Non-proportional Loading Paths Research Articles

Numerical analysis of damage and fracture in steel sheets undergoing non-proportional loading paths

The paper deals with the numerical analysis of damage and fracture mechanisms in steel sheets. Newly developed H-specimens are taken from thin sheets and are tested under different biaxial loading conditions. Results of numerical simulations of a series of biaxial experiments are presented showing also the effect of non-proportional loading paths on inelastic deformation behavior and on damage processes. In this context, an anisotropic continuum damage model is presented based on yield and damage conditions as well as evolution laws for plastic and damage strain rates. Different stress-state-dependent branches of the damage criteria are taken into account corresponding to various damage and failure processes on the micro-scale depending on the stress triaxiality and the Lode parameter. Experiments with the biaxially loaded H-specimen have been performed. Results for proportional and corresponding non-proportional loading histories are discussed. During the experiments strain fields in critical regions of the specimens are analyzed by digital image correlation (DIC) technique. Numerical simulations of the experiments have been performed and numerical results are compared with experimental data. Furthermore, based on the numerical analysis evolution of stress variables is examined and stress distributions in critical specimens areas are detected allowing prediction of stress-state-dependent damage and fracture mechanisms. The numerical results also demonstrate the efficiency of the experimental program and the new specimens geometries covering stress states in the shear-tension regime as well as the effect of the loading histories on inelastic deformation and damage behavior in steel sheets.

Tube Expansion by Single Point Incremental Forming: An Experimental and Numerical Investigation

In this paper, we revisit the formability of tube expansion by single point incremental forming to account for the material strain hardening and the non-proportional loading paths that were not taken into consideration in a previously published analytical model of the process built upon a rigid perfectly plastic material. The objective is to provide a new insight on the reason why the critical strains at failure of tube expansion by single point incremental forming are far superior to those of conventional tube expansion by rigid tapered conical punches. For this purpose, we replaced the stress triaxiality ratio that is responsible for the accumulation of damage and cracking by tension in monotonic, proportional loading paths, by integral forms of the stress triaxiality ratio that are more adequate for the non-proportional paths resulting from the loading and unloading cycles of incremental tube expansion. Experimental and numerical simulation results plotted in the effective strain vs. stress triaxiality space confirm the validity of the new damage accumulation approach for handling the non-proportional loading paths that oscillate cyclically from shearing to biaxial stretching, as the single point hemispherical tool approaches, contacts and moves away from a specific location of the incrementally expanded tube surface.

Fracture Estimation in Ship Collision Analysis—Strain Rate and Thermal Softening Effects

This study examined the effects of the strain rate and thermal softening on large-scale ductile fracture in ship collisions using a rate-dependent combined localized necking and fracture model. A Johnson–Cook type-hardening model, consisting of strain hardening, rate-sensitivity, and thermal softening terms, was adopted together with an associated flow rule. The temperature was treated as an internal state variable and was calculated from the plastic strain energy using a strain-rate-dependent weighting function under fully isothermal and adiabatic conditions. At every time increment, the fracture locus was updated based on the temporal strain rate, whereas the necking locus was coupled with the hardening law, which was dependent on both the strain rate and temperature. The damage indicator framework was used to consider the non-proportional loading paths. The dynamic shell-element failure model was verified through plate-panel penetration tests and applied to a large-scale ship collision analysis involving a struck ship/ship-shaped offshore installation and a supply vessel. The effects of the loading rate and impact energy were assessed in terms of the global behavior of the structure and observed failure modes.

