Non-proportionality Of Loading Research Articles

Topology optimization for infinite fatigue life of cyclic symmetric structures subjected to non-proportional loading

This paper presents a density based topology optimization method for infinite fatigue life constraints of non-proportional load cases, with a specific focus on parts with cyclic symmetry. Considering non-proportional loads in topology optimization significantly broadens the types of design problems that can be handled. The method estimates the local variation in Signed von Mises stress using a smooth min/max function and constrains the resulting stress amplitude using established stress based topology optimization methods. Accounting for non-proportionality of loading significantly increases the computation cost with respect to existing proportional methods, as the time-varying stress field needs to be computed. Inertia effects are neglected in the structural analysis. Therefore, a quasi-static analysis is used to obtain the stress history. To reduce the computational cost, advantage is taken of cyclic symmetric properties to reduce the number of necessary time steps to evaluate. This reduces the computational cost roughly proportional to the number of unique load time steps present in the repeated segments as opposed to a standard implementation. The method is tested on numerical examples in 2D and 3D for both proportional and non-proportional loads and was found to be locally accurate up to the accuracy of the constraint aggregation.

A Novel Damage Parameter for Fatigue Life Assessment under Non-Proportional Loading

As is well-known, the loading non-proportionality has a significant influence on fatigue process, since the rotation of principal stress/strain axes can lead to additional cyclic hardening. In order to accurately reflect the effect of additional cyclic hardening on the predicted lifetime, Vantadori has recently formulated a strain-based fatigue criterion which implements a novel strain factor (function of material constants and degree of non-proportionality) in the damage parameter relationship. By considering some experimental data available in the literature, the goal of the present paper is to discuss the accuracy of the present criterion in estimating fatigue lifetime of metallic components, especially under non-proportional loading.

A notch critical plane approach of multiaxial fatigue life prediction for metallic notched specimens

AbstractAn approach based on the local stress response is proposed to locate the fatigue critical point for metallic blunt notched specimens under multiaxial fatigue loading. According to the stress analysis, both stress gradient and gradient of loading nonproportionality exist at notch root. The plane in the vicinity of the notch that passes through the fatigue critical point and experiences the maximum shear stress amplitude is defined as the critical plane for notch specimens (CPN). Furthermore, the Susmel's fatigue damage parameter is modified to assess fatigue life of notched components by combining CPN and the theory of critical distance (TCD). The multiaxial fatigue test of the thin‐walled round tube specimens made of Ni‐base alloy GH4169 is carried out to verify the above approaches. In addition, test data of two kinds of materials are collected. The results show that the maximum absolute error of the fatigue critical point is 9.6° and the majority of the predicted life falls within the three‐time scatter band.

A kinematic hardening rule to investigate the impact of loading path and direction on ratcheting response of steel alloys

The present study investigates ratcheting response of 304, 1045, 1026 and 1018 steel samples undergoing butterfly loading paths with different loading sequences/ directions through the newly developed kinematic hardening rule of Ahmadzadeh–Varvani (A-V). Components of backstress unity vector and the normal vector to the yield surface in the dynamic recovery of the model account for loading non-proportionality. The MaCaulay brackets 〈dɛ¯p·a¯/|a¯|〉 enable the hardening rule to track different directions under multiaxial stress cycles. Coefficient γ2 controls the rate of ratcheting and is regulated by term (2−n¯.a¯/|a¯|) to further lower the ratcheting strain curve. Term 〈n¯.a¯/|a¯|〉1/2 in the dynamic recovery prevents ratcheting plastic shakedown as stress cycles progress. Constant γ2 introduces a gradual decrease in the magnitude of (a¯−b¯) as the stress cycles advances. The movement of tensor b¯ with the initial value of zero follows backstress (a¯)exponentially over stress cycles. The difference in the magnitudes of b¯ and a¯ curves in butterfly down BFD (forward) and butterfly up BFU (reverse) histories verifies the loading path accountability through the dynamic recovery in the A-V model. This difference is initially large and as the number of cycles increases, term (a¯−b¯) decays and then stays constant and unchanged. History BFD with higher magnitude of term (a¯−b¯) possessed higher ratcheting as compared with BFU history. The predicted ratcheting curves through the A-V model consistently agreed with experimental data obtained under different loading paths and directions.

A criterion for high-cycle fatigue life and fatigue limit prediction in biaxial loading conditions

This paper presents a criterion for high-cycle fatigue life and fatigue strength estimation under periodic proportional and non-proportional cyclic loading. The criterion is based on the mean and maximum values of the second invariant of the stress deviator. Important elements of the criterion are: function of the non-proportionality of fatigue loading and the materials parameter that expresses the materials sensitivity to non-proportional loading. The methods for the materials parameters determination uses three S–N curves: tension–compression, torsion, and any non-proportional loading proposed. The criterion has been verified using experimental data, and the results are included in the paper. These results should be considered as promising. The paper also includes a proposal for multiaxial fatigue models classification due to the approach for the non-proportionality of loading.

