Dorn Equation Research Articles

Activation entropy and enthalpy of creep in pure metals

Previously published results on the power law creep of a selection of fcc, bcc and hcp pure metals are discussed using a modified Stokes – Einstein equation originally describing Newtonian viscosity of spherical particles in a fluid. Activation energy is modified to include terms representing activation enthalpy, activation entropy and the effect of strain rate. It is similar to the Dorn equation for power law creep but with strain rate rather than stress as the independent variable. Curves of activation energy vs shear rate are linear rather than having a discontinuity as have those with stress as independent variable. All metals have linear curves of activation energy at zero rate vs temperature, the slope of which reflects the activation entropy and is similar for all metals. Intercepts which represent activation enthalpy are dependent on lattice structure.

High‐Temperature Creep Behavior of Selective‐Laser‐Melting‐Fabricated Lattice Structures Inspired from Euplectella Aspergillum

In this study, the lattice‐like arrangement of Euplectella aspergillum is used to derive bionic grid‐stiffened structures in the form of thin tubes. The bionic tubes are 3D printed with AlSi10Mg alloy using a selective laser melting process. High‐temperature creep behavior at 250 °C and stress (0.3–0.7) times yield strength of the tubes is studied under uniaxial constant compressive load. The effect of the stress level, temperature, and tube design on the creep performance and deformation behavior is studied. The creep performance of bioinspired tubes is compared with honeycomb lattice tubes. In the results, the dependence of creep life and creep rate on the design of the bionic tubes is shown. E. aspergillum–based thin tubes show higher creep life and lower creep rate as compared to honeycomb‐based tubes at all stress levels. Also, the creep curves of lattice structures based thin tubes are different than the solid material, especially in the primary and tertiary regions. The fractography analysis shows higher creep life in E. aspergillum–inspired thin tubes is due to the improvement in ductility at high temperature. The stress exponent value in the Mukherjee–Bird–Dorn equation shows that the mechanism of creep is diffusion. Herein, in the results, it is shown that the bioinspiration and additive manufacturing provide an opportunity to design and fabricate advanced iso‐grids structures.

Effect of powder reuse on tensile, compressive, and creep strength of Inconel 718 fabricated via laser powder bed fusion

This work is concerned with determining the effect of powder reusing during fabrication of Inconel 718 via laser powder bed fusion (LPBF) on tensile, compressive, and creep strength. Test specimens were machined from rods fabricated by LPBF using virgin powder and powder that had been used in the process for some time. The rods were treated by hot isostatic pressing (HIP) and then heat-treated as per AMS 2774. While grain structure and texture were not appreciably different in the samples made from virgin versus reused powder, dislocation density and dispersion strengthening were estimated higher in the samples made from reused powder than virgin powder due to multiple heating cycles experienced by the reused powder. As a result, samples made from reused powder were slightly stronger in tension and compression. Moreover, the distribution of solute atoms was such that the samples made from reused powder exhibited serrated yielding at higher temperatures than samples made of virgin powder. The creep strength was higher for the samples made from virgin powder up to 650°C but not at 700°C. Although the differences were not substantial, the results were rationalized based on the dislocation density, dispersion hardening, and underlying diffusion governing solute distribution and precipitates in the materials. The creep parameters were estimated using the measured data and then refined by a subsequent data fitting using the Mukherjee – Bird – Dorn equation. The quality of fits and established parameters for the creep behavior for the two material categories are presented and discussed.

Derivation of novel creep-integrated fatigue equations

This manuscript presents a systematic derivation of novel creep-integrated fatigue equations that possess the following unique characteristics: (i) creep damage and creep-fatigue interaction damage are embedded into the fatigue equation and requires no deliberate computation; (ii) the embedded creep damage is formulated from the constitutive equation of creep; and (iii) the integrated equation is capable of modelling the full range of creep fatigue from pure creep to pure fatigue. These characteristics ensure that these creep-integrated fatigue equations are fundamentally sound, easy to use, and applicable over a wide range of creep fatigue conditions. The creep-integrated fatigue equations based on three constitutive equations of steady-state creep: Dorn equation, Larson-Miller equation, and Manson-Haferd equation are derived and successfully validated using two metals of vastly different creep and fatigue characteristics – the SS316 stainless steel and the Sn37Pb solder.

