A neural-network approach to describe the scatter of cyclic stress–strain curves

In order to predict a product’s durability in the early phases of development it is necessary to know the stress–strain behaviour of the material, its resistance to fatigue and the loading states in the material. These parameters, however, tend to exhibit a considerable degree of uncertainty. Due to a lack of knowledge of the actual circumstances in which the product is used, during the early development phase, simulations based on statistical methods are used. The results of the experiments show that the cyclic stress–strain curves demonstrate not only a large amount of scatter, but also a dependence on the temperature, the size of the cross-section, the content of alloying elements, the loading rate, etc.This article presents a method for modelling cyclic stress–strain curve scatter using a hybrid neural network for an arbitrary selection of the influencing factors. In an example of the measured data for a high pressure die-cast aluminium alloy it is clear that the suggested method is suitable for describing cyclic stress–strain curves. The main advantage of a hybrid neural network in comparison with a conventional method is the neural network’s ability to precisely describe the influence of various factors, and their combinations, based on the form and scatter of the cyclic stress–strain curve families. Defining the model parameters, i.e., training the neural network, is a procedure that does not require any additional user interventions; however, it enables us to gather knowledge that would otherwise require a lot of research. Thus, the trained neural network is a robust tool that can be used to predict cyclic stress–strain curves for random values of influencing factors. The capabilities of the presented method are only limited by the quantity of the measured data used for the neural-network training.     

通过硬度确定板材大变形硬化曲线的方法研究

为了获得板材在大变形范围内的硬化性能,采用硬度试验与单向拉伸试验相结合的方法对材料的硬化行为进行了研究.以碳钢板为研究对象,通过测量断裂后试件不同部位的微氏硬度获得变形量与硬度之间的关系,并结合硬度与屈服应力的关系,获得了常规拉伸试验方法无法获得的大变形阶段的硬化行为.结果表明,通过该方法获得的硬化曲线变形范围广,结果可靠,能够满足塑性大变形的需要.     

Effective Parameters on the Stress–Strain Curve of Nylon 66/Clay Nanocomposite Using FEM

Abstract: In recent years, the study of polymer/clay nanocomposites has attracted major research and commercial interests due to their superior mechanical and thermal properties to those of the neat polymers. Numerical modelling is an advantageous approach to understand the material behaviour. In this work a symmetric two dimensional finite element model is developed to simulate the fully exfoliated Polymer/Clay nanocomposite behaviour and evaluate its’ before stress stress–strain curve. The Nylon 66/Clay in this study has a nonlinear elastic behaviour. The influence of volume fraction and aspect ratio of clay platelet (defined as the ratio of the particle length to the particle thickness) on the tensile behaviour of exfoliated Nylon 66/clay nanocomposite is investigated with the aid of numerical simulations. Results show good agreement with experimental data from literature.     

An estimation of true Ramberg‐Osgood curve parameters for materials with and without Luder's strain using yield and ultimate strengths

Engineering tensile stress–strain curves for metallic materials typically show two different behaviours, namely, with Luder's strain and without Luder's strain. Luder's strain is more common for ductile materials, whereas high‐strength steels deform without Luder's strain. Usually, the stress–strain curves of ductile steels exhibit ultimate load where necking starts to develop. On the other hand, steels with low ductility exhibit monotonic increase of the applied load till failure without necking. Recently, Kamaya proposed a method to estimate the Ramberg‐Osgood relationship parameters for true stress–strain curves on the basis of conventional yield and ultimate strengths. This method can be not accurate enough for ductile materials exhibiting Luder's strain. Hence, a more general procedure for the materials exhibiting Luder's strain is proposed. In addition, an inverse method for assessing an ‘apparent ultimate tensile stress’ (akin to the ultimate stress of ductile materials at point of zero slope) for materials with low ductility (due to quenching or carburizing) is suggested.     