Visualization of Multiaxial Stress/Strain State and Evaluation of Failure Life by Developed Analyzing Program under Non-Proportional Loading

This study presents definitions of principal stress/strain range and mean stress/strain introduced by utilizing Itoh-Sakane criterion for multiaxial loading including non-proportional loading, and shows the method of calculating the non-proportional factor which expresses the severity of non-proportional loading under the multiaxial 3D loading. This paper also shows a method of visually presenting the stress/strain, the non-proportionality of loading and the damage evaluation.

Modification of Zenner and Liu Criterion due to Non-Proportionality of Fatigue Load by Means of MCE Approach

The results of experimental verification of Zenner and Lius criterion confirm its efficiency in the case of estimation of fatigue life for proportional loading. Due to application of general damage parameter, the criterion is useful in case of various types of proportional loading and for multiple materials. However, for non-proportional loadings, of high non-proportionality level, the error of estimation of fatigue lives often exceeds scatter band 3. The hereby work presents a proposal for Zenner and Lius criterion modification based upon introduction of loading non-proportionality measure with minimum circumscribed ellipse method (MCE).

Analysis of Rails of a Ferry Boat under Wheels Contact Loading

The paper presents the effect of the discontinuity of the rails of a ferry boat and the presence of lower modulus insulation material at the gap to the variations of stresses in the insulated rail. The analysis consists of a three-dimensional wheel rail contact model based on the finite element method. One of the results shows that the maximum stress occurs in the subsurface of the railhead of the ferry boat. The ratio of the elastic modulus of the railhead and insulation material is found to alter the levels of stress concentration. Numerical result indicates that a higher elastic modulus insulating material can reduce the stress concentration in the railhead but will generate higher stresses in the insulation material, leading to earlier failure of the insulation material. A general subsurface crack propagation analysis methodology is used for the wheel and rail rolling contact. The fatigue damage in the wheel is calculated using a previously developed mixed-mode fatigue crack propagation model. The advantages of the proposed methodology are that it can accurately represent the contact stress of complex mechanical components and can consider the effect of loading non-proportionality. The effects of wheel diameter, vertical loading amplitude, initial crack size, location and orientation on stress intensity factor range are investigated using the proposed model. The prediction results of the proposed methodology are compared with in field observations. The contact elements were used to stimulate the interaction between a wheel and a railhead. Variations in contact stress fields at various locations of the rail are sensitive to the contact distance. The location of the maximum von Mises stress was shifted to the contact surface as the contact point moves close to the rail end. A higher stress, larger deflection and significant plastic deformation occurring at the rail from ferry boat may lead to deterioration at the rail end.

Cyclic deformation behavior of 316LN under non-proportional loading based on the analysis on the hysteresis loop and microstructure

The evolution of the effective and internal stresses of 316LN under total strain controlled non-proportional loadings is presented by partitioning the hysteresis loop, and the microstructure evolution was studied by transmission electron microscopy observation on specimens loaded with different cycles. Particular attention was paid to the two different contributions of the internal stress, intergranular and intragranular components, which were studied by taking into account quantitatively the heterogeneous dislocation structures. As compared with uniaxial loadings, the dislocation structures were observed with higher heterogeneity and density under non-proportional loadings, especially at the beginning of cycling. The dislocation rearrangement from planar structure to heterogeneous distribution occurred during the initial hardening stage led to not only an increase in the intragranular component which dominates the initial cyclic hardening, but also a decrease in the intergranular one which induces the following cyclic softening. Moreover, the hardening with the loading non-proportionality was also determined mostly by the intragranular component.

Effect of High Amplitude Loading on Fatigue Life Prediction of Steel Bridges

This paper presents a new fatigue model to predict life of steel bridges considering the effect of high amplitude loading. It consists of a modified strain-life curve and a new strain based damage index. Modified strain-life curve consists of Coffin-Manson relation in low cycle fatigue region and a new strain-life curve in high cycle fatigue region. The damage variable is based on a modified von Mises equivalent strain to account for the effects of loading nonproportionality and strain path orientation in multiaxial stress state. The proposed model was verified with experimental test results of two materials available on the literature. Then, it was illustrated with an old riveted wrought iron railway bridge. The obtained results verify the effectiveness of the proposed model over commonly used Miner's rule based life prediction of steel bridges.

