TY - JOUR
T1 - Macroscopic constitutive relations for elastomers reinforced with short aligned fibers
T2 - Instabilities and post-bifurcation response
AU - Avazmohammadi, Reza
AU - Ponte Castañeda, Pedro
N1 - Funding Information:
This material is based upon work supported by the National Science Foundation , which began under Grant no. CMMI-1068769 and was completed under Grant no. DMS-1108847 . The writing of the paper was completed while P.P.C. was visiting the University of Stuttgart with the support of the Humboldt Research Award.
Publisher Copyright:
© 2015 Elsevier Ltd
PY - 2016/12/1
Y1 - 2016/12/1
N2 - This paper is concerned with the characterization of the macroscopic response and possible development of instabilities in a certain class of anisotropic composite materials consisting of random distributions of aligned rigid fibers of elliptical cross section in a soft elastomeric matrix, which are subjected to general plane strain loading conditions. For this purpose, use is made of an estimate for the stored-energy function that was derived by Lopez-Pamies and Ponte Castañeda (2006b) for this class of reinforced elastomers by means of the second-order linear comparison homogenization method. This homogenization estimate has been shown to lose strong ellipticity by the development of shear localization bands, when the composite is loaded in compression along the (in-plane) long axes of the fibers. The instability is produced by the sudden, collective rotation of a band of fibers to partially release the high stresses that develop in the elastomer matrix when the composite is compressed along the stiff, long-fiber direction. Consistent with the mode of the impending instability, a lower-energy, post-bifurcation solution is constructed where “striped domain” microstructures consisting of layers with alternating fiber orientations develop in the composite. The volume fractions of the layers and the fiber orientations within the layers adjust themselves to satisfy equilibrium and compatibility across the layers, while remaining compatible with the imposed overall deformation. Mathematically, this construction is shown to correspond to the rank-one convex envelope of the original estimate for the energy, and is further shown to be polyconvex and therefore quasiconvex. Thus, it corresponds to the “relaxation” of the stored-energy function of the composite, and can in turn be viewed as a stress-driven “phase transition,” where the symmetry of the fiber microstructures changes from nematic to smectic.
AB - This paper is concerned with the characterization of the macroscopic response and possible development of instabilities in a certain class of anisotropic composite materials consisting of random distributions of aligned rigid fibers of elliptical cross section in a soft elastomeric matrix, which are subjected to general plane strain loading conditions. For this purpose, use is made of an estimate for the stored-energy function that was derived by Lopez-Pamies and Ponte Castañeda (2006b) for this class of reinforced elastomers by means of the second-order linear comparison homogenization method. This homogenization estimate has been shown to lose strong ellipticity by the development of shear localization bands, when the composite is loaded in compression along the (in-plane) long axes of the fibers. The instability is produced by the sudden, collective rotation of a band of fibers to partially release the high stresses that develop in the elastomer matrix when the composite is compressed along the stiff, long-fiber direction. Consistent with the mode of the impending instability, a lower-energy, post-bifurcation solution is constructed where “striped domain” microstructures consisting of layers with alternating fiber orientations develop in the composite. The volume fractions of the layers and the fiber orientations within the layers adjust themselves to satisfy equilibrium and compatibility across the layers, while remaining compatible with the imposed overall deformation. Mathematically, this construction is shown to correspond to the rank-one convex envelope of the original estimate for the energy, and is further shown to be polyconvex and therefore quasiconvex. Thus, it corresponds to the “relaxation” of the stored-energy function of the composite, and can in turn be viewed as a stress-driven “phase transition,” where the symmetry of the fiber microstructures changes from nematic to smectic.
KW - Finite deformations
KW - Hyperelastic composites
KW - Mechanical instabilities
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U2 - 10.1016/j.jmps.2015.07.007
DO - 10.1016/j.jmps.2015.07.007
M3 - Article
AN - SCOPUS:84939486997
SN - 0022-5096
VL - 97
SP - 37
EP - 67
JO - Journal of the Mechanics and Physics of Solids
JF - Journal of the Mechanics and Physics of Solids
ER -