Prediction of stress–strain relation using cone indentation: effect of friction

Atkins and Tabor's approach (J. Mech. Phys. Solids 1965; 13 : 149) for predicting uniaxial stress–strain relation of metals from cone indentation tests has been studied using numerical (finite element) simulation of cone indentation. Two indentation parameters, namely representative strain and constraint factor, which are central to the prediction approach, have been estimated using the simulation for cone indenters of different apical angles. The effect of specimen–indenter interface friction on these parameters has been characterized. It is shown that uncertainty in our knowledge of this friction condition has an important bearing on the prediction of the stress–strain curve. However, a good estimate of the stress–strain curve can be obtained by making reasonable assumptions about the nature of the friction at the specimen–indenter interface. The simulation results are found to agree well with those reported in the experimental study of Atkins and Tabor, when a coefficient of friction value typical for the specimen–indenter interface is used. Copyright © 2004 John Wiley & Sons, Ltd.     

Ratcheting of 304 Stainless Steel Alloys subjected to Stress‐Controlled and mixed Stress‐ and Strain‐Controlled Conditions evaluated by Kinematic Hardening Rules

The present study predicts ratcheting response of SS304 tubular stainless steel samples using kinematic hardening rules of Ohno–Wang (O–W), Chen‐Jiao‐Kim (C–J–K) and a newly modified hardening rule under various stress‐controlled, and combined stress‐ and strain‐controlled histories. The O–W hardening rule was developed based on the critical state of dynamic recovery of backstress. The C–J–K hardening rule further developed the O–W rule to include the effect of non‐proportionality in ratcheting assessment of materials. The modified rule involved terms , and in the dynamic recovery of the Ahmadzadeh–Varvani (A–V) model to respectively track different directions under multiaxial loading, account for non‐proportionality and prevent plastic shakedown of ratcheting data over multiaxial stress cycles. The O–W model persistently overestimated ratcheting strain over the multiaxial loading paths. The C–J–K model further lowered this overprediction and improved the predicted ratcheting curves. The predicted ratcheting curves based on the modified model closely agreed with experimental data under various loading paths.     

Notch ductile failure with significant strain‐hardening: The modified equivalent material concept

The equivalent material concept (EMC) assumes that the ductile material has a valid K‐based fracture toughness (K_Ic or K_c). For ductile materials with significant strain‐hardening, no valid K_Ic or K_c is determined by the standard experiments and, hence, EMC seems null. The modified EMC (MEMC) is proposed in this study by which a virtual K_c value is defined and computed for the ductile material with significant strain‐hardening. In this way, Mode I and mixed Mode I/II fracture behaviors of U‐notched aluminum alloy 5083 are assessed in the view points of experiments and theories. Several U‐notched rectangular samples are used for performing the experiments and obtaining the failure loads. Then, the MEMC is coupled with the maximum tangential stress and mean stress criteria and utilized to predict the failure loads theoretically. Finally, it is shown that both the MEMC‐stress‐based criteria can provide very good predictions of the test data.     

Influences of Friction Stir Welding and Post‐Weld Heat Treatment on Al–Cu–Li Alloy

In this paper, the influences of friction stir welding (FSW) and post‐weld heat treatment (PWHT) on the microstructures and tensile properties of Al–Cu–Li alloy are investigated. After FSW, strengthen loss occurred in the welding area. Remarkable softening occurs in the thermo‐mechanically affected zone (TMAZ) resulting from dissolution of Al₃Li (δ′) phases. Recrystallization and precipitation of ultra‐fine δ′ phases take place in the nugget zone (NZ) that lightens the softening degree of this zone. A noteworthy enhancement in the hardness and tensile strength of the joint is achieved after T8 re­aging treatment (3% ? pre‐deformation, 30 h at 152 °C). However, re‐solution treatment coupled with re‐aging treatment leads to ductility deterioration in the joint because coplanar slip of coarse Al₃Li phases induces severe stress concentration during plastic deformation.

