Kachanov Rabotnov Research Articles

Finite-Element Analysis of Waspaloy Using Sinh Creep-Damage Constitutive Model Under Triaxial Stress State

The creep deformation and damage evolution of nickel base superalloy (Waspaloy) at 700 °C are studied using the classic Kachanov–Rabotnov (KR) and a recently developed Sin-hyperbolic (Sinh) model. Uniaxial creep deformation and Bridgman rupture data collected from literature are used to determine the model constants and to compare the KR and the Sinh solutions. Finite-element (FE) simulations on a single eight-node element are conducted to validate the accuracy of the FE code. It is observed that KR cannot predict the creep deformation, damage, and rupture life of nickel base superalloys accurately using one set of constants for all the stress levels. The Sinh model exhibits a superior ability to predict the creep behavior using one set of constants for all the stress levels. Finite-element analysis (FEA) on 3D Bridgman notched Waspaloy specimen using the Sinh model is conducted. The results show that the Sinh model when combined with a representative stress equation and calibrated with experimental data can accurately predict the “notch effect” observed in the rupture life of notched specimen. Contour plots of damage evolution and stress redistribution are presented. It is demonstrated that the Sinh model is less stress sensitive, produces unconditional critical damage equal to unity at rupture, exhibits a more realistic damage distribution around the crack tip, and offers better crack growth analysis than KR.

Study on creep damage and life prediction of threaded connections at high temperature

In this study, Kachanov–Rabotnov model and stress relaxation damage constitutive equations deduced from Kachanov–Rabotnov model were applied to analyze the creep damage and to predict life for threaded connection structure at high temperature with finite element method. The parameters of Kachanov–Rabotnov model were obtained by fitting the results of creep experiments for titanium alloy at 650°C. Based on the experimental and finite element analysis results for standard specimen, a creep failure criterion was established. Then the influences of the external tensile load on the creep damage and life, as well as the stress relaxation on the initial preload, were studied. The analysis of stress relaxation for bolt shows that the stress relaxation has a remarkable effect on the bolt preload. The preloads decrease to a determined value with creep time and remain almost unchanged later. When the determined value is less than the required preload acting on the bolt, the structure will fail due to insufficient preload caused by stress relaxation.

Study on creep mechanical behaviour of P92 steel under multiaxial stress state

The creep mechanical behaviour of P92 steel at 650°C has been studied by experimental research and finite element analysis. During the creep of P92 steel, there existed the notched strengthening effect, which was influenced by the shapes of the notch and the nominal stress. Under the condition of the same notch depth, the creep life enhancement factor increased with decreasing notched radius or the increase of stress. The multiaxial stress caused by the notch effect had a significant influence on the evolution of the microstructure and resulted in a transforming tendency from ductile to brittle at the root of the notch. The fracture position varied with the shapes of the notch: the U shaped notch started to fracture at the root of the notch, while the C shaped notch in the centre of the specimen. The creep process of notched specimens was simulated by embedding Kachanov–Rabotnov creep damage constitutive model into the interface program of finite element software. The result showed that damage distribution of notched specimens varied during the process of creep. The maximum damage location at the end of creep depended on the notch shape: with larger notch radius the maximum damage location was in the centre, while smaller radius of notch specimens was near the notch root, which was consistent with the analysis of the fracture morphology.

Prediction of creep rupture life of a V-notched bar in DD6 Ni-based single crystal superalloy

The circumferential V-type notched round bar has been designed to investigate the effect of multi-axial stress state on creep rupture behavior of DD6 Ni-based single crystal superalloys at 1100°C. Creep test on both smooth and notched specimens were carried out under 160MPa and 200MPa on the minimum net-section. Time to creep fracture was found to be higher in notched specimens than in smooth specimens. Finite element (FE) analysis coupled with continuum damage mechanics (CDM) was carried out to understand the stress distribution and the creep damage evolution under uniaxial and multi-axial stress states. The creep rupture life of the smooth specimen was successfully predicted by the classical Kachanov–Rabotnov damage law. The creep rupture life of the notched specimen predicted by finite element calculation incorporating a multi-axial creep CDM model was in good agreement with the experimental results. Both the fracture morphology and FE damage analysis indicate that rupture initiates first at the notch root in the notched specimen and finally towards to the center location.

