Application of a three-dimensional finite-element method to strain field analyses |
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| Affiliation: | 1. CSIR – National Physical Laboratory (CSIR-NPL), Dr K. S. Krishnan Road, New Delhi 110012, India;2. Academy of Scientific & Innovative Research, CSIR-HRDC Campus, Ghaziabad, Uttar Pradesh 201002, India;1. Children''s Hospital of Eastern Ontario Research Institute, Ottawa, ON, Canada;2. Library Services, Children''s Hospital of Eastern Ontario, Ottawa, ON, Canada;3. Clinical Research Unit, Children''s Hospital of Eastern Ontario, Ottawa, ON, Canada;4. Department of Pediatrics, Children''s Hospital of Eastern Ontario, Ottawa, ON, Canada;5. Michael G. DeGroote School of Medicine, McMaster University, Hamilton, ON, Canada;6. Max Rady College of Medicine, University of Manitoba, Winnipeg, MB, Canada;7. Department of Biology, University of Ottawa, Ottawa, ON, Canada;8. Health Sciences Library, University of Ottawa, Ottawa, ON, Canada;1. Key Laboratory of Green Preparation and Application for Materials, Ministry of Education, Hubei Provincial Key Laboratory of Polymers, Department of Materials Science and Engineering, Hubei University, Wuhan, 430062, PR China;2. Physics Department, Oakland University, Rochester, MI, 48309-4401, USA |
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| Abstract: | I have programmed a three-dimensional finite-element model to spatially integrate distributed strains. The mathematics is based on Cobbold and Percevault (1983). The method finds the pre-strain configuration of a region by unstraining rectangular prisms into parallelepipeds, then rotating and translating them iteratively to minimize the interelement deviations. A numerical singular-value decomposition calculates the necessary rotations and the final cycle of further strains to make a best fit of the elements. The deviation of elements from holes formed by their nearest neighbors indicates the degree of compatibility of strain measurements.The program reproduces the rotations required in examples with analytic solutions. Even though only a single horizontal layer of elements is used, the fitting procedure correctly calculates rotations in all three dimensions from the constraints of strain compatibility.Three-dimensional strains measured in an Archean greenstone belt of northeastern Minnesota were integrated using a layer of nearly equant finite elements. Rotations calculated from the pre-tectonic configuration accord with field observations of fabrics indicating variable amounts of shear. Rotations about horizontal axes are minimal, precluding large vertical shears. An estimate of 40% true horizontal north-south shortening across the belt can be made from the undeformed configuration. |
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