Microbuckling of viscoplastic composites by the high-fidelity generalized method of cells micromechanics

The high-fidelity generalized method of cells (HFGMC) micromechanical analysis is employed for the prediction of microbuckling stresses of composite materials which are composed of elastic–viscoplastic constituents. The inelastic material behavior is represented by a unified viscoplasticity theory which can model non-proportional loading paths, namely it is capable of providing the effect of axial loading on the reduction of shear modulus which dominates the microbuckling. By applying an incremental procedure and in conjunction with the instantaneous stiffnesses of the viscoplastic material, an eigenvalue problem at each time step is established by the HFGMC micromechanical method, which is modified to include the buckling terms arising from the linearization of the nonlinear equations of equilibrium. These buckling terms are provided by HFGMC analysis which predicts the local stress field and the effective current tangent properties of the composite. The present approach is verified by comparison of its predictions with an exact solution that can be established in the special case of composites that consist of periodic viscoplastic layers. Applications are given for the prediction of the microbuckling stresses of bi-layered, continuously reinforced and aligned short-fiber viscoplastic composites. In addition, the microbuckling stresses of viscoplastic lattices and layered plates are presented.

Continuum Damage Mechanics Approach for Modeling Cumulative-Damage Model

In this study, we propose a novel cumulative-damage model based on continuum damage mechanics under situations where the mechanical components are subjected to variable loading. The equivalent completely reversed stress amplitude accounting for the effect of mean stress, stress gradients, loading history, and additional hardening behavior related to nonproportional loading paths on high-cycle fatigue under variable loading is elaborated. The effect of mean stress, stress gradients, loading history, and additional hardening behavior related to nonproportional loading paths is considered by averaging the superior limit of the intrinsic damage dissipation work in the critical domain. We developed a novel cumulative-damage model by introducing the equivalent completely reversed stress amplitude into the damage-evolution model. For better comparison, existing cumulative-damage models, including the Palmgren–Miner law, corrected Palmgren–Miner law, Morrow’s plastic work interaction rule, and Wang’s rule, were employed to predict the fatigue life under variable loading. The proposed model performed better, considering the error scatter band obtained by plotting the predicted and experimental fatigue life on the same coordinate system. The model precisely predicts fatigue life under variable loading and easily identifies its material constants.

The evolution of yield loci with finite plastic deformation along proportional and non-proportional loading paths in an annealed extruded AZ31 magnesium alloy

This investigation reports for the first time, the subsequent yield loci after finite plastic deformation based on 10με offset definition yield of an annealed extruded AZ31 magnesium alloy under proportional and non-proportional loading paths. The proportional loading paths comprise uniaxial tension, free end torsion, and combined tension-torsion radial paths. The tension loading followed by torsion, and torsion followed by tension are the non-proportional loading paths. Subsequent yield loci under proportional loading paths show a prominent nose in the loading direction and are almost flat in the unloading direction with evolution of significant isotropic hardening under tension and visible kinematic hardening under torsion loading path. Strong path-dependency is observed in subsequent yield loci under non-proportional loading directions along with hardening effects. All the yield loci are investigated in tension-torsion (σ₁₁-√3σ₁₂) stress space. The results are significantly different than observed for other metals, especially for aluminum alloys by Khan et al. (2009) and Iftikhar et al. (2021). Elastic moduli within each subsequent yield loci are also measured during these experiments with the increase in plastic strain. The observed small decreases in the elastic moduli with increasing plastic strains are very similar to the ones reported earlier (Iftikhar et al., 2021; Khan et al., 2009).

Discontinuous plastic flow in stainless steels subjected to combined loads at extremely low temperatures

In the present paper, the question of the mechanism of discontinuous plastic flow (DPF) occurring at extremely low temperatures (in the proximity of absolute zero), is for the first time raised in the context of kinematically controlled combined loads (independent control of displacement and rotation) and non-proportional loading paths. In order to identify the multiaxial stress state during DPF, a unique set-up for testing tubular samples under kinematically controlled traction and torsion in liquid helium (4.2 K) has been developed. The results of tests performed on grade 304 stainless steel thin-walled tubular samples subjected to combined loads (traction and torsion) in the proximity of absolute zero are for the first time reported. These novel results confirm the assumptions accepted when building the multiaxial constitutive model of discontinuous plastic flow, namely, the production of lattice barriers, the pile-ups of dislocations and the criterion of their collective failure, as well as the assumption that the serrations may be recorded by force and torque transducers independently. Thus, the numerically implemented model allows to reproduce the observed serrations, and to redistribute them between the loading directions.

