Multiaxial Loading Histories Research Articles

Taking Full Advantage of Nominal Stresses to Design Notched Components against Variable Amplitude Multiaxial Fatigue

The present paper is concerned with the use of the Modified Wöhler Curve Method (MWCM), applied in terms of nominal stresses, to estimate lifetime of notched components subjected to variable amplitude multiaxial fatigue loading. The MWCM is applied by defining the critical plane through that direction experiencing the maximum variance of the resolved shear stress: since the shear stress resolved along the above direction is a monodimensional quantity, fatigue cycles are directly counted by the classical Rain-Flow method. The performed validation exercise, based on an extensive experimental investigation, seems to strongly support the idea that the MWCM applied along with the classical nominal stress based approach is capable of accurately estimating fatigue damage also in notched components subjected to variable amplitude multiaxial load histories.

Fatigue life assessment under a complex multiaxial load history: an approach based on damage mechanics

ABSTRACTIn order to assess the fatigue behaviour of structural components under a complex (cyclic or random) multiaxial stress history, methods based on damage mechanics concepts can be employed. In this paper, a model for fatigue damage evaluation in the case of an arbitrary multiaxial loading history is proposed by using an endurance function which allows us to determine the damage accumulation up to the final failure of the material. By introducing an evolution equation for the endurance function, the final collapse can be assumed to occur when the damage D is complete, that is when D reaches the unity. The parameters of this model, which adopts the stress invariants and the deviatoric stress invariants to quantify the damage phenomenon, are determined through a Genetic Algorithm once experimental data on the fatigue behaviour of the material being examined are known for some complex stress histories. With respect to traditional approaches to multiaxial fatigue assessment, the proposed model presents the following advantages: (1) the evaluation of a critical plane is not necessary; (2) no cycle counting algorithm to determine the fatigue life is required, because it considers the progressive damage process during the fatigue load history; (3) the model can be applied to any kind of stress history (uniaxial cyclic loading, multiaxial in‐phase or out‐of‐phase cyclic loading, uniaxial or multiaxial random loading).

On the overall accuracy of the Modified Wöhler Curve Method in estimating high-cycle multiaxial fatigue strength

The aim of the present paper is to systematically investigate the accuracy of the so-called Modified Wöhler Curve Method (MWCM) in estimating high-cycle fatigue strength of plain and notched engineering materials damaged by in-service multiaxial load histories. In more detail, the MWCM, which is a bi-parametrical critical plane approach, postulates that initiation and Stage I propagation of fatigue cracks occur on those material planes experiencing the maximum shear stress amplitude (this being assumed to be always true independently from the degree of multiaxiality of the applied loading path). Further, the fatigue damage extent is hypothesised to depend also on the maximum stress perpendicular to the critical plane, the mean normal stress being corrected through the so-called mean stress sensitivity index (i.e., a material constant capable of quantifying the sensitivity of the assessed material to the presence of superimposed static stresses). In the present investigation, the overall accuracy of the MWCM in estimating high-cycle fatigue strength was checked through 704 endurance limits taken from the literature and generated, under multiaxial fatigue loading, by testing both plain and notched samples made of 71 different materials. Such a massive validation exercise allowed us to prove that the MWCM is highly accurate, resulting in 95% of the estimates falling within an error interval equal to ±15%.

A notch multiaxial-fatigue approach based on damage mechanics

The fatigue assessment of structural components under complex multiaxial stresses (cyclic or random stress histories) can be conveniently tackled by means of damage mechanics concepts. In the present paper, a model for notch fatigue damage evaluation in the case of an arbitrary multiaxial loading history is proposed by using an endurance function which quantifies the damage accumulation in the material up to the final failure. The material collapse can be assumed to occur when the damage is complete, that is, when the parameter D reaches the unity. In the case of notched structural components, such a damage parameter D must be evaluated by taking into account the stress value as well as the gradient effect at the notch root. The proposed model, which also employs the stress invariants and the deviatoric stress invariants to quantify the damage phenomenon, is calibrated through a Genetic Algorithm once experimental data on the fatigue behaviour of the material being examined are known for some uniaxial or complex stress histories. The model presents the advantages to be mechanically based and to not require any evaluation of a critical plane and any loading cycle counting algorithm to determine the fatigue life.

