  We introduce a data-driven fractional modeling framework aimed at complex
materials, and particularly bio-tissues. From multi-step relaxation experiments
of distinct anatomical locations of porcine urinary bladder, we identify an
anomalous relaxation character, with two power-law-like behaviors for
short/long long times, and nonlinearity for strains greater than 25%. The first
component of our framework is an existence study, to determine admissible
fractional viscoelastic models that qualitatively describe the linear
relaxation behavior. After the linear viscoelastic model is selected, the
second stage adds large-strain effects to the framework through a fractional
quasi-linear viscoelastic approach, given by a multiplicative kernel
decomposition of the selected relaxation function and a nonlinear elastic
response for the bio-tissue of interest. From single-relaxation data of the
urinary bladder, a fractional Maxwell model captures both short/long-term
behaviors with two fractional orders, being the most suitable model for small
strains at the first stage. For the second stage, multi-step relaxation data
under large strains were employed to calibrate a four-parameter fractional
quasi-linear viscoelastic model, that combines a Scott-Blair relaxation
function and an exponential instantaneous stress response, to describe the
elastin/collagen phases of bladder rheology. Our obtained results demonstrate
that the employed fractional quasi-linear model, with a single fractional order
in the range $\alpha = 0.25 - 0.30$, is suitable for the porcine urinary
bladder, producing errors below 2% without need for recalibration over
subsequent applied strains. We conclude that fractional models are attractive
tools to capture the bladder tissue behavior under small-to-large strains and
multiple time scales, therefore being potential alternatives to describe
multiple stages of bladder functionality.