Evolution of subsequent yield surfaces with plastic deformation along proportional and non-proportional loading paths on annealed AA6061 alloy: Experiments and crystal plasticity finite element modeling

This work presents an experimental and numerical investigation of annealed aluminum alloy AA6061 subjected to proportional and non-proportional loading paths at various finite levels of pre-strain. A careful mechanical characterization program is conducted for subsequent yield surfaces in the σ11−3σ12 stress space using an equivalent 10με offset linearity definition of yield. Electron backscatter diffraction (EBSD) is also used to characterize the microstructure of the material in the annealed condition. Each subsequent surface shows a very well-known shape of being nearly flat in unloading and having a prominent “nose” in the loading direction that deviates substantially from a generalized isotropic behavior. All proportional loading conditions show a rapid evolution of the mixed hardening effect that was characterized by the first load reversal in the first yield surface probe. Differences associated with the path-dependent effect in shifting the non-proportional yield surfaces are also observed in experiments. The experimental program is followed by Taylor-type crystal plasticity finite element (CPFE) simulations of the observed mechanical phenomenon. The Taylor-type CPFE model is calibrated using only the proportional uniaxial tension response and the EBSD imaging to predict all subsequent proportional and non-proportional loading paths. All CPFE show good predictive capabilities in capturing the experimental yield surface measurements for both proportional and non-proportional loading conditions. The CPFE simulations predict both the resulting shape and magnitude of the “nose” in the yield surfaces, as well as the mixed hardening, distortion, and yield surface translations. A systematic study is presented to investigate the effects of latent hardening on the yield surface predictions. This work highlights the promising results of advanced material model developments for larger-scale component simulations of lightweight aluminum alloys.

Modelling and experimental validation of material deformation at different zones of welded structural-steel under multiaxial loading

The fatigue life assessment of welds represents an engineering challenge regarding material inhomogeneity, welding residual stresses, notch stress effects and multiaxial loading conditions. In a weldment due to the non-uniform heating process the material in the weld metal and adjacent regions may exhibit heterogeneous mechanical properties. This makes the local stress-strain evaluations even more complicated under fatigue loading conditions. Therefore it is essential to characterize the variation of mechanical and particularly cyclic properties of the material in the vicinity of the weld notches. To improve the predictions of fatigue life estimation, numerical modelling techniques are complemented by experimental studies.The present work investigates the effects of weld induced microstructure variations in the multiaxial cyclic properties of the ferritic-pearlitic steels. A numerical model to describe the cyclic plastic deformation behavior of constructional steel S355J2H is established. Two material states present in the vicinity of a weld, the base material and the heat-affected zone (HAZ) are considered. Special attention was given to the non-proportional hardening of the material. Experimental results for both materials are used to calibrate the isotropic and kinematic hardening constants in the Chaboche plasticity model. The calibrated model is applied to multiaxial loading conditions. To account for non-proportional hardening effects, a constitutive motivated parameter by Tanaka is implemented. Therefore, first, the proportional results are reproduced and then the non-proportionality parameter is incorporated into the isotropic hardening law. The results of the calibration procedure for both proportional and non-proportional multiaxial load paths are discussed regarding its applicability in fatigue analysis and limitations of the model.

Evaluation of Localized Necking Models for Fracture Prediction in Punch-Loaded Steel Panels

This study compared the experimental test results on punch-loaded unstiffened and stiffened panels with numerical predictions using different localized necking modeling approaches with shell elements. The analytical models that were derived by Bressan–Williams–Hill (BWH) were used in their original form and extended version, which considers non-proportional loading paths while using the forming-severity concept and bending-induced suppression of through-thickness necking. The results suggest that the mesh size sensitivity depends on the punch geometry. Moreover, the inclusion of bending effects and the use of the forming-severity concept in the BWH criterion yielded improved estimations of fracture initiation with shell elements.

