A unified determinant-preserving formulation for compressible/incompressible finite viscoelasticity
This paper presents a formulation alongside a numerical solution algorithm to describe the mechanical response of bodies made of a large class of viscoelastic materials undergoing arbitrary quasistatic finite deformations. With the objective of having a unified formulation that applies to a wide ran...
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| Published in | Journal of the mechanics and physics of solids Vol. 177; p. 105312 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
England
Elsevier Ltd
01.08.2023
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0022-5096 1873-4782 |
| DOI | 10.1016/j.jmps.2023.105312 |
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| Abstract | This paper presents a formulation alongside a numerical solution algorithm to describe the mechanical response of bodies made of a large class of viscoelastic materials undergoing arbitrary quasistatic finite deformations. With the objective of having a unified formulation that applies to a wide range of highly compressible, nearly incompressible, and fully incompressible soft organic materials in a numerically tractable manner, the viscoelasticity is described within a Lagrangian setting by a two-potential mixed formulation. In this formulation, the deformation field, a pressure field that ensues from a Legendre transform, and an internal variable of state Fv that describes the viscous part of the deformation are the independent fields. Consistent with the experimental evidence that viscous deformation is a volume-preserving process, the internal variable Fv is required to satisfy the constraint detFv=1. To solve the resulting initial–boundary-value problem, a numerical solution algorithm is proposed that is based on a finite-element (FE) discretization of space and a finite-difference discretization of time. Specifically, a Variational Multiscale FE method is employed that allows for an arbitrary combination of shape functions for the deformation and pressure fields. To deal with the challenging non-convex constraint detFv=1, a new time integration scheme is introduced that allows to convert any explicit or implicit scheme of choice into a stable scheme that preserves the constraint detFv=1 identically. A series of test cases is presented that showcase the capabilities of the proposed formulation. |
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| AbstractList | This paper presents a formulation alongside a numerical solution algorithm to describe the mechanical response of bodies made of a large class of viscoelastic materials undergoing arbitrary quasistatic finite deformations. With the objective of having a unified formulation that applies to a wide range of highly compressible, nearly incompressible, and fully incompressible soft organic materials in a numerically tractable manner, the viscoelasticity is described within a Lagrangian setting by a two-potential mixed formulation. In this formulation, the deformation field, a pressure field that ensues from a Legendre transform, and an internal variable of state Fv that describes the viscous part of the deformation are the independent fields. Consistent with the experimental evidence that viscous deformation is a volume-preserving process, the internal variable Fv is required to satisfy the constraint det Fv=1. To solve the resulting initial-boundary-value problem, a numerical solution algorithm is proposed that is based on a finite-element (FE) discretization of space and a finite-difference discretization of time. Specifically, a Variational Multiscale FE method is employed that allows for an arbitrary combination of shape functions for the deformation and pressure fields. To deal with the challenging non-convex constraint det Fv=1, a new time integration scheme is introduced that allows to convert any explicit or implicit scheme of choice into a stable scheme that preserves the constraint det Fv=1 identically. A series of test cases is presented that showcase the capabilities of the proposed formulation. This paper presents a formulation alongside a numerical solution algorithm to describe the mechanical response of bodies made of a large class of viscoelastic materials undergoing arbitrary quasistatic finite deformations. With the objective of having a unified formulation that applies to a wide range of highly compressible, nearly incompressible, and fully incompressible soft organic materials in a numerically tractable manner, the viscoelasticity is described within a Lagrangian setting by a two-potential mixed formulation. In this formulation, the deformation field, a pressure field that ensues from a Legendre transform, and an internal variable of state Fv that describes the viscous part of the deformation are the independent fields. Consistent with the experimental evidence that viscous deformation is a volume-preserving process, the internal