Digital Volume Correlation Including Rotational Degrees of Freedom during Minimization by Tait S. Smith, Brian K. Bay, and Mark M. Rashid ABSTRACT--Digital volume correlation is a new experimental technique that allows the measurement of the full-field strain tensor in three dimensions. We describe the addition of rota- tional degrees of freedom into the minimization problem for digital volume correlation in order to improve the overall per- formance of the strain measurement. A parameterization of rotations that is particularly suited to the minimization problem is presented, based on the angle-axis representation of finite rotations. The partial derivative of both a normalized cross- correlation coefficient and the sum-of-squares correlation co- efficient are derived for use with gradient-based minimization algorithms. The addition of rotation is shown to greatly re- duce the measurement error when even small amounts of rigid body rotation are present in an artificially rotated test volume. In an aluminum foam sample loaded in compression, including rotational degrees of freedom produced smoother contours of minimum principal strain. Renderings of the alu- minum foam architecture in areas of low, medium and high rotation showed material deformation pattern in detail. KEY WORDS--Digital image correlation, experimental me- chanics, finite rotations, aluminum foam Introduction The computational modeling of complex materials has evolved to the point that new experimental techniques must be developed to validate the models. There is a gap between the amount of information needed in constitutive models and the current abilities of experimental mechanics to provide this information. 1 Microstructurally complex materials are being used in engineering applications because they can be designed to exhibit desirable material properties, such as high stiffness-to-weight ratio, high toughness, or high specific sur- face areas. To intelligently use these materials in design, they must be well characterized with predictive models. Therefore, there is an increasing need to provide very detailed informa- tion about the load-deformation characteristics of these types of materials. Recently, high-resolution computed tomogra- phy (IxCT) has been used to study the architectural details of certain types of materials. 2-4 Our work has extended the 2D digital image correlation technique into three dimensions by utilizing IxCT imaging of samples before and after defor- Tait S. Smith is a Postdoctoral Student, Orthopaedic Research Laboratories, University of California, Davis, Sacramento, CA, 95817, USA. Brian K. Bay is an Assistant Professor, Orthopaedic Research Laboratories, U.C. Davis, Sacramento, CA, 95817, USA. Mark M. Rashid is an Associate Professor, Department of Civil and Environmental Engineering, U.C. Davis, Davis, CA, 95616, USA. Original manuscript submitted: September 7, 2000. Final manuscript received: February 18, 2002. mation. Our method, digital volume correlation (DVC), is then used as part of a mathematical procedure that produces high-resolution, full-field strain information within the sam- ple volume. The basic sequence for application of three-dimensional digital volume correlation to strain field measurement is de- scribed in detail in Ref. 5 and involves three main steps: (1) generation of volume images of samples in unloaded and loaded states; (2) measurement of a discrete displacement vector field throughout the sample by a pattern-matching cor- relation procedure; and finally (3) calculation of the strain tensor field from the measured displacement vector field. In Ref. 5, step 2 (displacement vector measurement) was formu- lated in terms of the minimization of an objective function in which only translational degrees of freedom (DOF) appeared, and using a sum-of-squares correlation coefficient (see Ref. 5, eq (1)). This is in contrast to 2D image correlation where the pattern matching is typically formulated as a full affine trans- formation (displacement in x and y, rotation about z, and 3 strains). 6-8 Including rotation and strain improves the accu- racy of the pattern matching but usually these parameters are ignored during the strain field calculation which is only based on the displacements. The affine pattern-matching problem in 3D would involve 12 DOE In this paper we have not implemented the full 12 DOF formulation but we have included displacements and rotations (i.e., 6 DOF). The methodology is also ex- tended to encompass both a normalized cross-correlation co- efficient as well as a sum-of-squares correlation coefficient. The resulting rotations from the pattern-matching algorithm are discarded and only the displacement information used in the strain calculation; however, including rotations re- sults in a better displacement measurement. The gradients of the pattern-matching functions are also derived, with a view toward use of gradient-based minimization procedures. Com- parisons between correlation with and without rotational de- grees of freedom are made using an artificially rotated vol- ume data set, as well as an aluminum foam sample under compressive loading. Methods General Correlation Problem Computed tomography imaging produces a discretization of the 3D X-ray attenuation scalar field of an object. The values of this scalar field at discrete points is determined by averaging over a small volume called a voxel, which is the 3D extension of a pixel. Voxels are distributed in space on a 272 9 VoL 42, No. 3, September 2002