A novel energy-based model for predicting the fatigue life of the automobile sealant adhesively bonded butt joints

The evolution of dissipation energy of the uniaxial and multiaxial fatigue experiments on the automobile sealant adhesively bonded butt joints was investigated by the method of fatigue dissipation energy analysis and fatigue failure identification. The results demonstrated that the dissipation energy increased and the fatigue life decreased with the increase of stress amplitude, mean stress, and the decrease of cycle time. Meanwhile, the non-proportional loading path resulted in greater dissipation energy and shorter fatigue life than that of uniaxial and proportional loading path. Moreover, the existing energy-based model using the energy dissipated in different cyclic loading stages as the fatigue parameters were adopted to predict the fatigue life of the automobile sealant adhesively bonded butt joints, respectively. The practical limitations of the prediction model using the total dissipation energy as the fatigue parameter were evaluated. Therefore, a novel energy-based fatigue life prediction model was proposed which using the partial steady stage of dissipation energy curves as the parameters. The predicted results of the novel model and the comparison with that of the existing energy-based model indicated that the novel model had an excellent ability to predict the fatigue life, and was more practical in engineering application for the automobile sealant adhesively bonded butt joints.

Ductile fracture under proportional and non-proportional multiaxial loading

This paper revisits proportional and non-proportional experiments performed earlier by the authors, with the aim to establish the fracture envelope of the material. The experiments involved inflation of thin-walled AA6260-T4 tubes under axial force and internal pressure. By controlling the force/pressure ratio during the experiments, proportional (i.e., radial) and non-proportional (i.e., corner) loading paths were generated. These experiments are reviewed here. Since fracture is of interest, a computationally-efficient finite element model with solid element discretization around the anticipated fracture location and shell elements in the rest of the tube is described. The Yld2000-2D and Yld2004-3D yield criteria are calibrated and implemented in the simulations. In concert with earlier findings, these criteria are found to offer significantly better agreement with the experiments than von Mises. In particular, the local normals to the yield locus show discrepancies under 3°, while 8°–12° are typical for von Mises. This has significant impact in the prediction of the induced strain paths. The fidelity of the modeling framework is established by comparing the predictions of average stress-strain responses, as well as post-mortem local surface strains, to experiments. Then, the finite element models are used to establish the fracture locus of the material, under proportional loading. The fracture strains are found to increase monotonically as the triaxiality approaches zero from above, i.e., no spike is found at triaxiality of 1/3. The fracture strains also increase at higher triaxialities. The presence of the radial stress shifts that spike from triaxiality of 2/3 (equibiaxial tension) to lower values. By superposing the non-proportional results, it is found that the fracture locus is path-dependent.

Analysis of dissipation energy and ratchetting behaviors on multiaxial fatigue of adhesively bonded joints

The multiaxial fatigue experiments were conducted on the adhesive bonded hollow cylinder butt-joints under the stress-controlled mode. The effect of loading path, equivalent stress amplitude, equivalent mean stress and period on the multiaxial fatigue behaviors were discussed. The multiaxial fatigue damages of adhesively bonded joints were also analyzed from the evolution of ratchetting and dissipative energy behavior. The results showed that the loading paths had a great influence on dissipated energy and ratchetting strain, and the effects of non-proportional loading path on the ratchetting strain and dissipation energy were greater than that of the proportional loading path. Moreover, the equivalent stress amplitude, equivalent mean stress and period all had some influence on the dissipation energy and equivalent ratchetting strain. In addition, the dissipative energy and the ratchetting strain increased gradually with the number of cycle increase, which resulted in the fatigue life decreases with the increase of equivalent stress amplitude and mean stress, and the decreasing period. The effects of period on the multiaxial fatigue life under the elliptic loading path were insignificant for the adhesive bonding butt-joints.

