Extension of the Optimised Virtual Fields Method to Estimate Viscoelastic Material Parameters from 3D Dynamic Displacement Fields

In vivo measurement of the mechanical properties of soft tissues is essential to provide necessary data in biomechanics and medicine (early cancer diagnosis, study of traumatic brain injuries, etc.). Imaging techniques such as magnetic resonance elastography can provide 3D displacement maps in the bulk and in vivo, from which, using inverse methods, it is then possible to identify some mechanical parameters of the tissues (stiffness, damping, etc.). The main difficulties in these inverse identification procedures consist in dealing with the pressure waves contained in the data and with the experimental noise perturbing the spatial derivatives required during the processing. The optimised virtual fields method (OVFM) ( 26 ), designed to be robust to noise, presents natural and rigorous solution to deal with these problems. The OVFM has been adapted to identify material parameter maps from magnetic resonance elastography data consisting of 3D displacement fields in harmonically loaded soft materials. In this work, the method has been developed to identify elastic and viscoelastic models. The OVFM sensitivity to spatial resolution and to noise has been studied by analysing 3D analytically simulated displacement data. This study evaluates and describes the OVFM identification performances: Different biases on the identified parameters are induced by the spatial resolution and experimental noise. The well‐known identification problems in the case of quasi‐incompressible materials also find a natural solution in the OVFM. Moreover, an a posteriori criterion to estimate the local identification quality is proposed. The identification results obtained on actual experiments are briefly presented.     

Ultra‐High‐Speed Full‐Field Deformation Measurements on Concrete Spalling Specimens and Stiffness Identification with the Virtual Fields Method

Abstract: For one decade, spalling techniques based on the use of a metallic Hopkinson bar in contact with a concrete sample have been widely employed to characterise the dynamic tensile strength of concrete at strain rates ranging from a few tens to hundreds of s^?1. However, the processing method based on the use of the velocity profile measured on the rear free surface of the sample (Novikov formula) remains quite basic. In particular, the identification of the whole softening behaviour of the concrete material is currently out of reach. In the present paper, a new processing technique is proposed based on the use of the virtual fields method (VFM). First, a digital ultra‐high‐speed camera is used to record the pictures of a grid bonded onto the specimen. Then, images of the grid recorded by the camera are processed to obtain full‐field axial displacement maps at the surface of the specimen. Finally, a specific virtual field has been defined in the VFM equation to use the acceleration map as an alternative ‘load cell’. This method applied to three spalling tests with different impact parameters allowed the identification of Young's modulus during the test. It was shown that this modulus is constant during the initial compressive part of the test and decreases in the tensile part when microdamage exists. It was also shown that in such a simple inertial test, it was possible to reconstruct average axial stress profiles using only the acceleration data. It was then possible to construct local stress–strain curves and derive a tensile strength value.     

Single‐camera special stereo‐digital image correlation applied for accurate virtual fields method identification

A single‐camera special stereo‐DIC (SS‐DIC) is proposed for accurate virtual fields method (VFM) identification. The single‐camera SS‐DIC allows accurate surface 3D deformation measurements using a single colour camera and a specially designed colour separation device. It not only effectively eliminates the unavoidable out‐of‐plane movement/rotation due to unideal in‐plane loading but also delivers uniformly distributed measurement points that brings great simplicity and convenience for internal virtual work calculation in VFM. In addition, since only a single camera is used for stereovision, the proposed SS‐DIC system requires no complicated synchronisation devices. The effectiveness and practicality of the proposed method are evaluated by heterogeneous deformation experiments of a holed aluminium alloy and 304 stainless steel plate. Combined with a high‐speed colour camera, the proposed method is expected to be a simple and practical method for the calibration of material constitutive model under intermediate and high strain rate conditions using VFM.     

