Effective behavior of viscoelastic composites: comparison of Laplace–Carson and time-domain mean-field approach

The paper focuses on deriving the macroscale viscoelastic constitutive laws using asymptotic expansion method. Both the differential and integral form of the linear viscoelastic constitutive relation of the phases is used in deriving the effective incremental potential and effective constitutive rel...

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Vydáno v:	Archive of applied mechanics (1991) Ročník 92; číslo 8; s. 2371 - 2395
Hlavní autoři:	Pallicity, Tarkes Dora, Cruz-González, O. L., Otero, J. A., Rodríguez-Ramos, R.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2022 Springer Nature B.V Springer Verlag
Témata:	Algorithms Asymptotic methods Asymptotic series Boundary conditions Civil Engineering Classical Mechanics Constitutive relationships Engineering Engineering Sciences Fiber volume fraction Finite element method Fractions Homogenization Inclusions Materials Mathematical analysis Matériaux composites et construction Mechanics Original Poisson's ratio Synergistic effect Tensors Theoretical and Applied Mechanics Time domain analysis Viscoelasticity Asymptotic homogenization Viscoelastic composites Incremental variational method Laplace–Carson Finite Element
ISSN:	0939-1533, 1432-0681
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Abstract	The paper focuses on deriving the macroscale viscoelastic constitutive laws using asymptotic expansion method. Both the differential and integral form of the linear viscoelastic constitutive relation of the phases is used in deriving the effective incremental potential and effective constitutive relation, respectively. The integral form is handled by considering the correspondence principle and the Laplace–Carson (LC) transform. A closed-form expression for the effective viscoelastic properties in LC domain is obtained by means of the asymptotic homogenization method (AHM). In addition, AHM coupled with finite element simulation of a representative volume element with periodic boundary conditions is used (AHM + FE). The last step in both approaches is the numerical inversion to the time domain. Solution in time domain is obtained with numerical Laplace inversion algorithms. In case of the differential form, using variational approach, the effective incremental potential in time domain is directly obtained using mean-field method. Different homogenization approaches are exemplified for evaluation of the effective relaxation behavior of composite (viscoelastic matrix reinforced by unidirectional elastic fibers), and they are compared. In the approaches based on LC transform, effective modulus and Poisson’s ratio agree well with each other for any property contrast and fiber volume fraction. However, in case of relatively low property contrast, mean field overpredicts as compared to LC approaches in the fiber direction, whereas at relatively higher property contrast, it is vice versa. The difference increases at higher volume fractions due to synergistic effect of the error due to geometrical assumptions involved in the localization tensor and interaction effects of the fiber inclusions. A good agreement in all directions is observed among the three schemes at intermediate volume fractions and property contrast. This study serves as benchmark for further theoretical improvements and experimental investigations.
AbstractList	The paper focuses on deriving the macroscale viscoelastic constitutive laws using asymptotic expansion method. Both the differential and integral form of the linear viscoelastic constitutive relation of the phases is used in deriving the effective incremental potential and effective constitutive relation, respectively. The integral form is handled by considering the correspondence principle and the Laplace–Carson (LC) transform. A closed-form expression for the effective viscoelastic properties in LC domain is obtained by means of the asymptotic homogenization method (AHM). In addition, AHM coupled with finite element simulation of a representative volume element with periodic boundary conditions is used (AHM + FE). The last step in both approaches is the numerical inversion to the time domain. Solution in time domain is obtained with numerical Laplace inversion algorithms. In case of the differential form, using variational approach, the effective incremental potential in time domain is directly obtained using mean-field method. Different homogenization approaches are exemplified for evaluation of the effective relaxation behavior of composite (viscoelastic matrix reinforced by unidirectional elastic fibers), and they are compared. In the approaches based on LC transform, effective modulus and Poisson’s ratio agree well with each other for any property contrast