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Stress reconstruction based on experimentally acquired full-field strain measurements is computationally expensive when using conventional implicit stress integration algorithms. The computational burden associated with repetitive stress reconstruction is particularly relevant when inversely characterizing plastic material behaviour via inverse methods, like the nonlinear Virtual Fields Method (VFM). Spatial and temporal down-sampling of the available full-field strain data is often used to mitigate the computational effort. However, for metals subjected to non-linear strain paths, temporal down-sampling of the strain fields leads to erroneous stress states biasing the identification accuracy of the inverse method. Hence, a significant speedup factor of the stress integration algorithm is required to fully exploit the experimental data acquired by Digital Image Correlation (DIC). To this end, we propose an explicit stress integration algorithm that is independent on the number of images (i.e. strain fields) taken into account in the stress reconstruction. Theoretically, the proposed method eliminates the need for spatial and temporal down-sampling of the experimental full-field data used in the nonlinear VFM. Finally, the proposed algorithm is also beneficial in the emerging field of real-time DIC applications. [ABSTRACT FROM AUTHOR]

Full‐field optical experimental techniques such as Digital Image Correlation (DIC) enable measurement of information‐rich heterogeneous strain states. The aim of Material Testing 2.0 is to capitalise on this and design experimental setups for efficient material characterisation using inverse identification techniques such as the Virtual Fields Method (VFM). In that case, however, a priori knowledge of the constitutive model is required, which sometimes is not a trivial task. This limitation can be overcome by identifying multiple constitutive models using the same DIC strain fields and comparing their performance. In this work, we present a methodology to evaluate model performance and compare different constitutive models based on quantitative metrics. DIC experiments were performed on an S‐shaped high density polyethylene test sample under uniaxial load. The heterogeneous DIC strain fields were used in combination with the VFM to identify parameters of hyperelastic constitutive models with increasing complexity. Two deterministic metrics, the equilibrium gap indicator (EGI) and the reconstructed axial force ratio (RAFR), were defined and used to compare the performance of these constitutive models. The impact of spatial smoothing on DIC strain fields was studied by calculating the EGI and RAFR using a digital twin of the DIC experiments. EGI and RAFR metrics provide complementary information regarding the material behaviour, and both are necessary to make an informed choice. In this case study, hyperelastic constitutive models were found not to capture the material stiffness well in the small strain regime. A linear‐elastic contribution exponentially decaying with the equivalent shear stress was added to the hyperelastic models and the corresponding model parameters identified using the VFM. The elastic‐hyperelastic constitutive models were found to perform better than their purely hyperelastic counterparts. [ABSTRACT FROM AUTHOR]

Glass fiber-reinforced polymer composites are important structural materials and are widely used in structure engineering. In this study, a new V-notch non-standard tensile specimen is proposed. All the in-plane stiffness coefficients of glass fiber-reinforced polymer composites could be obtained by the virtual fields method with only one uniaxial tensile test. First, the special virtual fields method for inversion of elastic constitutive parameters of orthotropic materials in the uniaxial tensile test was introduced. The optimization of the geometrical design of the specimen was conducted using finite element simulation experiments. Batch modeling calculation was conducted within the designed range of the geometric parameters of the specimen. The generated ideal strain field data were substituted into the virtual fields method to invert and identify the stiffness coefficients. The optimized geometry of the specimen was determined according to the objective function of minimum error. Through a tensile experiment on glass fiber composites, the influence of specimen deformation on the identification results was assessed, and the load level suitable for parameter identification was determined. Based on the results, it can be concluded that the inversion identification accuracy meets the requirements of engineering measurement.

This work proposes a new method aiming at the direct identification of viscoelastic properties of materials with a Laplace formalism implemented in the Virtual Fields Method and named L-VFM. Using a single test, this formalism allows for a direct extraction of the different viscoelastic properties without any parametric description of their time dependency. The Laplace transform enables the use of theory of elasticity in the Laplace domain. The constitutive equations are expressed in the plane stress framework with the 2D plane stress stiffness coefficients. The conversion from the 2D plane stress stiffness coefficients to the bulk and shear moduli as well as Poisson’s ratio and Young’s modulus is realised in the Laplace domain. The inverse Laplace transform is then applied to these functions in order to obtain the temporal evolution of the material properties. The L-VFM changes the viscoelastic identification from a non-linear to a linear process.

Biphasic hyperelastic models have become popular for soft hydrated tissues, and there is a pressing need for appropriate identification methods using full‐field measurement techniques such as digital volume correlation. This paper proposes to address this need with the virtual fields method (VFM). The main asset of the proposed approach is that it avoids the repeated resolution of complex nonlinear finite element models. By choosing special virtual fields, the VFM approach can extract hyperelastic parameters of the solid part of the biphasic medium without resorting to identifying the model parameters driving the osmotic effects in the interstitial fluid. The proposed approach is verified and validated through three different examples: the first and second using simulated data and then the third using experimental data obtained from porcine descending thoracic aortas samples in osmotically active solution. [ABSTRACT FROM AUTHOR]

With the advent of camera‐based full‐field measurement techniques such as digital image correlation (DIC), researchers have been trying to exploit such rich data sets through the use of more complex test configurations than the standard ones (uniaxial tension/compression, bending etc.). This new paradigm in mechanical testing of materials has recently been christened 'Material Testing 2.0'. This paper provides a brief overview of this field, which is currently seeing a large increase in research effort. [ABSTRACT FROM AUTHOR]