A microplane-based anisotropic damage effective stress

The Kachanov–Rabotnov damage effective stress is a fundamental concept in continuum damage mechanics. Controversies remain over the multiaxial generalization of the Kachanov–Rabotnov effective stress, especially when anisotropic damage effects occur. From a microplane point of view, we show that an anisotropic Kachanov–Rabotnov effective stress cannot simply be defined as a Cauchy-like stress tensor, namely a symmetric tensor of order two, but is essentially a vector-valued orientation distribution function. The practical effective stress tensor is identified by the second-order fabric tensor of the vector-valued Kachanov–Rabotnov stress orientation distribution function. The proposed model encompasses a wide range of classical models and thus clarifies the presuppositions of the models at a microplane level. We also prove the positive definiteness of the fourth-order damage effect tensor and provide a sufficient condition for its Voigt symmetry.

Numerical investigation of frictional effect on measuring accuracy of different small specimen creep tests

Small specimen test methods have distinct advantages in life prediction of components operating at high temperatures. Yet some factors which can be ignored in traditional bulk material creep tests should be considered in the small specimens. In the present paper four different miniature specimens used to evaluate high-temperature creep properties of materials, such as small punch, impression, three-point bending and cantilever beam specimens, are discussed. Combined with Kachanov–Rabotnov creep damage constitutive equations, finite element models are established and applied to analyze the creep properties of small specimens. The effect of friction on the steady and tertiary creep properties is further investigated for four different small specimen tests. The results show that the friction existing between the specimen and clamp has a significant influence on the evaluation of creep properties and is more pronounced than that between the specimen and punch. In comparison with the small punch, impression, and three-point bending specimens with friction constraints the three-point bending specimens with fixed ends and the cantilever beam specimens give better results. Smaller ratio of characteristic punch size to gauge length, fixed constraint and smaller or no sliding between the punch and specimen are accordingly recommended.

A damage-based cohesive zone model of intergranular crack growth in a nickel-based superalloy

A cohesive zone model is utilized to simulate grain boundary interface decohesion under load reversals with hold time being imposed at the maximum load level. This study considers the role of creep–fatigue and environment, which have been introduced as scalar damage parameters. These parameters are coupled with the traction–displacement laws governing the grain boundary cohesion and surrounding time-dependent deformation field. The accumulation of reversible fatigue damage is described using a damage rate law as a function of the elastic potential energy of the interface while creep damage is introduced using a modified Kachanov–Rabotnov type law. The role of environment is modeled by considering the grain boundary dynamic embrittlement occurring as a consequence of oxygen diffusion and accumulation in the grain boundary fracture path. The separation of damage laws into creep, fatigue, and environment, allows the examination of the influence of the different mechanisms on the intergranular crack growth rate. Theoretical aspects of the model are discussed and finite element simulations of intergranular crack growth are performed on a nickel-based superalloy, IN100, at 700℃ to illustrate the main features of the model and its comparison with the experimental data.

Modelling of high-temperature inelastic behaviour of the austenitic steel AISI type 316 using a continuum damage mechanics approach

A conventional material behaviour model can be extended to taking into account varying thermo-mechanical loading conditions in wide stress range. The motivation for developing this model is given by the well documented failure case study of high-temperature components at unit 1 of the Eddystone fossil power plant (Pennsylvania, USA), which have operated for 130,520 h in creep–fatigue interaction conditions. The developed model basis is a creep constitutive law in the form of hyperbolic sine stress response function originally proposed by Nadai (1938). The constitutive law is extended to assume the damage process by the introduction of scalar damage parameter and appropriate evolution equation according to Kachanov–Rabotnov concept. The research task is the introduction into the constitutive model of a few additional material state variables, able to reflect hardening and recovery effects under cyclic loading conditions. The first variable is represented by the relatively fast saturating back-stress [Formula: see text] describing kinematic hardening. The second variable is represented by the relatively slow saturating parameter [Formula: see text] describing isotropic hardening. Evolution equations for [Formula: see text] and [Formula: see text] are formulated in a modified form originally proposed by Chaboche and based on the Frederick–Armstrong concept. The uniaxial modelling results are compared with cyclic stress–strain diagrams and alternative experimental data in the form of creep curves, tensile stress–strain diagrams, relaxation curves, etc., for the austenitic steel AISI type 316 at 600 °C in a wide stress range.

