Full-field stress computation from measured deformation fields: A hyperbolic formulation
Recent developments in imaging techniques and correlation algorithms enable measurement of strain fields on a deforming material at high spatial and temporal resolution. In such cases, the computation of the stress field from the known deformation field becomes an interesting possibility. This is kn...
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Published in | Journal of the mechanics and physics of solids Vol. 147; p. 104186 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
London
Elsevier Ltd
01.02.2021
Elsevier BV |
Subjects | |
Online Access | Get full text |
ISSN | 0022-5096 1873-4782 |
DOI | 10.1016/j.jmps.2020.104186 |
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Abstract | Recent developments in imaging techniques and correlation algorithms enable measurement of strain fields on a deforming material at high spatial and temporal resolution. In such cases, the computation of the stress field from the known deformation field becomes an interesting possibility. This is known as an inverse problem. Current approaches to this problem, such as the finite element update method, are generally over-determined and must rely on statistical approaches to minimize error. This provides approximate solutions in some cases, however, implementation difficulties, computational requirements, and accuracy are still significant challenges. Here, we show how the inverse problem can be formulated deterministically and solved exactly in two or three dimensions for large classes of materials including isotropic elastic solids, Newtonian fluids, non-Newtonian fluids, granular materials, plastic solids subject to co-directionality, and some other plastic solids subject to associative or non-associative flow rules. This solution is based on a single assumption of the alignment of the principal directions of stress and strain or strain rate. No further assumptions regarding incompressibility, pressure independence, yield surface shape or the hardening law are necessary. This assumption leads to a closed, first order, linear system of hyperbolic partial differential equations with variable coefficients. The solution of this class of problems is well established and hence the equations can be solved to give the solution for any geometry and loading condition, enabling broad applicability to a variety of problems. We provide a numerical proof-of-principle study of the plastic deformation of a two-dimensional bar with spatially varying yield stress and strain hardening coefficient. The results are validated against the solution of the corresponding forward problem – solved with a commercial finite element solver – indicating the solution is exact up to numerical error (the normalized root mean square error of the stress is 1.63×10−4). No model calibration or material parameters are required. The sensitivity of the solution to error in the input data is also analyzed. Interestingly, this solution procedure lends itself to a simple physical interpretation of stress propagation through the material. Finally, we provide some examples showing how this approach may be analytically applied to both solid and fluid mechanics problems.
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•An approach is developed to compute the full field stress from the strain.•Stress is obtained from a hyperbolic system of partial differential equations.•The equation is exact for isotropic materials that are elastic, plastic or viscous.•The approach is valid in two or three dimensions for any geometry and loading.•The approach is suitable for application to digital image correlation data. |
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AbstractList | Recent developments in imaging techniques and correlation algorithms enable measurement of strain fields on a deforming material at high spatial and temporal resolution. In such cases, the computation of the stress field from the known deformation field becomes an interesting possibility. This is known as an inverse problem. Current approaches to this problem, such as the finite element update method, are generally over-determined and must rely on statistical approaches to minimize error. This provides approximate solutions in some cases, however, implementation difficulties, computational requirements, and accuracy are still significant challenges. Here, we show how the inverse problem can be formulated deterministically and solved exactly in two or three dimensions for large classes of materials including isotropic elastic solids, Newtonian fluids, non-Newtonian fluids, granular materials, plastic solids subject to co-directionality, and some other plastic solids subject to associative or non-associative flow rules. This solution is based on a single assumption of the alignment of the principal directions of stress and strain or strain rate. No further assumptions regarding incompressibility, pressure independence, yield surface shape or the hardening law are necessary. This assumption leads to a closed, first order, linear system of hyperbolic partial differential equations with variable coefficients. The solution of this class of problems is well established and hence the equations can be solved to give the solution for any geometry and loading condition, enabling broad applicability to a variety of problems. We provide a numerical proof-of-principle study of the plastic deformation of a two-dimensional bar with spatially varying yield stress and strain hardening coefficient. The results are validated against the solution of the corresponding forward problem – solved with a commercial finite element solver – indicating the solution is exact up to numerical error (the normalized root mean square error of the stress is 1.63×10−4). No model calibration or material parameters are required. The sensitivity of the solution to error in the input data is also analyzed. Interestingly, this solution procedure lends itself to a simple physical interpretation of stress propagation through the material. Finally, we provide some examples showing how this approach may be analytically applied to both solid and fluid mechanics problems.
[Display omitted]
•An approach is developed to compute the full field stress from the strain.•Stress is obtained from a hyperbolic system of partial differential equations.•The equation is exact for isotropic materials that are elastic, plastic or viscous.•The approach is valid in two or three dimensions for any geometry and loading.•The approach is suitable for application to digital image correlation data. Recent developments in imaging techniques and correlation algorithms enable measurement of strain fields on a deforming material at high spatial and temporal resolution. In such cases, the computation of the stress field from the known deformation field becomes an interesting possibility. This is known as an inverse problem. Current approaches to this problem, such as the finite element update method, are generally over-determined and must rely on statistical approaches to minimize error. This provides approximate solutions in some cases, however, implementation difficulties, computational requirements, and accuracy are still significant challenges. Here, we show how the inverse problem can be formulated deterministically and solved exactly in two or three dimensions for large classes of materials including isotropic elastic solids, Newtonian fluids, non-Newtonian fluids, granular materials, plastic solids subject to co-directionality, and some other plastic solids subject to associative or non-associative flow rules. This solution is based on a single assumption of the alignment of the principal directions of stress and strain or strain rate. No further assumptions regarding incompressibility, pressure independence, yield surface shape or the hardening law are necessary. This assumption leads to a closed, first order, linear system of hyperbolic partial differential equations with variable coefficients. The solution of this class of problems is well established and hence the equations can be solved to give the solution for any geometry and loading condition, enabling broad applicability to a variety of problems. We provide a numerical proof-of-principle study of the plastic deformation of a two-dimensional bar with spatially varying yield stress and strain hardening coefficient. The results are validated against the solution of the corresponding forward problem – solved with a commercial finite element solver – indicating the solution is exact up to numerical error (the normalized root mean square error of the stress is 1.63×10−4). No model calibration or material parameters are required. The sensitivity of the solution to error in the input data is also analyzed. Interestingly, this solution procedure lends itself to a simple physical interpretation of stress propagation through the material. Finally, we provide some examples showing how this approach may be analytically applied to both solid and fluid mechanics problems. |
ArticleNumber | 104186 |
Author | Cameron, Benjamin C. Tasan, C.Cem |
Author_xml | – sequence: 1 givenname: Benjamin C. surname: Cameron fullname: Cameron, Benjamin C. email: bcam@mit.edu organization: Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, USA – sequence: 2 givenname: C.Cem surname: Tasan fullname: Tasan, C.Cem email: tasan@mit.edu organization: Department of Materials Science and Engineering, Massachusetts Institute of Technology, USA |
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SubjectTerms | Algorithms Computational fluid dynamics Correlation analysis Elastic material Finite differences Finite element method Fluid flow Fluid mechanics Forward problem Granular materials Imaging techniques Incompressibility Inverse problem Inverse problems Newtonian fluids Non Newtonian fluids Parameter sensitivity Partial differential equations Plastic deformation Strain hardening Strain rate Stress distribution Stress propagation Temporal resolution Visco-plastic material Yield stress |
Title | Full-field stress computation from measured deformation fields: A hyperbolic formulation |
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