variable Fv is required to satisfy the constraint detFv=1. To solve the resulting initial–boundary-value problem, a numerical solution algorithm is proposed that is based on a finite-element (FE) discretization of space and a finite-difference discretization of time. Specifically, a Variational Multiscale FE method is employed that allows for an arbitrary combination of shape functions for the deformation and pressure fields. To deal with the challenging non-convex constraint detFv=1, a new time integration scheme is introduced that allows to convert any explicit or implicit scheme of choice into a stable scheme that preserves the constraint detFv=1 identically. A series of test cases is presented that showcase the capabilities of the proposed formulation. This paper presents a formulation alongside a numerical solution algorithm to describe the mechanical response of bodies made of a large class of viscoelastic materials undergoing arbitrary quasistatic finite deformations. With the objective of having a unified formulation that applies to a wide range of highly compressible, nearly incompressible, and fully incompressible soft organic materials in a numerically tractable manner, the viscoelasticity is described within a Lagrangian setting by a two-potential mixed formulation. In this formulation, the deformation field, a pressure field that ensues from a Legendre transform, and an internal variable of state that describes the viscous part of the deformation are the independent fields. Consistent with the experimental evidence that viscous deformation is a volume-preserving process, the internal variable is required to satisfy the constraint det . To solve the resulting initial-boundary-value problem, a numerical solution algorithm is proposed that is based on a finite-element (FE) discretization of space and a finite-difference discretization of time. Specifically, a Variational Multiscale FE method is employed that allows for an arbitrary combination of shape functions for the deformation and pressure fields. To deal with the challenging non-convex constraint det , a new time integration scheme is introduced that allows to convert any explicit or implicit scheme of choice into a stable scheme that preserves the constraint det identically. A series of test cases is presented that showcase the capabilities of the proposed formulation. This paper presents a formulation alongside a numerical solution algorithm to describe the mechanical response of bodies made of a large class of viscoelastic materials undergoing arbitrary quasistatic finite deformations. With the objective of having a unified formulation that applies to a wide range of highly compressible, nearly incompressible, and fully incompressible soft organic materials in a numerically tractable manner, the viscoelasticity is described within a Lagrangian setting by a two-potential mixed formulation. In this formulation, the deformation field, a pressure field that ensues from a Legendre transform, and an internal variable of state Fv that describes the viscous part of the deformation are the independent fields. Consistent with the experimental evidence that viscous deformation is a volume-preserving process, the internal variable Fv is required to satisfy the constraint det Fv=1. To solve the resulting initial-boundary-value problem, a numerical solution algorithm is proposed that is based on a finite-element (FE) discretization of space and a finite-difference discretization of time. Specifically, a Variational Multiscale FE method is employed that allows for an arbitrary combination of shape functions for the deformation and pressure fields. To deal with the challenging non-convex constraint det Fv=1, a new time integration scheme is introduced that allows to convert any explicit or implicit scheme of choice into a stable scheme that preserves the constraint det Fv=1 identically. A series of test cases is presented that showcase the capabilities of the proposed formulation.This paper presents a formulation alongside a numerical solution algorithm to describe the mechanical response of bodies made of a large class of viscoelastic materials undergoing arbitrary quasistatic finite deformations. With the objective of having a unified formulation that applies to a wide range of highly compressible, nearly incompressible, and fully incompressible soft organic materials in a numerically tractable manner, the viscoelasticity is described within a Lagrangian setting by a two-potential mixed formulation. In this formulation, the deformation field, a pressure field that ensues from a Legendre transform, and an internal variable of state Fv that describes the viscous part of the deformation are the independent fields. Consistent with the experimental evidence that viscous deformation is a volume-preserving process, the internal variable Fv is required to satisfy the constraint det Fv=1. To solve the resulting initial-boundary-value problem, a numerical solution algorithm is proposed that is based on a finite-element (FE) discretization of space and a finite-difference discretization of time. Specifically, a Variational Multiscale FE method is employed that allows for an arbitrary combination of shape functions for the deformation and pressure fields. To deal with the challenging non-convex constraint det Fv=1, a new time integration scheme is introduced that allows to convert any explicit or implicit scheme of choice into a stable scheme that preserves the constraint det Fv=1 identically. A series of test cases is presented that showcase the capabilities of the proposed formulation. |
