A Constitutive Law for Characterizing the Response of Insect Trachea

Frances M. Davis · Raffaella De Vita · Jake Socha
Virginia Tech


Abstract. Insect cuticles are composite materials made of chitin nano-fibers embedded in a matrix of cross- linked protein, water, lipids, metal ions, and calcium carbonate. The chitin fibers are among the stiffest natural fibers, with a Young’s modulus estimated to be approximately 150 GPa. The mechanical performance of the tracheal system strongly depends on the properties of the chitin fibers, their arrangement, orientation, waviness, and interaction with the matrix. Cuticles have been classified as nonlinear elastic [1], anisotropic [2], and inhomogeneous [3] materials. Nothing is known about how the micro-structural properties of cuticles in the tracheal system influence their mechanical properties. The relatively small size of tracheal tubes, small stresses and strains involved, and apparent inhomogeneities represent major challenges encountered in characterizing their mechanical behavior. An integrated theoretical and computational approach is used to accurately characterize the mechanics of the tracheal systems.

Three-dimensional constitutive relations are developed that describe the mechanical response of the insect tracheae. The model is formulated by employing a structural approach [4, 5] and the trachea wall is modeled as a thin-walled non-linear elastic circular cylinder. The waviness of the chitin fibers is assumed to be distributed according to a Weibull distribution. The constitutive formulation also takes into account the possibility of residual stress in the in vitro trachea wall. The stress-free configuration is assumed to be an open sector, which can be described by a single angular parameter. A parametric study is performed on the resulting constitutive law to determine the influence of fiber angle, Weibull parameters, and residual stress state on the response of the trachea under combined axial stretch and inflation.

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Keywords: insect trachea, constitutive model, chitin, nonlinear elastic, pressure-radius relationship.