Identification of Johnson–Cook's Viscoplastic Model Parameters Using the Virtual Fields Method: Application to Titanium Alloy Ti6Al4V

Abstract: The identification of viscoplastic material parameters is addressed using a new powerful method: the virtual fields method (VFM). Contrary to classical procedures that are statically determined, the VFM is applied to heterogeneous mechanical fields. Without any hypotheses of homogeneity required, the exploitation of tests with the VFM is not limited to small levels of strains anymore and it can be taken advantage of the large amount of information available thanks to full‐field measurements. In the case of viscoplastic models, the characterisation of strain‐rate sensitivity with the VFM is attempted in this paper using only one test under high‐speed loading conditions, whereas several tests performed at different constant strain‐rates are required for the classical procedures. This article focuses on the development of the VFM for the characterisation of Johnson–Cook's (JC) viscoplastic model. To his aim a return‐mapping algorithm was developed according to the JC's model with an implicit Euler scheme implemented to integrate the constitutive relations. The whole viscoplastic behaviour of a Titanium alloy (Ti6Al4V) is successfully characterised by the VFM using only two tensile tests on notched flat specimens, with full‐field strain measurements by digital image correlation.     

A Multigrid PGD‐based Algorithm for Volumetric Displacement Fields Measurements

The use of a finite elements‐based Digital Volume Correlation (FE‐DVC) leads to lower measurement uncertainties in comparison to subset‐based approaches. However, the associated computing time may become prohibitive when dealing with high‐resolution measurements. To overcome this limitation, a Proper Generalised Decomposition solver was recently applied to 2D digital image correlation. In this paper, this method is extended to measure volumetric displacements from 3D digital images. In addition, a multigrid Proper Generalised Decomposition algorithm is developed, which allows to use different discretisations in each term of the decomposition. Associated to a coarse graining of the digital images, this allows to avoid local minima, especially in presence of large displacements. Synthetic and practical cases are analysed with the present approach, and measurement uncertainties are compared with standard FE‐DVC. Results show that such an approach reduces the computational cost (when compared to FE‐DVC) whilst maintaining lower measurement uncertainties than standard subset‐based DVC.     

Comparison of the identification performance of conventional FEM updating and integrated DIC

Full‐field identification methods are increasingly used to adequately identify constitutive parameters to describe the mechanical behavior of materials. This paper investigates the more recently introduced one‐step method of integrated digital image correlation (IDIC) with respect to the most commonly used two‐step method of finite element model updating (FEMU), which uses a subset‐based DIC algorithm. To make the comparison as objective as possible, both methods are implemented in the most equivalent manner and use the same FE model. Various virtual test cases are studied to assess the performance of both methods when subjected to different error sources: (1) systematic errors, (2) poor initial guesses for the constitutive parameters, (3) image noise, (4) constitutive model errors, and (5) experimental errors. Results show that, despite the mathematical similarity of both methods, IDIC produces less erroneous and more reliable results than FEMU, particularly for more challenging test cases exhibiting small displacements, complex kinematics, misalignment of the specimen, and image noise. Copyright © 2015 John Wiley & Sons, Ltd.     

Virtual Fields Method on Planar Tension Tests for Hyperelastic Materials Characterisation

Abstract: New methods are emerging in the material characterisation field with the aim of exploiting innovative full‐field strain measurement techniques. Besides experimental issues, also numerical procedures for inverse problems should adapt to a new philosophy: the large amount of data referred to local strains should be used in an appropriate way to obtain as much benefits as possible. In this context, an experimental and numerical procedure for the characterisation of hyperelastic materials is proposed. Planar tension tests have been performed on flat rubber specimens of different geometries. Strain maps obtained by means of a 2D Digital Image Correlation system are used to implement the virtual fields method, to estimate material dependent parameters of two of the most known hyperelastic constitutive laws: Ogden and 2nd order Mooney‐Rivlin models. Numerical results and comparisons with experimental data are shown, analysing also aspects concerning implementation of the numerical procedures and computational efficiency of the algorithms.     

