Variable Material Properties Method Research Articles

Variable Material Properties Approach: A Review on Twenty Years of Progress

This paper provides a critical review of the advancements made in the application of the variable material properties (VMP) method over the past two decades. The VMP method was originally proposed in 1997 (Jahed and Dubey, 1997, ASME J. Pressure Vessel Technol., 119(3), pp. 264–273; Jahed et al., 1997, Int. J. Pressure Vessels Piping, 71(3), pp. 285–291) and further developed in 2001 (Parker, 2001, ASME J. Pressure Vessel Technol., 123(3), p. 271) as an elastoplastic method for the analysis of axisymmetric problems. The model was originally developed as a boundary value problem to predict the spatial distribution of stress. However, since 1997, it has been extended to include thermal effects to solve thermomechanical residual stresses; time domain to solve creep of disks and cylinders; finite deformation to solve cylinders under large strains; numerical solutions to make them more efficient; and asymmetric hardening behavior to accommodate nonslip deformation modes. These advancements, made over the past 20 years, are reviewed in this paper, and future trends and frontiers are discussed.

Compressive residual stress field in functionally graded material rotating disc under thermo-mechanical loading

Residual stress fields in functionally graded material rotating disc under thermo-mechanical loading is obtained by using the variable material properties method. The mixture model of Tamura–Tomato–Ozawa is used to correlate the overall mechanical behavior to the behavior of the constitutive phases in the construction of a rotating disc. The elastic-linearly kinematic hardening plastic material model is used for the metallic phase and a linear perfectly elastic behavior is assumed for the ceramic part. A finite element method analysis by using ABAQUS software is performed to assess the validity of the proposed method. The effect of velocity and temperature on the residual stress fields are studied. A study also worked out to see the effect of different parameters upon the residual stress fields. The effect of reverse yielding on the transposition of the compressive residual stress zone is investigated.

Effective parameters on fatigue life of wire-wound autofrettaged pressure vessels

In this paper, fatigue life of wire-wound autofrettaged vessels is investigated. The mutual and simultaneous effects of cylinder thickness, autofrettage pressure, number of wire layers, wire-winding stress, and working pressure are studied in several cases. The vessel cylinder is made of high strength steel, DIN1.6959, and actual behavior in loading, unloading and reloading is considered. Modified Variable Material Properties method is used to calculate residual stresses in autofrettage process. For wire-winding, since tangent and/or Young's modulus could be changed in autofrettage unloading step or during reverse yielding according to material behavior, a capable approach is applied. Moreover, the fatigue life is determined by using ASME code equations for tick-walled pressure vessels. The results show that by combination of wire-winding and autofrettage techniques, in addition to ability of reducing production costs, infinite fatigue life could be accessible. Furthermore, optimum autofrettage pressure has significant effects on fatigue life and wire-winding efficacy.

Elastic–plastic analysis of functionally graded rotating disks with variable thickness and temperature-dependent material properties under mechanical loading and unloading

This work presents an analysis of the thermo-mechanical behavior of rotating discs made of functionally graded material (FGM) with variable thickness. The solutions are obtained by variable material property (VMP) theory. In this theory, the domain is divided into some finite sub-domains in the radial direction, in which the thermo-mechanical properties are assumed to be constant and the form of the elastic response is used to solve elastic–plastic problems. The results obtained by the VMP method are then compared with the results obtained by the finite element analysis using ANSYS software. In addition, the unloading and reverse yielding behavior of FG rotating disk are investigated and the residual stresses are then calculated with the same values of pressure and temperature by VMP theory and FE analysis. The results reveal that the mentioned methods are in very good agreement in both elastic and elasto-plastic states. Also, the effect of considering the temperature-dependent material properties is discussed. It is found that the results obtained by ignoring the temperature-dependent material properties lead to high discrepancies in comparison with those by considering that. Subsequently, the effect of various parameters including the disk geometry, temperature distribution, and boundary conditions on the stress behavior of disk is investigated. The results show that unlike the uniform rotating discs in which the yielding necessarily initiates from the inner radius, in the FG rotating discs, plasticity can be initiated from any point.

Parametric investigation of autofrettage process in thick spherical vessel made of functionally graded materials

In this paper, the effect of autofrettage process parameters on the ultimate pressure that functionally graded spherical vessel can tolerate are investigated. FGM properties and autofrettage pressure are considered as important parameters. Assumptions are variation of properties of FGM in radial direction, the residual stress in the absence of Bauschinger effect with the operation of variable material property method for bilinear material and power law fraction distribution for FG vessel calculated. The stress distribution in loading phase is computed using projection method and rule of mixture for FGM and elastic solution for thick spherical vessel. In unloading phase the material behavior is assumed to be isotropic and residual stress computed by using superposition method for loading and unloading phase. For reloading phase the rules of linear mixture employed for estimating the ultimate strength of FGM. By assuming functionally graded material properties and autofrettage process as effective parameters on amount of pressure capacity of autofrettaged vessel, the effects of parameters discussed separately. The results illustrate considerable effect of volume fraction used in FGM (up to 35% compared to full metal case) and inhomogeneity exponent (up to 154% compared to homogeneous case) on amount of ultimate pressure which is mentioned in results of parametric analyzing.