Left ventricle (LV) myocardial dysfunction has been recently investigated using the estimation of isotropic myocardial stiffness from magnetic resonance imaging (MRI). However, Myocardium is known to have a 3D complex geometry with anisotropic stiffness. The assessment of the anisotropy properties characterizes structural changes in myocardium as a consequence of heart failure (HF). From image data, the virtual field method (VFM) can determine material stiffness in a non-invasive manner. In the present work, the objective is to compare two inverse identification methods, given the isotropic and anisotropic models in the characterization of properties of myocardium in acute lymphoblastic leukemia (ALL) survivors using VFM and MRI. Two types of VFM approach are presented. Using the first, the virtual displacements (VFs) allow whole-field LV to be imposed into VFM formulation and caused to directly estimate two independent parameters from isotropic constitutive relation. With the second, anisotropic parameters are estimated using piece-wise (Finite element-based) VFM. The resulting values showed significant differences between the subjects in comparative study of leukemia survivors, and variance in estimated parameters by two different VFM approach. This approach would be an efficient tool to characterize early cardiac dysfunction. This work elucidates the benefits and shortcomings of using VFM to determine anisotropic parameters of LV myocardium in linear elastic and of using the FEM application to generate meshes of patient-specific LVs from MRI images. [ABSTRACT FROM AUTHOR]

Background: Characterizing the mechanical parameters of laser repaired components is an important step in evaluating their reliability. However, these mechanical parameters tend to exhibit gradient characteristics in a specific direction, which brings a great challenge in the operation of characterization. Objective: To address the above issue, an orthogonal-function-based virtual fields method (O-VFM) was developed in this study, which can simultaneously characterize the spatial distributions of Young's modulus E and Poisson's ratio v by an individual experiment. Methods: In this study, orthogonal functions were employed to parameterize the stiffness coefficients and design the virtual displacement and strain fields. And the accuracy of the O-VFM was investigated through finite element analysis (FEA). A dimensionless index was also proposed to quantitatively describe the difference in mechanical properties between the substrate, heat-affected zone (HAZ), and repaired zone (RZ). Results: Numerical experiments show that the proposed O-VFM can accurately determine the gradient characteristics of E and v, and the identification accuracy of E was less affected by the constant v assumption. The actual experiments on the laser repaired GH4169 show that the distributions of E and v in different zones can be simultaneously identified by O-VFM. The E and v in the substrate presented uniform properties; whereas the gradient characteristics were mainly concentrated in the HAZ and RZ. Moreover, the gradient characteristic of E decreased significantly at 650 ℃. Conclusions: The experimental results demonstrate that the proposed O-VFM is an efficient experimental technique for evaluating the heterogeneous mechanical properties of laser repaired components. [ABSTRACT FROM AUTHOR]

Thermomechanical parameters are important indicators for evaluating the mechanical properties of superalloys and generally include the coefficients of stiffness and thermal expansion at high temperatures. At present, there are few methods for simultaneously characterising the thermomechanical parameters of superalloys, especially single‐crystal superalloys. To satisfy the demand for simultaneously identifying the thermomechanical parameters of orthotropic superalloys, an optimised virtual fields method for decoupling the thermomechanical parameters was developed in this study by combining the self‐developed heat‐resistant grids and the sampling moiré method. First, several factors, including the oblique angle of the grids, image noise and thermomechanical coupling phenomena, were studied through numerical experiments to analyse their influences on the identification accuracy. Then, an optimised identification strategy was established. Finally, the thermomechanical parameters of Ni‐based polycrystalline and single‐crystal superalloys were successfully identified and comparatively studied. The identification results demonstrate that the proposed method is highly accurate and robust. This research will provide an effective way to accurately characterise the multiple parameters of superalloys at high temperatures. [ABSTRACT FROM AUTHOR]

Background: Residual stresses often cause distortion to occur when machining slender workpieces like those used in aeronautics. This distortion is undesirable and may even cause the workpiece to be scrapped. Identifying these residual stresses during machining is therefore crucial to limit their effect on the final geometry of the part. Objective: The aim of this work is to identify the through-thickness residual stress distribution by processing displacement/strain fields measured on the workpiece during machining. Methods: Digital Image Correlation is employed to measure, between successive milling passes, the displacement and strain fields on the lateral surface of the workpiece. These fields are then processed with the Virtual Fields Method to identify the through-thickness residual stress distribution. Compared to previous studies on this topic, no assumption is made concerning the real through-thickness displacement field. Results: Simulations performed with synthetic data provided by a finite element model show the feasibility of this approach and quantifies its robustness when displacements are affected by measurement noise. Results obtained with this approach in a real case are then compared with their counterparts obtained with another identification technique, which assumes that the workpiece behaves as a beam. Conclusion: This study shows that it is possible to measure the through-thickness distribution of residual stress in slender workpieces during machining by using a classic subset-based DIC software and the Virtual Fields Method to process the displacement/strain maps. This opens the way for future developments aimed for instance at updating in live machining sequences in order to obtain machined parts immune from distortions caused by residual stresses. [ABSTRACT FROM AUTHOR]

In this work, virtual fields method (VFM) is applied to extract the constitutive parameter of silicone elastomers under equi-biaxial and general biaxial tests; the influence of missing deformation data near specimen edges and noise introduced by digital image correlation (DIC) on parameter identification using VFM is investigated. The results indicate that the negative impact of the missing data and noise can be mitigated by scaling the load applied to the specimen boundary properly and extracting parameters based on the deformation fields of all loading steps through the least square method. [ABSTRACT FROM AUTHOR]

The accurate characterization of the anisotropic thermal-mechanical constitutive properties of structural sheet metals at elevated temperatures and under nonuniform stress/strain states is crucial for the precise hot plastic forming and structural behavior evaluation of an engineering sheet part. Traditional thermal-mechanical testing methods rely on the assumption of states homogeneity, leading to a large number of tests required for the characterization of material anisotropy and nonlinearity at various high temperatures. In this work, a highly efficient identification method is proposed that allows the simultaneous characterization of the anisotropic yielding, strain hardening and elasto-plasticity thermal softening material properties using the minimum number of tests, releasing the limitations of traditional identification methods. This is implemented by performing a digital image correlation and infrared thermography assisted heterogeneous high temperature test and processing the full-field measurement data based on the principle of virtual work. A double-notched tensile specimen configuration combined with a center-to-periphery temperature gradient is designed first to enable the high heterogeneity of stress/temperature states. Simulations of the heterogeneous tests are then performed to supply idealized reference data that can be used to evaluate the identification sensitivity of the proposed identification algorithm. After the numerical validation of the methodology, it is then applied to the TC4 titanium alloy sheet specimen using the self-developed experimental setup. The experimental identification results verify that the multiple anisotropic thermal-mechanical elasto-plasticity constitutive parameters can be accurately identified from the heterogeneous test with high computation efficiency, significantly simplifying the testing process in comparison to applying multiple traditional homogeneous tests. The current work provides an effective and convenient alternative to high temperature identification strategies used by the material processing community. [Display omitted] • A heterogeneous tensile test for anisotropic thermal-mechanical properties of sheet metals is proposed. • A wide range of temperatures and comprehensive stress states can be realized in a single test. • The anisotropic and thermal softening properties can be simultaneously identified from the coupled full-field data. • The proposed identification method has been successfully implemented on the TC4 alloy sheet. [ABSTRACT FROM AUTHOR]

