Modeling the impact behavior of AD85 ceramic under multiaxial loading

This paper presents an advanced constitutive model to describe the complex behavior of ceramic materials under impact loading conditions. The governing equations utilize a set of microphysically based constitutive relationships to model deformation and damage processes in a ceramic. The total strain is decomposed into elastic, plastic and microcracking components. The model parameters for AD85 ceramic were determined using the data from split Hopkinson bar and bar-on-bar experiments under uniaxial stress state and plate impact experiment under uniaxial strain state. To further validate the generality of the model parameters, modeling of a diagnostic ballistic experiment in which a steel projectile impacted a AD85 ceramic-front-faced thick aluminium plate, was considered. In this experiment, stress histories were measured in the target by embedded manganin and carbon stress gauges. The results from the numerical simulations of the ballistic experiment using a shock-wave propagation based finite element code, successfully matched the measured stress history.     

Stabilized cyclic stress-strain calculation for metals under multiaxial loading

A simple plasticity model for modeling the stabilized cyclic stress-strain responses is developed to consider the effect of non-proportional additional hardening. In the proposed model, the plastic modulus for uniaxial loading is extended to multiaxial loading by introducing the non-proportionality factor and the additional hardening coefficient. The two introduced factors take into account the effects of non-proportional additional hardening, not only on the shape of the loading path, but also on the material and its microstructure. And then, the basic Armstrong-Frederick nonlinear hardening rule is modified to model the evolution of the back stress. The consistency condition is enforced to obtain the relationship between the back stress and plastic modulus. The proposed model requires only six material constants for estimating the stabilized responses. Comparisons between the test results (30CrNiMo8HH steel, SA 333 Gr.6 steel, and 1 %CrMoV steel) and model predictions show that the proposed model predicts relatively accurate stress responses under both proportional and non-proportional loading paths.     

Microcrack propagation and fatigue lifetime under non-proportional multiaxial cyclic loading

Non-proportional multiaxial fatigue tests of tubular specimens were performed under purely alternating strain-controlled loading. Different loading paths with different phase shifts were applied. With increasing phase shift at the same equivalent load, the lifetime was found to increase. For lifetime prediction a model based on the Manson–Coffin law was developed. By including the hydrostatic loading part, it was possible to compare the results of the multiaxial fatigue tests with uniaxially ascertained results. To obtain more information about the microcrack behaviour under multiaxial non-proportional loading, sonic emission studies and fractographic analyses were performed. The results suggest a discontinuous microcrack propagation. Motivated by the good agreement between these observations and some microcrack propagation models known from literature, a simplified model was proposed for micro and short crack propagation. This model which is based on the J-integral range ΔJ yields a quite good agreement between the experimentally observed and the calculated lifetimes.     

Notch fatigue life prediction considering nonproportionality of local loading path under multiaxial cyclic loading

An innovative numerical methodology is presented for fatigue lifetime estimation of notched bodies experiencing multiaxial cyclic loadings. In the presented methodology, an evaluation approach of the local nonproportionality factor F for notched specimens, which defines F as the ratio of the pseudoshear strain range at 45° to the maximum shear plane and the maximum shear strain range, is proposed and discussed deeply. The proposed evaluation method is incorporated into the material cyclic stress‐strain equation for purpose of describing the nonproportional hardening behavior for some material. The comparison between multiaxial elastic‐plastic finite element analysis (FEA) and experimentally measured strains for S460N steel notched specimens shows that the proposed nonproportionality factor estimation method is effective. Subsequently, the notch stresses and strains calculated utilizing multiaxial elastic‐plastic FEA are used as input data to the critical plane‐based fatigue life prediction methodology. The prediction results are satisfactory for the 7050‐T7451 aluminum alloy and GH4169 superalloy notched specimens under multiaxial cyclic loading.     