A technique to estimate the local multiaxial elastic–plastic behavior from a purely elastic solution

In most fatigue applications, the nominal structural behavior is dominated by elastic deformation, but the fatigue lifetime is significantly influenced by plasticity around stress concentrations and flaws. Although the elastic–plastic behavior can be modeled with finite element analysis (FEA), the computational expense may be prohibitive, especially for variable amplitude loading with multiaxial stress states. To overcome this complexity, a local elastic–plastic estimate is explored that utilizes the purely elastic solution. The method is conceptually similar to previous work, but is adapted to be consistent for variable amplitude multiaxial cyclic loading histories. This approach combines a magnitude criterion such as Nueber’s rule, the Masing character of the pseudo-material method, and has the generality to adopt any appropriate multiaxial plasticity model. The assumptions of the current approximation are developed in a general manner, with the potential to adjust the constraint (i.e. direction alignment), magnitude, and the plasticity character as necessary.

An improved multiaxial method to estimate the elastic-plastic behavior from a purely elastic solution

In most fatigue applications, the nominal structural behavior is dominated by elastic deformation, but the fatigue lifetime is significantly influenced by plasticity around stress concentrations and flaws. Although the elastic-plastic behavior can be modeled with finite element analysis (FEA), the computational expense may be prohibitive, especially for variable amplitude loading with multiaxial stress states. To overcome this complexity, an elastic-plastic estimate is explored that utilizes the purely elastic solution. The method is conceptually similar to previous work, but is adapted to be consistent for variable amplitude multiaxial cyclic loading histories. This approach utilizes a magnitude criterion (i.e. Neuber [1] or ESED [2]), and an elastic alignment assumption to adjust the boundary conditions applied to the local material model. In this document, the effect of alignment is explored with a multiaxial ‘Box’ loading path.

Estimating fatigue lifetime of steel weldments locally damaged by variable amplitude multiaxial stress fields

The present investigation aims to prove that the so-called Modified Wöhler Curve Method applied along with the Theory of Critical Distances (formalised in the form of the Point Method) can successfully be used also to estimate fatigue lifetime of steel welded joints subjected to both uniaxial and proportional/non-proportional multiaxial variable amplitude load histories. The accuracy and reliability of our fatigue life estimation technique was systematically checked by reanalysing approximately 500 experimental results taken from the literature and generated by testing welded samples having not only different geometries but also different absolute dimensions. Such an extensive validation exercise allowed us to fully confirm the robustness of our design methodology as well as to show that, when the method itself is formalised for a probability of survival equal to 97.7%, the obtained estimates fully complies, from a statistical point of view, with the recommendations of the available standard codes.

Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method

We propose a general formulation – which we believe to be new – for the mean-field homogenization of inclusion-reinforced elasto-viscoplastic composites assuming small strains. Our proposal is based on an interplay between constitutive equations and numerical algorithms, and the key ideas behind it are the following. The evolution equations for inelastic strain and internal variables at the beginning of each time interval are linearized around the ending time of the same interval. The linearized equations are then numerically integrated using a fully implicit backward Euler scheme. The obtained algebraic equations lead to an incrementally affine stress–strain relation which involves two important terms. The first one is the algorithmic tangent operator, obtained by consistent linearization of the time discretized constitutive equations. The second term is a new one and called an affine strain increment. The proposal leads to thermoelastic-like relations directly in the time domain, and not in the Laplace–Carson (L–C) one. There is no need for viscoelastic-type integral rewriting of the evolution equations, for L–C transforms, or for numerical inversion back from L–C to time domains. The proposed method can be readily applied to sophisticated elasto-viscoplastic models with an arbitrary set of scalar or tensor internal variables, and is valid for multi-axial, non-monotonic and non-proportional loading histories. The theory is applied in detail to a well-known constitutive model, and verified against finite element simulations of representative volume elements or unit cells, for a number of composite materials.

