Simulation of mixed-mode I/III stable tearing crack growth events using the cohesive zone model approach

A cohesive zone model (CZM) approach is applied to simulate mixed-mode I/III stable tearing crack growth events in specimens made of 6061-T6 aluminum alloy and GM 6208 steel. The materials are treated as elastic–plastic following the \(J_{2}\) flow theory of plasticity, and the triangular cohesive law is employed to describe the traction-separation relation in the cohesive zone ahead of crack front. A hybrid numerical/experimental approach is employed in simulations using 3D finite element method. For each material, CZM parameter values are chosen by matching simulation prediction with experimental measurement (Yan et al. in Int J Fract 144:297–321, 2009), of the crack extension-time curve for the \(30^{\circ }\) mixed-mode I/III stable tearing crack growth test. With the same sets of CZM parameter values, simulations are performed for the \(60^{\circ }\) loading cases. Good agreements are reached between simulation predictions of the crack extension-time curve and experimental results. The variations of CTOD with crack extension are calculated from CZM simulations under both \(30^{\circ }\) and \(60^{\circ }\) mixed-mode I/III conditions for the aluminum alloy and steel respectively. The predictions agree well with experimental measurements (Yan et al. in Int J Fract 144:297–321, 2009). The findings of the current study demonstrate the applicability of the CZM approach in mixed-mode I/III stable tearing simulations and reaffirm the connection between CTOD and CZM based simulation approaches shown previously for mixed-mode I/II crack growth events.     

A new view on J-integrals in elastic–plastic materials

It is well known that the application of the conventional $J$ -integral is connected with severe restrictions when it is applied for elastic–plastic materials. The first restriction is that the $J$ -integral can be used only, if the conditions of proportional loading are fulfilled, e.g. no unloading processes should occur in the material. The second restriction is that, even if this condition is fulfilled, the $J$ -integral does not describe the crack driving force, but only the intensity of the crack tip field. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008), have derived a $J$ -integral, $J^{\mathrm{ep}} $ , which overcomes these restrictions: $J^{\mathrm{ep}} $ is able to quantify the crack driving force in elastic–plastic materials in accordance with incremental theory of plasticity and it can be applied also in cases of non-proportionality, e.g. for a growing crack. The current paper deals with the characteristic properties of this new $J$ -integral, $J^{\mathrm{ep}}$ , and works out the main differences to the conventional $J$ -integral. In order to do this, numerical studies are performed to calculate the distribution of the configurational forces in a cyclically loaded tensile specimen and in fracture mechanics specimens. For the latter case contained, uncontained, and general yielding conditions are considered. The path dependence of $J^{\mathrm{ep}} $ is determined for both a stationary and a growing crack. Much effort is spent in the investigation of the path dependence of $J^{\mathrm{ep}} $ very close to the crack tip. Several numerical parameters are varied in order to separate numerical and physical effects and to deduce the magnitudes of the crack driving force for stationary and growing cracks. Interpretation of the numerical results leads to a new, completed picture of the $J$ -integral in elastic–plastic materials where $J^{\mathrm{ep}} $ and the conventional $J$ -integral complement each other. This new view allows us also to shed new light on a long-term problem, which has been called the “paradox of elastic–plastic fracture mechanics”.     

Obtaining S–N curves from crack growth curves: an alternative to self-similarity

A crack growth model that allows us to obtain the S–N curves from the crack growth rate curves is presented in an attempt to harmonize the stress based and fracture mechanics approaches in lifetime prediction of long cracks propagation. First, using the Buckingham theorem, the crack growth rate curve $\frac{da}{dN}-\varDelta K$ is defined over all its range as a cumulative distribution function based on a normalized dimensionless stress intensity factor range $\varDelta K^+$ . Then, a relevant theorem is derived that provides an alternative to self-similarity allowing significant reduction of experimental planning. In this way, different $a-N$ crack growth curves for different stress ranges $\varDelta \sigma $ and initial crack lengths $a_0$ can be obtained from a particular crack growth curve under some conditions. The S–N field is obtained from the crack growth curves, showing the close relation between the fracture mechanics and stress approaches. Finally, the model is applied to a particular set of experimental data to obtain the crack growth rate curve and the S–N curves of a certain material for a subsequent fatigue lifetime assessment     

Investigation of Magnetic Memory Signals Induced by Dynamic Bending Load in Fatigue Crack Propagation Process of Structural Steel

