesting at the crack tip $\left(R={\epsilon }_{\min }/{\epsilon }_{\max }\right)$ was found to be $R=0.53$ from the finite element results. This would need to be taken into account when comparing accumulated damage from in-service conditions to fully reversed strain-life data. On the other hand, the cumulative damage model employed here captures the effect of variable load ratios as the cyclic constitutive model used is deformation-history dependent.

The in-service pressure vessel results are provided in tabulated form in Table 5. The table includes the calculated $CDP$ value and the estimated cycles to failure resulting from Eq. (14). The pressure vessel modeling results are provided Fig. 21a) and b) in the form of $CDP$ versus ${\text{N}}_{\text{f}}$ and Crack Depth versus ${\text{N}}_{\text{f}}$, respectively.

Table 5.

Results of pressure vessel study in which the internal crack was varied in size from 0.25 mm to 3.38 mm. Internal pressure fluctuates from 0MPa (0 psi) to 20.68MPa (3000 psi). CDP values marked with an asterisk result from loading conditions and boundary conditions that do not reach a critical volume of damage accumulation.

a (in)	a (m)	a (%)	CDP @ 0.095 mm3	Ni
0.010	0.00025	3.8%	8.25E-9*	2.10E + 16
0.021	0.00054	8.0%	4.07E-05	9,941,062
0.027	0.00068	10.0%	1.14E-04	731,696
0.067	0.00169	25.0%	1.05E-03	2690
0.100	0.00253	37.5%	2.50E-03	303
0.133	0.00338	50.0%	9.08E-03	12

Fig. 21.

Predicted cycles to crack extension for a pressure vessel having an internal, thumbnail shaped crack of varying size. The parametric study incorporated a finite element solid model of a DOT 3AA cylinder rated at 20.67 MPa (3000psi) experiencing pressurization/depressurization cycles of 0–20.67MPa (3000 psi). Crack sizes vary from 0.25 mm to 3.38 mm. (a) Calculated CDP versus cycles to crack extension. (b) Crack size (in % of wall thickness) versus cycles to crack extension.

Note that the predicted results for 3.8% through-thickness thumbnail shaped crack indicate that the critical volume of damage accumulation is not reached for the loading conditions and boundary conditions studied here. In which case, the modelling results indicate that the 3.8% crack would not extend under the conditions studied. The results in Fig. 21b) indicate that, for a DOT 3AA pressure vessel having 6.76 mm (0.266″) wall thickness experiencing pressure cycling between of 0 MPa and 20.68 MPa (3000 psi), a single trend can be used to correlate cycles to failure given an internal thumbnail shaped crack of known size between 8% and 50% of the wall thickness.

4. Discussion

This work has calibrated two plasticity constitutive models for the monotonic and cyclic response of a DOT 3AA 4130X pressure vessel steel. The cyclic plasticity model is a non-linear isotropic/kinematic model which captures the Bauschinger effect, cyclic hardening or softening with plastic shakedown, ratcheting, and relaxation of mean stress. The material models have been shown to capture both the deformation and deformation evolution well for the DOT 3AA 4130X material studied. Finite element solid models were also created to simulate monotonic tests, strain-controlled tests, in-service pressure vessels having thumbnail shaped cracks, and the three test methods approved for hydrogen compatibility testing in the ISO 11114–4 code. The finite element solid models were coupled to the appropriate constitutive models (monotonic or cyclic) to quantify deformation accumulation for each loading and boundary condition.

Ultimately, this work set out to accomplish two primary goals: (1) to create a finite element methodology to accurately predict the material deformation response of the in-service condition pressure vessel having a thumb-nail shaped crack and the three ISO 11114–4 test methods, and (2) to produce a numerical model that correlates the results of the ISO test methods to each other, as well as to the in-service conditions. The calibrated cumulative damage life relationship provided in Eq. (14) correlates the critical cumulative damage at the onset of crack extension (CDP) for a monotonic tensile test, the ISO 11114–4 test Method B, and six strain-controlled tests for the material to their respective cycles to failure. As such, this relationship may be used to compare these test methods to one-another, as well as predict cycles to failure for any $CDP$ value of interest, e.g. the $CDP$ value for a pressure vessel having thumbnail-shaped cracks of varying size.

This work was unable to utilize the cumulative damage model to predict results for the ISO test Method A. In which case, nothing can be said here about the efficacy of test Method A to correlate to in-service conditions. Likely one of the reasons why a single relationship was unable to correlate the calibrated $CDP$ values to cycles to failure for test Method A is the inability of the finite element method to predict very large-scale deformations with accuracy. The burst-disk deformations resulting from test Method A are likely beyond the ability of the finite element method to predict. While test Method A may be sound, and may perform equally as well as test Methods B and C at predicting failure in air, this work is unable to provide evidence either way.