and fiber volume fraction. However, in case of relatively low property contrast, mean field overpredicts as compared to LC approaches in the fiber direction, whereas at relatively higher property contrast, it is vice versa. The difference increases at higher volume fractions due to synergistic effect of the error due to geometrical assumptions involved in the localization tensor and interaction effects of the fiber inclusions. A good agreement in all directions is observed among the three schemes at intermediate volume fractions and property contrast. This study serves as benchmark for further theoretical improvements and experimental investigations. The paper focuses on deriving the macroscale viscoelastic constitutive laws using asymptotic expan-sion method. Both the differential and integral form of the linear viscoelastic constitutive relation of the phasesis used in deriving the effective incremental potential and effective constitutive relation, respectively. Theintegral form is handled by considering the correspondence principle and the Laplace–Carson (LC) transform.A closed-form expression for the effective viscoelastic properties in LC domain is obtained by means of theasymptotic homogenization method (AHM). In addition, AHM coupled with finite element simulation of arepresentative volume element with periodic boundary conditions is used (AHM + FE). The last step in bothapproaches is the numerical inversion to the time domain. Solution in time domain is obtained with numer-ical Laplace inversion algorithms. In case of the differential form, using variational approach, the effectiveincremental potential in time domain is directly obtained using mean-field method. Different homogenizationapproaches are exemplified for evaluation of the effective relaxation behavior of composite (viscoelastic matrixreinforced by unidirectional elastic fibers), and they are compared. In the approaches based on LC transform,effective modulus and Poisson’s ratio agree well with each other for any property contrast and fiber volumefraction. However, in case of relatively low property contrast, mean field overpredicts as compared to LCapproaches in the fiber direction, whereas at relatively higher property contrast, it is vice versa. The differenceincreases at higher volume fractions due to synergistic effect of the error due to geometrical assumptionsinvolved in the localization tensor and interaction effects of the fiber inclusions. A good agreement in alldirections is observed among the three schemes at intermediate volume fractions and property contrast. Thisstudy serves as benchmark for further theoretical improvements and experimental investigations.
Author	Pallicity, Tarkes Dora Cruz-González, O. L. Rodríguez-Ramos, R. Otero, J. A.
Author_xml	– sequence: 1 givenname: Tarkes Dora orcidid: 0000-0003-4445-0305 surname: Pallicity fullname: Pallicity, Tarkes Dora email: tarkes.pallicity@kit.edu organization: Chair for Continuum Mechanics, Institute of Engineering Mechanics (ITM), Karlsruhe Institute of Technology (KIT) – sequence: 2 givenname: O. L. surname: Cruz-González fullname: Cruz-González, O. L. organization: Aix-Marseille Université, CNRS, Centrale Marseille, LMA UMR 7031, iMAT, Institut Jean Le Rond d’Alembert, UMR 7190, Sorbonne Université – sequence: 3 givenname: J. A. surname: Otero fullname: Otero, J. A. organization: Escuela de Ingeniería y Ciencias, Tecnológico de Monterrey – sequence: 4 givenname: R. surname: Rodríguez-Ramos fullname: Rodríguez-Ramos, R. organization: Facultad de Matemática y Computación, Universidad de La Habana
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Keywords	Asymptotic homogenization Viscoelastic composites Incremental variational method Laplace–Carson Finite Element
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Snippet	The paper focuses on deriving the macroscale viscoelastic constitutive laws using asymptotic expansion method. Both the differential and integral form of the... The paper focuses on deriving the macroscale viscoelastic constitutive laws using asymptotic expan-sion method. Both the differential and integral form of the...
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SubjectTerms	Algorithms Asymptotic methods Asymptotic series Boundary conditions Civil Engineering Classical Mechanics Constitutive relationships Engineering Engineering Sciences Fiber volume fraction Finite element method Fractions Homogenization Inclusions Materials Mathematical analysis Matériaux composites et construction Mechanics Original Poisson's ratio Synergistic effect Tensors Theoretical and Applied Mechanics Time domain analysis Viscoelasticity
Title	Effective behavior of viscoelastic composites: comparison of Laplace–Carson and time-domain mean-field approach
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