Non-homogeneous Strain Fields Research Articles

A two-scale approach for assessing the role of defects in fatigue crack nucleation in metallic structures

Metal structures often exhibit macroscopic defects from which cracks can nucleate during cyclic loading. The current work presents a two-scale approach to enable the prediction of crack nucleation from such defects by taking into account local microstructure features. The geometrical description of the defect and associated non-homogeneous strain fields are modeled using a macroscale model which employs a continuum elastoplastic material model for cyclic deformation. The cyclic deformation of the microstructure near the defect is modeled using a mesoscale model which employs a crystal plasticity material model and uses multiple realizations to address the statistical microstructure variability. The boundary conditions of the mesoscale model are extracted from the macroscale model. By simulating the deformation of the microstructure using the strain fields near the defect and by introducing a fatigue indicator parameter for crack nucleation, along with the weakest-link based upscaling methodology, the developed approach enables the prediction of the distribution of crack nucleation life. The approach is used for analyzing different defects for crack nucleation by considering local grain orientations. The predictions are shown to not only capture phenomena such as scatter, size effects, etc. qualitatively, but also agree with a classical engineering approach and experimentally reported data sets quantitatively.

On the inverse identification methods for forming plasticity models using full-field measurements

The simulation of deep drawing processes and its quality is intrinsically dependent on the accuracy of the constitutive model in reproducing the mechanical behaviour of the sheet metal material. Today, the calibration of elastoplastic models – correspondent to the inverse identification of the material parameters – often uses full-field measurements, through Digital Image Correlation (DIC) techniques, to capture non-homogeneous strain fields and states, coupled with non-straightforward numerical inverse methodologies. In the last decade, new parameter identification methodologies, such as the Finite Element Model Updating (FEMU), the Constitutive Equation Gap (CEG) method, the Equilibrium Gap Method (EGM) and the Virtual Fields Method (VFM) have been developed and have proven to be effective for non-linear plasticity models. Nonetheless, the FEMU and the VFM have distinguished themselves from the others. More recently, supervised Machine Learning (ML) techniques have been also used as an inverse identification method. These artificial intelligence-based methods use large datasets of numerical tests to train an inverse model in which the input is the history of the strain field and loads during the test, and the output are directly the material parameters.The goal of this paper is to analyse, compare and discuss these inverse identification methods, with particular focus on the FEMU, VFM, and ML methodologies. A heterogeneous tensile-load test is considered to compare in detail the FEMU, VFM, and ML strategies.

The Use of Machine-Learning Techniques in Material Constitutive Modelling for Metal Forming Processes

Accurate numerical simulations require constitutive models capable of providing precise material data. Several calibration methodologies have been developed to improve the accuracy of constitutive models. Nevertheless, a model’s performance is always constrained by its mathematical formulation. Machine learning (ML) techniques, such as artificial neural networks (ANNs), have the potential to overcome these limitations. Nevertheless, the use of ML for material constitutive modelling is very recent and not fully explored. Difficulties related to data requirements and training are still open problems. This work explores and discusses the use of ML techniques regarding the accuracy of material constitutive models in metal plasticity, particularly contributing (i) a parameter identification inverse methodology, (ii) a constitutive model corrector, (iii) a data-driven constitutive model using empirical known concepts and (iv) a general implicit constitutive model using a data-driven learning approach. These approaches are discussed, and examples are given in the framework of non-linear elastoplasticity. To conveniently train these ML approaches, a large amount of data concerning material behaviour must be used. Therefore, non-homogeneous strain field and complex strain path tests measured with digital image correlation (DIC) techniques must be used for that purpose.

A novel specimen shape for measurement of linear strain fields by means of digital image correlation

Strains on the surface of engineering structures or biological tissues are non-homogeneous. These strain fields can be captured by means of Digital Image Correlation (DIC). However, DIC strain field measurements are prone to noise and filtering of these fields influences measured strain gradients. This study aims to design a novel tensile test specimen showing two linear gradients, to measure full-field linear strain measurements on the surface of test specimens, and to investigate the accuracy of DIC strain measurements globally (full-field) and locally (strain gauges’ positions), with and without filtering of the DIC strain fields. Three materials were employed for this study: aluminium, polymer, and bovine bone. Normalized strain gradients were introduced that are load independent and evaluated at two local positions showing 3.6 and 6.9% strain change per mm. Such levels are typically found in human bones. At these two positions, two strain gauges were applied to check the experimental strain magnitudes. A third strain gauge was applied to measure the strain in a neutral position showing no gradient. The accuracy of the DIC field measurement was evaluated at two deformation stages (at approx 500 and 1750 μstrain) using the root mean square error (RMSE). The RMSE over the two linear strain fields was less than 500 μstrain for both deformation stages and all materials. Gaussian low-pass filter (LPF) reduced the DIC noise between 25% and 64% on average. As well, filtering improved the accuracy of the local normalized strain gradients measurements with relative difference less than 20% and 12% for the high- and low-gradient, respectively. In summary, a novel specimen shape and methodological approach are presented which are useful for evaluating and improving the accuracy of the DIC measurement where non-homogeneous strain fields are expected such as on bone tissue due to their hierarchical structure.

