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Effective stiffness as a concave function of inhomogeneity properties

Hi everyone, I am a postdoc at Yale University. I am posting on some of my previous work on fiber networks and heterogeneous materials. We reported that effective stiffness is a concave function of inhomogeneity properties, and effective stiffness becomes smaller with increasing heterogeneity as an example consequence. 

 

Fiber networks are assemblies of connected rods or springs. We used them as models of biological polymers. In my work on networks, we considered the uniaxial testing on a network before and after adding a fiber. Using reciprocity, we derived a relationship between the change in effective stiffness or overall stiffness of the network and the stiffness of a single added fiber

ΔEnetwork=a/(c+1/krod),

where a describes the deformation of the network at the points where the new fiber was attached prior to its attachment, and c describes the network response function at the attachment points. Both a and c are positive. According to this relation

∂2Enetwork/∂k2rod≤0.

This relation indicates that networks with fibers having a distribution of stiffnesses are softer, at the same mean fiber properties. The reason is that the rate of change of Enetwork with krod becomes slower at higher krod. The decrease of Enetwork from softer fibers is larger than its increase by stiffer fibers. Algebraically an inequality may be derived from the Taylor series of Enetwork in the space of krod’s about the average of krod. The Taylor series is regarded as a weak contrast expansion in previous literature on continua. 

Later I extended this relation to heterogeneous continua. This resulted in a relationship for change of effective stiffness as a function of the mode of deformations of the inhomogeneity. This relationship is also a concave function. The result of continua becoming softer with increasing heterogeneity has been previously reported empirically to the best of my knowledge.

 

I plan to work on further applications of these relationships in biomaterials.

Referense: [preprint_networks, preprint_continua, and postprint_networks, postprint_continua]

 

 

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