Strain and Damage-Based Analytical Methods to Determine the Kachanov–Rabotnov Tertiary Creep-Damage Constants

In the power generation industry, the goal of increased gas turbine efficiency has led to increased operating temperatures and pressures necessitating nickel-base superalloy components. Under these conditions, the tertiary creep regime can become the dominant form of creep deformation. In response, the classical Kachanov–Rabotnov coupled creep-damage constitutive model is often used to predict the creep deformation and damage of Ni-base superalloys. In this model, the secondary creep behavior can be determined through analytical methods while the tertiary creep behavior is often found using trial and error or numerical optimization. Trial and error may produce no constants. Numerical optimization can be computationally expensive. In this study, a strain-based and damage-based approach to determine the tertiary creep behavior of nickel-base superalloys has been developed. Analytically determined constants are found for a given nickel-base superalloy. Creep deformation and damage evolution curves are compared. Methods to deal with stress dependence are introduced and studied.

One-dimensional Localization Solutions for Time-dependent Damage

This article presents analytical solutions for a class of one-dimensional time-dependent elasto-damage problems. The considered damage evolution law may be seen as a one-dimensional version of the Kachanov–Rabotnov creep damage model with classical loading–unloading conditions. We construct analytical solutions for the quasistatic one-dimensional problem. The evolution consists of a first regime, in which damage and strain grow uniformly, followed by a regime in which localization occurs. In the second regime, the uniqueness of the solution is lost and the deformation of the body is represented by a sequence of arbitrary alternate loading/unloading regions. Complex evolutions with progressive enlargement of the unloading regions in a finite number of steps are also constructed. We study analytically and numerically the features of the obtained bifurcated solutions. It is shown that, at every instant of time, a lower limit exists for the size of the localization zone. This lower limit is actually realized by the solution with successive unloadings constructed in this article. These features help us to understand the behavior of numerical solutions for time-dependent damage in the quasistatic approximation.

An Improved Anisotropic Tertiary Creep Damage Formulation

Directionally solidified (DS) Ni-base superalloys are commonly used as gas turbine materials to primarily extend the operational lives of components under high load and temperature. The nature of DS superalloy grain structure facilitates an elongated grain orientation, which exhibits enhanced impact strength, high temperature creep and fatigue resistance, and improved corrosion resistance compared with off-axis orientations. Of concern to turbine designers are the effects of cyclic fatigue, thermal gradients, and potential stress concentrations when dealing with orientation-dependent materials. When coupled with a creep environment, accurate prediction of crack initiation and propagation becomes highly dependent on the quality of the constitutive damage model implemented. This paper describes the development of an improved anisotropic tertiary creep damage model implemented in a general-purpose finite element analysis software. The creep damage formulation is a tensorial extension of a variation in the Kachanov–Rabotnov isotropic tertiary creep damage formulation. The net/effective stress arises from the use of the Rabotnov second-rank symmetric damage tensor. The Hill anisotropic behavior analogy is used to model secondary creep and tertiary creep damage behaviors. Using available experimental data for a directionally solidified Ni-base superalloy, the improved formulation is found to accurately model intermediate oriented specimen.

Characterization of the Creep Deformation and Rupture Behavior of DS GTD-111 Using the Kachanov–Rabotnov Constitutive Model

Creep deformation and rupture experiments are conducted on samples of the Ni-base superalloy directionally solidified GTD-111 tested at temperatures between 649°C and 982°C and two orientations (longitudinally and transversely oriented). The secondary creep constants are analytically determined from creep deformation experiments. The classical Kachanov–Rabotnov model for tertiary creep damage is implemented in a general-purpose finite element analysis (FEA) software. The simulated annealing optimization routine is utilized in conjunction with the FEA implementation to determine the creep damage constants. A comparison of FEA and creep deformation data demonstrates high accuracy. Using regression analysis, the creep constants are characterized for temperature dependence. A rupture prediction model derived from creep damage evolution is compared with rupture experiments.