     

Comparative analysis of fatigue life predictions in central notched specimens of Al–Mg–Si alloys

The fatigue behaviour of an Al–Mg–Si alloy was studied using notched specimens. Fatigue tests were conducted at two stress ratios R= 0 and R= 0.4 on thin plates with a central hole. Constant and block variable loading amplitudes were applied to the specimens using a servo‐hydraulic machine. The applicability of the local strain approach method to the prediction of the fatigue life was investigated for this type of discontinuity. Two methods, the equivalent strain energy density approach and a modified stress–strain intensity field approach, were used to predict the fatigue strength. For the second one an elastic–plastic finite element analysis was carried out in order to obtain the local strain and stress distributions near the notch root. Based on Miner's rule an equivalent stress was used to correlate the fatigue lives for the variable amplitude histories. The experimental results were compared with the predicted results obtained by the two methods investigated and better agreement was found with the stress–strain field intensity approach, while the strain energy approach gave more conservative results. Miner's rule gives a good correlation between the variable amplitude and constant amplitude results.     

Loading Rate Effects on Mechanical Response of Ag,Ni, Al and Cu at Nano‐Scale

Abstract: Molecular dynamics (MD) is now playing a unique as well as conspicuous role in probing in nano‐science and technology. In this paper, MD simulation of uniaxial tension of some face‐centred cubic (FCC) metals (namely, Ag, Ni, Al and Cu) at nano‐scale was carried out. The effects of different loading rates on the stress–strain curves at a constant temperature of 300 K were studied. The potential used in these simulations was Sutton–Chen, and Velocity Verlet formulation of Noise‐Hoover dynamic was also applied to have a constant temperature ensemble. The boundary condition was periodic. Eventually, MD simulations were carried out and it was concluded that by increasing the loading rate both maximum stress and strain at failure increase. The maximum engineering stress for Al, Ag, Cu and Ni can be rounded as 4, 9, 12 and 18 GPa, respectively, for all loading rates.     

On the computation of design derivatives for Huber–Mises plasticity with non‐linear hardening

This paper concerns design sensitivity analysis (DSA) for an elasto–plastic material, with material parameters depending on, or serving as, design variables. The considered constitutive model is Huber–Mises deviatoric plasticity with non‐linear isotropic/kinematic hardening, one which is applicable to metals. The standard radial return algorithm for linear hardening is generalized to account for non‐linear hardening functions. Two generalizations are presented; in both the non‐linearity is treated iteratively, but the iteration loop contains either a scalar equation or a group of tensorial equations. It is proven that the second formulation, which is the one used in some parallel codes, can be equivalently brought to a scalar form, more suitable for design differentiation. The design derivatives of both the algorithms are given explicitly, enabling thus calculation of the ‘explicit’ design derivative of stresses entering the global sensitivity equation. The paper addresses several issues related to the implementation and testing of the DSA module; among them the concept of verification tests, both outside and inside a FE code, as well as the data handling implied by the algorithm. The numerical tests, which are used for verification of the DSA module, are described. They shed light on (a) the accuracy of the design derivatives, by comparison with finite difference computations and (b) the effect of the finite element formulation on the design derivatives for an isochoric plastic flow. Copyright © 2003 John Wiley & Sons, Ltd.     

Two‐point constitutive equations and integration algorithms for isotropic‐hardening rate‐independent elastoplastic materials in large deformation

This paper presents alternative forms of hyperelastic–plastic constitutive equations and their integration algorithms for isotropic‐hardening materials at large strain, which are established in two‐point tensor field, namely between the first Piola–Kirchhoff stress tensor and deformation gradient. The eigenvalue problems for symmetric and non‐symmetric tensors are applied to kinematics of multiplicative plasticity, which imply the transformation relationships of eigenvectors in current, intermediate and initial configurations. Based on the principle of plastic maximum dissipation, the two‐point hyperelastic stress–strain relationships and the evolution equations are achieved, in which it is considered that the plastic spin vanishes for isotropic plasticity. On the computational side, the exponential algorithm is used to integrate the plastic evolution equation. The return‐mapping procedure in principal axes, with respect to logarithmic elastic strain, possesses the same structure as infinitesimal deformation theory. Then, the theory of derivatives of non‐symmetric tensor functions is applied to derive the two‐point closed‐form consistent tangent modulus, which is useful for Newton's iterative solution of boundary value problem. Finally, the numerical simulation illustrates the application of the proposed formulations. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical integration of a pressure‐dependent,non‐linear kinematic hardening constitutive model for large strain cyclic plasticity of metals

