Michel Grédiac

The Virtual Fields Method

The Virtual Fields Method (VFM) is one of the techniques developed to identify the parameters governing constitutive equations, the experimental data processed for this purpose being displacement or strain fields. It will be shown in this chapter that one of its main advantages is the fact that, in several cases, the sought parameters can be directly found from the measurements, without the need of calculating the stress with a numerical tool such as a finite element programme.

Special virtual fields for the direct determination of material parameters with the virtual fields method. 1–Principle and definition

This paper deals with the direct and simultaneous estimation of parameters used in some constitutive laws. Whole-field data captured in mechanical configurations which give rise to heterogeneous stress fields are processed. Since no analytical relationship is available between measured data and unknown parameters, a specific procedure based on a relevant use of the principle of virtual work is proposed. The main advantage is to provide directly the unknown parameters. The main features of the method are described in the paper.

A Numerical and Experimental Study of Woven Composite Pin-Joints

A numerical and experimental study was carried out to determine the stiffness and the bearing strength of bolted woven composite joints. The main objective was to investigate the possibility of predicting the properties of the joint from the properties of the material measured with standard tests. A refined finite element model was developed in which the nonlinearities due to both the material and the contact angle between the pin and the hole were taken into account. Particular attention was paid to account for the influence of the clearance which has been shown to be very significant. In conclusion, good agreement between experimental results and numerical predictions has been obtained.

Investigation of the grid method for accurate in-plane strain measurement

This paper deals with the accurate measurement of the in-plane strain components on a deformed specimen using the grid method. A crossed grid transferred on the specimen surface is used for this purpose. Images of this grid are captured with a CCD camera before and after deformation. The complete strain state is deduced by processing these images with a suitable procedure described in the paper. The originality of this procedure is twofold. The first one is to determine the phase derivatives directly from the images. The second one is to compensate local variations of the grid pitch as well as local rotations of the grid, because these two phenomena may corrupt the final strain maps if they are not taken into account. The metrological performance of this procedure is assessed and discussed in terms of accuracy, measurement noise and spatial resolution. Two types of tests are used for this purpose: a translation and a rotation. These tests simulate the effect of rigid-body motion in a strain field that is rigorously equal to zero. One example finally illustrates the procedure: a tensile test performed on an open-hole specimen.

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

This paper deals with the direct identification of mechanical parameters that govern the in-plane constitutive law of orthotropic materials. Those parameters are extracted from heterogeneous strain fields that occur in a short beam specimen tested in a Iosipescu fixture. The procedure used is the virtual fields method with special virtual fields. The case of linear elasticity is first addressed. It is shown that the parameters are directly extracted with this method: no iterative calculations are required. The stability is also discussed in different cases. A non-linear shear response is then considered. The parameter that governs this non-linearity is also directly identified with the special virtual fields.

Novel procedure for complete in-plane composite characterization using a single T-shaped specimen

This paper deals with the direct identification of the in-plane elastic properties of orthotropic composite plates from heterogeneous strain fields. The shape of the tested specimen is that of a T subjected to a complex stress state. As a result, the entire set of unknown parameters is directly involved in the strain and displacement responses of the sample. No exact analytical solution is available for such a geometry, and a specific strategy is used to identify the different stiffness components from the whole-field displacements measured over the tested specimen with a suitable optical method. The paper focuses mainly on the experimental aspects of the procedure, and an example of mechanical characterization of a fabric-reinforced composite plate is given.

Identification of the through-thickness moduli of thick composites from whole-field measurements using the Iosipescu fixture: theory and simulations

This paper presents a method to measure the four through-thickness moduli of a thick coupon subjected to a shear-bending load obtained with the Iosipescu fixture. The identification method relies on the global equilibrium of the specimen written with the principle of virtual work with different virtual fields. It requires the measurements of the in-plane strains over the whole surface of the specimen. The choice of a set of virtual fields is presented and validated using finite element simulated measurements. The stability of the method is checked by adding simulated experimental noise. The stability is found to be compatible with experimental implementation except for Poissons ratio.

Special virtual fields for the direct determination of material parameters with the virtual fields method. 3. Application to the bending rigidities of anisotropic plates

This paper deals with the direct identification of bending rigidities of thin anisotropic plates. These parameters are extracted from an heterogeneous strain field which takes place onto the top surface of a bent plate. The loading conditions are such that no closed-form solution is available for the deflection/slope/curvature fields. The procedure presently used is the virtual fields method with “special” virtual fields. It is shown that the unknown parameters are directly extracted with this method since no iterative calculations are required. The parameters are in fact directly equal to the virtual work of the applied loading with the special virtual displacement fields. The headlines of the method are recalled in the first part of the paper. They are then applied in the case of anisotropic bent plates. The accuracy and the stability of the procedure are finally discussed through some relevant examples.

Identification of the through-thickness properties of thick laminated tubes using the virtual fields method

This paper presents a method to determine the four through-thickness stiffnesses of thick laminated composites. Only one specimen submitted to one test is required. The procedure is based on a suitable use of the principle of virtual work with four independent virtual fields. This leads to a system of four linear equations where the through-thickness stiffnesses are the unknowns. The system is finally inverted to determine the stiffnesses. Finite element simulations have been carried out to validate the approach and to show its stability.

Closed-form solution for the cross-section warping in short beams under three-point bending

The transverse shear behavior of composite beams can be critical and therefore must be properly represented by the various models of structures usually employed either to predict their behavior or to identify experimentally their properties. In this work, a simple analytical approach based on higher-order theories is proposed that accounts for the cross-section warping in beams. Then the solution of a beam under three-point bending is solved and the accuracy of both displacement and strain distribution predictions is shown through comparisons with the FE analysis results. In this formulation, the cross-section locking at mid-span is ensured owing to the dependence of the transverse shear strain upon the position along the beam axis. Eventually, comparisons with experimental measurements demonstrate the ability of these simple analytical solutions to grasp the main phenomena which control the response of composite beams under three-point bending loading, i.e. the arising of transverse shear.