Global Bifurcation in Nonlinear Elasticity with an Application to Barrelling States of Cylindrical Columns

Healey, Timothy J.; Montes-Pizarro, Errol L.

doi:10.1007/1-4020-2308-1_27

Cited by 12 publications

(34 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, the recent results of [11] and [14], suggest possible strategies for studying sufficient conditions for the existence of local bifurcating branches; in this vein, another issue worth exploring concerns the stability analysis of possible bifurcating solutions which emerge in the elliptic range or at the border of the domain of strong ellipticity of the adjacent equilibrium partial differential system.…”

Section: Discussionmentioning

confidence: 99%

“…7 We now turn to the adjacent equilibrium equations (3.6-3.8) for the class of incremental deformations (5.1). First, using (3.9), (3.10), (5.1) and (5.2), we find, after some calculations, that (C(α)[grad u]) 11 = {2φ (Ĩ)λαa 1 + 4φ (Ĩ)(1+α 2 )(λαa 1 + λa 2 + κa 3 ) 7 In the classical Taylor instability problem of fluid dynamics, such modes of bifurcation are not observed. …”

Section: Periodic Bifurcationsmentioning

confidence: 99%

See 1 more Smart Citation

Shear driven planar Couette and Taylor-like instabilities for a class of compressible isotropic elastic solids

Fosdick

Foti

Fraddosio

et al. 2009

Z. Angew. Math. Phys.

View full text Add to dashboard Cite

We study the possibility for an isotropic elastic body to support forms of instability induced by shear stress states which are reminiscent of the planar Couette and the Taylor-Couette patterns observed in the flow of viscous fluids. Here, we investigate the emergence of bifurcating periodic deformations for an infinitely long compressible elastic block confined between and attached to parallel plates which are subject to a relative shear displacement. We specialize our analysis by considering a generalized form of the Blatz-Ko strain energy function and show through numerical representative examples that planar Couette modes are always preferred with respect to the twisting Taylor-Couette modes. Finally, we introduce a suitably restricted form of the strong ellipticity condition for the incremental elasticity tensor and discuss its significance in this bifurcation problem. (2000). 74B20 · 74B15 · 74G60. Mathematics Subject Classification

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Periodic Bifurcationsmentioning

confidence: 99%

Shear driven planar Couette and Taylor-like instabilities for a class of compressible isotropic elastic solids

Fosdick

Foti

Fraddosio

et al. 2009

Z. Angew. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…In general, we do not have an explicit linearization at a non-trivial solution, hence we cannot check the complementing condition directly along those branches. We refer here to the works in [12,18,19], and Simpson and Spector (On bifurcation in finite elasticity: buckling of a rectangular rod, unpublished manuscript) where the complementing condition is violated at least once along the trivial solution branch for GreenHadamard type materials and for incompressible materials in [19]. It is obvious that the trivial solution branch, which is explicitly known, does not "stop" at places where the complementing condition fails.…”

Section: Introductionmentioning

confidence: 94%

Local Bifurcation Analysis of a Second Gradient Model for Deformations of a Rectangular Slab

Montes-Pizarro

Negrón-Marrero

2006

J Elasticity

Self Cite

View full text Add to dashboard Cite

In this paper we carry out a derivation of the equilibrium equations of nonlinear elasticity with an added second-gradient term proportional to a small parameter ε > 0. These equations are given by a fourth order semilinear system of pdes. We discuss different types of possible boundary conditions for these equations. We then specialize the equations to a rectangular slab and study the linearized problem about a homogenous deformation. We show that these equations admit solutions representable as Fourier series in one of the independent variables. Furthermore, we obtain the characteristic equation for the eigenvalues (possible bifurcation points) for the linear problem and derive asymptotic representations for this equation for small ε. We used these expressions to show that in the limit as ε → 0 the characteristic equation for ε > 0 converges uniformly (in certain regions of the parameter space) to the corresponding characteristic equation for ε = 0. When the base material (ε = 0) is that of a Blatz-Ko type, we get conditions for the existence of eigenvalues of the linear problem with ε > 0 and small. Our numerical results in this case indicate that the number of bifurcation points is finite when ε > 0 and that this number monotonically increases as ε → 0. For the problem with ε > 0 we get conditions for the existence of local branches of non-trivial solutions.

show abstract

“…Fu (2001) gave a list of contributions to linear stability analysis. Healey & Montes-Pizarro (2003) presented rigorous local and global bifurcation results for a compression of a three-dimensional nonlinear cylinder leading to a barrelling state by using the generalized degree designed in Healey & Simpson (1998) to overcome the specific difficulties of three-dimensional nonlinear elasticity. Their results are the first global bifurcation results for a problem from three-dimensional nonlinear elastostatics not governed by ordinary differential equations.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation to a corner-like formation in a slender nonlinearly elastic cylinder: asymptotic solution and mechanism

Dai

Wang

2008

Proc. R. Soc. A.

View full text Add to dashboard Cite

In this paper, we study the corner-like formation in a slender nonlinearly elastic cylinder due to compression. Mathematically, this is a very challenging problem: one needs to study the bifurcation of the nonlinear field equations (complicated nonlinear partial differential equations) and show that there is a bifurcation leading to a solution with a corner-like profile. We also aim to obtain the asymptotic expression for this post-bifurcation solution. As far as we know, there is no available analytical method to obtain the post-bifurcation solution of nonlinear partial differential equation(s). Here we use a novel approach to tackle the present problem. Through a method of coupled series-asymptotic expansions, we manage to derive the normal form equation of the original nonlinear field equations, which can be written as a singular ordinary differential equation (ODE) system (the vector field has a singularity at one point). With welding end conditions, the problem is reduced to study the boundary-value problem of a singular ODE system. It seems that such a boundary-value problem of a singular ODE system was not formulated or studied before in the context of the present problem. With the help of phase planes, we manage to solve such a boundary-value problem. It turns out that there is a vertical singular line in phase planes, which causes the bifurcation to a corner-like solution. The expression for this post-bifurcation solution is also obtained. From the analytical results obtained, we reveal that the coupling effect of the material nonlinearity and geometrical size is the physical mechanism that causes the formation of a corner-like profile.

show abstract

Global Bifurcation in Nonlinear Elasticity with an Application to Barrelling States of Cylindrical Columns

Cited by 12 publications

References 30 publications

Shear driven planar Couette and Taylor-like instabilities for a class of compressible isotropic elastic solids

Shear driven planar Couette and Taylor-like instabilities for a class of compressible isotropic elastic solids

Local Bifurcation Analysis of a Second Gradient Model for Deformations of a Rectangular Slab

Bifurcation to a corner-like formation in a slender nonlinearly elastic cylinder: asymptotic solution and mechanism

Contact Info

Product

Resources

About