Metal magnetic memory effect, induced by applied stress under the excitation of the geomagnetic field, has attracted a lot of attentions due to its unique advantages of stress concentration identification and early damage detection for ferromagnetic materials. To further investigate the regularity of magnetic memory signals in the fatigue crack propagation process under the dynamic bending load, the surface magnetic field intensity \(H_{p}(y)\) of ferromagnetic structural steel was measured throughout the dynamic three-point bending fatigue tests; variation of \(H_{p}(y)\) and its maximum gradient \(K_{max}\) were studied; meanwhile the possibility of using \(K_{max}\) to predict the fatigue crack propagation was discussed. The results showed that \(H_{p}(y)\) was relatively stable at different loading cycles and its maximum value appeared at the fatigue crack area before the specimen fractured; instead the \(K_{max}\) increased exponentially with the increase of loading cycles, and an approximate linear relationship was found between \(K_{max}\) and crack length 2a. The cause for this phenomenon was also discussed.     

Accurate loading analyses of curved cracks under mixed-mode conditions applying the J-integral

In linear elastic fracture mechanics the path-independent J-integral is a loading quantity equivalent to stress intensity factors (SIF) or the energy release rate. Concerning plane crack problems, $J_k$ J k is a 2-dimensional vector with its components $J_1$ J 1 and $J_2$ J 2 . These two parameters can be related to the mode-I and mode-II SIFs $K_{\mathrm{I}}$ K I and $K_{\mathrm{II}}$ K II . To guarantee path-independence for curved crack geometries, an integration path along the crack faces must be considered. This paper deals with problems occurring at the numerical calculation of the J-integral in connection with the FE-method. Two new methods for accurately calculating values of $J_2$ J 2 for arbitrary cracks are presented.     

Derivation of constraint-dependent J–R curves based on modified T-stress parameter and GTN model for a low-alloy steel

In this paper, a modified load-independent $T$ -stress constraint parameter $\tau ^{*}$ was defined. The $\tau ^{*}$ of specimens with different crack-tip constraint levels at different $J$ -integrals was calculated, and its load-independence has been validated. Based on the modified constraint parameter $\tau ^{*}$ and the numerically calculated J–R curves by using the Gurson–Tvergaard–Needleman (GTN) model for the SENB specimens with different $a/W$ , the equations of constraint-dependent J–R curves for the A508 steel were obtained. The predicted J–R curves using the equations essentially agree with the experimental and calculated J–R curves. The transferability of the constraint-dependent J–R curves to the CT, SENT and CCT specimens was validated. The results show that the modified constraint parameter $\tau ^{*}$ and the GTN model can be effectively used to derive the constraint-dependent J–R curves for ductile materials.     

Mixed-mode crack-tip stress fields for orthotropic functionally graded materials

Quasi-static mixed mode stress fields for a crack in orthotropic inhomogeneous medium are developed using asymptotic analysis coupled with Westergaard stress function approach. In the problem formulation, the elastic constants E ₁₁, E ₂₂, G ₁₂, ν ₁₂ are replaced by an effective stiffness ${E=\sqrt {E_{11} E_{22}}}$ , a stiffness ratio ${\delta =\left({{E_{11}}\mathord{\left/ {\vphantom {{E_{11}} {E_{22}}}}\right. \kern-0em} {E_{22}}} \right)}$ , an effective Poisson’s ratio ${\nu =\sqrt {\nu_{12}\nu _{21}} }$ and a shear parameter ${k=\left({E \mathord{\left/ {\vphantom {E {2G_{12}}}}\right. \kern-0em} {2G_{12}}}\right)-\nu }$ . An assumption is made to vary the effective stiffness exponentially along one of the principal axes of orthotropy. The mode-mixity due to the crack orientation with respect to the property gradient is accommodated in the analysis through superposition of opening and shear modes. The expansion of stress fields consisting of the first four terms are derived to explicitly bring out the influence of nonhomogeneity on the structure of the mixed-mode stress field equations. Using the derived mixed-mode stress field equations, the isochromatic fringe contours are developed to understand the variation of stress field around the crack tip as a function of both orthotropic stiffness ratio and non-homogeneous coefficient.     