A new model for low cycle fatigue of metal alloys under non-proportional loading

Abstract This paper proposes a new model for low cycle fatigue of metal alloys based on material sensibility to non-proportionality of loading. The model was examined by application to the prediction of the fatigue life in proportional and non-proportional low cycle fatigue tests for type BT1-0 titanium alloy. Tests were carried out in accordance with the plan of bifactorial experiment with three levels Mises’ strain: 0.7%, 0.9%, 1.1% and at three levels of non-proportion factor expressing the severity of non-proportional straining – 0, 0.5, 1. All the tests were carried out at room temperature. It was determined that additional hardening for non-proportional path tests is insignificant for BT1-0. Also proposed model was examined by tests for SNCM 630, Inconel 718, type 08X18H10T stainless steel, type BT9 titanium alloy. The model successfully correlated most of all fatigue data for all materials. The results showed that the model gave almost the same error both at proportional and non-proportional loading.

Analysis of subsurface crack propagation under rolling contact loading in railroad wheels using FEM

A general subsurface crack propagation analysis methodology for the wheel/rail rolling contact fatigue problem is developed in this paper. A three-dimensional elasto-plastic finite element model is used to calculate stress intensity factors in wheels, in which a sub-modeling technique is used to achieve both computational efficiency and accuracy. Then the fatigue damage in the wheel is calculated using a previously developed mixed-mode fatigue crack propagation model. The advantages of the proposed methodology are that it can accurately represent the contact stress of complex mechanical components and can consider the effect of loading non-proportionality. The effects of wheel diameter, vertical loading amplitude, initial crack size, location and orientation on stress intensity factor range are investigated using the proposed model. The prediction results of the proposed methodology are compared with in-field observations.

Thermodynamic based model for the evolution equation of the backstress in cyclic plasticity

A nonlinear kinematic hardening rule is developed here within the framework of thermodynamic principles. The derived kinematic hardening evolution equation has three distinct terms: two strain hardening terms and a dynamic recovery term that operates at all times. The proposed hardening rule, which is referred in this paper as the FAPC (Fredrick and Armstrong–Phillips–Chaboche) kinematic hardening rule, shows a combined form of the Frederick and Armstrong backstress evolution equation, Phillips evolution equation, and Chaboche series rule. A new term is incorporated into the Frederick and Armstrong evolution equation that appears to have agreement with the experimental observations that show the motion of the center of the yield surface in the stress space is directed between the gradient to the surface at the stress point and the stress rate direction at that point. The model is further modified in order to simulate nonproportional cyclic hardening by proposing a measure representing the degree of nonproportionality of loading. This measure represents the topology of the incremental stress path. Numerically, it represents the angle between the current stress increment and the previous stress increment, which is interpreted through the material constants of the kinematic hardening evolution equation. This new kinematic hardening rule is incorporated in a material constitutive model based on the von Mises plasticity type and the Chaboche isotropic hardening type. Numerical integration of the incremental elasto-plastic constitutive equations is based on a simple semi-implicit return-mapping algorithm and the full Newton–Raphson iterative method is used to solve the resulting nonlinear equations. Experimental simulations are conducted for proportional and non-proportional cyclic loadings. The model shows good correlation with the experimental results.

A multiaxial cyclic plasticity model for non-proportional loading cases

Abstract The hardening behavior of metals to non-proportional loading and ratchetting effects is investigated here using the proposed model presented by Voyiadjis and Basuroychowdhury (1998) . The backstress evolution equations are modified here in order to account for non-proportionality of loading. The same general form of the kinematic hardening formulation defined previously by Voyiadjis and Basuroychowdhury (1998) is used here. The material constant, β ( i ) , is now expressed in a functional form, β (i) , to account for the variation in the loading direction. Numerical results are obtained using the proposed model for a series of plastic strain controlled cyclic tests due to multiaxial cyclic tests performed at room temperature on thin-walled tubural specimens of Type 316 stainless steel. The results are compared with the experimental values obtained by Tanaka, et al. (1985) . The drift correction due to the finite increments of stress or strain is corrected using an efficient approach that corrects the backstress only. The model is also tested for cyclic creep which occurs due to a non-zero mean stress. The cyclic creep occurs initially and saturates reaching a stable cycle.The modified proposed kinematic hardening rule and the associated bounding surface are used in conjunction with a robust plasticity model for metals. The hardening dependence on the plastic strain path is also investigated in this work.

A Two Surface Model for Transient Nonproportional Cyclic Plasticity, Part 1: Development of Appropriate Equations

A two surface stress space model is introduced with internal state variable repositories for fading memory of maximum plastic strain range and non-proportionality of loading. Evolution equations for isotropic hardening variables are prescribed as a function of these internal variables and accumulated plastic strain, and reflect dislocation interactions that occur in real materials. The hardening modulus is made a function of prior plastic deformation and the distance of the current stress point from the limit surface. The kinematic hardening rules of Mroz and Prager are used for the yield and limit surfaces, respectively. The structure of the model is capable of representing essential aspects of complex nonproportional deformation behavior, including direction of the plastic strain rate vector, memory of plastic strain range, cross-hardening effects, variation of hardening modulus, cyclic hardening or softening, cyclic racheting, and mean stress relaxation.