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Absolute Nodal Coordinates in Digital Image Correlation

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Abstract

A great deal of progress has been made in recent years in the field of global digital image correlation (DIC), where higher-order, element-based approaches were proposed to improve the interpolation performance and to better capture the displacement fields. In this research, another higher-order, element-based DIC procedure is introduced. Instead of the displacements, the elements’ global nodal positions and nodal position-vector gradients, defined according to the absolute nodal coordinate formulation, are used as the searched parameters of the Newton–Raphson iterative procedure. For the finite elements, the planar isoparametric plates with 24 nodal degrees of freedom are employed to ensure the gradients’ continuity among the elements. As such, the presented procedure imposes no linearization on the strain measure, and therefore indicates a natural consistency with the nonlinear continuum theory. To verify the new procedure and to show its advantages, a real large deformation experiment and several numerical tests on the computer-generated images are studied for the standard, low-order, element-based digital image correlation and the presented procedure. The results show that the proposed procedure proves to be accurate and reliable for describing the rigid-body movement and simple deformations, as well as for determining the continuous finite strain field of a real specimen.

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Acknowledgement

Operation partially financed by the European Union, European Social Fund.

Author information

Correspondence to J. Slavič.

Appendix

Appendix

The j-th element shape function matrix [20] is defined as

$$ \textbf{S}_j(\textbf{x}_j)=\textbf{S}_{ij}=\textbf{S}_j(x_{1ij},x_{2ij})=\left[s_{1ij}\,\textbf{I},~~s_{2ij}\,\textbf{I},~~\ldots,~~s_{12ij}\,\textbf{I}\right]~, $$
(36)

where I is 2 × 2 identity matrix, and

$$ \begin{array}{rll} s_{1ij}&=&-(\xi-1)\,(\eta-1)\,(2\,\eta^2-\eta+2\,\xi^2-\xi-1)~,\\ s_{2ij}&=&-a\,\xi\,(\xi-1)^2\,(\eta-1)~,\\ s_{3ij}&=&-a\,\eta\,(\eta-1)^2\,(\xi-1)~,\\ s_{4ij}&=&\xi\,(2\,\eta^2-\eta-3\,\xi+2\,\xi^2)\,(\eta-1)~,\\ s_{5ij}&=&-a\,\xi^2\,(\xi-1)\,(\eta-1)~,\\ s_{6ij}&=&a\,\xi\,\eta\,(\eta-1)^2~,\\ s_{7ij}&=&-\xi\,\eta\,(1-3\,\xi-3\,\eta+2\,\eta^2+2\,\xi^2)~,\\ s_{8ij}&=&a\,\xi^2\,\eta\,(\xi-1)~,\\ s_{9ij}&=&a\,\xi\,\eta^2\,(\eta-1)~,\\ s_{10ij}&=&\eta\,(\xi-1)\,(2\,\xi^2-\xi-3\,\eta+2\,\eta^2)~,\\ s_{11ij}&=&a\,\xi\,\eta\,(\xi-1)^2~,\\ s_{12ij}&=&-a\,\eta^2\,(\xi-1)\,(\eta-1)~,\\\quad \xi&=&x_{1ij}/a,~\eta=x_{2ij}/a~, \end{array} $$

where a is the rectangular plate size.

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Langerholc, M., Slavič, J. & Boltežar, M. Absolute Nodal Coordinates in Digital Image Correlation. Exp Mech 53, 807–818 (2013). https://doi.org/10.1007/s11340-012-9691-4

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Keywords

  • Absolute nodal coordinate formulation
  • Digital image correlation
  • Higher-order plate element
  • Large deformation