An Out-of-Plane Motion Compensation Strategy for Improving Material Parameter Estimation Accuracy with 2D Field Measurements

Abstract

In-plane surface displacements, when measured with 2D Digital Image Correlation (2D-DIC), are very sensitive to out-of-plane displacement components. Any out-of-plane motion of the surface can pollute the measured field by introducing artificial displacements. These displacements are difficult to separate from the underlying response of the surface and thereby limit the application of 2D-DIC in inverse problems where the test specimen has significant motion in the out-of-plane direction. In the context of inverse problems, we propose to partially relax this condition of no out-of-plane motion in 2D-DIC. With this approach, only the out-of-plane rigid-body motion of the specimen surface, which is initially in-plane, needs to be avoided while the requirement of surface deformations to be primarily in-plane is essentially waived. Compensation, based on the pinhole camera model, for out-of-plane displacements of the surface in response to applied load is included within the error function of the minimization problem. The improvements in material parameter estimation, obtained by using the proposed compensation strategy, are demonstrated by an example. The proposed technique makes it possible to utilize 2D-DIC with a simple conventional lens for an increased number of inverse problems; and in the process avoiding the computational and experimental difficulties associated with 3D measurement methods as well as the high cost and magnification limitations of a telecentric lens.

Appendix

Direct Estimation of Unknown Parameters

The two parameters extracted using inverse identification in this work are the Poisson’s ratio (PR) and coefficient of thermal expansion (CTE). Here we present a direct estimation of the unknown parameters in p to get a fair idea of the target values in our inverse problem. Since the material is nearly incompressible, the value of PR is expected to be very close to 0.5, hence no direct measurement is required.

The value of CTE however needs experimental evidence. For this purpose a temperature dependent CTE is measured for the same material using a 35 × 25 × 10 mm brick shape specimen. Two identical specimens are prepared for this purpose. Speckle pattern is applied on one specimen which is to be used for measuring temperature dependent strains using 2D-DIC; while the other is mounted with K-type thermocouple for precise temperature measurement. The specimens are placed in wooden boxes and covered with glass lids to avoid heat loss to the environment, Fig. 15. The wooden boxes containing the specimens are then put into an oven and heated. When a steady temperature of about 60 °C is achieved, the boxes are taken out and one of them is placed under a Nikon D90 camera, Fig. 16, while the other with thermocouple is just put next to this setup for continuous temperature measurement. Since the two blocks are of identical dimensions, and placed under same conditions, it is expected that they are at the same temperature.

Fig. 15

Brick shaped specimens for CTE measurement: (a) specimen for temperature measurement and (b) specimen for strain measurement

Fig. 16

Experimental setup for direct CTE measurement

By using time-lapse photography, images of the specimen surface are taken every minute until the temperature of the specimen stables at ambient temperature. The strains in the specimen are computed using OSM. Figure 17 shows the temperature and strain variation in the two specimens during test. The strain variation with respect to temperature and the estimated temperature dependent CTE are shown in Fig. 18. The temperature dependent CTE was calculated using equation (14)

Fig. 17

Temperature (a) and strain (b) variation in the specimen during CTE measurement experiment

Fig. 18

Strain (a) and CTE (b) variation with respect to temperature

$$ CTE{(T)}_i=\frac{\varepsilon_i}{T_i-{T}_o} $$

(14)

Where CTE(T)_i is the secant coefficient of thermal expansion, ε_i is the instantaneous strain and T_i and T₀ are instantaneous and initial temperatures. Due to presence of noise in strain data, it was smoothed by fitting to a third degree polynomial before using equation (14). From this experiment the CTE at 32 °C is estimated to be about 120 μm/deg.