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In this paper, we present a dimensional reduction to obtain a one-dimensional model to analyze localized necking or bulging in a residually stressed circular cylindrical solid. The nonlinear theory of elasticity is first specialized to obtain the equations governing the homogeneous deformation. Then, to analyze the nonhomogeneous part, we include higher-order correction terms of the axisymmetric displacement components leading to a three-dimensional form of the total potential energy functional. Details of the reduction to the one-dimensional form are given. We focus on a residually stressed Gent material and use numerical methods to solve the governing equations. Two loading conditions are considered. First, the residual stress is maintained constant, while the axial stretch is used as the loading parameter. Second, we keep the pre-stretch constant and monotonically increase the residual stress until bifurcation occurs. We specify initial conditions, find the critical values for localized bifurcation, and compute the change in radius during localized necking or bulging growth. Finally, we optimize material properties and use the one-dimensional model to simulate necking or bulging until the Maxwell values of stretch are reached. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the influence of residual stress on the stability of a solid circular cylinder subject to axial extension. The nonlinear theory of elasticity is used to derive the equations governing the linearized incremental deformations superimposed on a known finitely deformed configuration. Specialized to the neo-Hookean model with residual stress, the bifurcation analysis results in an exact bifurcation condition for zero and periodic modes, based on the Stroh formalism. An expansion technique is adopted to treat singularities in the governing equations. We investigate three loading scenarios, specifically, gradual increase in axial stretch with constant residual stress and gradual increase in residual stress with fixed axial length or fixed axial force. In each case, the zero mode, corresponding to a localized bifurcation, occurs first. An explicit bifurcation condition for the zero mode is derived, which accounts for the effect of geometric dimensions, the residual stress, and the axial elongation on the stability of the solid cylinder. The critical values of the residual stress and the axial force for localized necking or bulging to occur are identified. In particular, we show that a reduction in residual stress delays the onset of localized necking. At constant values of pre-stretch, with 1. 5 < λ < 2. 10692 , localized necking occurs at a critical residual stress, whereas localized bulging occurs when λ > 2. 10692. We use the Maxwell equal-area rule to characterize the "two-phase" deformation consisting of necked and bulged regions. It is shown that the Maxwell values of stretch identify the radii of the two regions, which are connected by a transition zone that translates along the cylinder. At the completion of "two-phase" deformation, the stretch is again uniform. [ABSTRACT FROM AUTHOR]

This paper reviews macroscopic aspects of the theory of magnetoelastostatics, starting with a brief summary of the experimental and theoretical contributions leading to the development of the current state-of-the-art. It offers some different perspectives than hitherto, with incompressible materials being the main concern. The use of the so-called total energy (density) functions is highlighted along with their associated total stress tensors and succinct forms of the constitutive equations. The symmetry of the total Cauchy stress tensor, which incorporates the non-symmetric Maxwell stress within the material, is emphasized and it is noted that the use of such a Maxwell stress, often appearing in the literature, is thereby avoided. The theory is illustrated for some simple prototype boundary-value problems, specifically the homogeneous deformation of an infinite slab of magnetoelastic material in the presence of a magnetic field and the non-homogeneous extension and inflation of an infinitely long circular cylindrical tube in the presence of either an axial or a circumferential magnetic field. [ABSTRACT FROM AUTHOR]

We place the Ogden model of rubber elasticity, published in Proceedings of the Royal Society 50 years ago, in the wider context of the theory of nonlinear elasticity. We then follow with a short interview of Ray Ogden FRS and introduce the papers collected for this Theme Issue. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'. [ABSTRACT FROM AUTHOR]

In the finite magnetoelastic deformation context, the Zeeman magnetic energy on its own is found to be inconsistent with the general theory of nonlinear magnetoelasticity. An appropriate additional term, depending on the definition of the reference configuration magnetization, is used to modify the total magnetoelastic energy to confer consistency. In particular, this confirms that the total Cauchy stress tensor is symmetric, in contrast to recent conclusions in the literature where the Zeeman magnetic energy was utilized. • Derivation of consistent magnetoelastic constitutive equation including the Zeeman magnetic energy. • Emphasizing the symmetry of the total Cauchy stress tensor. • Correcting errors in the analysis of hard-magnetic soft solids in the recent literature. [ABSTRACT FROM AUTHOR]