Mixed Finite Element Formulations for Polyconvex Anisotropic Material Formulations

Dietzsch, Julian; Groß, Michael

doi:10.23967/wccm-eccomas.2020.201

Cited by 3 publications

(10 citation statements)

References 12 publications

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“…The finite element discretization follows from the mixed principle of virtual power (see Reference [5,2]). Here, we need the complete internal energy, which consists of the assumed temperature field Θ, the entropy density field η as the corresponding Lagrange multiplier, the superimposed stress tensor S S S to derive an energy-momentum scheme, an independent mixed field C C C and the corresponding Lagrange multiplier S S S. The internal energy functional reads…”

Section: Finite Element Formulationmentioning

confidence: 99%

“…Furthermore, this leads to a less complex weak form. The superimposed fields (see Reference [2] and [5]), which have both variants in common, are given by…”

Section: Finite Element Formulationmentioning

confidence: 99%

“…On the other hand, we focus on numerically stable dynamic long-time simulations with locking free meshes, and thus use higher-order accurate energy-momentum schemes emanating from mixed finite element methods. Hence, we adapt the variational-based space-time finite element method in Reference [2] to the new material formulation, and additionally include independent fields to obtain well-known mixed finite elements [3,4,5]. As representative numerical example, Cook's cantilever beam is considered.…”

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confidence: 99%

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Energy-momentum time integration of gradient-based models for fiber-bending stiffness in anisotropic thermo-viscoelastic continua

Dietzsch

Groß²,

Kalaimani³

2021

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

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For our research, we are motivated by dynamic simulations of 3D fiber-reinforced materials in lightweight structures. In such materials, the material reinforcement is performed by fiber rovings with a separate bending stiffness, which can be modelled by a second order gradient of the deformation mapping. Therefore, we extend a thermo-viscoelastic Cauchy continuum for fiber-matrix composites with single fibers by an independent field for the gradient of the right Cauchy-Green tensor. On the other hand, we focus on numerically stable dynamic long-time simulations with locking free meshes, and thus use higher-order accurate energy-momentum schemes emanating from mixed finite element methods. Hence, we adapt the variational-based space-time finite element method to the new material formulation, and additionally include independent fields to obtain well-known mixed finite elements. As representative numerical example, Cook’s cantilever beam is considered. We primarily analyze the influence of the fiber bending stiffness, as well as the spatial and time convergence up to cubic order. Furthermore, we look at the influence of the physical dissipation in the material.

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Section: Finite Element Formulationmentioning

confidence: 99%

“…Furthermore, this leads to a less complex weak form. The superimposed fields (see Reference [2] and [5]), which have both variants in common, are given by…”

Section: Finite Element Formulationmentioning

confidence: 99%

mentioning

confidence: 99%

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Energy-momentum time integration of gradient-based models for fiber-bending stiffness in anisotropic thermo-viscoelastic continua

Dietzsch

Groß²,

Kalaimani³

2021

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

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show abstract

“…The finite element discretization follows from the mixed principle of virtual power (see Reference [4,5]). Here, we need the complete internal energy, which consists of the assumed temperature fieldΘ, the entropy density field η as the corresponding Lagrange multiplier, the superimposed stress tensorS S S to derive an energy-momentum scheme, an independent mixed fieldC C C and the corresponding Lagrangian multiplier S S S. The internal energy functional reads…”

Section: Finite Element Formulationmentioning

confidence: 99%

“…For both formulations we introduce an superimposed fieldH H H to obtain an energy-momentum scheme, as well. The superimposed fields (see Reference [5] and [4]) are given bỹ…”

Section: Finite Element Formulationmentioning

confidence: 99%

Energy-momentum time integration of gradient-based models for fiber-bending stiffness in anisotropic thermo-mechanical continua

Dietzsch¹,

Groß²,

Kalaimani³

2021

9th Edition of the International Conference on Computational Methods for Coupled Problems in Science and Engineering

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Incorporation of thermal effects into the energy‐momentum consistent elastic model for fiber‐bending stiffness in composites

Dietzsch

Kalaimani

Groß

2023

Proc Appl Math and Mech

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Owing to the wide acceptance of fiber reinforced composites in the lightweight structure industry, the need to study its thermo‐elastic behavior in dynamical multi‐body systems still remains an open question. Standard continuum models, by virtue of first‐order gradient of deformation, cannot capture curvature effects and hence appropriate extensions of the standard modeling techniques are indispensable to model the accurate physical stiffness. The fiber‐bending curvature are captured by introducing a higher‐order gradient of the deformation as an independent field in the standard continuum. This work explores in detail, a further extension of this continuum to understand the thermal effects on the bending stiffness of fiber bundle in fiber‐reinforced composites. Our proposed model incorporates the temperature of the composite and its bidirectional dependency with the higher‐order gradients of the deformation into the extended mechanical continuum model. The implementation of the model is pursued in the context of multi‐field mixed finite element method and dynamic variational principle. The principle of virtual power enables our extended non‐isothermal continuum to derive an energy‐momentum scheme of higher order. The resulting time‐integrator exhibits physically consistent locking‐ free numerical behavior in dynamical simulations. Our novel approach is illustrated by simple transient dynamic simulations with a hyperelastic, transversely isotropic, polyconvex material behavior. We investigate the conservation of total energy and total momentum in the framework of thermo‐elasticity and study the spatial and temporal convergence for long‐term dynamic simulations with coarser grids and increased accurate numerical solutions.

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Mixed Finite Element Formulations for Polyconvex Anisotropic Material Formulations

Cited by 3 publications

References 12 publications

Energy-momentum time integration of gradient-based models for fiber-bending stiffness in anisotropic thermo-viscoelastic continua

Energy-momentum time integration of gradient-based models for fiber-bending stiffness in anisotropic thermo-viscoelastic continua

Energy-momentum time integration of gradient-based models for fiber-bending stiffness in anisotropic thermo-mechanical continua

Incorporation of thermal effects into the energy‐momentum consistent elastic model for fiber‐bending stiffness in composites

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