Incorporation of asymmetric yield and hardening behaviour in axisymmetric elastoplastic problems

Within the context of total deformation plasticity, an axisymmetric model is developed for materials with asymmetric yield and hardening behaviours. The model employs the variable material properties (VMP) method which is an axisymmetric method of analysis widely used for problems such as pressure vessels. The original VMP method is applicable only to materials with symmetric hardening behaviour. In this study, the VMP method is extended to materials with asymmetric yield and hardening to examine the importance of considering the asymmetry within elastoplastic analyses. The asymmetric and isotropic Cazacu-Barlat yield criterion is employed. For residual stress calculations, actual unloading stress-strain curves obtained from experiments are directly incorporated into the proposed method. The results of the proposed model are first compared to the experimental results from a cylindrical magnesium specimen loaded under internal pressure and axial push/pull. The results are in very good agreement for the load cases where the stress state is uniaxial; the results deviate slightly for multi-axial stress states. An extensive comparison is made between the proposed asymmetric solution results and the results of symmetric solutions. The symmetric solutions adopt von Mises yield criterion and a symmetric hardening rule, e.g., isotropic and kinematic. The residual stress field results demonstrate that, for asymmetric materials, the results obtained from a symmetric solution can be significantly different under biaxial loading, and consideration of an asymmetric yield criterion along with an asymmetric hardening model is necessary. The loading and unloading responses of a thick-walled cylinder under internal pressure obtained from the proposed model are compared to those obtained from the LS-DYNA, MAT124 model. It is shown that the stress distributions are substantially different both after loading and unloading reversals.

Elasto-Plastic Stresses in a Functionally Graded Rotating Disk

A numerical method based on the extension of the variable material property method was developed to obtain the elasto-plastic stress field in a functionally graded (FG) rotating disk. The method was applied to estimate the stress field in a metal–ceramic functionally graded solid disk. To establish the validity of the proposed method, results were compared with finite element results. Unlike uniform rotating disks, where yielding starts from the disk center, plasticity in FG disks can originate at any point. The effect of different metal–ceramic grading patterns as well as the relative elastic moduli and densities of the ceramic and metallic constituents on the developed stresses were studied. Reinforcement of a metal disk with ceramic particles, in both elastic and plastic regimes, can significantly influence the mechanical response of the disk such as the stress distribution and the critical angular velocities corresponding to the onset of plasticity in the disk and plastic disk. Disks with increasing ceramic content from inner to outer radius showed a more uniform von Mises stress distribution for a fixed value of total ceramic content. In contrast, disks with decreasing ceramic content from inner to outer radius had the lowest outer edge displacement for a fixed value of total ceramic content.

Residual stress analysis of autofrettaged thick-walled spherical pressure vessel

In this study, residual stress distributions in autofrettaged homogenous spherical pressure vessels subjected to different autofrettage pressures are evaluated. Results are obtained by developing an extension of variable material properties (VMP) method. The modification makes VMP method applicable for analyses of spherical vessels based on actual material behavior both in loading and unloading and considering variable Bauschinger effect. The residual stresses determined by employing finite element method are compared with VMP results and it is demonstrated that the using of simplified material models can cause significant error in estimation of hoop residual stress, especially near the inner surface of the vessel. By performing a parametric study, the optimum autofrettage pressure and corresponding autofrettage percent for creating desirable residual stress state are introduced and determined.

Residual stresses in autofrettaged vessel made of functionally graded material

We used an extension of the Variable Material Property method for materials with varying elastic and plastic properties to evaluate the residual stresses in an autofrettaged thick vessel made of functionally graded metal–ceramic composite. It is shown that the reinforcement of the metal vessel by ceramic particles, with an increasing ceramic volume fraction from inner to outer radius, increases the magnitude of compressive residual stresses at the inner section of an autofrettaged vessel and thus, could lead to better fatigue life and load-carrying capacity of the vessel. A parametric study is carried out to highlight the role of ceramic particle strength and spatial distribution, as well as the autofrettage pressure on the induced residual stresses in a thick vessel.

General Variable Material Property Formulation for the Solution of Autofrettaged Thick-Walled Tubes With Constant Axial Strains

In this paper a general variable material property (VMP) formulation for the solution of thick-walled tubes with constant axial strains was developed and compared with the alternative VMP method that is called the Hencky program. The VMP method was initially developed for the analysis of plane stress and plane strain states. However, the actual autofrettage process is under constant axial strain, i.e., open-end and closed-end conditions. Results indicate very good agreement with the Hencky program. Our method is simple, accurate, and very efficient, so that the number of iterations for convergence reduces approximately to one-tenth of Hencky program iterations. The solution algorithm for plane strain, open-end, and closed-end conditions is the same.