Creep properties, creep deformation behavior, and microstructural evolution of 9Cr-3W-3Co-1CuVNbB martensite ferritic steel

Creep deformation behavior and microstructure evolution of G115 steel were systematically investigated for temperatures of 625–675°C under uniaxial tensile stress of 120–220MPa. The relationship between minimum creep rate and applied stress followed the Bird–Mukherjee–Dorn (BMD) equation. The modified BMD equation was proposed using threshold stress to elucidate the actual creep deformation mechanism. The values of the threshold stress were determined to be 177.8, 91.4 and 87.6MPa at 625, 650, and 675°C, respectively. The true creep activation energy and the true stress exponent were 275.76kJ/mol and 6, respectively. Thus, the dominant creep deformation mechanism was identified as dislocation climb. Three types of precipitates can be revealed after creep deformation: W-rich Laves, Nb-rich MX, and Cu-rich phases. The creep damage of G115 steel after creep deformation was characterized by martensite cracks and martensite fractures owing to the hardness and brittleness of the lath martensite structure. Further, a dense array of deep and equiaxed dimples appeared in the central region of fracture surfaces under the tested creep conditions. Ductile fracturing was the main fracture mechanism during creep deformation.

Determination of creep behaviour of Ti60 alloy by small punch creep test

The small punch (SP) creep test has distinct advantages in the creep property assessment of materials at elevated temperatures. However, there are few creep properties of Ti alloys obtained by the SP creep test in the current literature. In this paper, the SP creep behaviour of Ti60 alloy has been evaluated under various loads in the range 550–800 N over a temperature range 873–973 K. The SP creep curves obviously indicated the primary, secondary and tertiary stages of creep. The test results have been compared with those of conventional creep tests. The European Code of Practice (CoP) for Small Punch Testing, Dorn equation and Monkman–Grant relationship have also been used to analyse the results of the SP creep tests. The ratio of load of the SP creep tests to equivalent stress of conventional creep tests, the load exponent value of steady deflection rate and activation energy for creep deformation were estimated from the SP creep tests. In conclusion, it was found that dislocation creep may be the main mechanism that dominates the SP creep deformation of Ti60 alloy in the range of load and temperature.

Tensile creep behavior of Sn–Ag–Cu–Ni multicomponent lead-free solder alloy

In the process of electronic packaging, the dissolution of under bump metallizations, such as Cu and Ni, into liquid solder occurs during soldering, which can change the original solder to a multicomponent one. Under the trend of miniaturization, it is quite necessary to evaluate the properties of multicomponent solder with excessive Cu and Ni compositions. In this study, the tensile creep behavior of Sn–3.5Ag–2.0Cu–0.5Ni multicomponent lead-free solder alloy is investigated at three temperatures, i.e., 303, 348 and 393 K. The steady-rate creep rates are obtained in the range of 10−4–10−8 s−1, when the normalized stress, σ/E, is in the range of 10−4–10−3. Based on the Dorn equation, the apparent stress exponent (n a), threshold stress (σ th), and activation energy of creep (Q C) are calculated at the three temperatures. It is found that the Sn–3.5Ag–2.0Cu–0.5Ni solder alloy shows a better creep performance than pure tin and eutectic Sn–3.5Ag solder due to the strengthening effect of Ag3Sn and (Cu,Ni)6Sn5 IMC precipitations. The true stress exponent for creep is identified to be 7, indicating that the creep behave is controlled by the dislocation-pipe diffusion in the tin matrix.

