Finite element computation of large rubber membrane deformations

Fried, Isaac

doi:10.1002/nme.1620180503

Cited by 40 publications

(9 citation statements)

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“…A saccular cerebral aneurysm is here modelled as an axisymmetric membrane, which is hinged along its periphery and exposed to a (blood) pressure p. The membrane formulation used here is based on a work by Fried (1982) (also utilised by Kroon and Holzapfel (2007a) and Kyriacou et al (1996)), and a brief review of the formulation is provided below.…”

Section: Fe Formulation For An Axisymmetric Membranementioning

confidence: 99%

A theoretical model for fibroblast-controlled growth of saccular cerebral aneurysms

Kroon

Holzapfel

2009

Journal of Theoretical Biology

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A new theoretical model for the growth of saccular cerebral aneurysms is proposed by extending the recent constitutive framework of Kroon and Holzapfel (J. Theor. Biol., 247:775-787, 2007). The continuous turnover of collagen is taken to be the driving mechanism in aneurysmal growth. The collagen production rate depends on the magnitude of the cyclic deformation of fibroblasts, caused by the pulsating blood pressure during the cardiac cycle. The volume density of fibroblasts in the aneurysmal tissue is taken to be constant throughout the growth process. The growth model is assessed by considering the inflation of an axisymmetric membranous piece of aneurysmal tissue, with material characteristics representative of a cerebral aneurysm. The diastolic and systolic states of the aneurysm are computed, together with its load-free state. It turns out that the value of collagen pre-stretch, that determines growth speed and stability of the aneurysm, is of pivotal importance. The model is able to predict aneurysms with typical berry-like shapes observed clinically, and the predicted wall stresses correlate well with the experimentally obtained ultimate stresses of this type of tissue. The model predicts that aneurysms should fail when reaching a size of about 1.2-3.6 mm, which is smaller than what has been clinically observed. With some refinements, the model may, however, be used to predict future growth of diagnosed aneurysms.

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Section: Fe Formulation For An Axisymmetric Membranementioning

confidence: 99%

A theoretical model for fibroblast-controlled growth of saccular cerebral aneurysms

Kroon

Holzapfel

2009

Journal of Theoretical Biology

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“…The same assumption has been made by Papargyri-Pegiou (1995) and Papargyri-Pegiou and Stavrakakis (2000) for obtaining analytical and numerical solutions. Some examples of studies without this assumption include Fried (1982) who used finite element approach for large deformation of rubber membranes and Li and Steigmann (1995) who used tension field theory and Newton-Raphson method to numerically integrate the governing equations. An instance of finite element approach for large deformations can be found in a recent study by Papargyri-Beskou (2005).…”

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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“…Nonlinear formulations of membranes, particularly for rubber-like material, have been studied in the past (see e.g. Oden and Sato (1967) ;Haughton and Ogden (1978); Fried (1982); Wriggers and Taylor (1990); Ibrahimbegovic and Gruttmann (1993); Bonet et al (2000)). In this research, the computational membrane formulation developed by Sauer et al (2014) is employed.…”

Section: Membrane Theorymentioning

confidence: 99%