Analysis of the mechanical behavior of spatially reinforced composites with open holes

The paper deals with the experimental study of mechanical behavior and the common factors of deformation of structural composites in combined use of contactless optical video system and acoustic emission analysis and registration system. The experimental studies were conducted at the electromechanical testing system Instron 5989 in compliance with ASTM 5766 recommendations for the samples of spatially reinforced and fiber-laminated carbon fiber composites with open holes. The preform-based carbon plastic samples with spatially reinforced structure are obtained by 3D-weaving methods. The paper presents the results of mechanical strain tests of open-hole samples. The authors obtained stress and strain diagrams for carbon plastic of various structures and analyzed the evolution of non-homogeneous strain fields on the surface of 3D-reinforced composite samples of the studied structure. They observed the difference in mechanical behavior of spatially reinforced and fiber-laminated carbon fiber composites. The paper presents the images of the samples with the characteristic damages in the area of the opening. Using the acoustic emission signals recording data, the authors obtained the diagrams of dependence of cumulative power, which allowed to observe the diverse nature of damage accumulation for the studied groups of samples.

Mathematical data processing according to digital image correlation method for polymer composites

The paper analyses the numerical algorithms for experimental data processing using a contactless video system Vic-3D, designed for three-dimension analysis of displacement and strain fields, and digital image correlation method. The authors considered methodological issues of conducting an experiment using a video system. They suggested recommendations on the choice of parameters of calculation of correlation, the size of subset and step during the analysis of non-homogeneous displacement and strain fields in polymer composite materials through laminated fiberglass composite. The efficient parameters of mathematical data processing are identified according to digital image correlation method on the basis of building fields for one frame on the surface of laminated fiberglass reinforced plastic at various subset values and at fixed step value. The paper shows the impact of step value on the strain fields detail degree. The authors have identified the relation of the chosen parameters of experimental data processing using digital image correlation method with the scaled levels of consideration of composite materials strain processes. To evaluate the strains at various scale levels, the paper uses supplementary video system instruments: “virtual extensometer”, “rectangular area” and “line”. The authors obtained a longitudinal strain profile that allows evaluating the location of strain peak areas on the composite object surface.

Experimental characterization of the compressive mechanical behaviour of Ti6Al4V alloy at constant strain rates over the full elastoplastic range

Full range constant strain rate tests are required for accurately characterizing initial yield point, strength differential effect and direct identification of constitutive laws describing the plastic behavior of materials. These tests require the use of a closed-loop control in order to achieve the constant strain rate, however this feature is not available in many laboratories. A method for full range constant strain rate with testing machines that can be configured for user-defined displacements of the cross head prior to testing is presented here. Tests performed at a constant die speed include a variable strain rate response for the specimen involved. Significant deformation rate variation occurs between the elastic and plastic range with consequences for initial yield point identification. To overcome this drawback, appropriate user-defined displacements can be computed and applied, allowing for both tensile and compression tests to be performed at a constant strain rate. The method is validated using a compression test of Ti6Al4V alloy at room temperature, as well as a 3D digital image correlation (DIC) system exhibiting a constant strain rate value equal to 10−3 s−1, for both elastic and plastic ranges. A non-negligible inhomogeneous strain field was measured on the surface of the compression specimen using DIC and was corroborated by numerical modeling. Results identified the source of the non-homogeneous strain field, thereby proposing a quantitative indicator of plastic anisotropy. The initial yield stress and strain hardening rates of the alloy at several temperatures were obtained with both testing method, conventional constant cross-head speed, and the constant strain rate; these were then used to determine the influence of the small strain rate variations on the mechanical response of Ti6Al4V alloy.