Extended mean-field homogenization of viscoelastic-viscoplastic polymer composites undergoing hybrid progressive degradation induced by interface debonding and matrix ductile damage

In this contribution, a probabilistic micromechanics damage framework is presented to predict the macroscopic stress–strain response and progressive damage in unidirectional glass-reinforced thermoplastic polymer composites. Motivated by different failure modes observed experimentally, the damage mechanism in the vicinity of the fibers (namely, the interphase) is characterized by initiating and growing voids. The mechanisms can be formulated through a Weibull probabilistic density function. In contrast, the ductile progressive degradation of matrix initial stiffness is analyzed via the continuum damage theory. To accommodate different damage mechanisms in the matrix and the interphase, a three-phase Mori-Tanaka (MT) method and transformation field analysis approach (TFA) are established within a unified framework that allows simulation of both ductile and discrete damages in different phases. Moreover, the rate-dependent viscoelastic and viscoplastic response of the polymer matrix phase is modelled through a phenomenological model consisting of four Kelvin-Voigt branches and a viscoplastic branch under the thermodynamics framework. The reliability and efficiency of the modified mean-field damage model, based on TFA and Mori-Tanaka scheme, are assessed by comparing the simulated stress-strain response against full-field Abaqus simulations under both unidirectional and multiaxial nonproportional loading paths at different loading rates. The developed model provides an efficient alternative to the finite-element based full-field homogenization schemes or other mean-field micromechanics techniques that may be compared, as well as a framework for a potential extension of the theory for simulating damage evolution in composites with random reinforcement orientations.

Coupled Bilateral Hysteretic Behavior of High-damping Rubber Bearings under Non-proportional Plane Loading

ABSTRACT The hysteretic behavior of high-damping rubber bearings under different non-proportional plane loading paths is experimentally examined. Compared with unilateral reversal loading test results, non-proportional plane loading will make the rubber of high-damping rubber bearings have additional regional shear strain, thus leading to quite different mechanical properties and hysteretic behavior. A previously developed mathematical model is further modified for appropriately characterizing the unilateral and bilateral behavior of high-damping rubber bearings. The applicability and generalization of the further modified model are examined and demonstrated by comparing its predictions using a unique set and different sets of identified coefficients with the test results.

Prediction of limit strains during non-proportional load paths with a change in loading direction

Many different models have been published to predict failure after non-proportional load paths. Most of those models are phenomenological and heuristical models. They require a profound knowledge about the material. Examples are the enhanced Modified Maximum Force Criterion (eMMFC), the Polar Effective Plastic Strain-model (PEPS) or the Generalized Forming Limit Concept (GFLC). In addition to the load path, the loading direction has a significant influence on the formability of sheet metals. The mentioned models currently neglect this influence. By extending the GFLC-model by the parameter of loading direction, this influence is taken into account. By analyzing an acceptable number of bi-linear experiments, it is possible to calibrate the proposed model for a micro-alloyed steel HC340LA. Therewith an arbitrary load path with a change in loading direction can be evaluated. The results of this contribution show the effectiveness of this approach by different experiments.

The Effects of Nonproportional Biaxial Loading Paths on Ductile Fracture Initiation: A Void Growth Analysis

Abstract The effects of nonproportional biaxial loading paths on ductile fracture initiation toughness are studied in this article. To this end, the growth of a cylindrical void (hole) located in front of a mode I plane strain crack has been studied using large-deformation finite element analysis. A specific microstructural feature of a steel alloy was thoroughly studied by having a single void positioned at a fixed distance from the crack tip and void that was equal to 10 times the diameter of the void. In particular, the nonproportional biaxial loading path effects on the crack tip blunting, void growth, ligament reduction, and near-tip stress fields are investigated computationally. Under small-scale yielding conditions, one proportional loading and two nonproportional loading paths are applied to the modified boundary layer models, covering both low-constraint (negative T-stress) and high-constraint (positive T-stress) crack tip conditions. We have observed that the nonproportional load paths have a marked effect on the void growth, crack tip blunting, and the resulting ligament reduction. By applying the simple criteria for the coalescence of the crack tip and void, the ductile fracture initiation toughness is estimated. It is shown that the ductile fracture toughness is dependent on loading paths and constraint conditions (the T-stress ratios). Compared with the proportional biaxial loading case, different load sequences of the biaxial loads will result in either an increase or a decrease the fracture initiation toughness. This effect is particularly significant for specimen geometry with low-constraint conditions (i.e., negative T-stress ratios). Results from this study are of relevance to ductile fracture assessment of components or structures, such as pressure vessels that operate under nonproportional biaxial loading conditions.