Prediction of fatigue lifetime under multiaxial cyclic loading using finite element analysis

Life prediction for GH4169 superalloy thin tubular and notched specimens were investigated under proportional and nonproportional loading with elastic–plastic finite element analysis (FEA). A strain-controlled tension–torsion loading was carried out by applying the axial and circular displacements on one end of the specimen in the cylindrical coordinate system. Uniaxial cyclic stress–strain data at high temperature were used to describe the multi-linear kinematic hardening of the material. The comparison between FEA and experimental results for thin tubular specimen showed that the built model of FE is reliable. A fatigue damage parameter was proposed to predict the fatigue crack initiation life for notched specimen. The results showed that a good agreement was achieved with experimental data.     

Crack path evaluation on HC and BCC microstructures under multiaxial cyclic loading

In this paper the multiaxial loading path effect on the fatigue crack initiation, fatigue life and fracture surface topology are evaluated for two different crystallographic microstructures (bcc and hc): high strength low-alloy 42CrMo4 steel and the extruded Mg alloy AZ31B-F, respectively.A series of multiaxial loading paths were carried out in load control, smooth specimens were used. Experimental fatigue life and fractographic results were analyzed to depict the mechanical behavior regarding the different microstructures.A theoretical analysis was performed with various critical plane models such as the Fatemi–Socie, SWT and Liu in order to correlate the theoretical estimations with the experimental data. A new approach based on maximum stress concentration factors is proposed to estimate the crack initiation plane, estimations from this new approach were compared with the measured ones with acceptable results. To implement this new approach a virtual micro-notch was considered using FEM. Moreover, the multiaxial loading path effect on stress concentration factors is also studied. The obtained results clearly show the effect of the applied load conditions on local microstructures response.     

Effects of extrusion ratio on the ratcheting behavior of extruded AZ31B magnesium alloy under asymmetrical uniaxial cyclic loading

Magnesium alloys are increasingly used in the automotive and aerospace industries for weight reduction and fuel savings. The ratcheting behavior of these alloys is therefore an important consideration. The objective of this investigation was to study the effects of extrusion ratio on the ratcheting behavior of extruded AZ31B magnesium alloy. The experiments have shown that the extruded AZ31B Mg alloy presented the following characteristic behavior with increasing number of loading cycles: first an apparent cyclic softening was observed, then a cyclic hardening occurred, and finally a stable state was reached. This generic behavior can be explained by the fact that the variation trend of the maximum strain with the number of cycles differs from that of the minimum strain. The extrusion ratio did not influence the cyclic softening/hardening behavior or the final ratcheting strain variation trend of the extruded AZ31B Mg alloy with the mean stress and the peak stress. However, the extrusion ratio influenced the final ratcheting strain variation trend of the extruded AZ31B Mg alloy with the stress amplitude. Increasing the extrusion ratio also reduced the ratcheting strain and the effects of the load history on the ratcheting behavior of the extruded AZ31B Mg alloy.     

Effects of annealing treatment on the ratcheting behavior of extruded AZ31B magnesium alloy under asymmetrical uniaxial cyclic loading

The objective of this investigation is to study the effects of annealing treatment on the ratcheting behavior of extruded AZ31B magnesium alloy. First, the microstructures and monotonic tensile properties of the extruded and annealed alloys were assessed. The results showed that the grain size increased slightly with increasing annealing time until an annealing time of 6 h after which abnormal grain growth happened. Accordingly, the ultimate tensile strength of the Mg alloy decreased with increasing annealing time, while the tensile yield strength and elongation percentage of the Mg alloy increased with annealing time until the annealing time reached 2 h. The cyclic softening/hardening behavior of the annealed AZ31B Mg alloy was similar to that of the extruded alloy: first an apparent cyclic softening was observed, then a cyclic hardening occurred, and finally a stable state was reached. The annealing treatment delayed the occurrence of the cyclic hardening. It was also shown that the effects of the annealing time on the ratcheting strain of the Mg alloy depended of the loading path.     