As noted above, the calculated $CDP$ value for ISO test Method C under monotonic conditions predicts that the test will not result in crack extension in a single cycle. If one were to re-calculate the $CDP$ for a test in which, first the ISO test Method C was performed, then the specimen was unloaded to zero displacement, and the process was repeated until crack extension $(\text{R}=0)$, Eq. (14) predicts that crack extension would occur at or near 135 cycles. The cumulative damage parameter values for a pressure vessel having thumbnail shaped cracks of varying size were calculated and subsequently correlated to cycles to the onset of crack extension. The results indicate that a thumbnail shaped crack, having a depth of 3.75% of the wall thickness, would not extend when cycled between 0 MPa and 20.68 MPa (3000 psi). Similarly, an internal thumbnail shaped crack 8% of the total wall thickness would survive approximately 9.9 million cycles when operated under the conditions studied. At the other end of the life spectrum, an internal crack 37.5% in wall thickness would only survive approximately 300 cycles prior to crack extension.

Future work by the authors in this area includes calibration of the large scale monotonic and cyclic constitutive models to the DOT 3AA 4130X pressure vessel material response in gaseous hydrogen. Additionally, the $CDP$ damage model, Eq. (14), will also be calibrated to gaseous hydrogen test results for use in predicting hydrogen pressure vessel service lives.

5. Conclusions

1. Monotonic and cyclic constitutive models have been calibrated to a DOT 3AA 4130X pressure vessel material in laboratory air conditions.

2. Finite element solid models of the three test methods approved in ISO 11114–4: Test Methods for Selecting Steels Resistant to Hydrogen Embrittlement have been calibrated and validated to literature and experimental results.

3. A new cumulative damage model ($CDP$) is proposed to correlate the critical mechanical damage occurring over a critical volume to the cycles required to initiate crack extension. The $CDP$ has been shown to collapse the ISO test Method B, monotonic tensile test, and strain-controlled test results to a single $CDP$-Crack extension trend.

4. The $CDP$-Crack extension trend may be used to predict any loading conditions and boundary conditions that can be modeled in ABAQUS, including geometries with singularities.

5. A parametric study of pressure vessels having varying size internal thumbnail shaped cracks was performed to estimate the cycles to crack extension for a given pressurization and depressurization scheme in air. The results of the study indicate that non-destructive evaluation techniques with the capability to locate internal thumbnail shaped cracks before they extended to 10% of the vessel wall thickness would provide confidence that the vessel could experience approximately 500,000 more cycles prior to the onset of crack extension.

6. The deformations seen in test Method A do not fit the calibrated damage relationship (CDP) calculated in this work, and therefore correlations between the ISO test Method A and the other ISO test methods cannot be made at this time.

Fig. 5.

Force-cycle data and hysteresis loops collected during testing ($\text{Δ}\epsilon =0.0399$, $R=-1$, $\dot{\epsilon }=0.008$ s⁻¹).

Appendix A

To ensure that appropriate loading and boundary conditions were employed to model the ISO test Method A, the finite element solid model was coupled with the simplistic deformation model provided in [23]. The model was then loaded per the conditions listed in [17], and the model output was compared to that of the physical experimental results provided in the paper. Fig. A1 provides the maximum dome displacement predicted by the finite element model compared to the physical test results from [23].

The data provided in Fig. A1 indicates that the finite element model created to capture the deformation response occurring as a result of ISO test Method A performed well.

Fig. A1.

Finite element model predictions for Method A compared to experimental data [23]

Appendix B

To validate the ABAQUS models of the CT geometry, CT specimens having an effective length, $W=26.45$ mm and thickness, $B=3.06$ mm manufactured from the DOT 3AA 4130X pressure vessel material were fatigue tested in the high energy X-ray source at Argonne National Laboratory. Specific details of the synchrotron measurements, experimental apparatus, and results are provided in [24]. A finite element model of the CT specimen matching those used in the synchrotron measurement work, having a blunted crack tip with $\text{r}=5$ μm, used the cyclic plasticity model in ABAQUS to determine the efficacy of the geometry, loading conditions, boundary conditions, and calibrated deformation response. The predicted elastic strains in front of the crack tip (black dashed line in Fig. B1), are compared with the elastic strains measured by synchrotron X-rays (white dots). The plastic zone is indicated by the red dashed line.

Fig. B1 indicates that the two-dimensional plane strain elements, coupled with the blunted crack tip within the compact tension solid model, and the monotonic constitutive material model predicts the magnitude of the elastic strains well spatially. To predict the elastic strain magnitudes spatially, the finite element model must first accurately capture the size, shape, and magnitude of the inelastic strain region at the crack tip. While there are no methods to accurately measure the inelastic strains near a crack tip, the work presented here indicates that the size of the cyclic plastic zone in front of the crack tip, at the very least, is predicted well. This is indicated in Fig. B1 by the fact that the predicted elastic strains and the measured elastic strains converge just outside of the plastic zone predicted in the finite element model.

Fig. B1.

Predicted (black dashed line) and measured elastic crack tip strains (white dots) normal to the crack plane, where X [mm] is the distance, in mm, in front of the crack tip. The prediction uses a stress intensity factor value, $K$, of 27.5 MPa and crack length, $a$, of 11.551mm. The y-axis of the upper right figure is in micro-strain. The plastic zone is indicated by the red dashed line.