Fully integrated multi-scale modelling of damage and time-dependency in thermoplastic-based woven composites

In this work, a multi-scale model established from the concept of periodic homogenization is utilized to predict the cyclic and time-dependent response of thermoplastic-based woven composites. The macroscopic behaviour of the composite is determined from finite element simulations of the representative unit cell of the periodic microstructure, where the local non-linear constitutive laws of the components are directly integrated, namely, the matrix and the yarns. The thermoplastic matrix is described by a phenomenological multi-mechanisms constitutive model accounting for viscoelasticity, viscoplasticity and ductile damage. For the yarns, a hybrid micromechanical–phenomenological constitutive model accounting for anisotropic damage and anelasticity induced by the presence of a diffuse micro-crack network is utilized. The capabilities of the overall multi-scale model are validated by comparing the numerical predictions with experimental data. Further illustrative examples are also provided, where the composite undergoes time-dependent deformations under uni-axial and non-proportional multi-axial loading paths. The multi-scale model is also employed to analyze the influence of the local deformation processes on the macroscopic response of the composite.

Experimental investigation on multiaxial ratchetting behaviour and fatigue life of silicone seal adhesive bonding butt-joints

The multiaxial fatigue experiments were conducted on the silicone seal adhesive bonding hollow cylinder butt-joints under the stress-controlled mode at laboratory ambient temperature. The effect of loading path, equivalent stress amplitude, equivalent mean stress and cycle time on the multiaxial ratchetting behaviors and fatigue life were discussed. The experimental results showed that the effects of non-proportional loading path on the ratchetting strain were greater than that of the proportional loading path. Moreover, the effects of the normal stress on the ratchetting strain were relatively larger than that of the equivalent shear stress under the multiaxial loading paths. In addition, the ratchetting strain increases and fatigue life decreases with the increase of equivalent stress amplitude and mean stress, and the decreasing cycle time. The effects of cycle time on the multiaxial ratchetting strain and fatigue life under the elliptic loading path were insignificant for the silicone seal adhesive bonding butt-joints.

A recurrent neural network-accelerated multi-scale model for elasto-plastic heterogeneous materials subjected to random cyclic and non-proportional loading paths

An artificial Neural Network (NNW) is designed to serve as a surrogate model of micro-scale simulations in the context of multi-scale analyses in solid mechanics.The design and training methodologies of the NNW are developed in order to allow accounting for history-dependent material behaviors. On the one hand, a Recurrent Neural Network (RNN) using a Gated Recurrent Unit (GRU) is constructed, which allows mimicking the internal variables required to account for history-dependent behaviors since the RNN is self-equipped with hidden variables that have the ability of tracking loading history. On the other hand, in order to achieve accuracy under multi-dimensional non-proportional loading conditions, training of the RNN is achieved using sequential data. In particular the sequential training data are collected from finite element simulations on an elasto-plastic composite RVE subjected to random loading paths. The random loading paths are generated in a way similar to a random walking in stochastic process and allow generating data for a wide range of strain-stress states and state evolution.The accuracy and efficiency of the RNN-based surrogate model is tested on the structural analysis of an open-hole sample subjected to several loading/unloading cycles. It is shown that a similar accuracy as with a FE2 multi-scale simulation can be reached with the RNN-based surrogate model as long as the local strain state remains in the training range, while the computational time is reduced by four orders of magnitude.