A study on influence factors of small punch creep test by experimental investigation and finite element analysis

Small punch creep (SPC) tests for SUS304 and Cr5Mo specimens with dimension Ø10 mm × 0.5 mm were performed at different temperatures, load levels and atmospheres. Based on these tests, a finite element model (FEM) combined with modified Kachanov–Rabotnov (K–R) creep damage constitutive equations was established to simulate the ductile and creep damage of the round specimen during the test procedure. The validity of FEM was proved by the comparison of Center deflection–Time data yielded from finite element analysis (FEA) and experiment. Then, effects of sample thickness, load level, ceramic ball diameter, specimen diameter, temperature and protective atmosphere on SPC test were analyzed by experimental investigation and FEA. The results show that five of the six above-mentioned factors (except specimen diameter) influence the rupture time, center deflection, real creep strain and creep damage of SPC specimen strongly.

Modeling the Temperature Dependence of Tertiary Creep Damage of a Ni-Based Alloy

To capture the mechanical response of Ni-based materials, creep deformation and rupture experiments are typically performed. Long term tests, mimicking service conditions at 10,000 h or more, are generally avoided due to expense. Phenomenological models such as the classical Kachanov–Rabotnov (Rabotnov, 1969, Creep Problems in Structural Members, North-Holland, Amsterdam; Kachanov, 1958, “Time to Rupture Process Under Creep Conditions,” Izv. Akad. Nauk SSSR, Otd. Tekh. Nauk, Mekh. Mashin., 8, pp. 26–31) model can accurately estimate tertiary creep damage over extended histories. Creep deformation and rupture experiments are conducted on IN617 a polycrystalline Ni-based alloy over a range of temperatures and applied stresses. The continuum damage model is extended to account for temperature dependence. This allows the modeling of creep deformation at temperatures between available creep rupture data and the design of full-scale parts containing temperature distributions. Implementation of the Hayhurst (1983, “On the Role of Continuum Damage on Structural Mechanics,” in Engineering Approaches to High Temperature Design, Pineridge, Swansea, pp. 85–176) (tri-axial) stress formulation introduces tensile/compressive asymmetry to the model. This allows compressive loading to be considered for compression loaded gas turbine components such as transition pieces. A new dominant deformation approach is provided to predict the dominant creep mode over time. This leads to development of a new methodology for determining the creep stage and strain of parametric stress and temperature simulations over time.

Notch effect and its mechanism during creep rupture of nickel-base single crystal superalloys

In this research, the double notched plate mini-specimens have been designed to study the notch effect during creep. The experimental results show that the notch has strengthening effect. The scanning electron microscopy (SEM) photos of fractured surfaces reveal that void growth is the primary mechanism of creep rupture. Finite element analysis (FEA) with a modified form of Kachanov–Rabotnov damage law was carried out to simulate the damage evolution in the specimens. The creep damage calculations show that the creep is a process of stress redistribution. The 3-D voided unit cell model with the constant maximum principal stress was used to explore the mechanism of notch strength. The results show that the initial stress triaxility has remarkable influence on void growth. The greater the initial stress triaxility is, the slower the growth rate is. The stress state plays a main role in void deformation. The void grows remarkably in transverse direction when the stress triaxility is high. The void growth and coalescence is the main mechanism of creep rupture. The multiaxial stress state can inhibit the void growth, and this constraint effect is beneficial to the creep life.

Experiment and simulation of creep damage for duralumin alloy 2A12

In the present paper an experimental procedure and a finite element model were used to explore the possibilities of determining the creep-damage parameters of duralumin alloy 2A12 incorporated with modified Kachanov–Rabotnov (K–R) creep-damage constitutive equations. To study the phenomenon of creep damage of 2A12 a set of uniaxial tensile creep tests were carried out at 210 °C at different loads—1800, 2000 and 2200 N. From the experiment results and by using the optimum software ISIGHT, creep-damage parameters of duralumin alloy 2A12 based on the modified K–R creep-damage constitutive equations can be obtained by fitting the creep curves. Then, a finite element model was established by implementing a user subroutine in order to predict the creep damage. The optimum results were in agreement with the experimental results as well as the finite element method (FEM) results. The FEM results also showed that the creep damage at about the central part developed faster. An important conclusion obtained from the comparison was that the modified K–R creep-damage constitutive equations could be used to predict the creep damage and the residual strength as well as the optimum method could be used to treat the experimental results and calculate the creep-damage parameters of some metal materials.