A Gurson‐based constitutive model is presented, which includes non‐linear mixed isotropic–kinematic hardening and creep, and allows the analysis of problems involving arbitrarily large plastic strains. This model was developed with the main objective of allowing, on the basis of a single set of material parameters, the numerical simulation of all the main features of cold metal forming processes, which usually imply severe loading–unloading cycles with very large plastic strains, difficult to be correctly reproduced numerically. A suitable integration scheme of the rate equations is described and implemented into a finite element code. The results obtained are compared with some reference experimental ones; an application of the model for the simulation of wire drawing processes is also presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Extension of strain–life equation for low‐cycle fatigue of sheet metals using anisotropic yield criteria and distortional hardening model

The fatigue strain–life equation is in general applicable to isotropic materials. It was recently attempted to account for material anisotropy because of crystallographic texture in fatigue modelling. The proposed modification was limited to isotropic hardening. The present work is an extension of the previous work, wherein a general framework to model anisotropy using phenomenological yield criterion and anisotropic hardening is provided. Yld2004‐18p yield criterion and the so‐called homogenous anisotropic hardening model are used to demonstrate the anisotropic cyclic behaviour of low carbon steel. The proposed methodology can be utilized in applications including multiaxial fatigue modelling.     

A scaled boundary radial point interpolation method for 2‐D elasticity problems

The scaled boundary radial point interpolation method (SBRPIM), a new semi‐analytical technique, is introduced and applied to the analysis of the stress–strain problems. The proposed method combines the advantages of the scaled boundary finite element method and the boundary radial point interpolation method. In this method, no mesh is required, nodes are scattered only on the boundary of the domain, no fundamental solution is required, and as the shape functions constructed based on the radial point interpolation method possess the Kronecker delta function property, the boundary conditions of problems can be imposed accurately without additional efforts. The main ideas of the SBRPIM are introducing a new method based on boundary scattered nodes without the need to element connectivity information, satisfying Kronecker delta function property, and being capable to handle singular problems. The equations of the SBRPIM in stress–strain fields are outlined in this paper. Several benchmark examples of 2‐D elastostatic are analyzed to validate the accuracy and efficiency of the proposed method. It is found that the SBRPIM is very easy to implement and the obtained results of the proposed method show a very good agreement with the analytical solution. Copyright © 2017 John Wiley & Sons, Ltd.     

Modeling of cyclic stress–strain behavior and damage mechanisms under thermomechanical fatigue conditions

A multi-component model was applied to predict the cyclic stress–strain response of different alloys under thermomechanical fatigue conditions based upon isothermal hysteresis loops. A ductile AISI 304 L-type stainless steel and two high strength alloys, the near-α titanium alloy IMI 834 and the nickel-base superalloy IN 100, were chosen as test materials. These represent alloys with rather different dislocation slip modes, stress–strain characteristics and damage mechanisms. Model predictions are compared with experiments and the differences in cyclic stress–strain response and damage mechanisms under isothermal and thermomechanical fatigue conditions, respectively, are discussed based upon microstructural observations.     

An improved method for estimation of Ramberg–Osgood curves of steels from monotonic tensile properties

In the present study, first a method for estimating cyclic yield strength is improved and compared using the experimental data of 121 steels from literature. Correlations between cyclic deformation properties (K′ and n′) and monotonic tensile data are then investigated, and a simple method requiring only the monotonic tensile properties is developed for estimation of the Ramberg–Osgood curve. Prediction capability of the proposed method is not only evaluated using the aforementioned 121 steels, but also compared with several commonly used methods that are available in the literature. The proposed correlations are shown to predict the cyclic deformation properties of most of considered steels reasonable well.