  Soft biological tissues exhibit a remarkable resilience to large mechanical
loads, a property which is associated with the strain stiffening capability of
the biopolymer networks that structurally support the tissues. Yet, recent
studies have shown that composite systems such as tissues and blood clots
exhibit mechanical properties that contradict those of the polymer matrix -
demonstrating stiffening in compression, but softening in shear and tension.
The microscopic basis of this apparent paradox remains poorly understood. We
show that composite hydrogels and tissues do indeed exhibit non-linear elastic
stiffening in shear - which is governed by the stretching of the polymer chains
in the matrix - and that it is driven by the same mechanism that drives
compression stiffening. However, we show that the non-linear elastic stiffening
in composite hydrogels and tissues is masked by mechanical dissipation arising
from filler-polymer interactions known as the Mullins effect, and we introduce
a method to characterize the non-linear elasticity of the composites in
isolation from this overall strain softening response through large-amplitude
oscillatory shear experiments. We present a comprehensive characterization of
the non-linear elastic strain stiffening of composite hydrogels and soft
tissues, and show that the strain stiffening in shear and compression are both
governed by universal strain amplification factors that depend on essential
properties of the composite system, such as the filler concentration and the
filler-polymer interaction strength. These results elucidate the microscopic
mechanisms governing the non-linear mechanics of tissues, which provides design
principles for engineering tissue-mimetic soft materials, and have broad
implications for cell-matrix mechanotransduction in living tissues under
strain.
