Analytical solutions for bonded elastically compressible layers

Abstract

a b s t r a c tCompression of elastic layers bonded between parallel plates often find applications in the mechanical characterization of soft materials or the transfer-printing of nanomembranes with polymeric stamps. In addition, annular rubbery gaskets and sealers are often under uniaxial compression during service. Analysis of elastic layers under compression has been focused on nearly incompressible materials such as rubbers, and empirical assumptions of displacements were adopted for simplicity. For compressible materials, solutions obtained by the method of averaged equilibrium are sufficient for effective compression modulus but inaccurate for the displacement or stress fields whereas solutions obtained by the method of series expansion are considerably complicated. In this paper, we report full field, closed-form solutions for bonded elastic layers (disks, annuli, annuli with rigid shafts, infinitely long strips) in compression using separation of variables without any pre-assumed deformation profile. These solutions can satisfy the exact forms of the equilibrium equations and all essential boundary conditions as well as the weak form of the natural boundary conditions. Therefore the predicted stress, displacement, and effective modulus have found excellent agreement with finite element modeling (FEM) results over a wide range of Poisson's ratio and aspect ratio. The analytical and FEM results of the stress, displacement, and effective modulus are highly sensitive to Poisson's ratio, especially near 0.5. Therefore we also propose a viable means to simultaneously measure the intrinsic Young's modulus and Poisson's ratio of elastically compressible layers without camera settings. When Poisson's ratio approaches 0.5, our solutions can degenerate to classical solutions for incompressible elastic layers.

A dual‐porosity scheme for fluid/solid substitution

“…Tsai and Lee (1998) derived an expression for the effective compressional stiffness of a circular interlayer of radius r and thickness 2h confined between parallel rigid plates and traction-free boundary conditions on the edge of the interlayer. Qiao and Lu (2015) confirmed validity of the basic assumptions of the derivation for Poisson's ratio ν f → 0.5 (μ f → 0) using finite-element-method computations. Given that the pore fill is assumed much weaker than the mineral grains, we can assume here that the grains are rigid.…”

Poroelasticity of (bio)polymer networks during compression: theory and experiment

“…The shear stress dominates the network stress when the ratio of this radial force to the local increase in radial force due to bending of the gel network G∂ 2 z U, with z the vertical coordinate, is small, i.e., M/GS 2 1. The solution interpolates between initial volume-conserving (VC) compression, during which no significant outflow of fluid occurs and of which the network displacement field and fluid pressure are well known 38 , and pressurized compression in which the fluid pressure is maximal, see Figure 3C. The dominant part of the normal force F is found as…”

“…Small-pore fibrin gels exhibit a qualitatively different increase in normal force during compression, see the inset of Figure 2, as compared to a large-pore gel, see Figure 1. Initially, the normal force increases as one would expect when the volume of the gel is conserved, based on the normal force of a linear elastic volume-conserving solid 38 with a shear modulus equal to that of the uncompressed fibrin network G 0 , see the blue line in the inset of Figure 2. Afterwards, it increases supralinearly before it starts to relax due to fluid outflow, similar as with a large-pore gel.…”

Stress analysis for nanomembranes under stamp compression

“…Thus, we can adopt our previous solutions for an infinitely long elastic layer bonded and compressed between two rigid plates (Eqs. (44) and (48) in [29]):…”

“…Then an external compression characterized by the compressive strain is applied to the stamp to form an intimate contact with the NM such that the NM can later be peeled off from the donor substrate by the stamp. Due to the Poisson's effect, the vertically compressed stamp would expand laterally, resulting in shear stresses on the stamp-NM interface [29][30][31], which rubs the NM and may eventually lead to cracked NM ( Fig. 1b).…”