Determination of fragments of multiaxial service loading strongly influencing the fatigue of machine components

The paper presents a method for determination of multiaxial load segments from original service histories, where the loaded machine part is meaningfully subjected to fatigue damage. These load segments are directly separated from a service-loading history, which can be of a random character. Typical procedures used for fatigue life assessment under multiaxial random loading are implemented to perform this task. While reduction from the multiaxial stress state to the equivalent uniaxial one, application of the linear multiaxial fatigue failure criteria was proposed. The equivalent stress history is subjected to the rainflow cycle-counting method, which allows to determine the amplitudes and mean values of counted cycles, their occurrence moment and time of duration. Influence of the stress mean value was taken into consideration with the Morrow's model. On the assumption of the linear Palmgren–Miner hypothesis of damage accumulation and the stress–life fatigue characteristics of the material, the damage-time function was determined. The load segments, where the influence on the material fatigue was significant, were determined on the basis of the fixed damage-intensity level and the proposed damage-intensity function. The presented method was studied on a hook loaded with two independent forces. FEM program which has the possibility to perform fatigue analysis was used during the computation for determining the expected place of crack initiation. The service-loading course was compressed to shorter one, so that only a small decrease of the fatigue damage in comparison with full length of the original service loading was observed. The proposed method seems to be right for preparing multiaxial loading histories in order to cut down the fatigue tests, especially in the case when the correlation between particular loading channels is very important.

A unified expression of elastic–plastic notch stress–strain calculation in bodies subjected to multiaxial cyclic loading

In this paper, a simplified thermodynamics analysis of cyclic plastic deformation is performed in order to establish an energy transition relation for describing the elastic–plastic stress and strain behavior of the notch-tip material element in bodies subjected to multiaxial cyclic loads. Based on the thermodynamics analysis, it is deduced that in the case of elastic–plastic deformation, Neuber’s rule inevitably overestimates the actual stress and strain at the notch tip, while the equivalent strain energy density (ESED) method tends to underestimate the actual notch-tip stress and strain. According to the actual energy conversion occurring in the notch-tip material element during cyclic plastic deformation, a unified expression for estimating the elastic–plastic notch stress–strain responses in bodies subjected to multiaxial cyclic loads is developed, of which Neuber’s rule and the ESED method become two particular cases, i.e. upper and lower bound limits of the notch stress and strain estimations. This expression is verified experimentally under both proportional and non-proportional multiaxial cyclic loads and a good agreement between the calculated and the measured notch strains has been achieved. It is also shown that in the case of multiaxial cyclic loading, the unified expression distinctly improves the accuracy of the notch-tip stress–strain estimations in comparison with Neuber’s rule and the ESED method. The unified expression of the notch stress–strain calculation developed in this paper can thus provide a more logical approximate approach for estimating the elastic–plastic notch-tip stress and strain responses of components subjected to lengthy multiaxial cyclic loading histories for local strain approach-based fatigue-crack-initiation life prediction.

A three-dimensional phenomenological model for martensite reorientation in shape memory alloys

In this work, we propose a macroscopic phenomenological model that is based on the classical framework of thermodynamics of irreversible processes and accounts for the effect of multiaxial stress states and non-proportional loading histories. The model is able to account for the evolution of both twinned and detwinned martensite. Moreover, reorientation of the product phase according to loading direction is specifically accounted for. Towards this purpose the inelastic strain is split into two contributions deriving, respectively, from creation of detwinned martensite and reorientation of previously existing martensite variants. Computational tests demonstrate the ability of the model to simulate the main aspects of the shape memory response in a one-dimensional setting and some of the features that have been experimentally found in the case of multiaxial non-proportional loading histories. Experimental non-proportional loading paths have also been simulated and a good qualitative agreement between numerical and experimental response is observed.