This paper proposes an experimental approach to measure contact forces in a two-dimensional system composed of cylindrical particles using the Virtual Fields Method (VFM). This identification method relies on the knowledge of the strain distribution in the particles forming the discrete medium. Synthetic strain data provided by a finite element model (with added noise) were first used as input data for the identification procedure. It is shown that if the mechanical response of the constitutive material is known, the contact forces applied to a particle can be identified, since they are proportional to a weighted average of the strain components, the weights being the virtual strain components. Various strategies were tested to propose kinematically admissible fields for the virtual displacement. Identification technique robustness was studied with respect to various sources of error, such as noise in the strain field, missing data along the boundaries, and a possible spurious shift between real and virtual fields. An experimental application was then performed on a three-particle system subjected to confined compression by processing strain maps obtained using Localized Spectrum Analysis (LSA) for each particle. In addition to the VFM equations, particle equilibrium and Newton's third law of motion were taken into account to propose a relevant strategy for processing the experimental data. The results open up prospects for applications to granular materials composed of numerous particles. • The Virtual Fields Method is used to identify contact forces in a 2D discrete medium. • Four strategies are compared to select the most relevant virtual displacements. • Identification is robust with respect to noise and missing data along particle boundary. • Experimental strain maps are obtained by Localized Spectrum Analysis on a three-particle system. • Particle equilibrium and Newton's third law of motion are also taken into account for processing the experimental data. [ABSTRACT FROM AUTHOR]

The virtual fields method (VFM) has played a pivotal role in parameter identification of hyperelastic materials, spanning from rubbers to soft tissues. A key challenge in VFM lies in calculating the internal virtual work, which necessitates the computation of the stress and its conjugate virtual strain. In this study, we introduce a novel theoretical framework for VFM, which directly computes the internal virtual work based on the variation of elastic energy (VEE). The proposed method, namely VEE-VFM, eliminates the calculation of stress and virtual strain, making it remarkably user-friendly compared with the conventional approach. We extend VEE-VFM to both incompressible and compressible hyperelastic materials, including invariant-based and principal stretch-based models. In the end, we demonstrate the accuracy of VEE-VFM and its robustness against noise with numerical experiments. • Reformulate the virtual fields method (VFM) based on the variation of elastic energy. • Develop the reformulated VFM for compressible and incompressible hyperelastic models. • Present an alternative approach to calculating the variation of principal stretches. [ABSTRACT FROM AUTHOR]

Although full‐field measurement techniques have been well established, material characterization from these data remains challenging. Often, no closed‐form solution exists between measured quantities and sought material parameters. In this paper, a novel approach to determine the stiffness of thin curved membranes is proposed, based on the virtual fields method (VFM). Utilizing Kirchhoff‐Love shell theory, we show that the displacements can be decomposed into an in‐plane displacement and a rotation of the mid‐surface of the shell. Consequently, the strain tensor at the outer surface of the shell can then be decomposed into a membrane and a bending part. This allows for the VFM to be applied based only on data of the outer surface and on surfaces of arbitrary curvature. The method is first applied to simulated data. It is shown that the elastic modulus can be identified with less than 5% error if the thickness and Poisson ratio are known accurately. A 5% uncertainty in either the Poisson ratio or the thickness changes the identified value by 5%. Then, the method is applied on experimental data acquired on rubber samples having a dome‐like shape. Tensile tests are performed on the same samples, which permits to assess the linearized Young's modulus of this material for moderate strains (0–2.1%). Using regression analysis, a Young's modulus of 1.21 ± 0.08 MPa is found. Next, we performed pressurization tests on eight dome‐like shapes with pressures up to 4 kPa. The average Young's modulus obtained with the novel virtual fields method is 1.20 ± 0.13 MPa. The results are in good agreement with the ones from the tensile test. Future applications could benefit from this method to analyse more complex shapes, for example those found in biological structures like arteries or eardrums. [ABSTRACT FROM AUTHOR]

This paper presents a novel apparatus, combined with application of the Virtual Fields Method (VFM) for the high frequency characterisation of the mechanical response of ultra‐soft materials. The viscoelastic response is characterised under harmonic deviatoric loading at a range of frequencies using the dynamic VFM in the frequency domain. Obtaining useful high rate data on soft materials is challenging using typical test methodologies, as the low speed of sound in the material makes stress equilibrium difficult to obtain; additionally, the low stiffness results in small stresses and weak measurement signals. This novel apparatus and analysis method have been shown capable of obtaining important material characterisation data that would be impossible to access using currently existing test techniques. [ABSTRACT FROM AUTHOR]

This paper presents a method to derive the virtual fields for identifying constitutive model parameters using the Virtual Fields Method (VFM). The VFM is an approach to identify unknown constitutive parameters using deformation fields measured across a given volume of interest. The general principle for solving identification problems with the VFM is first to derive parametric stress field, where the stress components at any point depend on the unknown constitutive parameters, across the volume of interest from the measured deformation fields. Applying the principle of virtual work to the parametric stress fields, one can write scalar equations of the unknown parameters and solve the obtained system of equations to deduce the values of unknown parameters. However, no rules have been proposed to select the virtual fields in identification problems related to nonlinear elasticity and there are multiple strategies possible that can yield different results. In this work, we propose a systematic, robust and automatic approach to reconstruct the systems of scalar equations with the VFM. This approach is well suited to finite-element implementation and can be applied to any problem provided that full-field deformation data are available across a volume of interest. We also successfully demonstrate the feasibility of the novel approach by multiple numerical examples. Potential applications of the proposed approach are numerous in biomedical engineering where imaging techniques are commonly used to observe soft tissues and where alterations of material properties are markers of diseased states. [ABSTRACT FROM AUTHOR]