Data-based derivation of material response

This paper proposes a method to identify the strain–stress relation of non-linear elastic materials, without any underlying constitutive equation. The approach is based on the concept of Data Driven Computational Mechanics recently introduced by Kirchdoerfer and Ortiz (2016). From a collection of non-homogeneous strain fields, for example measured through Digital Image Correlation, the algorithm builds a database of strain–stress couples that sample the mechanical response of the material for the range of measured strains. The method is first derived for truss structures and then extended to the case of small-strain elasticity. The method accuracy, sensitivity to measurement noise and parameters are discussed using manufactured data.

High-resolution spatiotemporal strain mapping reveals non-uniform deformation in micropatterned elastomers

Micropatterns are generated on a vast selection of polymeric substrates for various applications ranging from stretchable electronics to cellular mechanobiological systems. When these patterned substrates are exposed to external loading, strain field is primarily affected by the presence of microfabricated structures and similarly by fabrication-related defects. The capturing of such nonhomogeneous strain fields is of utmost importance in cases where study of the mechanical behavior with a high spatial resolution is necessary. Image-based non-contact strain measurement techniques are favorable and have recently been extended to scanning tunneling microscope and scanning electron microscope images for the characterization of mechanical properties of metallic materials, e.g. steel and aluminum, at the microscale. A similar real-time analysis of strain heterogeneity in elastomers is yet to be achieved during the entire loading sequence. The available measurement methods for polymeric materials mostly depend on cross-head displacement or precalibrated strain values. Thus, they suffer either from the lack of any real-time analysis, spatiotemporal distribution or high resolution in addition to a combination of these factors. In this work, these challenges are addressed by integrating a tensile stretcher with an inverted optical microscope and developing a subpixel particle tracking algorithm. As a proof of concept, the patterns with a critical dimension of 200 µm are generated on polydimethylsiloxane substrates and strain distribution in the vicinity of the patterns is captured with a high spatiotemporal resolution. In the field of strain measurement, there is always a tradeoff between minimum measurable strain value and spatial resolution. Current noncontact techniques on elastomers can deliver a strain resolution of 0.001% over a minimum length of 5 cm. More importantly, inhomogeneities within this quite large region cannot be captured. The proposed technique can overcome this challenge and provides a displacement measurement resolution of 116 nm and a strain resolution of 0.04% over a gage length of 300 µm. Similarly, the ability to capture inhomogeneities is demonstrated by mapping strain around a thru-hole. The robustness of the technique is also evaluated, where no appreciable change in strain measurement is observed despite the significant variations imposed on the measurement mesh. The proposed approach introduces critical improvements for the determination of displacement and strain gradients in elastomers regarding the real-time nature of strain mapping with a microscale spatial resolution.

Research on non-uniform strain profile reconstruction along fiber Bragg grating via genetic programming algorithm and interrelated experimental verification

A new heuristic strategy for the non-uniform strain profile reconstruction along Fiber Bragg Gratings is proposed in this paper, which is based on the modified transfer matrix and Genetic Programming(GP) algorithm. The present method uses Genetic Programming to determine the applied strain field as a function of position along the fiber length. The structures that undergo adaptation in genetic programming are hierarchical structures which are different from that of conventional genetic algorithm operating on strings. GP regress the strain profile function which matches the ‘measured’ spectrum best and makes space resolution of strain reconstruction arbitrarily high, or even infinite. This paper also presents an experimental verification of the reconstruction of non-homogeneous strain fields using GP. The results are compared with numerical calculations of finite element method. Both the simulation examples and experimental results demonstrate that Genetic Programming can effectively reconstruct continuous profile expression along the whole FBG, and greatly improves its computational efficiency and accuracy.

Imaging of highly inhomogeneous strain field in nanocrystals using x-ray Bragg ptychography: A numerical study

X-ray ptychography is a lensless microscopy method able to provide extended field of view with spatial resolution above the diffraction limit. A series of intensity coherent diffraction patterns measured in the far field is used to obtain the numerical deconvolution between the sample scattering contrast and the illumination function. The measurements are performed with a finite-size beam spot scanned across the sample. The scan step, smaller than the beam size, ensures a high redundancy in the data set, which allows for the convergence of the iterative inversion algorithm. This work explores the possibility to use ptychography for the investigation of strained crystals by means of coherent x-ray Bragg diffraction, taking advantage of the high sensitivity to the atomic displacement fields. The Bragg diffraction scattering contrast is described by an effective complex-valued electron density, where the phase holds the information on the displacement field. The detailed two-dimensional numerical study of Bragg ptychography is presented, both for the known and unknown illumination cases. It demonstrates the high robustness of the ptychographical iterative engine for highly nonhomogeneous strain fields. In particular, the local information is extracted from the individual diffraction patterns to calculate the modulus and phase estimates of the electron density, which are further used to constrain the newly derived algorithm. From this work, it is foreseen that Bragg ptychography when experimentally feasible, will open the way to the nondestructive imaging of strain fields at the nanoscale.

