The virtual fields method for identifying viscoelastic properties based on stress-sensitivity virtual fields

The image-based inertial impact test has previously shown that inertial effects generated with high-strain-rate loading can be used to measure the dynamic constitutive properties of composites at strain rates on the order of 1600,{rm s}^{-1}. This work represents an important next step in exploring the potential of this concept with two tests presented where loading heterogeneity is exploited to measure the interlaminar shear modulus and stress–strain behaviour at high strain rates. The first test configuration used a short-beam with asymmetric loading to activate the shear behaviour. The virtual fields method was used to directly identify the interlaminar shear modulus from heterogeneous full-field maps of strain and acceleration. Simulated experiments were used to optimise the test configuration, select optimal smoothing parameters, and quantify uncertainty from grid rotation on the shear modulus identifications. The test was validated experimentally with three different virtual fields identifying an average shear modulus ranging from 5.7 to 5.9 GPa measured at 1600,{rm s}^{-1}, representing a 16–19% increase compared to quasi-static measurements. The shear modulus could also be identified from shear introduced into specimens tested in the standard, end-on interlaminar IBII configuration from slight in-plane misalignments of the impactor. The identified value of 5.6 GPa validates measurements from the first configuration and also demonstrates the capability to identify multiple interlaminar stiffness parameters from a single test.

Inferring material parameters of soft materials, especially soft tissues from experiments, has always been a challenge because of the inaccuracy of specimen geometry, difficulty in proper gripping and difficulty in obtaining reliable displacement data. If one is able to obtain full field data with Digital Image Correlation or other similar techniques, we show that in spite of poor quality experimental data, it is still possible to get the material parameters using Virtual Fields Method with carefully chosen virtual fields. We demonstrate the approach by two cases. The first is Ecoflex (silicone rubber) under biaxial deformations at controlled loading rates, and the second is rat skin samples with imperfect shape under biaxial deformations at controlled loading rates. In both cases, rather than trying to obtain homogeneous deformation conditions, we use a technique based on the Virtual Fields Method to extract the material properties from the deformation of the specimen collected by Digital Image Correlation (DIC) and the force load measured by the load sensor. In order to apply loads in two principal directions simultaneously, a custom biaxial set-up was built and mounted into a uniaxial Instron tensile set-up. We show here that in spite of significant inhomogeneity in the deformation, errors, and missing data in the measured displacement field, we are still able to recover the material properties of the soft solid by suitable choices of virtual fields.

The aim of this work is to identify parameters driving constitutive equations of materials with displacement field measurements carried out by image stereo-correlation during an unidirectional tensile test. We evaluate two identification techniques. The first one is the virtual fields method which consists in writing the principle of virtual work with particular virtual fields. It is generally used in the case of linear elasticity and it requires a perfect knowledge of the model in terms of boundary condition since the virtual fields used must be kinematically admissible. This method allows to determine parameters by a direct and fast calculation, without iterations. The second method is the finite element model updating method. It consists in finding constitutive parameters that achieve the best match between finite element analysis quantities and their experimental counterparts. This method is more adaptable than the virtual field method but it needs to spend more calculation time.

In this work, the sensitivity-based virtual fields have been applied to identify two anisotropic plasticity models (Hill48, Yld2000-2D) using a deep-notched tensile test performed on flat samples of cold-rolled sheet of DC04 steel. The material was characterised using the standard protocol to obtain the reference sets of parameters. Deformation data was obtained during deep-notched tests using stereo digital image correlation and the virtual fields method was employed to identify material parameters. It was found that the sensitivity-based virtual fields outperform the standard user-defined virtual fields in terms of accuracy.

Finite element-based virtual fields method with pseudo-real deformation fields for identifying constitutive parameters

Extension of the virtual fields method to elasto-plastic material identification with cyclic loads and kinematic hardening

Heterogeneous materials are widely applied in many fields. Owing to the spatial variation of its constitutive parameters, the mechanical characterization of heterogeneous materials is very important. The virtual fields method has been used to identify the constitutive parameters of materials. However, there is a limitation: constitutive parameters of one material have to be a priori; then, constitutive parameters of the other one can be identified. Aiming at this limitation, this article presents a method to identify the constitutive parameters of heterogeneous orthotropic bimaterials under the condition that constitutive parameters of both materials are all unknown from a single test. A constitutive parameter identification method of orthotropic bimaterials based on optimized virtual field and digital image correlation is proposed. The feasibility of this method is verified by simulating the deformation fields of a two‐layer material under three‐point bending load. The results of numerical experiments with FEM simulations show that the weighted relative error of the constitutive parameter is less than 1%. The results suggest that the variation coefficient‐to‐noise ratio can perform a priori evaluation of a confidence interval on the identified stiffness components. The results of numerical experiments with DIC simulations show that the weighted relative error is 1.44%, which is due to the noise in the strain data calculated by DIC method.

Special virtual fields for the direct determination of material parameters with the virtual fields method. 2––Application to in-plane properties

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.

An alternative to modal analysis for material stiffness and damping identification from vibrating plates

Applying the virtual fields method to determine the through-thickness moduli of thick composites with a nonlinear shear response

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.

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.

The identification of nonhomogeneous elastic property distributions has been traditionally achieved with well acknowledged optimization based inverse approaches, but when full-field displacement measurements are available, the virtual fields method (VFM) can be computationally more efficient by converting the large-scale optimization problem into multiple small-scale optimization problems. A possible downside of the VFM so far was not to take into account prior knowledge, which is often available and needed when there is a very large number of unknowns and the inverse problem is ill-posed. In this work, different approaches are proposed for introducing regularization into the VFM, aiming to penalize the local variations of identified stiffness properties in order to reduce the effects of uncertainty in the inverse problem resolution. The feasibility and accuracy of the regularized VFM are tested through several numerical and experimental datasets. It is shown that the main advantage of the novel VFM approaches is the low computational cost, as large-scale inverse problems with 10,000 unknown parameters can be solved within several seconds using a standard personal computer. Although the regularized VFM can successfully detect a stiff inclusion in a soft solid with high accuracy, regularization also introduces unexpected spurious effects in the results, blurring the interface between soft and stiff regions. We also observed that the regularization did not improve the smoothness significantly due to local effects of the small-scale optimization problem introduced in the proposed VFM method. Therefore, traditional regularization, which penalizes local variations of identified stiffness properties, can be combined with the VFM to solve inverse problems with a high computational efficiency, but supplemental regularization conditions will need to be adapted in the future to better delineate soft-stiff interfaces with this methodology.

The paper presents an experimental application of a method leading to the identification of the elastic and damping material properties of isotropic vibrating plates. The theory assumes that the searched para meters can be extracted from curvature and deflection fields measured on the whole surface of the plate at two particular instants of the vibrating motion. The experimental application consists in an original excitation fixture, a particular adaptation of an optical full-field measurement technique, a data preprocessing giving the curvature and deflection fields and finally in the identification process using the Virtual Fields Method (VFM). The principle of the deflectom etry technique used for the measurements is presented. First results of identification on an acrylic plate are presented and compared to reference values. The used theoretical principle only assumes the global equilibrium of the observed part of the coupon. Therefore the identification remains insensible to the parasitic effects of the boundary conditions. This is confirmed by the results of the last presented experimentation. Keywords: vibration, identification, Virtual Fields Method, damping, deflectometry, slope fields, grid method, image processing.