A probabilistic model for multiaxial high cycle fatigue

ABSTRACTA probabilistic framework developed to model multiaxial high cycle fatigue tests is proposed. Up to now, with a probabilistic point of view (i.e., Weibull law), models account for the stress heterogeneity effect by introducing the concept of effective volume. It is proposed to extend this concept to multiaxial load histories. It consists in introducing a factor representing the distribution of activated slip directions. The non‐proportionality effect is taken into account with no additional parameter with respect to traditional (i.e., Weibull) probabilistic approaches. The proposed model is validated on a large experimental data base and compared with other models.

A life prediction model for welded joints under multiaxial variable amplitude loading histories

This paper describes a simple damage model for fatigue life predictions of welded joints under nonproportional, constant, and variable amplitude loading histories. This model, an extension of Sonsino’s effective equivalent stress amplitude method, consists of four material parameters accounting for the severity of nonproportional loading paths, the material’s susceptibility to nonproportional hardening, the material’s fatigue life under shear versus normal stresses, and the mean stress effect. These four parameters can be obtained by fatigue testing for S–N curves under bending only, torsion only, and 90° out-of-phase loading. This paper investigates the validity of the fatigue damage model by comparing the analytical and the published experimental results for various welded joint configurations subjected to multiaxial constant amplitude loads. This paper also describes a procedure to predict fatigue lives of welded joints under nonproportional variable amplitude loading histories based on the linear cumulative damage rule, the proposed equivalent stress amplitude and the Wang–Brown reversal counting technique.

Determination of an HCF criterion by thermal measurements under biaxial cyclic loading

A probabilistic two-scale model was recently developed to describe high-cycle fatigue (HCF) properties of steels associated with an identification method of the scatter based on thermal measurements. Up to now, this theoretical framework was used for uniaxial load histories. It is proposed to modify the probabilistic two-scale model to account for the effect of a multiaxial and proportional load histories on HCF properties. This modification consists in the introduction of an equivalent activation stress in the process controlling the plasticity kinetics. Thermal measurements are performed under biaxial tension–compression load histories to identify and validate an HCF criterion.

Multiaxial fatigue of rubber: Part I: equivalence criteria and theoretical aspects

ABSTRACTThis paper investigates commonly used approaches for fatigue crack nucleation analysis in rubber, including maximum principal strain (or stretch), strain energy density and octahedral shear strain criteria. The ability of these traditional equivalence criteria, as well as a recent equivalence criterion (the cracking energy density) to predict multiaxial fatigue behaviour is explored. Theoretical considerations are also introduced relating to the applicability of various fatigue life analysis approaches. These include the scalar nature of traditional equivalence criteria, robustness of the criteria investigated for a wide range of multiaxial loadings, effects of crack closure and applications to non‐proportional multiaxial loadings. It is shown that the notion of a stress or strain amplitude tensor used for the analysis of multiaxial loading in metals is not appropriate in the analysis of rubber due to nonlinearity associated with finite strains and near incompressibility. Taken together, these considerations illustrate that traditional criteria are not sufficiently consistent or complete to permit confident analysis of arbitrary multiaxial loading histories, and that an analysis approach specific to the failure plane is needed. Of the three traditional criteria, maximum principal strain is shown to match most closely to the cracking energy density criterion, in terms of a failure locus in principal stretch space.

RELIABILITY ANALYSIS OF STIFFNESS DEGRADATION IN AUTOMOTIVE SPOT-WELDED JOINTS

This paper develops a reliability-based methodology for the evaluation of stiffness degradation of spot-welded joints under high mileage. A global-local finite element analysis is used, with the loads on the detailed three-dimensional joint model coming from finite element analysis of the entire car model with proving ground loads. Probabilistic fatigue crack propagation analysis is developed for multi-axial variable amplitude loading history on the joint. Multiple spot welds contribute to the stiffness of the joint. Hence the problem is addressed through system reliability techniques. The effect of spot-weld separation on joint stiffness, and on global vehicle stiffness, is incorporated. This results in the computation of the statistics of vehicle stiffness degradation with mileage.

