Nonlinear Material Parameters Research Articles

On dislocation-based artificial neural network modeling of flow stress

Computational design of materials processes has received great interests during the past few decades. Successful designs require accurate assessment of material properties, which can be influenced by the internal microstructure of materials. This work aim to develop a novel computational model based on dislocation structures to predict the flow stress properties of metallic materials. To create sufficient training data for the model, the flow stress of a precipitation–hardening aluminum alloy was measured by characterizing the dislocation structure of specimens from interrupted mechanical tests using a high resolution electron backscatter diffraction technique. The density of geometrically necessary dislocations was calculated based on analysis of the local lattice curvature evolution in the crystalline lattice. For three essential features of dislocation microstructures – substructure cell size, cell wall thickness, and density of geometrically necessary dislocations – statistical parameters of their distributions were used as the input variables of the predictive model. An artificial neural network (ANN) model was used to back-calculate the in situ non-linear material parameters for different dislocation microstructures. The model was able to accurately predict the flow stress of aluminum alloy 6022 as a function of its dislocation structure content. In addition, a sensitivity analysis was performed to establish the relative contribution of individual dislocation parameters in predicting the flow stress. The success of this approach motivates further use of ANNs and related methods to calibrate and predict inelastic material properties that are often too cumbersome to model with rigorous dislocation-based plasticity models.

Identification of elastic-plastic mechanical properties for bimetallic sheets by hybrid-inverse approach

Analysis, evaluation and interpretation of measured signals become important components in engineering research and practice, especially for material characteristic parameters which can not be obtained directly by experimental measurements. The present paper proposes a hybrid-inverse analysis method for the identification of the nonlinear material parameters of any individual component from the mechanical responses of a global composite. The method couples experimental approach, numerical simulation with inverse search method. The experimental approach is used to provide basic data. Then parameter identification and numerical simulation are utilized to identify elasto-plastic material properties by the experimental data obtained and inverse searching algorithm. A numerical example of a stainless steel clad copper sheet is considered to verify and show the applicability of the proposed hybrid-inverse method. In this example, a set of material parameters in an elasto-plastic constitutive model have been identified by using the obtained experimental data.

Rapid evaluation of alkali–silica reactivity of aggregates using a nonlinear resonance spectroscopy technique

A new nonlinear acoustic technique — Nonlinear Impact Resonance Acoustic Spectroscopy (NIRAS) — is developed and used to characterize the alkali-reactivity of different aggregates. Cementitious materials such as mortar and concrete exhibit a hysteretic and nonlinear elastic behavior in their constitutive relations. This hysteretic nonlinearity is associated with interfacial debonding between the different constituents, and it changes with the progress of damage such as that induced by the alkali–silica reaction (ASR). One of the consequences of the hysteretic nonlinear property of these materials is the decrease in resonance frequencies, with increased excitation amplitude. This shift in the resonance frequency as a function of the material nonlinearity parameter can be used to directly characterize the damage state of the material. This research tracks the variation of the nonlinearity parameter during a standard accelerated mortar bar test (AMBT) to assess the potential for alkali-reactivity of aggregates. The results show that the NIRAS technique is more sensitive than conventional linear acoustic methods and is capable of accurately characterizing the reactivity of the aggregates examined. Furthermore, the results show advantages over standard expansion measurements for differentiating various aggregates having similar levels of reactivity, particularly at early test ages. These changes in the nonlinearity parameter are benchmarked against results from a petrographic analysis. Thus, the proposed NIRAS is a promising technique for the rapid identification of alkali-reactive aggregates.