Accurate material description is crucial to achieve high-quality results in computational analysis software. Phenomenological constitutive laws generalize the material behaviour observed in simple mechanical tests. The resulting empirical expressions contain parameters that need to be calibrated through an inverse optimization process. Advancements in Digital Image Correlation (DIC) techniques have enabled the extraction of non-uniform multi-axial displacement fields, facilitating the development of heterogeneous mechanical specimens and unlocking access to richer material behaviour data. Despite these advancements, constitutive models are still limited by mathematical formulations and biases due to simplifying assumptions. Artificial Neural Networks (ANNs), as universal function approximators, could potentially drive a paradigm shift in the field. ANNs do not require explicit pre-formulations, hence avoiding the need for identifying unknown parameters. Moreover, ANNs are able to implicitly learn patterns purely from data, allowing to circumvent the biases induced by the simplifying assumptions of first-order principles. However, the training of ANNs for implicit constitutive modelling is not straightforward, given the impossibility of obtaining direct measurements of stress. This paper explores the integration of Recurrent Neural Networks (RNNs) with the Virtual Fields Method (VFM) for material modelling. This approach uses global force and displacement data to indirectly train the neural network, with the equilibrium being evaluated globally through the VFM loss function. The proposed method is (i) computationally efficient by not requiring Finite Element Analysis (FEA), (ii) compatible with both full-field measurements and numerically generated data, and (iii) able to handle experimental boundary conditions. • A novel method integrates RNNs with the VFM for implicit material modelling. • The RNN is trained without labels, using only displacement/strain and global force. • The approach is real-test oriented, compatible with both DIC and numerical data. • The methodology is independent of the boundary conditions. • The method is computationally more efficient than FEA indirect training. [ABSTRACT FROM AUTHOR]

In vibroacoustics, inverse methods use the vibratory response of a structure to identify either a load or a structural parameter. The Force Analysis Technique (FAT) and the Virtual Fields Method (VFM), are two inverse methods that have been used in the past to identify loads or structural parameters of flexural beams or plates. The Corrected Force Analysis Technique (CFAT) is another inverse method that corrects the singularity of FAT, making this approach more accurate at higher frequencies. In this study, this principle is applied to the VFM, using polynomial interpolation of the displacement field, in order to adapt the method so that at each frequency, the accuracy of the method is improved. The accuracy of the Frequency-Adapted VFM is demonstrated using numerical simulations. A formal comparison between FAT, CFAT and the VFM is also proposed. • Comparison of the Virtual Fields Method with the Force Analysis Technique. • Application of the Virtual Fields Method in the high-frequency domain. • Application of the methods on a Euler–Bernoulli beam using simulated data. • Identification of a dynamic force or structural parameter using these methods. [ABSTRACT FROM AUTHOR]

A proper understanding of material mechanical properties is important in designing and modelling of components. As a part of a study on the structural integrity, the Digital Image Correlation technique was used to obtain the full-field strain distribution during a tensile test of the specimens. The displacement maps were analyzed using Matlab scripts to compute local stress-strain variations. Consequently the local proof stress values were extracted. In this study, the local extraction of the elastic and plastic properties of Al6061 alloy has been carried out using both the uniform stress method and the virtual fields method involving digital image correlation technique. In uniform stress methodology, full range stress–strain curves are obtained using the whole field strain measurement using Digital Image Correlation. The parameters investigated are Young's modulus, Poisson's ratio, yield strength, strain hardening exponent and strength coefficient. Recently, the virtual fields method is gaining a lot of popularity in domain characterization as it is robust, accurate and faster. Young's modulus, Poisson's ratio, yield strength, strength coefficient and strain hardening exponent are the parameters extracted using both uniform stress method and virtual fields method. The parameter variation obtained by both uniform stress method and the virtual fields method compares very well. Due to various advantages associated with virtual fields method, it is generally recommended for material mechanical properties extraction.

The characterization of material anisotropic properties at high strain rates remains a challenging work due to the limitations of the conventional dynamic testing methods. In this study, a novel inverse identification strategy is proposed based on the dynamic virtual fields method (VFM) to enable a simultaneous identification of the anisotropic constitutive parameters from a single high speed impact test of a non-uniform specimen. This is implemented by taking advantage of the heterogeneous full-field strain and inertial acceleration data, in which the restrictions of one-dimensional stress wave propagation and homogeneous deformation states for the conventional methods are released. Specifically, the dynamic VFM-based identification algorithm is developed first. Then, several virtual impact tests with different specimen configurations are designed to provide distinct stress/strain distributions. Next, the simulated full-field strain and acceleration data are utilized to extract the input target orthotropic stiffness components. The results show that the identification accuracy of the anisotropic parameters is highly dependent on the heterogeneity of the stress/strain distributions. Also, the identification results are unstable for different time steps but can be significantly improved using a minimization algorithm on multiple time steps. Finally, the influences of noise, impact speed and loading mode are analyzed in the sensitivity study.