High temperature tensile deformation behavior of Grade 92 steel

Candidate structural materials for advanced reactors need to have superior high temperature strength and creep-rupture properties among other characteristics. The ferritic–martensitic Grade 92 steel (Fe–9Cr–2W–0.5Mo, wt.%) is considered such a candidate structural material. Tensile tests were performed at temperatures of 600, 650 and 700°C in the strain rate range of 10−5–10−3s−1. After analyzing the tensile results using the Bird–Mukherjee–Dorn (BMD) equation, a stress exponent of about 9.5 and an activation energy of about 646kJ/mol were obtained. In the light of high values of the stress exponent and activation energy, the threshold stress concept was used to elucidate the operating high temperature deformation mechanism. As a result of this modification, the true activation energy and stress exponent of the high temperature deformation in Grade 92 steel were found to be about 245kJ/mol and 5, respectively. Thus, the dominant high temperature deformation mechanism was identified as the high temperature climb of edge dislocations and the appropriate constitutive equation was developed.

Determination of creep property of 1.25Cr0.5Mo pearlitic steels by small punch test

Small punch creep tests (SP-C) have been developed to determine creep data of a 1.25Cr0.5Mo pearlitic steel. The test results have been compared with those of conventional creep tests. The comparison shows that overall shapes of the SP-C tests’ creep curves are similar to those of conventional creep tests. For the same creep rupture life, ratio of load of SP-C tests to equivalent stress of conventional creep tests is approximately 2.7. Some empirical formulae such as Larson-Miller law, Dorn equation and Monkman–Grant correlation, which are widely used for conventional creep tests, have also been used to analyze the results of SP-C tests. The analytic results are in good consistency with those obtained from conventional creep tests. Therefore, it can be concluded that small punch creep tests can be potentially used as a non-destructive testing method to evaluate creep properties of in-service structural components for remnant life assessment.

Creep deformation and rupture behaviour of 9Cr–1W–0.2V–0.06Ta Reduced Activation Ferritic–Martensitic steel

This paper presents the creep deformation and rupture behaviour of indigenously produced 9Cr–1W–0.2V–0.06Ta Reduced Activation Ferritic–Martensitic (RAFM) steel for fusion reactor application. Creep studies were carried out at 773, 823 and 873 K over a stress range of 100–300 MPa. The creep deformation of the steel was found to proceed with relatively shorter primary regime followed by an extended tertiary regime with virtually no secondary regime. The variation of minimum creep rate of the material with applied stress followed a power law relation, έm = Aσn, with stress exponent value ‘n’ decreasing with increase in temperature. The product of minimum creep rate and creep rupture life was found to obey the modified Monkman–Grant relation. The time to onset of tertiary stage of deformation was directly proportional to rupture life. TEM studies revealed relatively large changes in martensitic sub-structure and coarsening of precipitates in the steel on creep exposure as compared to thermal exposure. Microstructural degradation was considered as the prime cause of extended tertiary stage of creep deformation, which was also reflected in the damage tolerance factor λ with a value more than 2.5. In view of the microstructural instability of the material on creep exposure, the variation of minimum creep rate with stress and temperature did not obey Dorn's equation modified by invoking Lagneborg and Bergman's concepts of back stress.

The Investigation of Creep Behavior for NiAl-28Cr-5.5Mo-0.5Hf-0.02wt.%P Alloy at High Temperature

The Microstructure and creep behavior for NiAl-28Cr-5.5Mo-0.5Hf-0.02wt.%P alloy at high temperature have been investigated in this paper. The results reveal that the high temperature creep behavior of the NiAl-28Cr-5.5Mo-0.5Hf-0.02wt.%P alloy is characterized by transient primary creep and dominant steady-state creep as well as ternary creep behavior. The primary creep can be described by Garofalo equation and the steady-state creep can be depicted by Dorn equation. The creep mechanisms are viscous glide of dislocations at lower and middle testing temperatures and dislocation climb at higher temperature. No change of the microstructure for the testing alloy indicates that the creep fracture is controlled by the formation and propagation of cavities and cracks, and the creep fracture behavior obeys Monk man-Grant relationship.