Effect of DIC Spatial Resolution,Noise and Interpolation Error on Identification Results with the VFM

The use of experimental tests that involve full‐field measurements to characterize mechanical material properties is becoming more widespread within the engineering community. In particular digital image correlation (DIC) on white light speckles is one of the most used tools, thanks to the relatively low cost of the equipment and the availability of dedicated software. Nonetheless the impact of measurement errors on the identified parameters is still not completely understood. To this purpose, in this paper, a simulator able to numerically simulate an experimental test, which involves DIC is presented. The chosen test is the Unnotched Iosipescu test used to identify the orthotropic elastic parameters of composites. Synthetic images are generated and then analysed by DIC. Eventually the obtained strain maps are used to identify the elastic parameters with the Virtual Fields Method (VFM). The numerical errors propagating through the simulation procedure are carefully characterized. Besides, the simulator is used to compare the performances of DIC and the grid method in the identification process with the VFM. Finally, the influence of DIC settings on the identification error is studied as a function of the camera digital noise level, in order to find the best testing configuration.     

Addendum to ‘Characterising the Strain and Temperature Fields in a Surrogate Bone Material Subject to Power Ultrasonic Excitation’

Recently, a very interesting article was published in Strain where a rigid polyurethane foam specimen was submitted to longitudinal vibrational excitation in the ultrasonic range. The authors showed that it was possible to measure time‐resolved strain response maps by combining digital image correlation and ultra‐high‐speed imaging. The objective of this discussion is to propose further analysis of the data published in that article, showing that it is possible to extract meaningful values for Young's modulus by using the acceleration field in the specimen as a load cell. The aim here is not to provide a complete solution to this problem but to alert the readers on the possibilities offered by this kind of test. This method is an interesting alternative where the energy is input repeatedly instead of in one go as in impact‐based tests. Full‐field vibration measurements have already been used in the past to identify stiffnesses but only in bending and at much lower strain rates. This article shows that the method can be extended to cover a much wider strain rate range. Finally, only global stiffness values were identified then, whereas here, maps of stiffnesses can be derived.     

基于Brown构形场的有限体积法在聚合物流动模拟中的应用

本文基于微观-宏观方法,提出了模拟聚合物流动的FVMBCF (Finite Volume Method Based on Brownian Configuration Fields)方法.为了验证该方法的有效性和计算结果的可靠性,基于Hooke哑铃模型,模拟了平面Poiseuille流动;同时,基于Hooke、FENE...     

Towards Material Testing 2.0. A review of test design for identification of constitutive parameters from full‐field measurements

Full‐field optical measurements like digital image correlation or the grid method have brought a paradigm shift in the experimental mechanics community. While inverse identification techniques like finite element model updating or the virtual fields method have been the object of significant developments, current test methods, inherited from the age of strain gauges or linear variable displacement transducers, are generally not well adapted to the rich information provided by these new measurement tools. This paper provides a review of the research dealing with the design and optimization of heterogeneous mechanical tests for the identification of material parameters from full‐field measurements, christened here Material Testing 2.0 (MT2.0).     

Modelling Strategies for Property Identification Based on Full‐Field Surface Displacement Data

Abstract: The use of full‐field displacement measurements in mechanical testing provides detailed response information that can be used, in conjunction with modelling and optimisation, for precise material property identification. One limitation of this technique is that the collection of response data and the sectioning of a specimen to reveal the material microstructure are both destructive tests and mutually exclusive, as the displacement measurement occurs only on the exposed surface. Therefore, modelling of an experiment to interpret a full‐field experiment requires assumptions about the structure of the material below the visible surface. This study evaluates the effects of several possible modelling assumptions on the errors in model‐predicted response and on the resulting material property estimates. A 3‐D microstructural model, for which the subsurface grain geometry and orientations are known, provides the basis for comparison of several common modelling assumptions based on the grain geometry and orientations on the visible surface of a specimen.     