Full-Potential Calculation of Structural, Electronic, and Thermodynamic Properties of Fluoroperovskite \text{ CsMF}_{3} (M?=?Be and Mg)

The structural and electronic properties of the cubic fluoroperoveskite $\text{ CsBeF}_{3}$ and $\text{ CsMgF}_{3}$ have been investigated using the full-potential-linearized augmented plane wave method within the density functional theory. The exchange-correlation potential was treated with the local density approximation and the generalized gradient approximation. The calculations of the electronic band structures show that $\text{ CsBeF}_{3 }$ has an indirect bandgap, whereas $\text{ CsMgF}_{3}$ has a direct bandgap. Through the quasi-harmonic Debye model, in which the phononic effects are considered, the effect of pressure $P$ and temperature $T$ on the lattice parameter, bulk modulus, thermal expansion coefficient, Debye temperature, and the heat capacity for $\text{ CsBeF}_{3}$ and $\text{ CsMgF}_{3}$ compounds are investigated for the first time.     

Modal analysis of the dynamic crack growth and arrest in a DCB specimen

This paper discusses dynamic crack growth and arrest in an elastic double cantilever beam (DCB) specimen, simulated using the Bernoulli–Euler beam theory. The specimen is made from two different materials. The section of interest, where the dynamic crack growth takes place, is made from a material, the fracture energy of which will be denoted \(2\Gamma _1 \) . The initial crack grows slowly in a starter material with a fracture energy \(2\Gamma _{0} \;(\Gamma _{0} >\Gamma _1 )\) , while opposed displacements on both arms of the specimen are continuously increased. As the crack reaches the material interface at \(\ell =\ell _c \) , the loading displacement is instantly suspended, and the crack suddenly propagates through the test zone, until it stops at \(\ell =\ell _A \) . During this process, the energy \(2\Gamma _1 (\ell _A -\ell _c )\) is dissipated. The beam motion and the fracture process during the fast crack growth stage are investigated, based on the balance energy associated to the Griffith criterion. The motion equations are approximated using a modal decomposition up to order \(N\) of the beam deflection (the analysis has been performed up to \(N=10\) but in most cases \(N=5\) is sufficient to obtain an accurate solution). This process leads to a set of N second order differential equations whose unknowns are the mode amplitudes and their derivatives, and another equation the unknowns of which are the current crack length \(\ell (t)\) , velocity \(\dot{\ell }(t)\) and acceleration \(\ddot{\ell }(t)\) . To demonstrate the accuracy of this method, it is first tested on a one dimensional peeling stretched film problem, with an insignificant bending energy. An exact solution exists, accurately approximated by the modal solution. The method is then applied to the DCB specimen described above. Despite the rather crude nature of the Bernoulli–Euler model, the results crack kinematics, and specially the arrest length, correspond well to those obtained by the combined use of finite elements and cohesive zone models, even for a few modes. Moreover, for the basic mode \(N=0\) (also referred to as Mott solution), even if the crack kinematics is not accurately reproduced, the prediction of the crack arrest length remains correct for moderate ratios. Some parametric studies about the beam geometry and the initial crack velocity are performed. The relative crack arrest \(\ell _A /\ell _c \) appears to be almost insensitive to these parameters, and is mainly governed by the ratio \(R=\Gamma _{0} /\Gamma _1 \) which is the key parameter to predict the crack arrest.     

Measurements of Microstructural, Mechanical, Electrical, and Thermal Properties of an Al–Ni Alloy

The Al–7.5 wt% Ni alloy was directionally solidified upwards with different temperature gradients, $G$ ( $0.86\,\text{ K}~{\cdot }~ \text{ mm}^{-1}$ to $4.24\,\text{ K}~{\cdot }~\text{ mm}^{-1})$ at a constant growth rate, $V$ ( $8.34\,\upmu \text{ m}~{\cdot }~\text{ s}^{-1})$ . The dependence of dendritic microstructures such as the primary dendrite arm spacing ( $\lambda _{1}$ ), the secondary dendrite arm spacing ( $\lambda _{2}$ ), the dendrite tip radius ( $R$ ), and the mushy zone depth ( $d$ ) on the temperature gradient were analyzed. The dendritic microstructures in this study were also compared with current theoretical models, and similar previous experimental results. Measurements of the microhardness (HV) and electrical resistivity ( $\rho $ ) of the directionally solidified samples were carried out. Variations of the electrical resistivity ( $\rho $ ) with temperature ( $T$ ) were also measured by using a standard dc four-point probe technique. And also, the dependence of the microhardness and electrical resistivity on the temperature gradient was analyzed. According to these results, it has been found that the values of HV and $\rho $ increase with increasing values of $G$ . But, the values of HV and $\rho $ decrease with increasing values of dendritic microstructures ( $\lambda _{1}, \lambda _{2}, R,$ and $d$ ). It has been also found that, on increasing the values of temperature, the values of $\rho $ increase. The enthalpy of fusion ( $\Delta {H}$ ) for the Al–7.5 wt%Ni alloy was determined by a differential scanning calorimeter from a heating trace during the transformation from solid to liquid.     

