Stable extreme damping in viscoelastic two-phase composites with non-positive-definite phases close to the loss of stability

• We performed a continuum-mechanics analysis of linear viscoelastic composite systems with non-positive definite (often called negative-stiffness) phases.
• Examples include an instructive 1-D spring-dashpot system, a three-dimensional two-phase solid of a coated spherical particle, and particle–matrix composites.
• All results are based on exact solutions and not on composite bounds or estimates.
• Composite property fine-tuning can lead to extreme damping increases (well beyond the damping of each constituent).
• In contrast to extreme stiffness, such extreme damping is thermodynamically stable (and not an effect of local resonance).

By investigating the effective response of linear viscoelastic composites, we demonstrate that stiff systems can exhibit stable extreme increases in overall damping if one of the composite phases loses positive-definiteness of its elasticities. While non-positive-definite elastic moduli (often referred to as negative stiffness) are thermodynamically unstable in unconstrained homogeneous solids, the geometric constraints among constituents in a composite can provide sufficient stabilization. Allowing for negative-stiffness phases in principle expands the range of attainable composite properties and promises extremely high composite stiffness and damping (significantly beyond those of the composite base materials) if the composite is appropriately tuned. This, however, raises questions of stability. In particular, the resulting high damping in stiff composites so far has only been shown to be stable in simple structural and elementary spring-dashpot systems, and therefore has remained a key open question for general composite materials. Studying successively the examples of a spring-dashpot model, a two-phase solid, and a general particle–matrix composite, we demonstrate that a non-positive-definite phase may indeed result in stable extreme damping, which is in line with recent experimental findings.