Experimental Investigations of Fracture Process Using DIC in Plain and Reinforced Concrete Beams under Bending

The fracture behaviour of concrete and reinforced concrete beams under quasi‐static three‐point bending was comprehensively investigated with experiments at laboratory scale. The eight various concrete mixes were tested. The influence of the shape, volume and size of aggregate particles and reinforcement on concrete fracture under bending was studied. Displacements on the surface of concrete beams were measured by means of the digital image correlation (DIC) technique. Attention was paid to the formation of a localized zone and its characteristics. In order to avoid the effect of the search patch size and the cut‐off value at displacement and strain profiles, a consistent method was proposed to determine uniformly and accurately the width of a localized zone. Measured surface displacements from DIC were fitted by the error function ERF, whereas surface strains calculated from displacements were fitted by the usual normal distribution (Gauss) function. The width of a localized zone preceding a macro‐crack grew strongly with increasing maximum aggregate size and slightly with diminishing aggregate volume. It did not depend on the aggregate roughness and reinforcement presence.     

Cross‐Correlation and Differential Technique Combination to Determine Displacement Fields

Abstract: This study presents a method to measure the displacement fields on the surface of planar objects with sub‐pixel resolution, by combining image correlation with a differential technique. First, a coarse approximation of the pixel level displacement is obtained by cross‐correlation (CC). Two consecutive images, taken before and after the application of a given deformation, are recursively split in sub‐images, and the CC coefficient is used as the similarity measure. Secondly, a fine approximation is performed to assess the sub‐pixel displacements by means of an optical flow method based on a differential technique. To validate the effectiveness and robustness of the proposed method, several numerical tests were carried out on computer‐generated images. Moreover, real images from a static test were also processed for estimating the displacement resolution. The results were compared with those obtained by a commercial digital image correlation code. Both methods showed similar and reliable results according to the proposed tests.     

Digital Image Correlation and Noise‐filtering Approach for the Cracking Assessment of Massive Reinforced Concrete Structures

This paper describes the use of Digital Image Correlation (DIC) techniques for the cracking assessment of reinforced concrete (RC) massive beams and walls. DIC is known to provide accurate and detailed information on displacement and strain fields. Non‐contact measurements can be used to evaluate concrete cracking of destructive tests carried out on a wide range of specimen scales. When applied to large RC structures tested outdoors or in difficultly controllable conditions, DIC‐based methods may lead to erroneous results. In this study a post‐processing procedure is presented to cope with noisy full‐field measurements. The proposed cracking assessment approach is validated on a large experimental campaign. Four points bending tests are carried out on RC beams: firstly on full‐scale rectangular beams and then on mock‐ups scaled down by 1/3. In addition, fours RC walls are tested under in‐plane cyclic shear up to failure. Digital images taken throughout the tests are processed by DIC techniques to provide in‐plane displacement and strain fields. Full‐field measurements are post‐processed by the noise‐filtering technique and the cracks patterns are identified. Crack widths are measured and compared with measurements obtained from conventional point‐based sensors (linear variable differential transformer LVDT and fibre‐optic FO transducers). The proposed DIC‐based post‐processing provides accurate estimation of cracks width for most of the tests. The analyses carried out on the two groups of RC beams show a scale‐effect on the cracks width.     

Characterising the Strain and Temperature Fields in a Surrogate Bone Material Subject to Power Ultrasonic Excitation

The behaviour of the cancellous bone surrogate material, rigid polyurethane foam (PUF), subject to power ultrasonic vibration excitation has been studied, with the purpose of identifying a methodology to investigate the effects that ultrasonic surgical devices have on biological tissue materials. To characterise the vibrational response to ultrasonic excitation, non‐contact measurement of the full in‐plane displacement field of PUF plate specimens was performed by combining the use of an ultra‐high speed camera and 2D digital image correlation. To investigate the thermal response, an infrared camera was used in real time to detect the temperature field. The measured surface displacement and strain fields of the PUF specimens and the thermal response are compared with data from an analytical model, and two different finite element models using Abaqus and PZFlex . The close agreement between calculated and measured data provides initial confidence in the use of the models for predicting the effects of ultrasonic excitation on tissue materials. The measurement data demonstrate the success of the experimental method for measuring vibrational responses in a hard tissue surrogate material at the ultrasonic frequencies associated with power ultrasonic surgical devices.     