Moisture content effect on the fracture characterisation of Pinus pinaster under mode I

A moisture content effect on fracture characterisation of Pinus pinaster under mode I is addressed. The double cantilever beam test is selected for mode I loading, based on specimens scaled down to the growth ring level. Specimens are stabilised using aqueous solutions at several equilibrium moisture contents ( \( M_{\text{e}} \) ) ranging from 0 % to about 13 %. The strain energy release rate ( \( G_{\text{I}} \) ) is evaluated by applying the compliance-based beam method, from which the Resistance-curve is determined directly from load–displacement (P–δ) data without crack length measurement. The crack tip opening displacement in mode I ( \( w_{\text{I}} \) ) is determined by post-processing the displacements at the initial crack tip determined from digital image correlation. \( G_{\text{I}} \) and \( w_{\text{I}} \) are then combined for the direct identification of the cohesive law, \( \sigma_{\text{I}} = {\text{f}}(w_{\text{I}} ) \) , which is assumed in cohesive zone modelling. Experimentally, crack propagation consistently occurs in the earlywood layer. The increase of \( M_{\text{e}} \) does not show any influence on the initiation stage of the fracture process zone. However, a statistical correlation exists for the critical ( \( G_{\text{I,max}} \) ) and at maximum load ( \( G_{{{\text{I,}}P_{ \hbox{max} } }} \) ) values of \( G_{\text{I}} \) with regard to \( M_{\text{e}} \) . Consistently, the area under the identified cohesive curve increases with \( M_{\text{e}} \) , although high scatter and low correlation between maximum cohesive strength ( \( \sigma_{\text{Iu}} \) ) and \( M_{\text{e}} \) are observed. The methodology is also validated using finite element analysis including cohesive elements and taking into account the growth rings heterogeneity. The numerical results show that the identification of the cohesive law is insensitive to the variability of the growth ring structure observed experimentally.     

Atomistic and continuum modelling of temperature-dependent fracture of graphene

This paper presents a comprehensive molecular dynamics study on the effects of nanocracks (a row of vacancies) on the fracture strength of graphene sheets at various temperatures. Comparison of the strength given by molecular dynamics simulations with Griffith’s criterion and quantized fracture mechanics theory demonstrates that quantized fracture mechanics is more accurate compared to Griffith’s criterion. A numerical model based on kinetic analysis and quantized fracture mechanics theory is proposed. The model is computationally very efficient and it quite accurately predicts the fracture strength of graphene with defects at various temperatures. Critical stress intensity factors in mode I fracture reduce as temperature increases. Molecular dynamics simulations are used to calculate the critical values of $J$ integral ( $J_\mathrm{IC}$ ) of armchair graphene at various crack lengths. Results show that $J_\mathrm{IC}$ depends on the crack length. This length dependency of $J_\mathrm{IC}$ can be used to explain the deviation of the strength from Griffith’s criterion. The paper provides an in-depth understanding of fracture of graphene, and the findings are important in the design of graphene based nanomechanical systems and composite materials     

Simulation of fatigue crack growth with a cyclic cohesive zone model

Fatigue crack growth is simulated for an elastic solid with a cyclic cohesive zone model (CZM). Material degradation and thus separation follows from the current damage state, which represents the amount of maximum transferable traction across the cohesive zone. The traction–separation relation proposed in the cyclic CZM includes non-linear paths for both un- and reloading. This allows a smooth transition from reversible to damaged state. The exponential traction–separation envelope is controlled by two shape parameters. Moreover, a lower bound for damage evolution is introduced by a local damage dependent endurance limit, which enters the damage evolution equation. The cyclic CZM is applied to mode I fatigue crack growth under \(K_{\mathrm{I}}\) -controlled external loading conditions. The influences of the model parameters with respect to static failure load \(K_{\mathrm{0}}\) , threshold load \(\varDelta K_{\mathrm{th}}\) and Paris parameters \(m, C\) are investigated. The study reveals that the proposed endurance limit formulation is well suited to control the ratio \(\varDelta K_{\mathrm{th}}/K_{\mathrm{0}}\) independent of \(m\) and \(C\) . An identification procedure is suggested to identify the cohesive parameters with the help of Wöhler diagrams and fatigue crack growth rate curves.     

