Identification of Elastic Orthotropic Material Parameters Based on ESPI Measurements

Parameter identification methods, which integrate optimization techniques and numerical methods such as the finite element method (FEM), offer an alternative tool for material characterisation. The most common approach is to determine the optimal estimates of the model parameters by minimizing a selected measure-of-fit between the responses of the system and the model. The possibility is studied of retrieving the four independent elastic engineering constants for an orthotropic medium, based on the measurement of a heterogeneous displacement field. In the present case a tensile test is performed on a perforated specimen. The responses of the system, i.e. the surface displacements are measured with an Electronic Speckle Pattern Interferometer. Strains are subsequently calculated, based on the measured displacement field. A finite element model of the perforated specimen is made. The difference between the experimental and numerical strains is minimized in a least squares sense by updating the values of the parameters. The obtained material (or model) parameters are very well in agreement with the traditionally determined ones. INTRODUCTION Material properties determined from measurements based on the use of test samples with a well-defined standardized geometry and loading in a laboratory may be strongly different from those of actual structural components manufactured in factory or used in engineering. In some cases, it is unacceptable to take out a test sample from a larger structure because of the disruption of the internal coherence of the material [1]. Therefore, material parameter identification methods, which integrate optimization techniques and numerical methods such as the finite element method (FEM), offer an alternative tool. The most common approach is to determine the optimal estimates of the model parameters by minimizing a selected measure-of-fit between the responses of the system and the model. The elastic material parameters corresponding to an orthotropic material can be identified from laboratory tests by classical and already well-known experiments [2]. In the present study a method is proposed for the identification of the four independent elastic in-plane material parameters (E1, E2, G12 and v12) for an orthotropic medium based on surface measurements of a specimen subjected to heterogeneous deformation. In the present case a tensile test is performed on a perforated specimen. The responses of the system, i.e. the surface displacements are measured with an Electronic Speckle Pattern Interferometer. Strains are subsequently calculated, based on the measured displacement field, which is subjected to noise. The calculated strains are not only influenced by the experimental noise on the displacements, but also by the chosen gauge length, i.e. the length over which the strains are calculated and the location in which they are determined. A method for the calculation of strain, dealing with these problems, is presented. A finite element model of the perforated specimen serves as numerical counterpart for the experimental set-up. The difference between the experimental and numerical strains (εx, εy and j’ai mis un autre gamma xy ) is minimized in a least squares sense by updating the values of the four independent elastic moduli. The sensitivities used to obtain the parameter updates are determined analytically, using the stresses calculated during the previously converged step. This choice allows a convergence rate that is three times faster than a routine based on finite difference sensitivities. The optimization routine used, is based on a simple Newton-Raphson algorithm. A more sophisticated routine is not necessary since the problem has a good convergence rate. ELECTRONIC SPECKLE PATTERN INTERFEROMETER (ESPI) The ESPI technique is based on the scattered reflection of incident light on a rough surface which in the present case is the surface of the specimen illuminated by a monochromatic laser [3]. The experimental set-up is shown in figure1. Figure 1: set-up for tension test measured by ESPI Figure 2: Principle of in-plane displacement measurement The scattered light creates a so-called speckle pattern as shown in figure 3a and 3b. When using two identical wave’s symmetrically incident on an object surface, a camera aligned perpendicular to the reflecting surface will visualize an interferential image due to the combination of the two speckle patterns. The correlation fringes are obtained by subtracting the image of the reference speckle and the speckle image obtained after the displacement or the deformation of the object (figure 3c). In order to understand the formation of fringes, consider the intensities of the beams Ibefore before deformation and Iafter, the intensity after deformation in each point of the image plane. cosυ I I 2 I I y x, I 2 1 2 1 before (1) Δυ υ cos I I 2 I I y x, I 2 1 2 1 after (2) I1: scattered intensity of first beam I2: scattered intensity of second beam : phase difference between both beams : phase change caused by current displacement The intensity at any position (x,y) in the image plane may be approximated [4] by the following expression: Δυ 2 1 sin Δυ 2 1 υ sin I I 4 y x, I 2 1 final (3) A high pass filter is used to eliminate the part of expression 3 containing which can be considered as noise when displacements are to be measured. Hereafter, the corresponding wrapped phase map is obtained (figure 3d). The relation between and the displacement component u(x,y) is given by expression 4: y)sinα u(x, λ 4π Δυ (4) In equation (4) u(x,y) is the resolved component of the surface displacement in the x or y direction, depending on the orientation of the incident laser beams. λ is the wavelength of the used light source and α represents the angle of incidence of the laser beams as shown in figure 2. a) b) c) d) e) Figure 3: (a) Speckle before deformation; (b) Speckle after deformation; (c) resulting fringe pattern by subtraction; (d) wrapped phase map; (e) unwrapped phase map describing the vertical displacement field In the current set-up a CCD camera with 752 x 582 pixels is used and the wave length of the laser diodes is 780nm. The sensitivity of the system with respect to the measured displacements lies between 0. 05μm and 1μm. STRAIN CALCULATION In the framework of the Mixed Numerical Experimental Methods, an adequate procedure is required to determine the experimental strains and compare them with the strains based on a Finite element Analysis of the experiment. In this paragraph a method for strain calculation based on a measured displacement field is presented. It permits to deal with the corrupted displacement data and calculates the experimental strains immediately at the position of the integration points in the FE model. A direct matching of experimental and numerical strains is possible because they are calculated using the same routine. An additional advantage is that the rigid body motions, induced by the tension machine, can be omitted. The displacement data obtained by ESPI is linked to a well structured and dense grid of data points (figure 4). Figure 4: Comparison between experimental data and FEM mesh Figure 5: Triangular 3 node element in an experimental displacement field The idea of the present method for strain calculation is based on the smoothening of the experimental displacement field with the entire FE mesh. The aim is to determine the displacement values at the nodes of the finite element mesh, based on the measured displacements. Once these nodal displacement values are determined, the strains can be calculated using the FEM-routine. The numerical displacement values in an arbitrary point within a given element (figure 5) can be calculated using the three interpolation functions related to the element nodes: