Identification of orthotropic elastic constants using the Eigenfunction Virtual Fields Method

The Virtual Fields Method (VFM – Pierron and Grediac, 2012), an inverse method based on the principle of virtual work (PVW), is being increasingly used to estimate mechanical properties of materials from full-field deformations obtained from techniques such as Digital Image Correlation, moiré and speckle interferometry and grid methods. By making specific choices for virtual fields (VFs) in PVW, one obtains a system of algebraic equations, which is then solved for the unknown material constants. Recently, a new variant of VFM, known as the Eigenfunction Virtual Fields Method (EVFM) has been proposed (Subramanian, 2013). In EVFM, principal components of the measured (i.e. true) strain fields are used to systematically generate VFs. We extend EVFM to orthotropic elastic materials in this work, and estimate the relevant material parameters from full-field strain data generated from a finite-element model of an unnotched Iosipescu test. Varying levels of Gaussian white noise are added to the synthetic strain data to evaluate the sensitivity of EVFM to input noise. It is observed that for low to moderate noise, the material properties estimated by the proposed method are relatively insensitive to noise. However, when noise levels are high, the proposed method yields large variance in some of the computed properties when compared to the state-of-the-art optimized piecewise continuous VFM (Toussaint et al., 2006; Pierron and Grediac, 2012). Some of the large variance in properties estimated from noisy data using EVFM is traced to the sensitivity of the third dominant eigenfunction and modifications to the proposed method to address this issue are suggested.     

In-situ mechanical properties identification of 3D particulate composites using the Virtual Fields Method

This paper presents an identification procedure based on the Virtual Fields Method (VFM) for identifying in situ mechanical properties of composite materials constitutive phases from 3D full-field measurements. The new procedure, called the Regularized Virtual Fields Method (RVFM), improves the accuracy of the VFM thanks to the imposition of mechanical constraints derived from an appropriate homogenization model. The developed algorithms were validated through virtual experiments on particulate composites. The robustness of both the VFM and the RVFM was assessed in the presence of noisy strain data for various microstructures. A study was also carried out to investigate the influence of the size of region of interests on the reliability of the identified parameters. Accordingly, the optimum size of region of interest was determined based on full-field measurement requirements and accuracy of the identified parameters. This study enables determining, a priori, the required magnification level of 3D images for composites of any mechanical and morphological characteristics.     

A principle of virtual work for combined electrostatic and mechanical loading of materials

The equations governing mechanics and electrostatics are formulated for a system in which the material deformations and electrostatic polarizations are arbitrary. A mechanical/electrostatic energy balance is formulated for this situation in terms of the electric enthalpy, in which the electric potential and the electric field are the independent variables, and charge and electric displacement, respectively, are the conjugate thermodynamic forces. This energy statement is presented in the form of a principle of virtual work (PVW), in which external virtual work is equated to internal virtual work. The resulting expression involves an internal material virtual work in which (1) material polarization is work-conjugate to increments of electric field, and (2) a combination of Cauchy stress, Maxwell stress and a product of polarization and electric field is work-conjugate to increments of strain. This PVW is valid for all material types, including those that are conservative and those that are dissipative. Such a virtual work expression is the basis for a rigorous formulation of a finite element method for problems involving the deformation and electrostatic charging of materials, including electroactive polymers and switchable ferroelectrics. The internal virtual work expression is used to develop the structure of conservative constitutive laws governing, for example, electroactive elastomers and piezoelectric materials, thereby determining the form of the Maxwell or electrostatic stress. It is shown that the Maxwell or electrostatic stress has a form fully constrained by the constitutive law and cannot be chosen independently of it. The structure of constitutive laws for dissipative materials, such as viscoelastic electroactive polymers and switchable ferroelectrics, is similarly determined, and it is shown that the Maxwell or electrostatic stress for these materials is identical to that for a material having the same conservative response when the dissipative processes in the material are shut off. The form of the internal virtual work is used further to develop the structure of dissipative constitutive laws controlled by rearrangement of material internal variables.     

Simultaneous identification of stiffness and damping properties of isotropic materials from forced vibrating platesIdentification simultanée de rigidités et d'amortissements de matériaux isotropes à partir de plaques en vibrations forcées

This paper presents a novel methodology for the identification of damping of isotropic plates. It relies on forced inertial excitation of a clamped plate and full-field curvature measurements using a suitable optical technique. Using the Virtual Fields Method, it is shown that the damping parameter is easily related to the curvature field, even on a non-resonant plate. This paper opens a totally new field of investigation for damping identification. To cite this article: A. Giraudeau, F. Pierron, C. R. Mecanique 331 (2003).     

Interior formulation of axisymmetric Levinson plate theory

In this study, we show that the axisymmetric Levinson plate theory is exclusively an interior theory and we provide a consistent variational formulation for it. First, we discuss an annular Levinson plate according to a vectorial formulation. The boundary layer of the plate is not modeled and, thus, the interior stresses acting as surface tractions do work on the lateral edges of the plate. This feature is confirmed energetically by the Clapeyron's theorem. The variational formulation is carried out for the annular Levinson plate by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the tractions based on the interior stresses on the inner and outer lateral edges of the Levinson plate. The obtained plate equations are consistent with the vectorially derived Levinson equations. Finally, we develop an exact plate finite element both by a force-based method and from the total potential energy of the Levinson plate.     

Fundamental solutions of critical wedge angles for one-dimensional piezoelectric quasicrystal wedge

Two problems of a one-dimensional (1D) piezoelectric quasicrystal (QC) wedge are investigated, i. e., the two sides of the wedge subject to uniform tractions and the wedge apex subject to the concentrated force. By virtue of the Stroh formalism and Barnett-Lothe matrices, the analytical expressions of the displacements and stresses are derived, and the generalized solutions for the critical wedge angles are discussed. Numerical examples are given to present the mechanical behaviors of the wedge in each field. The results indicate that the effects of the uniform tractions and the concentrated force on the phonon field displacement are larger than those on the phason field.     

再论变形体虚功原理

在现有结构力学教材中,关于虚功原理的表述大体上可以分为4大类: 虚功原理是一充分必要性命题; 虚功原理是一充分必要性命题但只证明必要性部分; 虚功原理仅仅是必要性命题,但它有2种``应用'------虚位移原理和虚力原理; 虚功原理仅仅是必要性命题,它和虚位移原理、虚力原理是有异同的. 即使是最近出版的``十一五'规划教材中,也还存在虚功原理是一充分必要性命题的提法. 为此,再次给出原理的正确阐述和证明,指出需要对虚功原理和虚位移原理加以区分,阐明它们之间的异同.     

Boundary Value Problems of Plane Elasticity Involving Orientations of Displacements and Tractions

This paper presents unconventional formulations of boundary problems of plane elasticity formulated in terms of orientations of tractions and displacements on a closed contour separating internal and external domains as the boundary conditions. These are combined with the conditions of continuity of tractions or displacements across the boundary. Therefore the magnitudes of neither tractions nor displacements are assumed on the contour. Four boundary value problems for both external and internal domains are investigated by analyzing the solvability of the corresponding singular integral equations. It is shown that all considered problems can have non-unique solutions expressed as linear combinations of particular solutions and, hence, contain free arbitrary parameters, the number of which is finite and determined by the contour orientations of tractions and/or displacements     

General Anisotropy Identification of Paperboard with Virtual Fields Method

This work extends previous efforts in plate bending of Virtual Fields Method (VFM) parameter identification to include a general 2-D anisotropic material. Such an extension was needed for instances in which material principal directions are unknown or when specimen orientation is not aligned with material principal directions. A new fixture with a multi-axial force configuration is introduced to provide full-field strain data for identification of the six anisotropic stiffnesses. Two paper materials were tested and their Q _ij compared favorably with those determined by ultrasonic and tensile tests. Accuracy of VFM identification was also quantified by variance of stiffnesses. The load fixture and VFM provide an alternative stiffness identification tool for a wide variety of thin materials to more accurately determine Q ₁₂ and Q ₆₆.     