The incremental Virtual Fields Method and prestretching method applied to rubbers under uniaxial medium‐strain‐rate loading

Conventional dynamic experiments on rubbers have several limitations including low signal‐to‐noise ratio and a long time period during which the specimen is not in static equilibrium, which causes difficulties separating constitutive material behaviour from specimen response. In order to overcome these limitations, we build on previous research in which the Virtual Fields Method (VFM) is applied to dynamic tensile experiments. A previous study has demonstrated that the VFM can be used to identify the material parameters of a hyperelastic model for a given rubber based on optical measurements of wave propagation in the rubber, eliminating the need for force measurements by instead using acceleration fields as a “virtual load cell.” In order for us to successfully characterise the strain hardening in the material, large deformations are required, and these were achieved by applying static preloads to the specimen before the dynamic loading. In order for us to then apply the VFM, measurements of the static force, or strain, or both, are required. This paper explores different methods for applying the VFM, in particular, comparing the use of a static force measurement, as in the previous research, to methods that only require strain fields in order to apply the incremental equation of motion. Finite element method simulations were conducted to compare the identification sensitivity to experimental error sources between the 2 VFM implementations; the experimental data used in the previous studies were then applied to the incremental VFM. A further experimental comparison is provided between constitutive parameters obtained in tensile experiments using the VFM and compressive measurements from a modified split Hopkinson bar technique equipped with a piezoelectric force transducer. Finally, there is a discussion of the effects of preloading and relaxation in the material.     

Mode I–III Decomposition of the J‐integral from DIC Displacement Data

This paper presents the implementation of the decomposition method on digital image correlation (DIC) obtained displacement fields to obtain J‐integral results (J) and respective stress intensity factors (SIFs). DIC is increasingly used with the J‐integral approach in experimental mechanics to obtain J estimates from complex fracture processes. In this approach, the decomposition method is applied to DIC displacement fields for the first time. Here, displacement fields are separated before stresses and strains are computed, so that subsequent computation of separate J or SIF components may follow the classical full‐field J‐integral approach. The sensitivity of the decomposition method to experimental errors is investigated using synthetically generated errors imposed on crack tip displacement fields (Williams' series), from which improvements to the procedure are proposed. The method is experimentally tested on PMMA Arcan specimens under mode I, II, and III, and mixed‐mode I–III loading. Test results were compared to fracture toughness values obtained from ASTM tests and literature with close agreement.     

Variability,heterogeneity, and anisotropy in the quasi‐static response of laser sintered PA12 components

Additive manufacturing (AM) receives an increasing industrial interest thanks to its advantages in the economic production of highly complex and small‐series components. Especially laser sintering (LS) is in this context of particular interest for the production of plastic components, as it is generally deemed the most robust AM technology for polymer parts and therefore is expected to enable AM for functional components in the near future. However, to date, designers are often confronted with a severe lack of knowledge on the possible mechanical behavior of AM components. More specifically, the unit‐to‐unit variability, heterogeneity (within‐part variation), and anisotropy of the mechanical properties very often prove to be substantial and therefore require more elaborated studies in order to take these effects into account in the engineering of reliable components. Moreover, typical experimental results that are used for the determination of the elastic stiffness tensor are subject to variability, caused by the influence of the difference in thermal history between produced parts. This work therefore focuses first on the identification and quantification of the variability and heterogeneity in the quasi‐static response of laser sintering‐polyamide 12 (LS‐PA12) components. Second, also the anisotropy in this quasi‐static response is studied. For the first part, uniaxial tensile tests are performed and the variability on the quasi‐static properties is quantified by means of statistical analysis. Also, the elastic stiffness tensor is identified based on these tests. Next, the heterogeneity in the tested specimens is investigated by means of digital image correlation. Finally, in order to study the anisotropy in the quasi‐static properties, the Virtual Fields Method is applied to determine the variability in the elastic stiffness tensor of the LS‐PA12 material. A variability with a coefficient of variance of up to 6.5% on Young's modulus was measured. It was also found that the production planning has an important influence on the homogeneity of the mechanical properties of the produced parts. Finally, the Virtual Fields Method showed that, contrary to most literature on the topic, the elastic properties of LS‐PA12 material is best described using an isotropic material model.