Current high strain rate testing techniques typically rely on the split‐Hopkinson bar (SHB). The early response in an SHB test is corrupted by inertia making it difficult to accurately characterise the transition from elasticity to plasticity for metals. Therefore, a new test method is required. This article is the second in a two‐part series which aims at developing a new high strain rate test for elasto‐plasticity identification using the image‐based inertial impact (IBII) method. The goal of this article is to validate the new method experimentally using IBII tests on aluminium 6082‐T6 (minimal rate sensitivity) and stainless steel 316L (rate sensitive). Comparison of the quasi‐static and dynamic stress–strain curves for the aluminium case showed minimal difference providing experimental validation of the method. The same comparison for the steel showed that the method was able to detect rate sensitivity. [ABSTRACT FROM AUTHOR]

Current high strain rate testing procedures generally rely on the split Hopkinson bar (SHB). In order to gain accurate material data with this technique, it is necessary to assume the test sample is in a state of quasi‐static equilibrium so that inertial effects can be neglected. During the early portion of an SHB test, it is difficult to satisfy this assumption making it challenging to investigate the elastic–plastic transition for metals. With the development of ultra‐high speed imaging technology, the image‐based inertial impact (IBII) test has emerged as an alternative to the SHB. This technique uses full‐field measurements coupled with the virtual fields method to identify material properties without requiring the assumption of quasi‐static equilibrium. The purpose of this work is to develop the IBII method for the identification of elasto‐plasticity in metals. In this paper (part 1), the focus is on using synthetic image deformation simulations to analyse identification errors for two plasticity models, a simple linear hardening model and a modified Voce model. Additionally, two types of virtual fields are investigated, a simple rigid body virtual field and the recently developed sensitivity‐based virtual fields. The results of these simulations are then used to select optimal processing parameters for the experimental data analysed in part 2. [ABSTRACT FROM AUTHOR]

The isotropic and anisotropic behaviors are considered as the important formats of the constitutive behaviors, and can also be called the global properties. To improve the identification ability of virtual fields method (VFM) when the global properties are unknown, this paper proposes the strain correlation method (SCM) to determine the global properties before the parameter identification using the VFM. Firstly, the basic principle of SCM is described in detail. Then, the feasibility and accuracy of SCM are verified through the numerical experiments based on the three-point bending configuration and the real experiment of polymethyl methacrylate (PMMA). The influence of the additive Gaussian white noise, local errors in the strain fields, and missing data at the specimen edges on the characterization results are evaluated. The results show that the SCM has good noise immunity and lower accuracy requirements for the strain fields. As an application, the mechanical properties of Ti-6Al-4V alloys fabricated by selective laser melting (SLM) are characterized by the SCM. The results show that the alloys are isotropic, and the isotropic VFM is utilized to determine the mechanical parameters. By using the SCM, the accuracy of identification results can be improved for the isotropic or bidirectional reinforced orthotropic materials when using VFM. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

Background: Experimental, fully three-dimensional mechanical characterization of opaque materials with arbitrary geometries undergoing finite deformations is generally challenging. Objective: We present a promising experimental method and processing pipeline for acquiring and processing full-field displacements and using them toward inverse characterization using the Virtual Fields Method (VFM), a combination we term MR-u. Methods: Silicone of varying crosslinker concentrations and geometries is used as the sample platform. Samples are stretched cyclically to finite deformations inside a 7T MRI machine. Synchronously, a custom MRI pulse sequence encodes the local displacement in the phase of the MR image. Numerical differentiation of phase maps yields strains. Results: We present a custom image processing scheme for this numerical differentiation of MRI phase-fields akin to convolution kernels, as well as considerations for gradient set calibration for data fidelity. Conclusions: The VFM is used to successfully determine hyperelastic material properties, and we establish best practice regarding virtual field selection via equalization. [ABSTRACT FROM AUTHOR]

A key limitation of current moderate and high strain rate test methods is the need for external force measurement. For high loading rate hydraulic machines, ringing in the load cell corrupts the force measurement. Similarly, the analysis of split-Hopkinson bar tests requires the assumption that the specimen is in a state of quasi-static equilibrium. Recently, image-based inertial test methods have shown that external force measurement is not required if full-field measurements are available and inertial effects are significant enough. In this case the load information is provided by the acceleration fields which are derived from full-field displacement measurements. This article describes a new image-based inertial test method that can be used for simultaneous quasi-static and high strain rate stiffness identification on the same test sample. An experimental validation of the new test method is provided using PMMA samples. A major advantage of this new test method is that it utilises a standard tensile test machine and the only specialist equipment that is required is an ultra-high speed camera. [ABSTRACT FROM AUTHOR]

In this study, pressure distributions were reconstructed from phase-locked surface deformation measurements on a thin plate. Slope changes on the plate surface were induced by an external flow interacting with the specimen and measured with a highly sensitive deflectometry setup. The Virtual Fields Method (VFM) was used to obtain pressure reconstructions from the processed surface slopes and the plate material constitutive mechanical parameters. The applicability of the approach in combination with phase-locked measurements is demonstrated using a synthetic jet setup generating a periodic flow in air. Phase-averaging slope data allows mitigating random noise effects and resolving low-range differential pressure amplitudes despite the turbulent flow. The size of the spatial structures of the investigated low amplitude flow events identified in full-field with the present method are O (1) mm , which is beyond the capabilities of other available surface pressure measurement techniques. Challenges and limitations in achieving the metrological performance for resolving the observed surface slopes of O (0.1) mm km − 1 are described and improvements for future applications are discussed. [ABSTRACT FROM AUTHOR]

In the present work Image-Based Inertial Impact (IBII) tests are performed on Ti6Al4V material. The IBII test uses an impact on the edge of the specimen to generate a short pulse that loads the specimen. Three specimen geometries have been tested: a classic rectangular specimen, and two specimen geometries with stress concentrating geometries (i.e. a hole and notches) to enhance high levels of plastic strain. Full-field measurement of the acceleration and strain are successfully used in combination with the Virtual Fields Method (VFM) to identify the strain rate sensitivity parameter of the Johnson-Cook model. The strain/strain rate spectra covered by each specimen are analysed. Finally, the influence of the virtual field used in the identification process is discussed as well as the simultaneous identification of the Johnson-Cook model strain rate sensitivity parameter and the strain rate threshold parameter. [ABSTRACT FROM AUTHOR]