동력학 시뮬레이션에 의한 다축 랜덤하중 하에서 자동차 서브프레임의 고 되풀이수 피로손상 평가

A FEM-based analytical approach was used to evaluate the multiaxial high cycle fatigue damage of an automotive sub-frame. Elastic Multi Body Simulation (MBS) has been applied in order to determine the multiaxial load histories. The stresses due to these loads have been given by FE computation. These results have been used as the input for the multiaxial fatigue analysis. For the assessment of multiaxial high cycle fatigue damage, the signed von Mises, the signed Tresca, the absolute maximum principal stress and critical<br/> plane methods have been employed. In addition, the biaxiality ratio, a_e, the absolute maximum principal stress,<br/> σ₁ and the angle, ω_p' between σ₁ and the local x-axis, have been calculated to evaluate the stress state at each node.

Damage tolerance reliability analysis of automotive spot-welded joints

Abstract This paper develops a damage tolerance reliability analysis methodology for automotive spot-welded joints under multi-axial and variable amplitude loading history. The total fatigue life of a spot weld is divided into two parts, crack initiation and crack propagation. The multi-axial loading history is obtained from transient response finite element analysis of a vehicle model. A three-dimensional finite element model of a simplified joint with four spot welds is developed for static stress/strain analysis. A probabilistic Miner's rule is combined with a randomized strain-life curve family and the stress/strain analysis result to develop a strain-based probabilistic fatigue crack initiation life prediction for spot welds. Afterwards, the fatigue crack inside the base material sheet is modeled as a surface crack. Then a probabilistic crack growth model is combined with the stress analysis result to develop a probabilistic fatigue crack growth life prediction for spot welds. Both methods are implemented with MSC/NASTRAN and MSC/FATIGUE software, and are useful for reliability assessment of automotive spot-welded joints against fatigue and fracture.

Calculation of elasto-plastic strains and stresses in notches under multiaxial loading

A method for calculating elasto-plastic notch tip strains and stresses in bodies subjected to multiaxial loading has been presented. The method has been formulated in terms of strain energy density relationships. Two approximate formulae are derived based on the analysis of strain energy density in the notch tip region. The two formulae represent the lower and upper limits of the band within which the actual elasto-plastic notch tip strains can be found. All necessary relationships are derived for a general multiaxial stress state. The calculated notch tip strain and stress components are compared with experimental and finite element data obtained for a variety of loading conditions and materials. This method may be particularly useful for stress/strain analysis of notched components subjected to lengthy multiaxial cyclic loading histories.

Fatigue of old metallic railroad bridges: Varona, J. M., Guterrez-Solana, F. and Gonzalez, J. J. Proc. Conf. Fatigue Design 1992, Helsinki, Finland, 19–22 May 1992, Vol 2 (Technical Research Centre of Finland, Vuorimiehentie 5, PO Box 42, SF-02151 Espoo, Finland) 1992 pp 197–213

Most fatigue models must somehow identify and count individual load events before quantifying the damage induced by each one of them, making multiaxial fatigue damage calculations under non-proportional variable amplitude loadings a challenging and laborious task in practical applications. Moreover, to apply such models it is usually necessary to use semi-empirical methods to evaluate the non-proportionality of the load path of each event, through path-equivalent ranges obtained e.g. using a convex enclosure or the MOI (Moment Of Inertia) method. To avoid this burden, a new approach called the Incremental Fatigue Damage methodology is proposed in this work to continuously accumulate multiaxial fatigue damage under actual service loads, without requiring path-equivalent range estimators or rainflow counters. This new approach is not based on questionable Continuum Damage Mechanics concepts, or on the integration of some unrealistic scalar damage parameter based on elastoplastic work. Instead, inspired on multiaxial plasticity procedures, a framework of nested damage surfaces is introduced, allowing the calculation of fatigue damage even for general 6D multiaxial load histories. In this way, fatigue damage itself can be continuously integrated along the load path, considering damage parameters adopted by traditional fatigue models and following reliable procedures well tested in engineering practice. The proposed approach is experimentally validated by non-proportional tension-torsion tests on tubular 316L stainless steel specimens.