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Research Papers

On Rate Boundary Conditions for Soft Tissue Bifurcation Analysis

[+] Author and Article Information
Nir Emuna

Faculty of Aerospace Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: emuna@campus.technion.ac.il

David Durban

Faculty of Aerospace Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: aer6903@technion.ac.il

1Corresponding author.

This work is based in part on a Ph.D. thesis, Emuna, Nir, 2014, “Bending Theory of Functionally Graded Beams,” The Technion, Israel.Manuscript received March 12, 2018; final manuscript received July 29, 2018; published online October 1, 2018. Assoc. Editor: Guy M. Genin.

J Biomech Eng 140(12), 121010 (Oct 01, 2018) (10 pages) Paper No: BIO-18-1137; doi: 10.1115/1.4041165 History: Received March 12, 2018; Revised July 29, 2018

Mechanical instability of soft tissues can either risk their normal function or alternatively trigger patterning mechanisms during growth and morphogenesis processes. Unlike standard stability analysis of linear elastic bodies, for soft tissues undergoing large deformations it is imperative to account for the nonlinearities induced by the coupling between load and surface changes at onset of instability. The related issue of boundary conditions, in context of soft tissues, has hardly been addressed in the literature, with most of available research employing dead-load conditions. This paper is concerned with the influence of imposed homogeneous rate (incremental) surface data on critical loads and associated modes in soft tissues, within the context of linear bifurcation analysis. Material behavior is modeled by compressible isotropic hyperelastic strain energy functions (SEFs), with experimentally validated material parameters for the Fung–Demiray SEF, over a range of constitutive response (including brain and liver tissues). For simplicity, we examine benchmark problems of basic spherical patterns: full sphere, spherical cavity, and thick spherical shell. Limiting the analysis to primary hydrostatic states we arrive at universal closed-form solutions, thus providing insight on the role of imposed boundary data. Influence of selected rate boundary conditions (RBCs) like dead-load and fluid-pressure (FP), coupled with constitutive parameters, on the existence and levels of bifurcation loads is compared and discussed. It is argued that the selection of the appropriate type of homogeneous RBC can have a critical effect on the level of bifurcation loads and even exclude the emergence of bifurcation instabilities.

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Figures

Grahic Jump Location
Fig. 1

Sections through the center of full sphere, spherical cavity embedded in an infinite medium, and thick shell loaded on the surface by compressive traction σ < 0. The thick shell is subjected to equal tractions on both surfaces. The colored area represents the domain of each body. The blurred edges of the cavity represent the boundary as r → ∞.

Grahic Jump Location
Fig. 2

The effect of stiffening parameter α ((a)–(c)) and of compressibility level ν0 ((d)–(f)) on the critical volume ratio Jcr ((a) and (d)), normalized stress σcr/μ0 ((b) and (e)), and mode number ncr ((c) and (f)) in the full sphere geometry. In panels ((a)–(c)), the compressibility level is fixed on ν0 = 0.45, while in panels ((d)–(f)) the stiffening level is fixed on α = 5. Arrows with labels indicate the rate boundary condition used to generate the critical curves: VNTR, FL, and VTR.

Grahic Jump Location
Fig. 3

The effect of stiffening parameter α ((a)–(c)) and of compressibility level ν0 ((d)–(f)) on the critical volume ratio Jcr ((a) and (d)), normalized stress σcr/μ0 ((b) and (e)), and mode number ncr ((c) and (f)) in the spherical cavity geometry. In panels ((a)–(c)), the compressibility level is fixed on ν0 = 0.45, while in panels ((d)–(f)) the stiffening level is fixed on α = 5. Arrows with labels indicate the rate boundary condition used to generate the critical curves: VNTR, FL, and VTR.

Grahic Jump Location
Fig. 4

Critical volume ratio Jcr as a function of thickness ratio A/B of shells (A/B ≪ 1 for very thick shells and A/B → 1 for thin shells) subjected to identical rate boundary conditions on both surfaces: (a) VNTR, (b) FL, and (c) VTR. The red line is the envelope joining the lowest volume ratio obtained for mode numbers 2 ≤ n ≤ 7 over the entire range of A/B. Circles with number indicate a transition between two critical modes. An arrow with number indicates the mode associated with the curve.

Grahic Jump Location
Fig. 5

Critical volume ratio Jcr as a function of thickness ratio A/B of shells (A/B ≪ 1 for very thick shells and A/B → 1 for thin shells) subjected to combination of different rate boundary conditions on the opposing surfaces: ((a) and (b)) FP and VNTR, ((c) and (d)) FL and VNTR, ((e) and (f)) VTR and FP. The convention “IN–OUT” is used to denote combinations of conditions on the inner and outer surfaces. The red line is the envelope joining the lowest volume ratio obtained for mode numbers 2 ≤ n ≤ 7 over the entire range of A/B. Circles with number indicate a transition between two critical modes. An arrow with number indicates the mode associated with the curve. Note the limited range of thickness ratio for the combination FP–VTR (f). For this combination, modes 2 ≤ n ≤ 19 were used to reveal behavior near the thin shell limit.

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