Timothy J. Healey

A group-theoretic approach to computational bifurcation problems with symmetry

Abstract Bifurcation of solution branches in static or steady problems of nonlinear mechanics is often associated with the underlying symmetry of the physical system. The use of group-theoretic methods in local bifurcation theory for problems with symmetry is well known. In this paper it is shown that the exploitation of symmetry via group invariance also yields an efficient computational approach to global bifurcation problems. These techniques are illustrated in the analysis of a lattice-dome structure with hexagonal symmetry. The methodology leads to a drastic reduction in numerical effort in the determination of several global solution branches, and enables the accurate computation of numerous singular points.

Material Symmetry and Chirality in Nonlinearly Elastic Rods

We treat certain classes of material symmetry in straight nonlinearly elastic rods in the presence of a uniform helical microstructure. In particular, we consider rods with chirality or “handedness”. This is a natural setting for manufactured ropes and cables and for biological filaments such as DNA strands. First we propose a novel definition of transverse material symmetry, enabling, e.g., a clear distinction to be made between hemitropic and isotropic rods. The former category can be realized from a simple spatial average of uniform helical symmetry. We show that hemitropic rods naturally exhibit mechanical coupling between extension and twist. Next we obtain an explicit representation theorem for the stored energy of rods with uniform helical symmetry (without averaging). We also study rods with dihedral-helical symmetry. This characterizes most manufactured ropes and cables, which are typically composed of two or more uniformly wound helical strands. Finally, we treat prismatic rods with transverse dihedral symmetry.

A group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell

Abstract The accurate prediction of the buckling load of thin shell structures is an important yet elusive goal. It is particularly important in the aerospace industry where thin shell members are commonly used as structural elements. Due to a lack of adequate analytical results, current practices in industry put heavy reliance on experimental testing and empirical data to supplement theoretical analysis (see D. Bushnell, Computerized Buckling Analysis of Shells, Martinus Nijhoff, Dordrecht, 1985). This paper focuses on recent results of the group theoretic approach to a numerical, global postbuckling analysis of a perfect, axially compressed cylindrical shell with “built-in” end conditions. The “built-in” end conditions obviate the existence of a “trivial” membrane solution branch. The example of an axially compressed cylindrical shell was chosen because it is well known that for thin shells, the primary axisymmetric solution branch is riddled with closely spaced, symmetry-breaking bifurcation points. In a numerical arc-length continuation scheme, the close proximity of the bifurcation points on the primary path manifests itself in severe ill-conditioning of the tangent stiffness matrix. Group theory helps one systematically find an “optimal” set of basis vectors, or symmetry modes, which reflect the symmetry of a given solution path. The immediate payoff in using these symmetry modes as basis vectors is that the tangent stiffness matrix block-diagonalizes and the numerical ill-conditioning is avoided. Thus, an efficient and accurate technique for computing solution branches of a specific type and a subsequent diagnosis for symmetry-breaking bifurcations is made relatively simple. Understanding the global behavior of the perfect structure is crucial in identifying critical imperfections and will ultimately diminish the heavy reliance on expensive experimental verification.

Global Bifurcations and Continuation in the Presence of Symmetry with an Application to Solid Mechanics

A group-theoretic approach to global bifurcation and continuation for one-parameter problems with symmetry is presented. The basic theme is the construction of a reduced problem, having solutions with specified symmetries, that can be analyzed by global or local techniques. A global analysis of a general class of reduced problems via well-established continuation techniques shows that symmetry is preserved on global continua of solutions. The approach is illustrated in the analysis of large post-buckling solutions of a nonlinearly elastic ring with

The role of the spinodal region in one-dimensional martensitic phase transitions

O(2)

Bifurcation to pear-shaped equilibria of pressurized spherical membranes

symmetry under uniform hydrostatic pressure, and yields several new results. Specific symmetries of global bifurcating solution branches are enumerated, which enables a detailed qualitative analysis.

Hidden Symmetry of Global Solutions in Twisted Elastic Rings

Abstract A common approach in modeling martensitic phase transitions in the framework of continuum mechanics involves a nonconvex energy. This paper analyzes the influence of the spinodal region, or the region where the energy density is concave, on the resulting equilibria. We compare a one-dimensional model with a degenerate spinodal region to models with a finite spinodal region. In all models we consider an elastic bar with a nonconvex energy placed on a rigid elastic foundation, to mimic elastic interactions between different phases in higher dimensions. Interfacial energy is modeled by a strain-gradient term. We find that when the spinodal region is small, global minima are not affected, and the minimum energy as a function of the overall strain exhibits nonsmooth oscillations associated with sudden finite phase nucleation. However, a sufficiently wide spinodal region results in the partial smoothening of the global minimum energy and infinitesimal phase nucleation in the interior of the bar. This involves gradual growth of a pretransitional nucleus with strain in the spinodal region. We show a hysteresis path using an energetic strategy of switching between branches of local minima.

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

Abstract We study the problem of non-spherical, axisymmetric equilibria of an inflated, spherical membrane. We model the membrane as a two-dimensional elastic body characterized by a general class of strain-energy functions, and we consider a general class of loading devices, including (soft) pressure control and (hard) control of the total mass of gas enclosed by the membrane as special cases. Employing tools of modern bifurcation theory, we illuminate the precise necessary and sufficient conditions for bifurcation from the spherical state to an axisymmetric “pear-shaped” state, and we perform a local post-bifurcation analysis. In particular, for a large class of physically reasonable strain-energy functions, we demonstrate the existence of an isola bifurcation (closed loop of non-spherical solutions), which is consistent with experimental observations.

MULTIPLE HELICAL PERVERSIONS OF FINITE, INTRISTICALLY CURVED RODS

Summary. We investigate global equilibria of twisted, isotropic elastic rings. The high degree of symmetry in the problem leads to nonisolated solutions. We prove that all solutions are flip-symmetric, and from this we can globally isolate all solution branches. We derive possible other choices for boundary conditions systematically and show that other conditions necessarily lead to spurious solutions. We show global computations performed by the Parallel Simplex Algorithm.

Bifurcation and metastability in a new one-dimensional model for martensitic phase transitions

We present rigorous local and global bifurcation results for a concrete example from 3-dimensional nonlinear elastostatics - the problem of barrelling of compressed cylindrical columns. We use standard tools of bifurcation theory for the local analysis, already producing results that are rare in our field. For the global part we employ the generalized degree designed by Healey and Simpson to overcome the specific difficulties of 3-dimensional nonlinear elasticity. Ours are the first global bifurcation results for a problem from 3-dimensional nonlinear elastostatics not governed by ordinary differential equations. Moreover, our approach to the barrelling problem provides a paradigm for the solution of a large class of problems in nonlinear elastostatics concerning bifurcation from a homogeneously deformed state.