Local strains and mesoscopic mechanisms of crack growth in 25Kh1M1F steel

Fatigue crack growth characterization requires non-homogeneous strain field measurements, the use of specific tech- niques is necessary. In this paper, the efficiency of the automated grid method for measuring in-plane displacements in the presence of high strain gradient is discussed. Vectors fields of displacement and strain gradient near crack tip have been stud- ied: macrostrain localization, mesostrain concentration at a crack tip in a 25Kh1M1F steel, and micromechanisms of fracture on a crack front.

An experimental numerical study of the response of a long fibre Bragg grating sensor neara crack tip

The use of fibre Bragg grating (FBG) sensors and the associated signal interrogation andanalysis methods for structural monitoring and micro-mechanics is an important area ofresearch. Long FBG sensors are very well suited for relatively non-invasive detection andmeasurements of complex strain distributions typically arising when cracks ordelaminations are present. However, when the fibre is embedded in a non-homogeneousstrain field, the interpretation of its response as a function of the position along the sensoris complicated. Thus, accurate strain data rely on the methods used to decode thewavelength changes and deduce the associated strain distribution. In this paper, theresponse of a 13 mm long FBG sensor embedded normal to the crack plane in a compacttension specimen of epoxy material is analysed. To introduce highly non-uniform strainsalong the fibre, a natural crack is grown by cyclic loads and two configurations areinvestigated: (a) when the fibre is ahead of the crack front, (b) when the fibre is behind thecrack front. In both cases, analysis of the signals is carried out using the conventionalT-matrix and a method based on optical low-coherence reflectometry (OLCR) and inversescattering which provides a direct reconstruction of the axial strains in the fibre withoutany assumptions.The results of the measurements and analysis demonstrated that the predicted spectra from theT-matrix method are not in good agreement with the recorded ones. These differences areattributed to the high strain gradients along the fibre due to its proximity to the crackfront. The results from the OLCR-based method directly provide the non-uniform strainsalong the fibre. The experimental measurements with the latter method are also in goodagreement with a three-dimensional numerical model of the actual experimentalconfiguration.

Characterization of a high birefringence fibre Bragg grating sensor subjected to non-homogeneous transverse strain fields

Several theoretical and experimental studies have been carried out in recent years to characterize the spectral response of an optical fibre Bragg grating (FBG) subjected to non-homogeneous axial strain fields. However, few works have considered the evolution of the spectrum when a FBG fabricated in a high birefringence fibre is subjected to non-homogeneous transverse loading. In this paper, a physical model is described and a numerical simulation based on the piecewise-uniform approach is also used to calculate the response of a FBG fabricated in a high birefringence fibre under some typical non-homogeneous transverse strain fields. Finally, simulated results are analysed and compared with a FBG fabricated in a conventional monomode fibre.

Characterization of residual strains in an epoxy block using an embedded FBG sensor and the OLCR technique

Cylindrical specimens made of epoxy and centrally located glass fibre are used to characterize residual stresses due to curing and post-curing of the epoxy. The fibre contains a Bragg grating which is used as a strain sensor along the fibre axis. A novel optical low-coherence reflectometry technique combined with a layer-peeling algorithm allows the direct reconstruction of the optical period of the grating and provides the strain distribution due to polymerisation of the epoxy along the fibre without any assumption. For the purpose of analysis, the polymerisation process is considered as an equivalent thermo-elastic problem. Using the experimental data as an input to a finite element model of the fibre-epoxy specimen, a non-homogeneous strain field is predicted in the whole specimen. Particularly, the experimental data are in good agreement with finite element simulations and a theoretical model [8] along the embedded fibre.

High strain gradient measurements by using digital image correlation technique

When material characterization requires nonhomogeneous strain field measurements, the use of specific techniques is necessary. In this paper, the efficiency of the digital image correlation method for measuring in-plane displacements in the presence of high strain gradient is discussed. Three types of strain gradient have been studied: strain localization around a hole in a composite laminate, strain concentration at a crack tip in a TiAl alloy, and strain gradient on a polymer neck front. These three applications concern various materials, various types of loading, and various gradient values. Results demonstrate the technique capabilities for a wide range of cases.