Identification of Material Parameters of PVC Foams using Digital Image Correlation and the Virtual Fields Method

This paper presents an effective methodology to characterize all the constitutive (elastic) parameters of an orthotropic polymeric foam material (Divinycell H100) in one single test using Digital Image Correlation (DIC) in combination with the Virtual Fields Method (VFM). A modified Arcan fixture is used to induce various loading conditions ranging from pure shear or axial loading in tension or compression to bidirectional loading. A numerical optimization study was performed with different loading angles of the Arcan test fixture and off-axis angles of the principal material axes. The objective is to identify the configuration that gives the minimum sensitivity to noise and missing data on the specimen edges, which are the two major issues when identifying the stiffness components from actual DIC measurements. Two optimized Arcan test configurations were chosen. The experimental results obtained for these two optimized test configurations show a significant improvement of the measurement accuracy compared with a pure shear load configuration. The larger sensitivity of the pure shear test to missing data as opposed to the tensile test is also evident from the experimental data and confirms the analysis from the optimization study. The recovery of missing data along the specimen edges is a promising way to further improve the identification results.     

A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior

This paper is the sequel of a companion Part I paper devoted to the constitutive equations and to the quasi-static behavior of a second strain gradient material model with second velocity gradient inertia. In the present Part II paper, a multi-cell homogenization procedure (developed in the Part I paper) is applied to a nonhomogeneous body modelled as a simple material cell system, in conjunction with the principle of virtual work (PVW) for inertial actions (i.e. momenta and inertia forces), which at the macro-scale level takes on the typical format as for a second velocity gradient inertia material model. The latter (macro-scale) PVW is used to determine the equilibrium equations relating the (ordinary, double and triple) generalized momenta to the inertia forces. As a consequence of the surface effects, the latter inertia forces include (ordinary) inertia body forces within the bulk material, as well as (ordinary and double) inertia surface tractions on the boundary layer and (ordinary) inertia line tractions on the edge line rod; they all depend on the acceleration in a nonstandard way, but the classical laws are recovered in the case of no higher order inertia. The classical linear and angular momentum theorems are extended to the present context of second velocity gradient inertia, showing that the extended theorems—used in conjunction with the Cauchy traction theorem—lead to the local force and moment (stress symmetry) motion equations, just like for a classical continuum. A gradient elasticity theory is proposed, whereby the dynamic evolution problem for assigned initial and boundary conditions is shown to admit a Hamilton-type variational principle; the uniqueness of the solution is also discussed. A few simple applications to wave propagation and dispersion problems are presented. The paper indicates the correct way to describe the inertia forces in the presence of higher order inertia; it extends and improves previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137]. Overall conclusions are drawn at the end of the paper.     

Relative displacements of the two sides of a plane crack

We consider a two dimensional deformation of an infinite anisotropic elastic material which contains a plane crack. When tractions are specified over the crack faces, we give a formula for the corresponding relative displacements of the two sides of the crack. We also consider the inverse problem of determining the elastic constants by measuring the tractions and the corresponding relative displacements on the crack.     

Full-Field Strain Measurement and Identification of Composites Moduli at High Strain Rate with the Virtual Fields Method

The present paper deals with full-field strain measurement on glass/epoxy composite tensile specimens submitted to high strain rate loading through a split Hopkinson pressure bar (SHPB) device and with the identification of their mechanical properties. First, the adopted methodology is presented: the device, including an Ultra-High Speed camera, and the experimental procedure to obtain relevant displacement maps are described. The different full-field results including displacement, strain and acceleration maps for two mechanical tests are then addressed. The last part of the paper deals with an original procedure to identify stiffnesses on this dynamic case only using the actual strain and acceleration maps (without the applied force) by using the Virtual Fields Method. The results provide very promising values of Young’s modulus and Poisson’s ratio on a quasi-isotropic glass-epoxy laminate. The load reconstructed from the moduli and strains compares favourably with that from the readings.     

Une formulation variationnelle du problème de contact avec frottement de Coulomb

A variational relationship is proposed as the weak form of the large deformation contact problem with Coulomb friction. It is a mixed relationship involving both the displacements and the multipliers; the weighting functions are the virtual displacements and the virtual multipliers. It is shown that the proposed weak form is equivalent to the strong form of the initial/boundary value contact problem and the multipliers are equal to the contact tractions. To cite this article: A. Le van, T.H.T. Nguyen, C. R. Mecanique 336 (2008).     

Stiffness and Damping Identification from Full Field Measurements on Vibrating Plates

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 parameters 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 deflectometry technique used for the measurements is presented. First results of identification on an acrylic plate are presented and compared to reference values. Results are discussed and improvements of the method are proposed.     

