Spatial symmetries and invariances play an important role in the behaviour of materials and should be respected in the description and modelling of material properties. The focus here is the class of physically symmetric and positive definite tensors, as they appear often in the description of materials, and one wants to be able to prescribe certain classes of spatial symmetries and invariances for each member of the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly 'higher' spatial invariance class. We formulate a modelling framework which not only respects these two requirements$-$positive definiteness and invariance$-$but also allows a fine control over orientation on one hand, and strength/size on the other. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear space of physically symmetric tensors, we consider it advantageous to widen the notion of mean to the so-called Fr\'echet mean on a metric space, which is based on distance measures or metrics between positive definite tensors other than the usual Euclidean one. It is shown how the random ensemble can be modelled and generated, independently in its scaling and orientational or directional aspects, with a Lie algebra representation via a memoryless transformation. The parameters which describe the elements in this Lie algebra are then to be considered as random fields on the domain of interest. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties$-$scaling, orientation, and prescribed material symmetry$-$on the desired quantities of interest.

翻译：空间对称性和不变性在材料行为中起着重要作用，应在材料特性的描述和建模中予以尊重。本文聚焦于物理对称且正定的张量类，这类张量在材料描述中经常出现，我们需要能够为整个集合的每个成员指定特定类别的空间对称性和不变性，同时要求集合的平均值或期望值服从可能"更高"的空间不变性类别。我们建立了一个建模框架，该框架不仅尊重这两个要求——正定性和不变性——同时还能在方向上以及强度/大小上实现精细控制。由于正定张量集不是线性空间，而是物理对称张量线性空间中的一个开放凸锥，我们认为将均值概念扩展为度量空间上的所谓弗雷歇均值是有利的，该均值基于正定张量之间不同于通常欧几里得距离的距离度量或指标。本文展示了如何通过李代数表示和无记忆变换，在缩放和方向/定向方面独立地对随机集合进行建模和生成。然后，描述该李代数中元素的参数被视为感兴趣区域上的随机场。作为示例，针对具有高度材料各向异性的骨骼——人类近端股骨，建立了二维和三维稳态热传导模型，其中采用随机热导率张量，数值结果显示，将不同的材料不确定性（缩放、定向和规定材料对称性）纳入本构模型对感兴趣量的影响显著不同。