A multi-scale framework is proposed for the prediction of the macroscopic hygro-elastic properties of oak wood. The distinctive features of the current multi-scale approach are that: (i) Four different scales of observation are considered, which enables the inclusion of heterogeneous effects from the nano-, micro-, and meso-scales in the effective constitutive behavior of oak at the macro-scale, (ii) the model relies on three-dimensional material descriptions at each considered length scale, and (iii) a moisture-dependent constitutive assumption is adopted at the nano-scale, which allows for recovering the moisture dependency of the material response at higher scales of observation. In the modeling approach, oak wood is assumed as homogeneous at the macro-scale. The meso-scale description considers the cellular structure of individual growth rings with three different densities. At the micro-scale, the heterogeneous nature of cell walls is described by the characteristics of the primary and secondary cell wall layers. Finally, the nano-scale response is determined by cellulose micro-fibrils embedded in a matrix of hemicellulose and lignin. The oak properties at the four length scales are connected via a three-level homogenization procedure, for which, depending on the geometry of the fine-scale configuration, an asymptotic homogenization procedure or Voigt averaging procedure is applied at each level to determine the effective hygro-elastic properties at the corresponding coarse scale. In addition, the moisture adsorption isotherms at each scale are constructed from a volume-weighted averaging of the moisture adsorption characteristics at the scale below. The computational results demonstrate that the macro-scale moisture-dependent, hygro-elastic behavior of oak wood is predicted realistically, thereby revealing the influence of the material density, the micro-fibril orientation, and the hygro-elastic properties from the underlying scales. The computed macro-scale properties of oak are in good agreement with experimental data reported in the literature.

提出了一个多尺度框架来预测橡木的宏观湿弹性特性。当前多尺度方法的显着特征是：（i）考虑了四种不同的观测尺度，这使得能够在橡木的有效本构行为中包含纳米、微米和介观尺度的异质效应在宏观尺度上，(ii) 该模型依赖于每个考虑的长度尺度上的三维材料描述，(iii) 在纳米尺度上采用依赖于水分的本构假设，这允许恢复更高观察尺度下的材料响应。在建模方法中，假设橡木在宏观尺度上是同质的。中观尺度描述考虑了具有三种不同密度的单个年轮的细胞结构。在微观尺度上，细胞壁的异质性是通过初生和次生细胞壁层的特征来描述的。最后，纳米级响应由嵌入半纤维素和木质素基质中的纤维素微纤维决定。四个长度尺度上的橡木特性通过三级均质化程序连接起来，其中，根据精细尺度配置的几何形状，在每个级别应用渐近均质化程序或 Voigt 平均程序来确定有效湿度-相应粗尺度下的弹性特性。此外，每个尺度的水分吸附等温线是根据以下尺度的水分吸附特性的体积加权平均值构建的。计算结果表明，橡木的宏观尺度水分依赖性湿弹性行为是真实预测的，从而揭示了材料密度、微纤维取向和底层尺度的湿弹性特性的影响。计算得到的橡木宏观特性与文献报道的实验数据非常吻合。以及来自底层鳞片的湿弹性特性。计算得到的橡木宏观特性与文献报道的实验数据非常吻合。以及来自底层鳞片的湿弹性特性。计算得到的橡木宏观特性与文献报道的实验数据非常吻合。