Life estimation for high-strength materials under cyclic loading in the multiaxial stress state

The life of high-strength materials under cyclic loading in the multiaxial stress state is experimentally estimated using the deformation-kinetic criterion. Translated from Problemy Prochnosti, No. 2, pp. 139–143, March–April, 2009.     

Cyclic uniaxial and multiaxial loading with yield surface distortion consideration on prediction of ratcheting

In this study, the yield surface distortion was incorporated in the cyclic plasticity modeling as well as its center movement. The combination of Chaboche’s model and the yield surface distortion model of Baltov was used in a set of uniaxial and multiaxial loadings. The variation of the stress amplitude and the mean stress and different multiaxial loadings such as tension-torsion tests were studied. It was shown that the consideration of the distortion of the yield surface via the distortion parameter and its sign in modeling has an important effect on the plastic strain increment determination and so on the ratcheting rate. The combined model was applied to the experimental results. It was shown that the combination of the nonlinear kinematic hardening model of Chaboche and the yield surface distortion leads to a good estimation of the ratcheting strain increment in different uniaxial and multiaxial tests.     

Long life fatigue under multiaxial loading

A life prediction model in the field of high-cycle (i.e. long-life) fatigue is presented in this paper. The proposed model applies in the case of constant amplitude multiaxial proportional and non-proportional loading. The problems of the fatigue limit criterion and of the fatigue life prediction are both addressed and comparisons with experimental data are shown. Some limited discussion of the stress gradient effect is also offered. Although the particular model developed here is better suited for ferritic steels, it is explained in the paper that the methodology used to obtain this model can be adequately adapted to derive mathematically consistent models for other classes of metallic materials.     

A plasticity model for multiaxial cyclic loading and ratchetting

Summary The nonlinear behavior of metals when subjected to monotonic and cyclic non-proportional loading is modeled using the proposed hardening rule. The model is based on the Chaboche [1], [2] and Voyiadjis and Sivakumar [3], [4] models incorporating the bounding surface concept. The evolution of the backstress is governed by the deviatoric stress rate direction, the plastic strain rate, the backstress, and the proximity of the yield surface from the bounding surface. In order to ensure uniqueness of the solution, nesting of the yield surface with the bounding surface is ensured. The prediction of the model in uniaxial cyclic loading is compared with the experimental results obtained by Chaboche [1], [2]. The behavior of the model in multiaxial stress space is tested by comparing it with the experimental results in axial and torsional loadings performed by Shiratori et al. [5] for different stress trajectories. The amount of hardening of the material is tested for different complex stress paths. The model gives a very satisfactory result under uniaxial, cyclic and biaxial non-proportional loadings. Ratchetting is also illustrated using a non-proportional loading history.     

A coupled plasticity correction approach to estimating notch root strains under multiaxial cyclic loading

Based on pseudo strain approach, a coupled plasticity correction approach is proposed to calculate conveniently local strains for blunt notches under multiaxial cyclic loading. A detailed algorithm of the method is presented. Compared with conventional pseudo strain based approach, the proposed method does not require the plasticity model twice. The applicability of the method was evaluated by the measured notch root strains for a notched round shaft subjected to combined axial–torsion multiaxial cyclic loading. The results showed that the proposed method was acceptable and the calculated axial strain range was more accurate than the calculated shear strain range.     

Fatigue behavior of laminated composites with a circular hole under in-plane multiaxial loading

A modified fiber failure fatigue model is presented for characterizing the behavior of laminated composites with a central circular hole under in-plane multiaxial fatigue loading. The analytical model presented is based on minimum strength model and fiber failure criterion under static loading available in the literature. The analysis starts with the determination of location of a characteristic curve around the hole and the stress state along the characteristic curve under in-plane multiaxial fatigue loading. Number of cycles to failure and location of failure are determined under given fatigue loading condition. Based on ply-by-ply analysis, ultimate fatigue failure and the corresponding number of cycles are determined. Analytical predictions are compared with the experimental results for uniaxial and multiaxial fatigue loading cases. A good match is observed. Further, studies are carried out for different in-plane biaxial tension–tension and biaxial compression–compression loading cases.     