Crystallization of polymers under strain: from molecular properties to macroscopic models

The crystallization of thermo-plastic polymers under strain is considered both theoretically and experimentally. The thermo-mechanical model presented here is performed in the framework of the so-called generalized standard materials. In our model we couple in a very natural way the kinetics of crystallization with the mechanical history experienced by the polymer. The viscoelastic properties of the polymer are described using molecular theories. Therefore, in this model of strain-induced crystallization, the kinetics of crystallization is explicitly linked to the polymer chain conformation. Our model is intended to be valid for both for shearing and elongation, or any other complex strain field. Two different viscoelastic molecular models are considered here, corresponding to Maxwell and Pom–Pom constitutive equations. The model is implemented in a dedicated finite element code and the case of injection molding is considered. To validate our strain-induced crystallization model, which explicitly takes into account the molecular conformation, experiments investigating the material behavior at the molecular scale are required. Several measurement techniques can be used to achieve this task, including infrared spectroscopy, optical polarimetry, X-ray scattering or diffraction, etc. In this paper, the wide angle X-ray diffraction (WAXD) is used to investigate the crystalline texture of the polymer. We consider here the case of poly(ethylene terephthalate) (PET) subjected to a biaxial elongation above its T g. The strain field is determined using a home-developed image correlation technique that allows us to infer all the strain components at each point of the specimen, even in the case of a non-homogeneous strain field. To minimize the effect of quiescent crystallization, specimens are quickly heated with infrared and the temperature was regulated during the test. At the end of the deformation process, the specimens were quenched to room temperature. Their microstructure was later investigated using the WAXD technique. In order to undertake local and accurate WAXD measurements Synchrotron radiation facilities are used.

WAXD study of induced crystallization and orientation in poly(ethylene terephthalate) during biaxial elongation

This study presents an experimental investigation into the strain-induced crystalline microstructure, under biaxial elongation in poly(ethylene terephthalate) (PET), using wide angle X-ray diffraction (WAXD). We examined how the microstructure of a polymer subjected to a complex strain field evolves in terms of its crystalline ratio, its molecular orientation and the size of its crystallite. PET injection-molded specimens have been subjected to biaxial elongation tests, both equibiaxial and sequential, at different drawing speeds, draw ratios and temperatures above and close to T g. The strain field was determined using a home-developed image correlation technique that has allowed us to determine all the strain components at each point of the specimen, even with a non-homogeneous strain field. To minimize the effect of quiescent crystallization, specimens are quickly heated with infrared and the temperature was regulated during the test. At the end of the deformation process, the specimens were quenched to room temperature. Their microstructures were later investigated, using both differential densimetry and WAXD with a synchrotron beam. Influences of strain rate, temperature and strain path sequence on the size of the crystallites and their orientation are evaluated.

Experimental verification of response of embedded optical fiber Bragg grating sensors in non-homogeneous strain fields

One of the advantages of optical fiber sensors is their ease of embedment within a structure for non-destructive strain monitoring. In particular, Bragg grating sensors are written directly into an optical fiber hence remaining unobtrusive. In addition, several gratings can be written in series along a single fiber, permitting sensing at discrete points throughout the strain field. However, in regions of strong strain gradients, measuring the strain at discrete points may not be sufficient. One solution is to write a Bragg grating longer than the strain region of interest and use the change in its spectral response to determine the applied strain field as a function of position along the fiber. This paper presents an experimental verification of the response of an embedded optical fiber Bragg grating (OFBG) to applied non-homogeneous strain fields. Optical fiber Bragg grating sensors were embedded in four epoxy specimens of different forms so as to apply known strain functions along the gauge length when the specimen is under uniaxial tension. The complete spectral response of the Bragg gratings was then measured as a function of increasing load. The results are compared with analytical calculations, based on the piecewise-uniform period assumption for chirped gratings. Finally, the use of these spectra is discussed as possible basis functions for the resolution of an arbitrary applied strain distribution.

Prediction of composite properties from a representative volume element

A vigorous mechanics foundation is established for using a representative volume element (RVE) to predict the mechanical properties of unidirectional fiber composites. The effective elastic moduli of the composite are determined by finite element analysis of the RVE. It is paramount in such analyses that the correct boundary conditions be imposed such that they simulate the actual deformation within the composite; this has not always been done previously. In the present analysis, the appropriate boundary conditions on the RVE for various loading conditions are determined by judicious use of symmetry and periodicity conditions. The non-homogeneous stress and strain fields within the RVE are related to the average stresses and strains by using Gauss theorem and strain energy equivalence principles. The elastic constants predicted by the finite element analysis agree well with existing theoretical predictions and available experimental data.