A Nonlinear Viscoelasticity Theory Based on Green-Naghdi Kinematic Assumptions and Generalized Strains

Core Concepts

The authors develop a nonlinear viscoelasticity theory based on the kinematic assumptions of the Green-Naghdi type and the concept of generalized strains within the framework of Hill's hyperelasticity.

Abstract

The key highlights and insights of the content are:

Motivation and Background:

The authors aim to generalize the finite deformation linear viscoelasticity models to the nonlinear regime.
They observe that the existing finite deformation linear viscoelasticity models, such as the Holzapfel-Simo model, implicitly adopt the kinematic assumption of the Green-Naghdi type.
The authors discuss the pros and cons of the multiplicative decomposition approach and the additive decomposition approach in modeling inelasticity.

Kinematic Assumptions and Generalized Strains:

The authors adopt the kinematic assumptions of the Green-Naghdi type, introducing a viscous deformation-like tensor Γ and the associated viscous strain Ev.
They utilize the concept of generalized strains, which allows for the description of material nonlinearity within the framework of Hill's hyperelasticity.
Various generalized strain families, such as the Seth-Hill, Curnier-Rakotomanana, Baˇzant-Itskov, and Curnier-Zysset strains, are presented and discussed.

Hyperelasticity of Hill's Class:

The authors construct the hyperelastic strain energy function based on Hill's hyperelasticity framework, which maintains the quadratic functional form while describing nonlinear response using generalized strains.
They introduce the concept of generalized Hill's hyperelasticity with multiple terms, which allows for improved fitting of experimental data compared to the conventional Hill's hyperelasticity model.

Constitutive Theory:

The authors derive the constitutive relations based on the Helmholtz free energy, which consists of an equilibrium part and a non-equilibrium part.
They show that the non-equilibrium stress vanishes in the equilibrium state, ensuring a well-posed model.

Computational Aspects:

The authors address the consistent linearization, constitutive integration, and modular implementation of the proposed nonlinear viscoelasticity theory.

Numerical Examples:

The authors provide a suite of numerical examples to demonstrate the capability of the proposed model in characterizing viscoelastic material behaviors at large strains.

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Key Insights Distilled From

A continuum and computational framework for viscoelastodynamics: III. A nonlinear theory

by Ju Liu,Jiash... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03090.pdf

Deeper Inquiries

How can the proposed nonlinear viscoelasticity theory be extended to account for multiple relaxation processes

To extend the proposed nonlinear viscoelasticity theory to account for multiple relaxation processes, we can introduce additional internal state variables ${\Gamma}_{\alpha}$, where $\alpha$ ranges from 1 to M. Each internal state variable represents a distinct relaxation process in the material. The constitutive relations can be expanded to include the contributions from each relaxation process, allowing for a more comprehensive description of the material's viscoelastic behavior. By incorporating multiple relaxation processes, the model can capture a wider range of material responses, especially in complex viscoelastic materials with multiple relaxation mechanisms.

What are the potential limitations or challenges in applying the Green-Naghdi type kinematic assumptions to model specific material behaviors, such as anisotropic or rate-dependent materials

Applying the Green-Naghdi type kinematic assumptions to model specific material behaviors, such as anisotropic or rate-dependent materials, may pose certain limitations and challenges.
For anisotropic materials, the assumption of additive decomposition may not fully capture the complex directional dependencies present in the material's response. Anisotropic materials exhibit varying mechanical properties in different directions, and the additive decomposition approach may oversimplify the material behavior, leading to inaccuracies in the model predictions. Alternative kinematic assumptions or more sophisticated models may be required to accurately capture the anisotropic behavior of materials.
In the case of rate-dependent materials, the Green-Naghdi type kinematic assumptions may struggle to account for the time-dependent behavior of the material. Rate-dependent materials exhibit stress and strain rate dependencies that evolve over time, requiring a more dynamic and comprehensive modeling approach. The additive decomposition approach may not adequately capture the evolving material response under varying loading rates, leading to discrepancies between the model predictions and experimental observations. Incorporating time-dependent terms or additional variables to account for the rate-dependent behavior may be necessary to improve the model's accuracy.

Can the connections between the multiplicative decomposition approach and the additive decomposition approach be further explored to provide deeper insights into the underlying mechanics

Exploring the connections between the multiplicative decomposition approach and the additive decomposition approach can provide valuable insights into the underlying mechanics of material behavior. By comparing and contrasting these two kinematic assumptions, researchers can gain a deeper understanding of the fundamental principles governing deformation and stress in materials.
Investigating how these two approaches converge or diverge in modeling different material behaviors can help identify the strengths and limitations of each method. Understanding the conditions under which one approach may be more suitable than the other can guide researchers in selecting the most appropriate kinematic assumption for specific material systems.
Furthermore, exploring the mathematical and physical implications of transitioning between these two approaches can shed light on the underlying mechanisms of deformation and relaxation in viscoelastic materials. By elucidating the connections between multiplicative and additive decompositions, researchers can enhance their modeling capabilities and develop more robust and accurate viscoelasticity theories.