有限变形微极线性粘弹性：热力学一致的推导、有限元实现及其在三维立方点阵中的应用

《International Journal of Engineering Science》：Micropolar linear viscoelasticity at finite strain: Thermodynamically-consistent derivation, finite element implementation, and application to 3D cubic lattices

　　

This is a technical paper developing a thermodynamically-consistent theory for micropolar viscoelasticity at finite strain, applied to architected lattice materials. Here are the key elements:

## Core Contributions

1. **Novel Theory**: First thermodynamically-consistent finite strain micropolar continuum formulation for linear viscoelasticity, accounting for both macroscopic and microscopic dissipation.

2. **Numerical Implementation**: Complete nonlinear finite element framework with implicit time integration, detailed stress update algorithms, and consistent tangent operators.

3. **Application**: Applied to 3D cubic lattice structures with parameters calibrated from direct numerical simulation (DNS).

## Background & Motivation

**Why micropolar theory?** Architected materials (lattice structures, cellular solids) manufactured by 3D printing have microstructural features where cell size ? is comparable to structural size L (0.1 < ?/L < 1). Classical continuum mechanics lacks intrinsic length scales and cannot capture this size effect.

**3M Continuum Hierarchy** (Eringen & Suhubi):
- **Micromorphic**: 9 additional DOF (unsymmetric microdeformation tensor)
- **Microstretch**: 4 additional DOF (1 scalar microstretch + 3 microrotations)
- **Micropolar**: 3 additional DOF (3 microrotation components) - chosen as simplest extension with fewest parameters

## Theoretical Framework

- Extends Eringen's micropolar continuum theory to finite strain viscoelasticity
- Linear viscoelasticity in the sense of Govindjee & Reese (1997) - refers to evolution equations for internal variables, not the overall model being linear
- Maxwell and Kelvin-Voigt models for micropolar constitutive behavior
- Based on compressible neo-Hookean hyperelastic potential

**Key Balance Equations** (weak form):
- Linear momentum: ∫Ωt wui, j tj,j dv = 0
- Angular momentum: ∫Ωt (-wφi,j mji + wφi εikl tkl) dv = 0

## Notation

- Summation over repeated indices implied
- Spatial derivatives: (Y),i = ?Y/?xi
- Material derivatives: (Y),I = ?Y/?XI
- Material time derivative: ? = DY/Dt
- Uppercase (X, E): reference configuration
- Lowercase (x, e): current configuration

## Paper Structure

| Section | Content |
|---------|---------|
| 2 | Thermodynamically-consistent theory derivation |
| 3 | Numerical implementation (stress update, tangent operators) |
| 4 | Specific constitutive equations (neo-Hookean base) |
| 5 | Numerical study with simple loading conditions |
| 6 | Application to 3D cubic lattice structures |
| 7 | Summary and future directions |

## Related Work Context

- Micropolar hyperelasticity at finite strain exists (Bauer et al., 2012; Schasching, 2024; Thatipelli et al., 2026)
- Finite strain viscoelasticity is well-established classically (Coleman & Noll, 1961; Holzapfel, 1996; Lubliner, 1985; Simo, 1987; Govindjee & Reese, 1997)
- Wang et al. (2022) developed finite strain viscoelastic director model for nematic liquid crystal elastomers (different physics)
- Glaesener et al. (2021) proposed FE2 framework for viscoelastic truss-based architected materials (explicitly resolves microstructure, computational overhead)

## Significance

This work fills a critical gap: prior to this paper, no general micropolar viscoelastic approach existed at finite strain. It enables efficient continuum-scale simulation of architected lattice materials with rate-dependent behavior while capturing length-scale effects through microrotational degrees of freedom.

具体而言，我们在应力松弛条件下针对以下简单标准载荷工况研究了尺寸效应：拉伸、压缩、直接剪切、弯曲和扭转。

**在简单三维立方晶格结构中的应用**

与经典连续介质建模方法不同，微极连续介质模型能够考虑在存在宏观旋转梯度的载荷工况下所观测到的长度尺度效应，而这些效应进而导致微旋转梯度，从而引起偶应力。特别是对于施加弯曲或扭转的结构的这一特性，在前文的载荷工况研究中已得到证实，即在弯曲和扭转条件下观察到了显著的尺寸效应。随着结构尺寸??的增加……

**总结与展望**

我们提出了一个新型框架，用于模拟表现出粘弹性行为、大变形以及微极长度尺度效应（即微旋转梯度）的粘弹性材料。该推导基于基本热力学论证，因而在热力学上是自洽的。在该框架内，粘弹性（耗散）过程被分解为宏观部分和微观部分，二者可以独立建模。基于这一新型框架，一个简单的本构……

**CRediT作者贡献声明**

A. Dummer：写作——审稿与编辑，写作——初稿，可视化，验证，软件，项目管理，方法论，调查，资金获取，形式分析，数据整理，概念构思。R.A. Regueiro：写作——审稿与编辑，验证，监督，资源，项目管理，方法论，形式分析。

**利益冲突声明**

作者声明，他们没有已知可能影响本文所报告工作的竞争性财务利益或个人关系。

**致谢**

本研究感谢Max Kade基金会（纽约）通过Max Kade奖学金提供的资金支持。RAR感谢美国能源部、国家核安全管理局预测科学学术联盟项目（PSAAP）的部分资助，资助编号为DE-NA0003962。作者感谢Matthias Neuner教授提供了关于有限应变下微极理论数值实现的代码。本工作利用了……

A. Dummer | R.A. Regueiro