Three-dimensional mixed mode I+II+III singular stress field at the front of a {\varvec{(}}\mathbf{111 }{\varvec{)}[}\bar{\mathbf{1 }}\bar{\mathbf{1 }}\mathbf 2 {\varvec{]}\times [}\mathbf 1 \bar{\mathbf{1 }}\mathbf 0 {\varvec{]}} crack weakening a diamond cubic mono-crystalline plate with crack turning and step/ridge formation

A heretofore-unavailable mixed Frobenius type series, in terms of affine-transformed x-y coordinate variables of the Eshelby–Stroh type, is introduced to develop a new eigenfunction expansion technique. This is used, in conjunction with separation of the z-variable, to derive three-dimensional mixed-mode I+II+III asymptotic displacement and stress fields in the vicinity of the front of a semi-infinite through-thickness $(111)[\bar{{1}}\bar{{1}}2]\times [1\bar{{1}}0]$ crack weakening an infinite diamond cubic mono-crystalline plate. Crack-face boundary conditions and those that are prescribed on the top and bottom (free, fixed or lubricated) surfaces of the diamond cubic mono-crystalline plate are exactly satisfied. Explicit expressions for the mixed mode I+II+III singular stresses in the vicinity of the front of the through-thickness crack, are presented. Most important mixed modes I+II+III response is elicited even though the far-field loading is only mode I or II or III or any combination thereof. Finally, atomistic modeling of cracks requires consideration of both the long range elastic interactions and the short range physico-chemical reactions, such as bond breaking. The Griffith-Irwin approach does not take the latter into account, and nano-structural details such as bond orientation must be accounted for. A new mixed-mode I+II+III crack deflection criterion elucidates the formation of steps and/or triangular ridges on the crack path. The planes of a multiply deflected crack are normal to the directions of broken bonds. Additionally, the mixed-mode (I+II+III) crack deflection and ridge formation are found to be strongly correlated with the elastic stiffness constants, ${c}^{\prime }_{14}$ and ${c}^{\prime }_{56}$ , of the diamond cubic single crystal concerned.     

DEM analysis of the influence of the intermediate stress ratio on the critical-state behaviour of granular materials

The critical-state response of granular assemblies composed of elastic spheres under generalised three-dimensional loading conditions was investigated using the discrete element method (DEM). Simulations were performed with a simplified Hertz–Mindlin contact model using a modified version of the LAMMPS code. Initially isotropic samples were subjected to three-dimensional stress paths controlled by the intermediate stress ratio, \(b=[(\sigma '_{2}-\sigma '_{3})/\) \((\sigma '_{1}-\sigma '_{3})]\) . Three types of simulation were performed: drained (with \(b\) -value specified), constant volume and constant mean effective stress. In contrast to previous DEM observations, the position of the critical state line is shown to depend on \(b\) . The data also show that, upon shearing, the dilatancy post-peak increases with increasing \(b\) , so that at a given mean effective stress, the void ratio at the critical state increases systematically with \(b\) . Four commonly-used three-dimensional failure criteria are shown to give a better match to the simulation data at the critical state than at the peak state. While the void ratio at critical state is shown to vary with \(b\) , the coordination number showed no dependency on \(b\) . The variation in critical state void ratios at the same \(p'\) value is apparently related to the directional fabric anisotropy which is clearly sensitive to \(b\) .     

Ternary codes from some reflexive uniform subset graphs

We examine the ternary codes \(C_3(A_i+I)\) from matrices \(A_i+I\) where \(A_i\) is an adjacency matrix of a uniform subset graph \(\Gamma (n,3,i)\) of \(3\) -subsets of a set of size \(n\) with adjacency defined by subsets meeting in \(i\) elements of \(\Omega \) , where \(0 \le i \le 2\) . Most of the main parameters are obtained; the hulls, the duals, and other subcodes of the \(C_3(A_i+I)\) are also examined.     

