On Axisymmetrical Deformations of Nonlinear Membranes

Yang, Wensheng; Feng, William W.

doi:10.1115/1.3408651

Cited by 151 publications

(79 citation statements)

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“…Let (X, Y) and (x, y) be the Cartesian coordinates of a point near the center of the plate before and after deformation, respectively. The nondimensional curvature is given by (Yang and Feng, 1970) In these expressions, prime denotes differentiation with respect to X, and H is the undeformed plate thickness. To avoid end effects, the curvature was computed near the center of the plate.…”

Section: Appendix A: Implementation In Comsol Appendix B: Computationmentioning

confidence: 99%

Theoretical study of Beloussov’s hyper-restoration hypothesis for mechanical regulation of morphogenesis

Taber

2007

Biomech Model Mechanobiol

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Computational models were used to explore the idea that morphogenesis is regulated, in part, by feedback from mechanical stress according to Beloussov's Hyper-restoration (HR) Hypothesis. According to this hypothesis, active tissue responses to stress perturbations restore, but overshoot, the original (target) stress. To capture this behavior, the rate of growth or contraction is assumed to depend on the difference between the current and target stresses. Stress overshoot is obtained by letting the target stress change at a rate proportional to the same stress difference. The feasibility of the HR Hypothesis is illustrated by models for stretching of epithelia, cylindrical bending of plates, invagination of cylindrical and spherical shells, and early amphibian development. In each case, an initial perturbation leads to an active mechanical response that changes the form of the tissue. The results show that some morphogenetic processes can be entirely self-driven by HR responses once they are initiated (possibly by genetic activity). Other processes, however, may require secondary mechanisms or perturbations to proceed to completion.

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Section: Appendix A: Implementation In Comsol Appendix B: Computationmentioning

confidence: 99%

Theoretical study of Beloussov’s hyper-restoration hypothesis for mechanical regulation of morphogenesis

Taber

2007

Biomech Model Mechanobiol

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“…Let (X, Y) and (x, y) be the Cartesian coordinates of a point near the center of the beam before and after deformation, respectively. The nondimensional curvature is given by (Yang and Feng, 1970) In these expressions, prime denotes differentiation with respect to X, and H is the undeformed beam thickness. To avoid end effects, the curvature was computed near the middle of the beam.…”

Section: Appendix A: Computation Of Rotation Tensormentioning

confidence: 99%

Computational modeling of morphogenesis regulated by mechanical feedback

Ramasubramanian

Taber

2007

Biomech Model Mechanobiol

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Mechanical forces cause changes in form during embryogenesis and likely play a role in regulating these changes. This paper explores the idea that changes in homeostatic tissue stress (target stress), possibly modulated by genes, drive some morphogenetic processes. Computational models are presented to illustrate how regional variations in target stress can cause a range of complex behaviors involving the bending of epithelia. These models include growth and cytoskeletal contraction regulated by stress-based mechanical feedback. All simulations were carried out using the commercial finite element code ABAQUS, with growth and contraction included by modifying the zero-stress state in the material constitutive relations. Results presented for bending of bilayered beams and invagination of cylindrical and spherical shells provide insight into some of the mechanical aspects that must be considered in studying morphogenetic mechanisms.

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“…We non-dimensionalise the above quantities by multiplying them with λ 3 /C 1 (Pearce et al, 2011, Yang andFeng, 1970). The resulting non-dimensional meridional and circumferential Cauchy stresses are as follows.…”

Section: Cauchy Stressesmentioning

confidence: 99%

“…Axisymmetric deformations of membranes with simply connected geometries were studied analytically through a direct integration method (Yang and Feng, 1970) while those of non-simply connected geometries, such as a toroid, had to be studied through perturbation methods and certain approximations. Studying non-axisymmetric problems can be even more challenging but they are prevalent in practical applications, for example, see the papers by Grossman (1991aGrossman ( ,b, 1994 on the rim supports for inflatable reflectors for space applications.…”

Section: Introductionmentioning

confidence: 99%

Limit points in the free inflation of a magnetoelastic toroidal membrane

Reddy

Saxena

2017

International Journal of Non-Linear Mechanics

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One common phenomenon native to inflation of membranes is the elastic limit-point instability-a bifurcation point at which the membrane begins to deform enormously at the slightest increase of pressure. In the case of magnetoelastic materials, there is another possible phenomenon which we call magnetic limitpoint instability, a state referring to the non-existence of an equilibrium state -either stable or unstable. In this work, we are concerned with such instabilities in an incompressible isotropic magnetoelastic toroidal membrane with an initial circular cross-section. A non-uniform magnetic field is generated using a circular current carrying loop placed inside the membrane in addition to inflation by a uniform hydrostatic pressure. An energy formulation based on magnetization is used to model the magneto-mechanical coupling along with a Mooney-Rivlin constitutive model for the elastic strain energy density. Computations show that the magnetic field strongly influences the location of elastic limit points and in some cases can cause them to vanish. Multiple equilibrium states are obtained as solutions of the governing equations and a criterion based on second variation is employed to determine their stability. Existence and dependence of magnetic limit point on the magnetic field is demonstrated. While the quantitative results obtained here are specific to the toroidal geometry, the deformation behaviour can be generalised to any magnetoelastic membrane.

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On Axisymmetrical Deformations of Nonlinear Membranes

Cited by 151 publications

References 0 publications

Theoretical study of Beloussov’s hyper-restoration hypothesis for mechanical regulation of morphogenesis

Theoretical study of Beloussov’s hyper-restoration hypothesis for mechanical regulation of morphogenesis

Computational modeling of morphogenesis regulated by mechanical feedback

Limit points in the free inflation of a magnetoelastic toroidal membrane

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