Modeling of a Long-Term Creep Curve of Alloy 617 for a High Temperature Gas-Cooled Reactor

This study aimed to model the long-term creep curves above 105 hours by implementing a nonlinear least square fitting (NLSF) of the Kachanov-Rabotnov (K-R) model. For this purpose, the short-term creep curves obtained from a series of creep tests at 950oC were used. In the NLSF of their full creep curves, the K-R model represented a poor match to the experimental curves, but the modified K-R one revealed a good agreement to them. The Monkman-Grant (M-G) strain represented the behavior of a stress dependency, but the 􀁏 parameter was constant with a stress independency. The 􀁏 value in the modified K-R model was 2.78. Long-term creep curves above 105 hours from short-term creep data were modeled by the modified K-R model.

Creep Curve Modeling of Hastelloy-X Alloy by Using the Nonlinear Regression Method in the Kachanov-Rabotnov Creep Model

To design HTGR components for up to 1000oC, their creep curves are necessary during a design process. In this study, the full creep curves were modeled by the nonlinear least square fitting method using the Kachanov-Rabotnov (K-R) creep model. A series of creep data was obtained experimentally under various stress levels for Hastelloy-X at 950oC, and the data was used to model the creep curves. The K-R model gave a poor description of modeling creep curves, but the modified K-R one, which has another variable, K in the K-R model, was in better agreement than the K-R one. It was found that the λ parameter in the K-R model was constant regardless of the stress variations. The λ value was about 3.9 for the K-R model and about 5.8 for the modified one.

Effective Stress and Vector-valued Orientational Distribution Functions

The original Kachanov—Rabotnov damage variable is inherently microplane-based and should be expressed by a scalar-valued orientation distribution function (ODF), and the corresponding effective stress is a vector-valued ODF. The analysis of vector-valued ODFs is established in this article in the spirit of the analysis of scalar-valued ODFs by Kanatani, K. (1984a). Distribution of Directional Data and Fabric Tensors, International Journal of Engineering Science, 22: 149—164 and Kanatani, K. (1984b). Stereological Determination of Structural Anisotropy, International Journal of Engineering Science, 22: 531—546. Explicit expansions of vector-valued ODFs up to the sixth-order have been developed, and the relationship of fabric tensors of different order is addressed. The fabric tensors and expansion of the Kachanov—Rabotnov effective stress vector can be fully determined by the scalar-valued ODF characterizing the microplane damage. The second-order effective stress and damage tensors of Murakami, S. (1988). Mechanical Modeling of Material Damage, ASME Journal of Engineering Materials and Technology, 55: 280—286 are related to the second-order expansion of the Kachanov—Rabotnov effective stress vector. Therefore, the analysis of vector-valued ODFs furnishes a rigorous and unified mathematical basis for the damage theory of Kachanov, L.M. (1958). Time of the Rupture Process under Creep Conditions, Izv. Akad. Nauk., USSR. Otd. Tekhn. Nauk. 8: 26—31, Rabotnov, I.N. (1963). On the Equations of State for Creep, Progress in Applied Mechanics, the Prager Anniversary, 8: 307—315, and Murakami, S. (1988). Mechanical Modeling of Material Damage, ASME Journal of Engineering Materials and Technology, 55: 280—286.

Creep damage evolution in a modeling specimen of nickel-based single crystal superalloys air-cooled blades

Plate specimens both with holes and without holes were used to model air-cooled turbine blades. Experimental study on the creep behavior of a second generation nickel-based single crystal has been performed at 980 °C with these modeling specimens in two crystallographic orientations [0 0 1] and [1 1 1]. The results of creep tests show that crystallographic orientations and cooling holes have remarkable influence on the stress rupture behavior. The scanning electron microscopy (SEM) observation on the fractured surface shows that the cavitation-controlled damage is the main mechanism inducing fracture. So the finite element analysis (FEA) with a modified form of Kachanov–Rabotnov damage law was carried out to simulate the damage evolution in modeling specimen with cooling holes in crystallographic orientation [0 0 1]. The FEM results reveal that the existence of cooling holes causes the stress and strain concentrations near the holes. The creep damage is localized in the cooling holes region, where the fracture will happen easily. The numerical result is coincident with the fracture appearance of the specimen.