Spherical and rod-like Gd 2 O 3 :Eu 3?+? nanophosphors—Structural and luminescent properties

A comparative study of spherical and rod-like nanocrystalline Gd $_{\boldsymbol 2}$ O $_{\boldsymbol 3}$ :Eu $^{\boldsymbol{3+}}$ (Gd $_{\boldsymbol{1\cdot92}}$ Eu $_{\boldsymbol{0\cdot08}}$ O $_{\boldsymbol 3}$ ) red phosphors prepared by solution combustion and hydrothermal methods have been reported. Powder X-ray diffraction (PXRD) results confirm the as-formed product in combustion method showing mixed phase of monoclinic and cubic of Gd $_{\boldsymbol 2}$ O $_{\boldsymbol 3}$ :Eu $^{\boldsymbol{3+}}$ . Upon calcinations at 800 $^{\boldsymbol\circ}$ C for 3?h, dominant cubic phase was achieved. The as-formed precursor hydrothermal product shows hexagonal Gd(OH) $_{\boldsymbol 3}$ :Eu $^{\boldsymbol{3+}}$ phase and it converts to pure cubic phase of Gd $_{\boldsymbol 2}$ O $_{\boldsymbol 3}$ :Eu $^{\boldsymbol{3+}}$ on calcination at 600 $^{\boldsymbol \circ}$ C for 3?h. TEM micrographs of hydrothermally prepared cubic Gd $_{\boldsymbol 2}$ O $_{\boldsymbol 3}$ :Eu $^{\boldsymbol{3+}}$ phase shows nanorods with a diameter of 15?nm and length varying from 50 to 150?nm, whereas combustion product shows the particles to be of irregular shape, with different sizes in the range 50?C250?nm. Dominant red emission (612?nm) was observed in cubic Gd $_{\boldsymbol 2}$ O $_{\boldsymbol 3}$ :Eu $^{\boldsymbol{3+}}$ which has been assigned to $^{\boldsymbol 5}{\bf \textit{D}}_{\boldsymbol 0}$ $\boldsymbol \to$ $^{\boldsymbol 7}{\bf \textit{F}}_{\boldsymbol 2}$ transition. However, in hexagonal Gd(OH) $_{\boldsymbol 3}$ :Eu $^{\boldsymbol{3+}}$ , emission peaks at 614 and 621?nm were observed. The strong red emission of cubic Gd $_{\boldsymbol 2}$ O $_{\boldsymbol 3}$ :Eu $^{\boldsymbol{3+}}$ nanophosphors by hydrothermal method are promising for high performance display materials. The variation in optical energy bandgap ( $\boldsymbol{E}_{\boldsymbol{\rm g}}$ ) was noticed in as-formed and heat treated systems in both the techniques. This is due to more ordered structure in heat treated samples and reduction in structural defects.     

Containerless Measurements of Density and Viscosity for a Cu_{48}Zr_{52} Liquid

The densities of solid and liquid Cu \(_{48}\) Zr \(_{52}\) and the viscosity of the liquid were measured in a containerless electrostatic levitation system using optical techniques. The measured density of the liquid at the liquidus temperature (1223 K) is (7.02 \(\pm \) 0.01) g \(\cdot \) cm \(^{-3}\) and the density of the solid extrapolated to that temperature is (7.15 \(\pm \) 0.01) g \(\cdot \) cm \(^{-3}\) . The thermal expansion coefficients measured at 1223 K are (6.4 \(\pm \) 0.1) \(\,\times \,10^{-5}\) K \(^{-1}\) in the liquid phase and (3.5 \(\pm \) 0.3) \(\,\times \,10^{-5}\) K \(^{-1}\) in the solid phase. The viscosity of the liquid, measured with the oscillating drop technique, is of the form \(A\exp \left[ \left( {{E}_{0}}+{{E}_{1}}\left( 1/T-1/{{T}_{0}} \right) \right) \times \left( 1/T-1/{{T}_{0}} \right) \right] \) , where \({{T}_{0}}=1223\) K, \(A= (0.0254 \pm 0.0004)\) Pa \(\cdot \) s, \({{E}_{0}}\) =  (8.43 \(\pm \) 0.26) \(\,\times \,10^3\) K and \({{E}_{1}}\) =  (1.7 \(\pm \) 0.2) \(\,\times 10^7\) K \(^{2}\) .