Ultrasonic harmonic generation from fatigue-induced dislocation substructures in planar slip metals and assessment of remaining fatigue life

An analytical model is presented of the microelastic-plastic nonlinearities resulting from the interactions of a stress perturbation with dislocation substructures and cracks that evolve during cyclic fatigue of planar slip metals. The interactions are quantified by a material nonlinearity parameter β extracted from acoustic (ultrasonic) harmonic generation measurements. The β parameter for a given fatigue state is highly sensitive to the volume fractions of active persistent Luders bands (PLBs) and PLB internal stresses, as well as to the densities, loop lengths, and dipole heights of the dislocation monopoles and dipoles that form the PLBs. The β parameter is predicted to increase monotonically with the increase in the hardness of the metal during cyclic loading, thus allowing an unambiguous assessment of the remaining life of the material. The model is applied to the calculation of β as a function of percent full fatigue life of IN100 nickel-base superalloy. The theoretical predictions are in good agreement with experimental measurements reported in the literature of IN100 samples fatigued in strain-controlled, low cycle, fully reversed loading.

Eigenvalue analysis for a crack in a power-law material

A nonlinear eigenvalue problem related to determining the stress and strain fields near the tip of a transverse crack in a power-law material is studied. The eigenvalues are found by a perturbation method based on representations of an eigenvalue, the corresponding eigenfunction, and the material nonlinearity parameter in the form of series expansions in powers of a small parameter equal to the difference between the eigenvalues in the linear and nonlinear problems. The resulting eigenvalues are compared with the accurate numerical solution of the nonlinear eigenvalue problem.

Transient Electrothermal Analysis of Multilevel Interconnects in the Presence of ESD Pulses Using the Nonlinear Time-Domain Finite-Element Method

Comprehensive electrothermal analysis of multilevel interconnects under electrostatic discharge (ESD) stress is carried out using the proposed nonlinear time-domain finite-element method (FEM). The technological, structural, and material parameters used in the analysis correspond to the advanced CMOS process of 90-, 65-, 45, and 32-nm nodes assessed by the International Technology Roadmap for Semiconductors. In order to enhance the computation efficiency and to reduce the memory cost, the preconditioned conjugated gradient technique combined with the element-by-element approximate factorization method is introduced to handle the sparse matrices formed by FEM. The nonlinear material parameters including the temperature-dependent electrical and thermal conductivities are treated rigorously. The transient temperature distributions, the maximum temperatures, and the temperature rise time of 3- and 4-level interconnect structures under the injection of ESD pulses with various waveforms are obtained and discussed.

Large Deflections of a Non-linear Cantilever Functionally Graded Beam

The analysis of the large deformation of a non-linear cantilever functionally graded material (FGM) beam is made. When subjected to an end moment, explicit expressions for deflection and rotation are derived for a functionally graded beam with work hardening of power law. The effects of the gradient distribution of Young’s modulus and the material non-linearity parameter on the deflections of the FGM beam are analyzed. Our results show that depth-dependent Young’s modulus and material non-linearity have a significant influence on the deflections of the beam, and a FGM beam can bear larger applied load than a homogeneous beam. Moreover, to determine an optimal gradient distribution, an optimum design of a beam of a lighter weight and larger stiffness is given. The influence of the geometric non-linearity of the beam is also studied. Large and small deformation theories predict nearly the same deflections with 5% error when rotation is less than 45°, and the predictions based on the small deformation theory are overestimated to exceed 10% when rotation is greater than 60°.

Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a 2, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter $f_{\rm max}^2 $ describing maximum stretch of the polymer chains.