The aim of the present study is to explore the combination of a biaxial test with a cruciform specimen and the virtual fields method to develop an efficient strategy for simultaneous identification of material parameters related with hardening and anisotropy in plasticity. In a first step, three cruciform geometries are evaluated as potential candidates to generate an experimental database for the calibration of the classical Hill'48 yield criterion and Swift's hardening law. In a second step, the geometry with the best results is used to calibrate YLD2000-2d yield criterion and Swift's hardening law. Numerical results are used as virtual experimental full-field measurements, which allows the comparison of the identified solution with the input material parameters. The accuracy of the identified material parameters is thoroughly assessed through the analysis of flow stress curve evolution, normalised yield stresses and plastic anisotropy coefficients, and for the last step, yield locus prediction. The results show the potential of this combination to identify simultaneously the material parameters related to hardening and anisotropy with a single test. [ABSTRACT FROM AUTHOR]

This work presents a methodology for reconstructing full-field surface pressure information from deflectometry measurements on a thin plate using the Virtual Fields Method (VFM). Low-amplitude mean pressure distributions of the order of few O (100) Pa from an impinging air jet are investigated. These are commonly measured point-wise using arrays of pressure transducers, which require drilling holes into the specimen. In contrast, the approach presented here allows obtaining a large number of data points on the investigated specimen without impact on surface properties and flow. Deflectometry provides full-field deformation data on the specimen surface with remarkably high sensitivity. The VFM allows extracting information from the full-field data using the principle of virtual work. A finite element model is employed in combination with artificial grid deformation to assess the uncertainty of the pressure reconstructions. Both experimental and model data are presented and compared to show capabilities and restrictions of this method. [ABSTRACT FROM AUTHOR]

Featured Application: 3D deformation measurement and elasticity quantification of biological tissue and biomaterials. The three-dimensional (3D) mechanical property characterization of biological tissues is essential for physiological and pathological studies. A digital volume correlation (DVC) and virtual fields method (VFM) based 3D optical coherence elastography (OCE) method is developed to quantitatively measure the 3D full-field displacements, strains and elastic parameters of layered biomaterials assuming the isotropy and homogeneity of each layer. The integrated noise-insensitive DVC method can obtain the 3D strain tensor with an accuracy of 10%. Automatic segmentation of the layered materials is realized based on the full field strain and strain gradient. With the strain tensor as input, and in combination with the segmented geometry, the Young's modulus and Poison's ratio of each layer of a double-layered material and a pork specimen are obtained by the VFM. This study provides a powerful experimental method for the differentiation of various components of heterogeneous biomaterials, and for the measurement of biomechanics. [ABSTRACT FROM AUTHOR]

A proper understanding of material mechanical properties is important in designing and modelling of components. As a part of a study on the structural integrity, the Digital Image Correlation technique was used to obtain the full-field strain distribution during a tensile test of the specimens. In this study, the elastic and plastic properties of Al6061 alloy has been carried out using both the uniform stress method and the virtual fields method involving digital image correlation technique. In uniform stress methodology, full range stress-strain curves are obtained using the whole field strain measurement using Digital Image Correlation. Recently, the virtual fields method is gaining a lot of popularity in domain characterization as it is robust, accurate and faster. Young's modulus, Poisson's ratio, yield strength, strength coefficient and strain hardening exponent are the parameters extracted using both uniform stress method and virtual fields method. The parameter variation obtained by both uniform stress method and the virtual fields method are compared very well. The Virtual Fields Method provides a theoretically sound basis for developing robust optimization constructs to estimate local material properties, including spatial variations in hardening exponent and strength coefficient. Due to various advantages associated with virtual fields method, it is generally recommended for material mechanical properties extraction. [ABSTRACT FROM AUTHOR]

Thermal barrier coatings (TBCs) are widely applied in thermal components to protect metallic components. Owing to the complex layered structure of TBCs and difficult preparation of coating, the mechanical characterization of TBCs should be of primary importance. With regard to TBCs, this study deals with the constitutive parameters identification of bi-material. Considering the complex construction and boundary of bi material, the virtual fields method (VFM) was employed in this study. A methodology based on the optimized virtual fields method combined with moiré interferometry was proposed for the constitutive parameters identification of bi-material. The feasibility of this method is verified using simulated deformation fields of a two-layer material subjected to three point bending loading. As an application, the deformation fields of the TBC specimens were measured by moiré interferometry. Then, the mechanical parameters of the coating were identified by the proposed method. The identification results indicate that Young's modulus of the TBC top coating is 89.91  GPa, and its Poisson's ratio is 0.23. [ABSTRACT FROM AUTHOR]

This paper investigates modifications to the Brazilian disk and compact tension C(T) standard test geometries to allow full-field extraction of elastic tensile and toughness properties. A central notch has been introduced in the Brazilian disk to allow for the extraction of fracture properties, while the C(T) sample has been lengthened to allow for the extraction of tensile properties. Full-field displacements, measured on samples prepared from PMMA (Polymethyl methacrylate) using digital image correlation are analysed under the assumption of linear elasticity. First, the Virtual Fields Method is applied to estimate Young’s Modulus (E) and Poisson’s Ratio (v). These tensile properties are then used to determine estimates for the peak stress intensity factor (SIF) prior to fracture through a non-linear least squares field fitting approach. Test results show that a modified disk sample performed well in both analysis approaches with relative errors obtained as: 2.1% in E, 12.9% in v and 1% in SIF. A modified C(T) sample obtained relative errors of: 1.5% in E, 40.2% in v and 1% in SIF. Furthermore it is shown that field fitting approaches are less sensitive to error in v than error in E. The presented evidence aims to guide future experiments utilizing full-field techniques to study material properties in the framework of fracture mechanics, in particular in applications where limited material for obtaining tensile properties is available. [ABSTRACT FROM AUTHOR]