Electromagnetoelastic behaviors of functionally graded piezoelectric

Analytical studies on electromagnetoelastic behaviors are presented for the functionally graded piezoelectric material (FGPM) solid cylinder and sphere placed in a uniform magnetic field and subjected to the external pressure and electric loading. When the mechanical, electric and magnetic properties of the material obey an identical power law in the radial direction, the exact displacements, stresses, electric potentials and perturbations of magnetic field vector in the FGPM solid cylinder and sphere are obtained by using the infinitesimal theory of electromagnetoelasticity. Numerical examples also show the significant influence of material inhomogeneity. It is interesting to note that selecting a specific value of inhomogeneity parameter can optimize the electromagnetoelastic responses, which will be of particular importance in modern engineering designs.     

On the use of simulated experiments in designing tests for material characterization from full-field measurements

The present paper deals with the use of simulated experiments to improve the design of an actual mechanical test. The analysis focused on the identification of the orthotropic properties of composites using the unnotched Iosipescu test and a full-field optical technique, the grid method. The experimental test was reproduced numerically by finite element analysis and the recording of deformed grey level images by a CCD camera was simulated trying to take into account the most significant parameters that can play a role during an actual test, e.g. the noise, the failure of the specimen, the size of the grid printed on the surface, etc. The grid method then was applied to the generated synthetic images in order to extract the displacement and strain fields and the Virtual Fields Method was finally used to identify the material properties and a cost function was devised to evaluate the error in the identification. The developed procedure was used to study different features of the test such as the aspect ratio and the fibre orientation of the specimen, the use of smoothing functions in the strain reconstruction from noisy data, the influence of missing data on the identification. Four different composite materials were considered and, for each of them, a set of optimized design variables was found by minimization of the cost function.     

Variational formulation of the static Levinson beam theory

In this communication, we provide a consistent variational formulation for the static Levinson beam theory. First, the beam equations according to the vectorial formulation by Levinson are reviewed briefly. By applying the Clapeyron's theorem, it is found that the stresses on the lateral end surfaces of the beam are an integral part of the theory. The variational formulation is carried out by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the stresses on the end surfaces of the beam. This external virtual work contributes to the boundary conditions in such a way that artificial end effects do not appear in the theory. The obtained beam equations are the same as the vectorially derived Levinson equations. Finally, the exact Levinson beam finite element is developed.     

3D Heterogeneous Stiffness Reconstruction Using MRI and the Virtual Fields Method

The first extension of the virtual fields method to the reconstruction of heterogeneous stiffness properties from 3D bulk full-field displacement data is presented in this paper. Data are provided by Magnetic Resonance Imaging (MRI). Two main issues are addressed: 1. the identification of the stiffness ratio between two different media in a heterogeneous solid; 2. the reconstruction of stiffness heterogeneities buried in a heterogeneous solid. The approach is based on a finite element discretization of the equilibrium equations. It is tested on experimental full-field data obtained on a phantom with the stimulated echo MRI technique. The phantom is made of a stiff spherical inclusion buried within a lower modulus material. Preliminary independent tests showed that the material of the inclusion was four times stiffer than the surrounding material. This ratio value is correctly identified by our approach directly on the phantom with the MRI data. Moreover, the modulus distribution is promisingly reconstructed across the whole investigated volume. However, the resulting modulus distribution is highly variable. This is explained by the fact that the approach relies on a second order differentiation of the data, which tends to amplify noise. Noise is significantly reduced by using appropriate filtering algorithms.     

Application of the reciprocity theorem to scattering of surface waves by a cavity

Scattering of surface waves by a cylindrical cavity at the surface of a homogenous, isotropic, linearly elastic half-space is analyzed in this paper. In the usual manner, the scattered field is shown to be equivalent to the radiation from a distribution of tractions, obtained from the incident wave on the surface of the cavity. For the approximation used in this paper, these tractions are shifted to tractions applied to the projection of the cavity on the surface of the half-space. The radiation of surface waves from a normal and a tangential line load, recently determined by the use of the reciprocity theorem, is employed to obtain the field scattered by the cavity from the superposition of displacements due to the distributed surface tractions. The vertical displacement at some distance from the cavity is compared with the solution of the scattering problem obtained by the boundary element method (BEM) for various depths and widths of the cavity. Comparisons between the analytical and BEM results are graphically displayed. The limitations of the approximate approach are discussed based on the comparisons with the BEM results.