A stress recovery procedure for solving geometrically non-linear problems in the mechanics of a deformable solid by the finite element method
A stress recovery procedure is presented for non-linear and linearized problems, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the Lagrange variational equation written in the initial configuration using an asymmetric P...
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| Veröffentlicht in: | Journal of applied mathematics and mechanics Jg. 74; H. 6; S. 710 - 720 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Kidlington
Elsevier Ltd
2010
Elsevier |
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| ISSN: | 0021-8928, 0021-8928 |
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| Abstract | A stress recovery procedure is presented for non-linear and linearized problems, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the Lagrange variational equation written in the initial configuration using an asymmetric Piola–Kirchhoff stress tensor. Vectors of the forces reduced to the mesh points are constructed using the displacements at the mesh points found by solving this equation and for the known stiffness matrices of the elements. On the other hand, these forces at the mesh points are defined in terms of unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The values of the Piola–Kirchhoff stress tensor of the first kind at the mesh points are determined using the values found for the distributed forces on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations for the initial configuration. The linearized representation of this tensor enables all the derivatives of the increment in the strain vector with respect to the coordinates to be found without invoking the operation of differentiation. The particular features of the use of the stress recovery procedure are demonstrated for a plane problem in the non-linear theory of elasticity. |
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| AbstractList | A stress recovery procedure is presented for non-linear and linearized problems, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the Lagrange variational equation written in the initial configuration using an asymmetric Piola–Kirchhoff stress tensor. Vectors of the forces reduced to the mesh points are constructed using the displacements at the mesh points found by solving this equation and for the known stiffness matrices of the elements. On the other hand, these forces at the mesh points are defined in terms of unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The values of the Piola–Kirchhoff stress tensor of the first kind at the mesh points are determined using the values found for the distributed forces on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations for the initial configuration. The linearized representation of this tensor enables all the derivatives of the increment in the strain vector with respect to the coordinates to be found without invoking the operation of differentiation. The particular features of the use of the stress recovery procedure are demonstrated for a plane problem in the non-linear theory of elasticity. |
| Author | Rogovoi, A.A. Stolbova, O.S. |
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| Keywords | Finite element method Stiffness matrix Asymmetry Non linear elasticity Variational calculus Displacement(deformation) Integral equation Differentiation Non linear effect Modeling Lagrange equation Fredholm equation |
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| References | Rogovoy (bib0005) 1997; 63. Novokshanov, Rogovoi (bib0040) 2005; 4 Oden (bib0015) 1972 Lurie (bib0020) 2005 Lurie (bib0025) 1991 Rogovoi (bib0045) 2005; 46 Novokshanov, Rogovoi (bib0035) 2002; 4 Rogovoi, Stolbova (bib0010) 2010; 74 Truesdell (bib0030) 1972 Rogovoy (10.1016/j.jappmathmech.2011.01.011_bib0005) 1997; 63. Truesdell (10.1016/j.jappmathmech.2011.01.011_bib0030) 1972 Lurie (10.1016/j.jappmathmech.2011.01.011_bib0025) 1991 Rogovoi (10.1016/j.jappmathmech.2011.01.011_bib0010) 2010; 74 Novokshanov (10.1016/j.jappmathmech.2011.01.011_bib0040) 2005; 4 Oden (10.1016/j.jappmathmech.2011.01.011_bib0015) 1972 Lurie (10.1016/j.jappmathmech.2011.01.011_bib0020) 2005 Novokshanov (10.1016/j.jappmathmech.2011.01.011_bib0035) 2002; 4 Rogovoi (10.1016/j.jappmathmech.2011.01.011_bib0045) 2005; 46 |
| References_xml | – volume: 4 start-page: 122 year: 2005 end-page: 140 ident: bib0040 article-title: Evolutionary constitution relations for finite viscoelastic deformations publication-title: Izv Rass Akad Nauk MTT – volume: 46 start-page: 138 year: 2005 end-page: 149 ident: bib0045 article-title: Constitutive relations for finite elasto-inelastic deformations publication-title: Zh Prikl Mekh Tekhn Fiz – volume: 4 start-page: 77 year: 2002 end-page: 95 ident: bib0035 article-title: The construction of evolutionary constitutive relations for finite deformations publication-title: Izv Ross Akad Nauk MTT – year: 1972 ident: bib0015 article-title: Finite. Elements of Nonlinear Continua – year: 2005 ident: bib0020 article-title: Theory of Elasticity – year: 1972 ident: bib0030 article-title: A First Course in Rational Continuum mechanics – volume: 74 start-page: 478 year: 2010 end-page: 488 ident: bib0010 article-title: Stress recovery procedure for solving boundary value problems in the mechanics of a deformable solid by the finite element method publication-title: Prikl Mat Mekh – volume: 63. start-page: 1121 year: 1997 end-page: 1137 ident: bib0005 article-title: The stress recovery procedure for the finite element method publication-title: Comp. Struct. – year: 1991 ident: bib0025 article-title: Non-Linear Theory of Elasticity – volume: 4 start-page: 122 year: 2005 ident: 10.1016/j.jappmathmech.2011.01.011_bib0040 article-title: Evolutionary constitution relations for finite viscoelastic deformations publication-title: Izv Rass Akad Nauk MTT – volume: 74 start-page: 478 issue: 3 year: 2010 ident: 10.1016/j.jappmathmech.2011.01.011_bib0010 article-title: Stress recovery procedure for solving boundary value problems in the mechanics of a deformable solid by the finite element method publication-title: Prikl Mat Mekh – volume: 46 start-page: 138 issue: 5 year: 2005 ident: 10.1016/j.jappmathmech.2011.01.011_bib0045 article-title: Constitutive relations for finite elasto-inelastic deformations publication-title: Zh Prikl Mekh Tekhn Fiz – year: 1972 ident: 10.1016/j.jappmathmech.2011.01.011_bib0030 – volume: 63. start-page: 1121 issue: 6 year: 1997 ident: 10.1016/j.jappmathmech.2011.01.011_bib0005 article-title: The stress recovery procedure for the finite element method publication-title: Comp. Struct. doi: 10.1016/S0045-7949(96)00405-1 – year: 2005 ident: 10.1016/j.jappmathmech.2011.01.011_bib0020 – year: 1991 ident: 10.1016/j.jappmathmech.2011.01.011_bib0025 – year: 1972 ident: 10.1016/j.jappmathmech.2011.01.011_bib0015 – volume: 4 start-page: 77 year: 2002 ident: 10.1016/j.jappmathmech.2011.01.011_bib0035 article-title: The construction of evolutionary constitutive relations for finite deformations publication-title: Izv Ross Akad Nauk MTT |
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| SubjectTerms | Exact sciences and technology Finite element method Fundamental areas of phenomenology (including applications) Mathematical analysis Nonlinearity Physics Recovery Shape functions Solid mechanics Static elasticity (thermoelasticity...) Stress tensors Stresses Structural and continuum mechanics Vectors (mathematics) |
| Title | A stress recovery procedure for solving geometrically non-linear problems in the mechanics of a deformable solid by the finite element method |
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