• A multi-parameter identification strategy for the SENT specimen was proposed, which can simultaneously characterize the elastic constants and stress intensity factors through a single experiment. • The proposed strategy was based on the virtual fields method (VFM), which was the new development of VFM for SENT testing in linear elastic fracture mechanics. • The finite element (FE) schema was applied to generate the virtual fields, which automatically satisfied the complex boundary conditions and made this approach more universal and programmable. The elastic constants are significant indicators to evaluate the damage status during fatigue, whereas the stress intensity factors (SIFs) are essential parameters to study the fatigue crack growth rate. However, due to different configurations of the specimens, tests on these two types of parameters are typically performed independently. To simultaneously characterize the elastic constants and SIFs through a single experiment, the virtual fields method (VFM) for single edge notched tension (SENT) specimens was developed in this study. The proposed approach combines the full-field measurements and virtual fields generated by the finite element method (FEM), which can automatically satisfy the complex boundary conditions and effectively minimize the influence of the plastic zone near the crack tip. The feasibility of this approach was verified by simulation experiments and several key factors involving identification accuracy were investigated. In the application, the SENT testing of Ni-based superalloys was implemented by the digital image correlation (DIC) technique, and the identification results were compared with FEM, the least-squares method (LSM), and theoretical values. The results of the current study demonstrate the excellent accuracy and robustness of this approach, indicating its broad application in the engineering field. [ABSTRACT FROM AUTHOR]

Orthotropic stiffness components of Pinus pinaster Ait. wood are simultaneously determined by means of a heterogeneous plate bending test. The proposed inverse identification approach couples full-field slope measurements provided by deflectometry with the virtual fields methods. Wooden plates oriented in the longitudinal–radial and longitudinal–tangential material planes were manufactured. A procedure was implemented to allow suitable specular reflective coating of the wooden plates, required in the deflectometry technique. Reconstructed curvature fields, applied load and plate dimensions were input in virtual fields methods for material parameter identification, assuming Kirchhoff–Love classical plate theory. Several virtual fields and load cases were analysed to address the identifiability of the method. The values of the orthotropic elastic constants obtained from the proposed approach were found in good agreement with regard to reference ones for the same species and determined from classical tensile, compression and shear mechanical tests. [ABSTRACT FROM AUTHOR]

In this paper, an effective general method is proposed to determine the optimal virtual fields and extract the elastic constants and the thermal expansion coefficients of orthotropic composite materials from the coupled thermo-mechanical deformation fields simultaneously. First, a practical analysis procedure is implemented based on the assumed material parameters with orthotropic in-plane properties to determine the effective virtual fields for three-point bending test configuration. Six thermo-mechanical parameters of three kinds of orthotropic composite materials are decoupled and identified by the determined effective virtual fields to verify its independence on material parameters. Second, the influences of specimen dimensions and external loadings on identified material parameters are investigated to validate the applicability of effective virtual fields under different test configurations. Finally, the optimal virtual fields are determined by means of a numerical method and its capacity for decreasing the noise influence is investigated quantitatively. The research work will provide an effective method to determine the optimal virtual fields for decoupling the thermo-mechanical coupled deformation of orthotropic composite materials. [ABSTRACT FROM AUTHOR]

This paper describes an approach for identifying the magnitude and location of both stationary and transient mechanical loadings applied to a thin rectangular simply supported plate. Full-field deflectometry measurements and the virtual fields method are used with the local equilibrium equation of the plate in the time domain to solve the force reconstruction problem, whereas previous work by the authors used this last equation in the frequency domain. As a result, it is possible to reconstruct load time history in addition to magnitude and location. Experimental results of this complete identification are presented for two different instrumented mechanical exciters: electrodynamic shaker and impact hammer for stationary and transient excitations, respectively. The approach is then applied to determine the location and time of multiple unknown transient excitations produced by a set of impacting metal marbles. [ABSTRACT FROM AUTHOR]

Abstract: Magnetic resonance elastography (MRE) has been used to estimate isotropic myocardial stiffness. However, anisotropic stiffness estimates may give insight into structural changes that occur in the myocardium as a result of pathologies such as diastolic heart failure. The virtual fields method (VFM) has been proposed for estimating material stiffness from image data. This study applied the optimised VFM to identify transversely isotropic material properties from both simulated harmonic displacements in a left ventricular (LV) model with a fibre field measured from histology as well as isotropic phantom MRE data. Two material model formulations were implemented, estimating either 3 or 5 material properties. The 3‐parameter formulation writes the transversely isotropic constitutive relation in a way that dissociates the bulk modulus from other parameters. Accurate identification of transversely isotropic material properties in the LV model was shown to be dependent on the loading condition applied, amount of Gaussian noise in the signal, and frequency of excitation. Parameter sensitivity values showed that shear moduli are less sensitive to noise than the other parameters. This preliminary investigation showed the feasibility and limitations of using the VFM to identify transversely isotropic material properties from MRE images of a phantom as well as simulated harmonic displacements in an LV geometry. [ABSTRACT FROM AUTHOR]

Moiré interferometry is an effective full-field deformation measurement technique and has been utilized for mechanical behavior analysis of materials and structures. For isotropic materials, Moiré patterns can be obtained by performing standard tests, such as, tensile and bending tests, to calculate the displacement and strain. Then, the mechanical properties can be characterized. However, standard tests are not sufficient to characterize the mechanical parameters of anisotropic materials due to the complexity of their material properties. Thus, in this work, Moiré interferometry was combined with the Virtual Fields Method to obtain the four in-plane elastic constants (Q11, Q22, Q12, and Q66) of orthotropic materials in the form of a diametrically compressed disk. Firstly, according to finite element method simulation results, optimized parameters can be achieved when the principal direction of the material does not coincide with the loading direction, making the loading configuration antisymmetric. Therefore, Moiré interferometry experiment was simulated to demonstrate the feasibility of measurement in the antisymmetric configuration. Finally, the Q11, Q22, Q12 and Q66 values of a unidirectional carbon fiber composite were measured in a real Moiré interferometry experiment using the proposed method, yielding results that agreed closely with those obtained using the strain gauges. [ABSTRACT FROM AUTHOR]