Non-destructive evaluation of concrete in damaged and undamaged states

This paper presents findings on nonlinear ultrasonic testing of concrete. The study was focused on testing cubic concrete specimens. It consisted of the non-destructive evaluation of concrete cubes using nonlinear ultrasonic technique with different frequency transducers. Concrete cubes of 150 mm × 150 mm × 150 mm in size and three distinct w/c 40%, 50%, and 60% were examined in damaged and undamaged states. Prior to inducing damages, the concrete cubes were ultrasonically evaluated in undamaged conditions. The damages were induced into the concrete cubes under compression testing. The transducer used at the wave-transmitting end had a broadband frequency of 100 kHz. However, at the receiving end, transducers of 200 and 300 kHz broadband frequency were used. The test results obtained for each concrete cube, in both damaged and undamaged conditions, were plotted and compared. The ratios of harmonic generations, nonlinear material parameters, and attenuation or loss in fundamental amplitude were calculated. The performance obtained by using transducers with different frequency as receivers in measuring the higher harmonics was also examined. The nonlinear ultrasonic test method using low frequency transducer as receiver exhibited greater sensitivity over high frequency transducer. Similarly, the third harmonic ratio was found more sensitive to concrete cracking than the second harmonic ratio. As concrete is a high attenuation material, the change in fundamental amplitude or attenuation was also found sensitive to using transducers with different frequency. The test results were also analyzed to find correlation between w/c, wave attenuation and higher harmonic generation based nonlinear parameters. These findings are very helpful for understanding the behavior of damaged concrete sections under nonlinear ultrasound testing.

Nonlinear dislocation dynamics at ultrasonic frequencies

An analytical model of the effects of the Peierls–Nabarro barrier stress on the nonlinear dynamics of dislocation motion in crystalline solids resulting from a perturbative ultrasonic wave is derived. The nonlinearity is quantified by a material nonlinearity parameter β extracted from measurements of the amplitudes of the fundamental and harmonically generated ultrasonic waveforms. The β parameter is found to be functionally dependent on the magnitude of the Peierls–Nabarro barrier stress, the dislocation loop length, the shear modulus, and the Burgers vector of the crystal. The parameter is shown to exhibit a Bessel function oscillatory dependence on the stress amplitude of the fundamental ultrasonic wave resulting directly from the Peierls–Nabarro barrier stress. A sharp increase in the magnitude of β is shown to occur at low ultrasonic amplitudes where the dislocation motion is confined between adjacent lattice planes bounding the unperturbed dislocation. The generalization of the model to polycrystalline solids predicts a significant reduction in the β oscillations that results in a dramatic hooklike shape of the β versus stress amplitude curve at small values of the ultrasonic stress amplitude. Experimental observations of the hooklike shape (known as the Buck hook) have been reported in literature but the phenomenon has been previously unexplained. The present model shows that the Buck hook is a consequence of dislocation dynamics at low ultrasonic drive amplitudes.

Characterisation of nonlinear material parameters of foams based on indentation tests

Closed cell polymeric foams, for example EVA foams, are widely used in engineering, sport and biomedical fields. The mechanical behaviours of foam is highly nonlinear, and the foams are often under complex loading conditions in service over different spectrums of strain. Foam testing and determination of the nonlinear foam constitutive parameters are important to predict their performance in service and to aid materials development and product design. Current testing approaches normally involve the use of conventional compression or shear tests, which require well defined sample shapes and sizes. In this work, the use of continuous indentation test and inverse FE modelling method to determine the nonlinear material parameters was systematically investigated. This potentially could provide a much more simple testing approach with the capacity for in situ situation where standard testing specimens are not attainable. The results were directly compared to conventional compression test and combined compression–shear tests of EVA foams. The accuracy and validity of each method were comparatively studied and discussed. Results showed that the indentation approach produced comparable results to the combined compression–shear tests, while uniaxial compression test could not provide accurate material data to simulate complex loading conditions.

Eigenvalues of the antiplane-shear crack problem for a power-law material

This paper discusses the problem of finding the eigenvalue spectrum in determining the stress and strain fields at the tip of an antiplane-shear crack in a power-law material. It is shown that the perturbation method provides an analytical dependence of the eigenvalue on the material nonlinearity parameter and the eigenvalue of the linear problem. Thus, it is possible to find the entire spectrum of eigenvalues and not only the eigenvalue of the Hutchinson-Rice-Rosengren problem.

