  A new relaxation approach is proposed which allows for the description of
stress- and strain-softening at finite strains. The model is based on the
construction of a convex hull replacing the originally non-convex incremental
stress potential which in turn represents damage in terms of the classical
$(1-D)$ approach. This convex hull is given as the linear convex combination of
weakly and strongly damaged phases and thus, it represents the homogenization
of a microstructure bifurcated in the two phases. As a result thereof, damage
evolves in the convexified regime mainly by an increasing volume fraction of
the strongly damaged phase. In contrast to previous relaxed incremental
formulations in G\"urses and Miehe [16] and Balzani and Ortiz [2], where the
convex hull has been kept fixated after construction, here, the strongly
damaged phase is allowed to elastically unload upon further loading. At the
same time, its volume fraction increases nonlinearly within the convexified
regime. Thus, strain-softening in the sense of a decreasing stress with
increasing strain can be modeled. The major advantage of the proposed approach
is that it ensures mesh-independent structural simulations without the
requirement of additional length-scale related parameters or nonlocal
quantities, which simplifies an implementation using classical material
subroutine interfaces. In this paper, focus is on the relaxation of
one-dimensional models for fiber damage which are combined with a microsphere
approach to allow for the description of three-dimensional fiber dispersions
appearing in fibrous materials such as soft biological tissues. Several
numerical examples are analyzed to show the overall response of the model and
the mesh-independence of resulting structural calculations.
