  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.