The virtual fields method (VFM) is generally used with two-dimensional digital image correlation (2D-DIC) or grid method (GM) for identifying constitutive parameters. However, when small out-of-plane translation/rotation occurs to the test specimen, 2D-DIC and GM are prone to yield inaccurate measurements, which further lessen the accuracy of the parameter identification using VFM. In this work, an easy-to-implement but effective “special” stereo-DIC (SS-DIC) method is proposed for accuracy-enhanced VFM identification. The SS-DIC can not only deliver accurate deformation measurement without being affected by unavoidable out-of-plane movement/rotation of a test specimen, but can also ensure evenly distributed calculation data in space, which leads to simple data processing. Based on the accurate kinematics fields with evenly distributed measured points determined by SS-DIC method, constitutive parameters can be identified by VFM with enhanced accuracy. Uniaxial tensile tests of a perforated aluminum plate and pure shear tests of a prismatic aluminum specimen verified the effectiveness and accuracy of the proposed method. Experimental results show that the constitutive parameters identified by VFM using SS-DIC are more accurate and stable than those identified by VFM using 2D-DIC. It is suggested that the proposed SS-DIC can be used as a standard measuring tool for mechanical identification using VFM. [ABSTRACT FROM AUTHOR]

An inverse approach based on the virtual fields method (VFM) is presented to identify the material hardening parameters under dynamic deformation. This dynamic-VFM (D-VFM) method does not require load information for the parameter identification. Instead, it utilizes acceleration fields in a specimen’s gage region. To investigate the feasibility of the proposed inverse approach for dynamic deformation, the virtual experiments using dynamic finite element simulations were conducted. The simulation could provide all the necessary data for the identification such as displacement, strain, and acceleration fields. The accuracy of the identification results was evaluated by changing several parameters such as specimen geometry, velocity, and traction boundary conditions. The analysis clearly shows that the D-VFM which utilizes acceleration fields can be a good alternative to the conventional identification procedure that uses load information. Also, it was found that proper deformation conditions are required for generating sufficient acceleration fields during dynamic deformation to enhance the identification accuracy with the D-VFM. [ABSTRACT FROM AUTHOR]

An investigation into the use of a novel curved-beam composite specimen is conducted to measure the interlaminar (through-thickness) tensile properties of carbon-fiber-reinforced plastic. A combination of a numerical model and full-field displacement/strain measurement with moiré interferometry is utilized in this study. Through-thickness material properties are identified from the measured displacement distribution using the virtual fields method. Because of the shape and the loading condition of the proposed curved composite beam, both tensile and shear stresses exist in the through-the-thickness direction. Therefore, the interlaminar tensile modulus, as well as the interlaminar shear modulus, can be evaluated. The measurement results by moiré interferometry provide the material properties through inverse analysis. [ABSTRACT FROM PUBLISHER]

Fused deposition modelling (FDM), a widely used rapid prototyping process, is a promising technique in manufacturing engineering. In this work, a method for characterizing elastic constants of FDM-fabricated materials is proposed. First of all, according to the manufacturing process of FDM, orthotropic constitutive model is used to describe the mechanical behavior. Then the virtual fields method (VFM) is applied to characterize all the mechanical parameters $$(Q_{11}$$ , $$Q_{22}$$ , $$Q_{12}$$ , $$Q_{66})$$ using the full-field strain, which is measured by digital image correlation (DIC). Since the principal axis of the FDM-fabricated structure is sometimes unknown due to the complexity of the manufacturing process, a disk in diametrical compression is used as the load configuration so that the loading angle can be changed conveniently. To verify the feasibility of the proposed method, finite element method (FEM) simulation is conducted to obtain the strain field of the disk. The simulation results show that higher accuracy can be achieved when the loading angle is close to $$30^{\circ }$$ . Finally, a disk fabricated by FDM was used for the experiment. By rotating the disk, several tests with different loading angles were conducted. To determine the position of the principal axis in each test, two groups of parameters $$(Q_{11}$$ , $$Q_{22}$$ , $$Q_{12}$$ , $$Q_{66})$$ are calculated by two different groups of virtual fields. Then the corresponding loading angle can be determined by minimizing the deviation between two groups of the parameters. After that, the four constants $$(Q_{11}$$ , $$Q_{22}$$ , $$Q_{12}$$ , $$Q_{66})$$ were determined from the test with an angle of $$27^{\circ }$$ . [ABSTRACT FROM AUTHOR]

The paper aims at providing an overview of the research activities performed in the two past decades in the field of materials characterisation under dynamic loadings (i.e. from 10 − 3 s − 1 to 10 + 3 s − 1 for structural crashworthiness and impact applications) and identification of parameters of constitutive behaviour and damage models. The different testing devices to load the material sample on the expected strain rate range are presented and discussed first, including the different experimental measurement techniques applied to analyse the stress-strain curves. From the normalised direct approach, two different numerical approaches, based on inverse problem resolution techniques, are introduced and discussed: the well-known Finite Element Model Updating method and the most advanced one based on the Virtual Fields Method, that enables to take the full advantages of full-field measurement techniques, such as the Digital Image Correlation method, into account. Applications for different materials or structures (including riveted and welded joints) and models, such as viscoplasticity and damage, are given to illustrate these advanced methods. [ABSTRACT FROM AUTHOR]

Full-field deformation response of a copper bar impacted in a three point bending setup is studied. Ultra-high speed photography (frame rate: 5 × 10 6 fps) in conjunction with digital image correlation is utilized to study the in-plane deformation kinematics of the impacted specimen over 25.4 μs duration after the impact. Distributions of temporal derivatives, i.e. strain rate, velocity and acceleration, over the area of interest are computed using the in-plane displacement and strain fields measured from image correlation. Complex deformation patterns resulting from wave propagation and inertial loading are studied and discussed. Having deformed the specimen well into the plastic regime, the idea initially proposed by Pierron et al. (2011) is extended to identify visco-plastic constitutive response of the examined material by incorporating an analysis based on the method of virtual fields. Accordingly, a strain rate sensitive constitutive model for the copper specimen is identified by reconstructing the inertia and the impact forces using full-field acceleration and reaction force measurements, respectively. [ABSTRACT FROM AUTHOR]