Theoretical and numerical analyses of rotating discs of non-uniform thickness and density

Theoretical and numerical methods are used for stress–strain analysis of rotating discs with non-uniform thickness and density. An elastic–linear hardening material is assumed. A theoretical solution employs a technique called variable material properties (VMP) theory. A numerical solution is based on solving the governing differential equation using Runge–Kutta's method for elastic and plastic regimes. Finite element modelling of the problem has also been carried out using commercially available software. The results of the three methods are presented and compared and generally show good agreement. The suggested VMP method provides reliable means for complex discs for which there are no exact solutions. Although a specific disc profile has been studied here, the VMP method shows no restriction for the solution of any other disc profiles.

Minimum weight design of inhomogeneous rotating discs

There are numerous applications for gas turbine discs in the aerospace industry such as in turbojet engines. These discs normally work under high temperatures while subjected to high angular velocities. Minimizing the weight of such items in aerospace applications results in benefits such as low dead weights and lower costs. High speed of rotation causes large centrifugal forces in a disc and simultaneous application of high temperatures reduces disc material strength. Thus, the latter effects tend to increase deformations of the disc under the applied loads. In order to obtain a reliable disc analysis and arrive at the corresponding correct stress distribution, solutions should consider changes in material properties due to the temperature field throughout the disc. To achieve this goal, an inhomogeneous disc model with variable thickness is considered. Using the variable material properties method, stresses are obtained for the disc under rotation and a steady temperature field. In this paper this is done by modelling the rotating disc as a series of rings of different but constant properties. The optimum disc profile is arrived at by sequentially proportioning the thicknesses of each ring to satisfy the stress requirements. This method vis-à-vis a mathematical programming procedure for optimization shows several advantages. Firstly, it is simple iterative proportioning in each design cycle not requiring involved mathematical operations. Secondly, due to its simplicity it alleviates the necessity of certain simplifications that are common in so-called rigorous mathematical procedures. The results obtained, compared to those published in the literature show agreement and superiority. A further advantage of the proposed method is the independence of the end results from the initially assumed point in the iterative design routine, unlike most methods published so far.

Optimum design of inhomogeneous non-uniform rotating discs

Gas turbine discs work mostly at high temperature gradients and are subjected to high angular velocity. High speed results in large centrifugal forces in discs and simultaneous high temperature reduces disc material strength. Thus, the latter effects increase deformation of the disc automatically. In order to attain a certain and reliable analysis for disc and to derive corresponding stress distribution, solution should consider changes in material specifications caused by temperature. To achieve this goal, an inhomogeneous disc model with variable thickness is considered. Using the variable material properties method [J. Pressure Vessel Technol. 119 (1997) 264; Thermoplastic analysis of inhomogeneous rotating disk with variable thickness, Proceedings of the EMAS Conference of Fatigue, Camdridge, England, April 2000; Int. J. Pressure Vessels Piping 71 (3) (2001) 285] stresses are obtained for a disc under rotation and steady temperature field. Optimization process begins following stress calculation. Inscribed hypersphere method is used to obtain optimized disc profile. Using the above method, the disc profile is optimized and final solutions are obtained. It is also shown that the solutions of optimization process regarding different initial profiles for disc of similar specifications are unique when inscribed hypersphere radii in the last solution stages are equal. Results are compared to published data and are found to be in good agreement with them.

Actual Unloading Behavior and Its Significance on Residual Stress in Machined Autofrettaged Tubes

A simple method for obtaining actual unloading behavior of high strength steels is proposed. Based on this method loading-unloading stress-strain curves for NiCrMoV125 steel is obtained. It is shown that the actual unloading behavior tends to be nonlinear as soon as unloading occurs. Using the variable material properties (VMP) method, residual stresses induced by autofrettage process of a tube made of this type of steel are calculated. Significance of employing actual unloading behavior is demonstrated by comparing the results with ideal models such as isotropic hardening and bilinear models. There is at least 30% over estimation of compressive residual stress at the bore if ideal models are used. Also, the process of material removal is simulated by using commercial FEM software to evaluate the recommended scenarios of removal on redistribution of residual stresses at the bore. Resulting stresses after machining are also calculated by the VMP method using actual behavior. It is shown that the final residual stress field differs significantly if actual unloading behavior is used.

Variable material property method in the analysis of cold-worked fastener holes

Based on a general axisymmetric method of elastic-plastic analysis presented by Jahed and Dubey, elastic-plastic boundaries and residual stress fields induced by cold expansion of fastener holes is predicted. The method uses a linear elastic solution to construct an elastic-plastic solution. The material parameters are treated as field variables and their spatial distributions are obtained as part of the solution. This method uses the actual loading-unloading behaviour of the material and therefore is capable of predicting an accurate residual stress field. Results obtained here are compared with available experimental and finite element results. The agreement of the results with experimental measurements is very good. It is shown that employment of the actual unloading material curve can make a significant change in residual field prediction.