Creep behaviour of 14Cr–15Ni–Ti stainless steel at 923 K

Creep behaviour of 14Cr–15Ni–Ti austenitic stainless steel has been investigated in the 20% cold worked condition at 923 K at various stress levels. The rupture lives were in the range 100–14,000 h. The stress dependence of minimum creep rate obeyed a power law and exhibited two slope behaviour with stress exponents of 5 at low stresses and 10 at high stresses. The two slope behaviour is also reflected in the apparent activation energy values in the low and high stress regimes. The high value of stress exponent is rationalized using modified Dorn equation invoking the concept of resisting stress. The resisting stress in the high stress regime is attributed to dislocation precipitate interaction. Creep ductility of the steel was low due to the presence of creep damage in the form of cracks. The precipitation of titanium carbides during creep provides strengthening to the matrix and prevents recovery of the cold worked structure.

Influence of the Substrate on the Creep of SN Solder Joints

The creep rate of Sn solder joints is noticeably affected by joint metallization. Cu|Sn|Cu joints have significantly higher creep rates than Ni|Sn|Cu joints, which, in turn, have higher creep rates than Ni|Sn|Ni joints. Replacing Ni by Cu on both substrates increases the creep rate at 333 K (60 °C) by roughly an order of magnitude. The increased creep rate appears with no apparent change in the dominant creep mechanism; the change in the constitutive equation for creep (the Dorn equation) is in the pre-exponential factor. The decreased creep rate on substituting Ni is accompanied by an increase in the hardness of the polygranular solder but a decrease in the nanohardness of the grain interiors. The source of the strong influence of the Ni substrate appears to be the introduction of an array of Ni3Sn4 intermetallic precipitates along the grain boundaries. These precipitates inhibit grain boundary sliding, boundary reconfiguration, and grain growth during creep. The intermediate creep rate of the asymmetric Ni|Sn|Cu joint has two causes: a decrease in grain boundary mobility due to precipitate decoration and a restriction in the free volume of the joint due to rapid intermetallic growth from the substrate on the Ni side. The sources of this anomalous intermetallic growth are discussed.

The Interaction Between an Imposed Current and the Creep of Idealized Sn-Ag-Cu Solder Interconnects

The work reported here concerns the effect of an imposed current on the creep of simple Sn-Ag-Cu interconnects. The samples employed were double-shear specimens that contained paired solder joints, 400 μm × 400 μm in cross-section, 200 μm in thickness, on Cu. Different representative microstructures were prepared by electromigration and isothermal aging. Samples were tested with and without an imposed current, and at a variety of temperatures. These tests consistently yield two unexpected results. First, the relative increase in creep rate with current was nearly the same over a range of temperatures and variety of starting microstructures. Second, when tests were done at the same temperature (including the effect of Joule heating), the rate of creep was lower under imposed current than under isothermal conditions. These results are explained in the light of new data that show that the temperature within the joint is almost constant, even under a relatively high current density of 5500 A/cm2. Given constant temperature and a microstructure that includes interfacial voids, the current depletes the joint of vacancies, lowering the␣average creep rate, and introducing observable heterogeneities in the creep pattern. The usual Dorn equation then provides a very useful basis for evaluating the change of creep rate with current.

Towards a Correlation between High-Temperature Creep and Volume Diffusion for Equiatomic NiTi Alloys

Classically a master curve as Dorn's equation is applied for elucidating stationary creep behaviour within high temperature range (T > 0.6 Tm). As the diffusion of both 63Ni and 44Ti have been measured in an equiatomic NiTi, an effective choice of creep-relevant diffusion coefficient D may be possible. Moreover, creep measurements in the same temperature range performed can be found in the literature. The correlation does not permit to establish precisely what D coefficient to integrate in the Dorn's equation.