Nonlinear Lamb waves for the detection of material nonlinearity

This paper reports on a method that uses nonlinear Lamb waves to detect material nonlinearity. Lamb waves are well suited for the interrogation of thin metallic layers which act as waveguides, giving Lamb waves great potential in nondestructive evaluation applications. However, measuring nonlinear Lamb waves and extracting the information necessary for nondestructive evaluation is complicated by the inherent dispersive and multi-mode nature of Lamb waves. This paper presents a procedure that overcomes these difficulties and develops a reliable and robust measurement methodology. By using hybrid wedge generation and laser interferometric detection in combination with signal processing in the time–frequency domain, it is possible to make relative measurements of material nonlinearity parameters, which are an indicator of plasticity-driven damage.

Nonlinearly viscoelastic analysis of asphalt mixes subjected to shear loading

This study presents the characterization of the nonlinearly viscoelastic behavior of hot mix asphalt (HMA) at different temperatures and strain levels using Schapery’s model. A recursive-iterative numerical algorithm is generated for the nonlinearly viscoelastic response and implemented in a displacement-based finite element (FE) code. Then, this model is employed to describe experimental frequency sweep measurements of two asphalt mixes with fine and coarse gradations under several combined temperatures and shear strain levels. The frequency sweep measurements are converted to creep responses in the time domain using a phenomenological model (Prony series). The master curve is created for each strain level using the time temperature superposition principle (TTSP) with a reference temperature of 40°C. The linear time-dependent parameters of the Prony series are first determined by fitting a master curve created at the lowest strain level, which in this case is 0.01%. The measurements at strain levels higher than 0.01% are analyzed and used to determine the nonlinear parameters. These parameters are shown to increase with increasing strain levels, while the time–temperature shift function is found to be independent of strain levels. The FE model with the calibrated time-dependent and nonlinear material parameters is used to simulate the creep experimental tests, and reasonable predictions are shown.

Nonlinear constitutive models from nanoindentation tests using artificial neural networks

This paper presents a new approach using Artificial Neural Networks (ANNs) models to simulate the response during nanohardness tests of a variety of materials with nonlinear behavior. The ANNs continuous input and output variables usually include material parameters, indentation deflection, and resisting force. Different ANN models, including dimensionless input/output variables, are generated and trained with discrete finite-element (FE) simulations with different geometries and nonlinear material parameters. Only the monotonic loading part of the load–displacement indentation response is used to generate the trained ANN models. This is a departure from classical indentation simulations or tests where typically the unloading portion is used to determine the stiffness and hardness. The experimental part of this study includes nanoindentation tests performed on a silicon (Si) substrate with and without a nanocrystalline copper (Cu) film. The new ANN models are used to back-calculate (inverse problem) the in situ nonlinear material parameters for different copper material systems. The results are compared with available data in the literature. The proposed FE–ANN modeling approach is very effective and can be used in calibrating and predicting the in situ inelastic material properties using the monotonic part of the indentation response and for depths above 50 nm where the overall resisting force represents a continuum response.

Unified Description of Nonlinear Effects in High Temperature Superconductor Microwave Devices

We experimentally demonstrate the equivalence of different manifestations of nonlinear response in high temperature superconductor (HTS) microwave devices. Using a combination of analytical and numerical analysis, we show that the results of intermodulation distortion measurements, harmonic generation measurements, and power-dependent resonator measurements of different coplanar waveguide structures patterned onto the same HTS thin-film sample all yield approximately the same values for the nonlinear penetration depth. The extraction of an underlying nonlinear material parameter that is independent of the specific device geometry and experimental configuration will allow our results to be quantitatively compared with other nonlinear measurements, and will therefore help in determining the dominant source(s) of nonlinear response in HTS microwave devices.