| ArticleNumber | 105312 |
| Author | Wijaya, Ignasius P.A. Lopez-Pamies, Oscar Masud, Arif |
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| References | Gent (b14) 1996; 69 Zener (b45) 1948 Lockett (b29) 1972 Anand, Kwack, Masud (b2) 2013; 72 Ghosh, Lopez-Pamies (b15) 2021; 56 Bergstrom, Boyce (b5) 1998; 46 Chockalingam, Roth, Henzel, Cohen (b10) 2021; 146 Hosseini-Farid, Ramzanpour, Ziejewski, Karami (b20) 2019; 116 Schöberl (b38) 1997; 1 Ghosh, Shrimali, Kumar, Lopez-Pamies (b16) 2021; 154 Shrimali, Ghosh, Lopez-Pamies (b39) 2021 Kumar, Lopez-Pamies (b24) 2016; 344 Halphen, Nguyen (b18) 1975; 14 Oyen, Cook (b34) 2003; 18 Green, Rivlin (b17) 1957; 1 Hochbruck, Ostermann (b19) 2010; 31 Beatty (b4) 2003; 70 Budday, Sommer, Haybaeck, Steinmann, Holzapfel, Kuhl (b7) 2017; 60 Weir (b43) 1951; 46 Birzle, Wall (b6) 2019; 94 Croom, Jin, Carroll, Long, Li (b11) 2019; 169 Chen, Ravi-Chandar (b9) 2022; 158 Ipsen, Rehman (b22) 2008; 30 Sidoroff (b40) 1974; 13 Lawson (b27) 1966; 3 Arruda, Boyce (b3) 1993; 41 Idiart, Lopez-Pamies (b21) 2012; 340 Lubachevsky, Stillinger, Pinson (b31) 1991; 64 Krieger (b23) 1972; 3 Simo (b41) 1992; 99 Reese, Wriggers, Reddy (b37) 2000; 75 Eleni, Perivoliotis, Dragatogiannis, Krokida, Polyzois, Charitidis, Ziomas, Gettleman (b12) 2013; 28 Wood, Martin (b44) 1964; 68A Le Tallec, Rahier, Kaiss (b28) 1993; 109 Pipkin, Rogers (b35) 1968; 16 Masud, Truster (b33) 2013; 267 Lopez-Pamies (b30) 2010; 338 Mao, Lin, Zhao, Anand (b32) 2017; 100 Freed, Rajagopal (b13) 2016; 227 Kwack, Masud, Rajagopal (b26) 2017; 83 Caylak, Mahnken (b8) 2012; 90 Kwack, Masud (b25) 2010; 48 Stickel, Powell (b42) 2005; 37 Ziegler, Wehrli (b46) 1987; 25 Ahn, Kim (b1) 2010; 14 Reese, Govindjee (b36) 1998; 35 Ahn (10.1016/j.jmps.2023.105312_b1) 2010; 14 Stickel (10.1016/j.jmps.2023.105312_b42) 2005; 37 Ziegler (10.1016/j.jmps.2023.105312_b46) 1987; 25 Masud (10.1016/j.jmps.2023.105312_b33) 2013; 267 Mao (10.1016/j.jmps.2023.105312_b32) 2017; 100 Ghosh (10.1016/j.jmps.2023.105312_b16) 2021; 154 Schöberl (10.1016/j.jmps.2023.105312_b38) 1997; 1 Beatty (10.1016/j.jmps.2023.105312_b4) 2003; 70 Arruda (10.1016/j.jmps.2023.105312_b3) 1993; 41 Idiart (10.1016/j.jmps.2023.105312_b21) 2012; 340 Reese (10.1016/j.jmps.2023.105312_b36) 1998; 35 Croom (10.1016/j.jmps.2023.105312_b11) 2019; 169 Kwack (10.1016/j.jmps.2023.105312_b25) 2010; 48 Halphen (10.1016/j.jmps.2023.105312_b18) 1975; 14 Lawson (10.1016/j.jmps.2023.105312_b27) 1966; 3 Le Tallec (10.1016/j.jmps.2023.105312_b28) 1993; 109 Shrimali (10.1016/j.jmps.2023.105312_b39) 2021 Freed (10.1016/j.jmps.2023.105312_b13) 2016; 227 Ipsen (10.1016/j.jmps.2023.105312_b22) 2008; 30 Birzle (10.1016/j.jmps.2023.105312_b6) 2019; 94 Ghosh (10.1016/j.jmps.2023.105312_b15) 2021; 56 Reese (10.1016/j.jmps.2023.105312_b37) 2000; 75 Bergstrom (10.1016/j.jmps.2023.105312_b5) 1998; 46 Eleni (10.1016/j.jmps.2023.105312_b12) 2013; 28 Gent (10.1016/j.jmps.2023.105312_b14) 1996; 69 Weir (10.1016/j.jmps.2023.105312_b43) 1951; 46 Zener (10.1016/j.jmps.2023.105312_b45) 1948 Pipkin (10.1016/j.jmps.2023.105312_b35) 1968; 16 Sidoroff (10.1016/j.jmps.2023.105312_b40) 1974; 13 Chen (10.1016/j.jmps.2023.105312_b9) 2022; 158 Chockalingam (10.1016/j.jmps.2023.105312_b10) 2021; 146 Hosseini-Farid (10.1016/j.jmps.2023.105312_b20) 2019; 116 Green (10.1016/j.jmps.2023.105312_b17) 1957; 1 Krieger (10.1016/j.jmps.2023.105312_b23) 1972; 3 Kwack (10.1016/j.jmps.2023.105312_b26) 2017; 83 Caylak (10.1016/j.jmps.2023.105312_b8) 2012; 90 Lockett (10.1016/j.jmps.2023.105312_b29) 1972 Simo (10.1016/j.jmps.2023.105312_b41) 1992; 99 Budday (10.1016/j.jmps.2023.105312_b7) 2017; 60 Wood (10.1016/j.jmps.2023.105312_b44) 1964; 68A Hochbruck (10.1016/j.jmps.2023.105312_b19) 2010; 31 Anand (10.1016/j.jmps.2023.105312_b2) 2013; 72 Lubachevsky (10.1016/j.jmps.2023.105312_b31) 1991; 64 Oyen (10.1016/j.jmps.2023.105312_b34) 2003; 18 Kumar (10.1016/j.jmps.2023.105312_b24) 2016; 344 Lopez-Pamies (10.1016/j.jmps.2023.105312_b30) 2010; 338 |
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| SubjectTerms | Elastomers Finite deformations Stabilized finite elements Stable ODE solvers |
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| Title | A unified determinant-preserving formulation for compressible/incompressible finite viscoelasticity |
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