Lateral behavior of concrete under uniaxial compressive cyclic loading

In reinforced concrete structures under seismic loading, concrete is subjected to compressive cyclic stress. Although cyclic stress–strain response has been described before, the cyclic behavior of strains in the direction orthogonal to loading has not been characterized yet. Such behavior can be of great importance for evaluating the efficiency of the confinement under cyclic loading. For this purpose an experimental program on cylindrical specimens of concrete strength from 35 to 80 MPa subjected to uniaxial cyclic compression was carried out. Stress versus longitudinal and lateral strains curves have been obtained both for the hardening and softening branches under monotonic and cyclic loading. Governing parameters of the lateral behavior are identified and correlated to describe the response of the lateral strain. Additionally, an analytical model to obtain the lateral deformations of concrete under cyclic uniaxial compression has been formulated and verified experimentally. Finally, some examples are presented in order to illustrate the applicability of the proposed model and its possible incorporation into a 3D constitutive cyclic model.     

Elasto-plastic behavior of thermoplastic composite laminates under cyclic loading

An experimental and numerical study of the elasto-plastic behavior of thermoplastic matrix composite laminates under static and cyclic loads is presented. Off-axis and angle ply specimens cut from laminates of poly(ether ether ketone) (PEEK) reinforced with continuous carbon fibers have been tested under cyclic sinusoidal tensile loads and the hysteresis loops have been monitored. A micro mechanical model, which includes a parabolic criteria based on the plastic behavior of the matrix, has been adopted to study the composite non-linear behavior and a correlation between plastic deformation and a strong rise of damping and temperature at high stresses is outlined. Good agreement is shown between theory and experimental results. The mathematical mdoel presented here can be used to predict the visco-elastic-plastic response of the material at high stresses and its influence in the fatigue damage.     

Energy dissipation under multiaxial thermomechanical fatigue loading

The paper presents an approach to the energy dissipation calculation under arbitrary multiaxial thermomechanical fatigue (TMF) loading. In such an approach the total area of plastic hysteresis loops is taken as a measure of dissipated energy. The calculation is based on the concept of the developed temperature dependent Prandtl type operator. Energy dissipation is associated to irreversible dislocation movements represented by slider shifts of three independent operators. The dissipated energy is then obtained continuously at any time by collecting dissipated energy increments of each operator. It is shown that the multiaxial operator approach gives us the same total energy of plastic deformation as compared to the classical integration approach. Furthermore, the presented approach enables to automatically split the obtained dissipated energy between the “true” dissipated energy and the elastically “stored” energy. In order to satisfy the request for a minimum number of dedicated material tests, the approach assumes fixed principal directions. Therefore, the proportional as well as non-proportional loading conditions are addressed in the same manner, which is currently the main deficiency of the approach.     

A plasticity model for calculating stress–strain sequences under multiaxial nonproportional cyclic loading

There is increasing demand for analytical methods that estimate the fatigue life of engineering components and structures with a high degree of accuracy. The fatigue life is determined by the stress–strain sequences at the critical locations. Therefore, these sequences have be calculated with sufficient accuracy for arbitrary nonproportional cyclic loading. Based on the experience with a variety of material models following macroscale continuum mechanics approaches, an improved set of constitutive equations is proposed. The stress–strain behaviour of a commercial structural steel has been investigated experimentally. Firstly, the results of this experimental study serve to identify the material parameters comprised in the model. Secondly, the predicted stress–strain paths are compared to their experimentally determined counterparts as well as to paths predicted by other models. The overall accuracy of the proposed model is quite satisfying, especially as far as calculated amplitudes are concerned.