Quantitative assessment of fatigue damage accumulation in wavy slip metals from acoustic harmonic generation

A comprehensive, analytical treatment is presented of the microelastic–plastic nonlinearities resulting from the interaction of a stress perturbation with dislocation substructures and cracks that evolve during cyclic fatigue of wavy slip metals. The interaction is quantified by a material nonlinearity parameter β extracted from acoustic harmonic generation measurements. The contribution to β from the substructures is obtained from the Cantrell model. The contribution to β from cracks is obtained by applying the Paris law to the Nazarov–Sutin crack nonlinearity equation. The nonlinearity parameter resulting from the two contributions is predicted to increase monotonically by hundreds of percent during fatigue from the virgin state to fracture. The increase in β during the first 80–90% fatigue life is dominated by the evolution of dislocation substructures, while the last 10–20% is dominated by crack growth. Application of the model to aluminium alloy 2024-T4 in stress-controlled loading at 276 MPa yields excellent agreement between theory and experiment.

A Constitutive Analysis of Extensional Flow of EVA Nanocomposites

Abstract Linear and nonlinear oscillatory and extensional flow was studied for polymer layered silicate nanocomposites of organically modified bentonite in ethylene-vinyl acetate copolymer with 18 wt.% of vinyl acetate (EVA18). It was found that the rheological properties of EVA18 nanocomposites were distinctly different from the pure copolymer. The elastic response at low frequencies was significantly enhanced in comparison to that of pure EVA18. The linear to nonlinear transition for stress relaxation measurements and the damping function were examined. The relaxation spectrum was shifted toward the longer relaxation time scales for EVA18 nanocomposites, while the dependence of the damping for EVA18 nanocomposites was much stronger than that of the EVA18. In contrast, the uniaxial extensional viscosity of EVA18 nanocomposites gave weaker strain-hardening properties. The prediction of strain-hardening behaviour for EVA18 and EVA18 nanocomposites through relaxation spectrum and the damping function based on Kaye-Bernstein-Kearsley-Zapas (K-BKZ) model is discussed. A simplified estimation of the nonlinear material parameter ß in the K-BKZ model is proposed to predict more accurately the extensional viscosity for EVA18 and EVA18 nanocomposites. Experimental data and detailed predictions are also presented in this paper.

A novel meso-mechanical model for concrete fracture

Concrete is a composite material and at meso-level, may be assumed to be composed of three phases: aggregate, mortar-matrix and aggregate-matrix interface. It is postulated herein that although non-linear material parameters are generally used to model this composite structure by finite element method, linear elastic fracture mechanics principles can be used for modelling at the meso level, if the properties of all three phases are known. For this reason, a novel meso-mechanical approach for concrete fracture which uses the composite material model with distributed-phase for elastic properties of phases and considers the size effect according to linear elastic fracture mechanics for strength properties of phases is presented in this paper. Consequently, the developed model needs two parameters such as compressive strength and maximum grain size of concrete. The model is applied to three most popular fracture mechanics approaches for concrete namely the two-parameter model, the effective crack model and the size effect model. It is concluded that the developed model well agrees with considered approaches.

Substructural organization, dislocation plasticity and harmonic generation in cyclically stressed wavy slip metals

Organized substructural arrangements of dislocations formed in wavy slip, face–centred–cubic metals during cyclic stress–induced fatigue are shown analytically to engender a substantial nonlinearity in the microelastic–plastic deformation resulting from an impressed stress perturbation. The non–Hookean stress–strain relationship is quantified by a material nonlinearity parameter βthat for a given fatigue state is highly sensitive to the volume fractions of veins and persistent slip bands (PSBs), PSB internal stresses, dislocation multipole configurations, dislocation loop lengths, dipole heights and the densities of secondary dislocations in the substructures. The effects on β of vacancy, microcrack and macrocrack formation are also addressed. The connection between β and acoustic harmonic generation is obtained. The model is applied to calculations of β for fatigued polycrystalline nickel as a function of per cent life to fracture. For cyclic stress–controlled loading at 241 MPa, the model predicts a monotonic increase in β of ca. 360% over the fatigue life. For strain–controlled loading at a total strain of 1.75 × 10 −3 , a monotonic increase in β of ca. 375% over the fatigue life is predicted.