Creep Rupture Studies of Zirlo Tubing Using Ring Tensile Specimen

A burst rupture test has been mainly used for evaluating the burst properties of internal pressurized tubes. Although the burst creep test provides accurate results, its complicated and material-wasting process makes it difficult to perform this test. In the burst test, it is known that the hoop stress is a main reason of burst, so it can be expected that the burst rupture properties are strongly related with the hoop creep rupture properties. A ring test is occasionally used for obtaining the hoop directional properties of tube-shaped structures. In this study, the creep rupture properties of Zirlo tubing are investigated at temperature ranging from 365 to 570 using the ring specimens. An analysis of the fractography was performed and the estimation of service life with Larson-Miller parameter was conducted. Finally, In view of the Dorn equation of power-law, the creep mechanism was determined and discussed.

Small punch test method assessment for the determination of the residual creep life of service exposed components

The small punch test, also known as miniature disk bend test, is a relatively new method for the creep test of specimens of small dimensions. Experiments have shown that the small punch test can be used to describe the time to failure by means of an equation of the Dorn type, in which stress is replaced by load. The activation energies calculated using such a modified equation are somewhat smaller than the activation energies calculated from the results obtained using conventional creep testing methods. However, the constant value of the activation energy and the almost constant value of the load exponent in the modified Dorn equation confirm the hypothesis about the almost iso-stress nature of the small punch creep test. A finite element analysis of the small punch test has been carried out taking into account the constitutive theta -projection concept of the law of creep of the experimental steel 14 MoV 6 3, obtained from conventional, constant load creep test data. The viscosity parameters were evaluated using the theta -projection concept, and numerical simulation was performed by prescribing the experimentally determined history of the displacement of the punch. An incremental timestepping procedure was used, which included updating of a state variable, which monitors the evolution of creep in variable load problems. There was found to be good agreement between the calculated and actual loads for the secondary stage of creep.

Diffusion creep of intermetallic TiAl alloys

The creep behaviour of γ-TiAl with L1 0 structure without second phases, γ-TiAl with precipitated particles of α 2-Ti 3Al with D0 19 structure, and γ-TiAl with the H-phase Ti 2AlC has been studied at low stresses in the temperature range 900–1200°C. The obtained data allow the construction of creep deformation mechanism maps for the studied alloys which may be used for an extrapolation of the observed creep behaviour. At higher stresses dislocation creep occurs in all alloys, which is well described by the Dorn equation with stress exponents in the range 3–5. Extended Coble creep with threshold stress was observed only for the studied two-phase alloys. A strong temperature dependence of the threshold stress for Coble creep was found for the TiAl alloy with carbide particles.

Quantitative assessment of superplastic deformation behavior in a commercial 5083 alloy

This paper describes the constitutive equation for the superplastic commercial 5083 (Al–Mg–Mn–Cr) alloy effective in a temperature range of 773–843 K and at a strain-rate range from 10 −5 to 4×10 −3 s −1. The superplastic properties have been investigated using velocity jump tests. The analysis of the experimental results reveals that the true stress exponent is two, and the true activation energy for superplastic flow is similar to the lattice self-diffusion of aluminum (142 kJ mol −1). All mechanical data can be represented by a single constitutive equation for superplasticity according to the Dorn equation.

Constant Stress Creep and Apparent Activation Energy Analysis of a Ni-Base Superalloy between 1023 and 1123 K

New steady state creep data of Inconel 738 lc at 1023, 1073 and 1123 K at stresses between 250 and 580 MPa are reported. At each temperature analyzed, two regions of different slope in the log a vs log d diagram were obtained related to different deformation mechanisms. The data were interpreted in terms of the Dorn's relationship and the Brown and Ashby's approach. Apparent activation energy were also measured and analyzed considering the variation with temperature of all the parameters of the Dorn's equation. This kind of approach could explain the high values of apparent activation energies obtained in precipitation hardened alloys.