Numerical studies of loss of ellipticity near singularities in an elastic material

Numerical solutions of anti-plane shear crack problems and screw dislocation problems are presented for materials in which the equilibrium equation varies in type locally from elliptic to hyperbolic as a result of deformation. These results show the emergence of surfaces of discontinuity in the displacement field in some materials. In other materials they show a chaotic mixture of elliptic phases at intermediate distances from the singularity. A statistical analysis applied to the numerical solutions demonstrates the role of elastic stability in the mechanics of these deformations.     

Localization of deformation and loss of macroscopic ellipticity in microstructured solids

Localization of deformation, a precursor to failure in solids, is a crucial and hence widely studied problem in solid mechanics. The continuum modeling approach of this phenomenon studies conditions on the constitutive laws leading to the loss of ellipticity in the governing equations, a property that allows for discontinuous equilibrium solutions. Micro-mechanics models and nonlinear homogenization theories help us understand the origins of this behavior and it is thought that a loss of macroscopic (homogenized) ellipticity results in localized deformation patterns. Although this is the case in many engineering applications, it raises an interesting question: is there always a localized deformation pattern appearing in solids losing macroscopic ellipticity when loaded past their critical state?In the interest of relative simplicity and analytical tractability, the present work answers this question in the restrictive framework of a layered, nonlinear (hyperelastic) solid in plane strain and more specifically under axial compression along the lamination direction. The key to the answer is found in the homogenized post-bifurcated solution of the problem, which for certain materials is supercritical (increasing force and displacement), leading to post-bifurcated equilibrium paths in these composites that show no localization of deformation for macroscopic strain well above the one corresponding to loss of ellipticity.     

On loss of ellipticity in second-gradient hyper-elasticity of fibre-reinforced materials

This paper establishes a three-dimensional hyper-elasticity framework for studying the manner in which fibre bending stiffness affects current knowledge regarding the presence of weak discontinuity surfaces in unconstrained fibre-reinforced solid materials. This is achieved by considering and studying the loss of ellipticity of a new set of incremental partial differential equations, which emerges from the second-gradient, hyper-elasticity theory developed in [11] and yields its conventional theory counterpart as a particular case. It is accordingly seen that, besides the conventional acoustic tensor met in relevant symmetric hyper-elasticity studies, where fibres are assumed perfectly flexible, some new, higher-order acoustic tensor is involved and becomes dominant in the present situation. Nevertheless, the manner becomes also clear in which the present analysis reduces to and, hence, connects with loss of ellipticity concepts met in conventional hyper-elasticity. No particular form is assigned to the strain energy density of the material, which is kept general throughout this paper. Considerable elucidation of the outlined new issues and concepts is however achieved by focusing attention on plane deformations of hyper-elastic solids reinforced by a single family of straight fibres. This development concludes with a specific application which relates the present analysis with kink band formation in unidirectional fibrous composites containing fibres resistant in bending.     

On tensile instabilities and ellipticity loss in fiber-reinforced incompressible non-linearly elastic solids

Loss of ellipticity and associated failure in fiber-reinforced non-linearly elastic solids is examined for uniaxial plane deformations. We consider separately fiber reinforcement that either endows the material with additional stiffness only in the fiber direction or introduces additional stiffness under shear deformations. In the first case it is shown that loss of ellipticity under tensile loading in the fiber direction corresponds to a turning point of the nominal stress and requires concavity of the Cauchy stress–stretch curve. For the second example loss of ellipticity occurs after the nominal stress maximum and prior to a turning point of the Cauchy stress.     

A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media

The conditions for the strong ellipticity of the equilibrium equations of compressible, isotropic, nonlinearly elastic solids (established by Simpson and Spector [1]) are expressed in terms of the stored-energy function regarded as a function of the principal stretches. The applicability of this reformulation is illustrated with the help of two specific examples.     

Strong ellipticity conditions for the differential operator of the finite isotropic elasticity

Summary TheStrong-Ellipticity conditions of the operator expressing the problem of a state of finite elastic deformation superimposed on a distorted configuration are examined. Sufficient criteria ofLocal Strong-Ellipticity, depending only on the so called Elastic Moduli of the material and on the principal components of stretch in the initial distorted configuration are supplied.

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Sommario Vengono esaminate le condizioni diEllitticità Forte dell'operatore traducente il problema di uno stato di deformazione elastica sovrapposto a una configurazione deformata. In particolare sono dati criteri sufficienti diEllitticità Forte Locale, dipendenti solo dai cosiddetti moduli elastici del materiale e le componenti principali di dilatazione pura nella configurazione iniziale deformata.

     

On the principles of material indifference and local equivalence

Meccanica - In terms of classical physics and in connection with constitutive equations, a suitable version of the local equivalence principle (basic for the construction of general relativity) and...     

Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation

In a recent paper we examined the loss of ellipticity and its interpretation in terms of fiber kinking and other instability phenomena in respect of a fiber-reinforced incompressible elastic material. Here we provide a corresponding analysis for fiber-reinforced compressible elastic materials. The analysis concerns a material model which consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. The assessment of loss of ellipticity can be cast in terms of the eigenvalues of the acoustic tensors associated with the isotropic and anisotropic parts of the strain-energy function. For the anisotropic part, two different reinforcing models are examined and it is shown that, depending on the choice of model and whether the fiber is under compression or extension, loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. Under compression the associated failure interpretations include fiber kinking and fiber splitting, while under extension fiber de-bonding and matrix failure are included.     

A simple derivation of necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials in plane strain

Necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials were first given by Knowles and Sternberg [3,4] by means of a lengthy calculation. Since then Aubert and Tahraoui [1] have shown the necessity of these conditions and simplified one of them using a different but still complicated method. The purpose of this note is to show how the conditions can be derived in a very simple way.     

Feasibility of laser-induced plasma spectroscopy for measurements of equivalence ratio in high-pressure conditions

In this paper, experimental results obtained with laser-induced plasma spectroscopy to retrieve local compositions are presented for an ambient pressure up to 5.0 MPa in a still cell. Well-controlled mixtures of gases are introduced and plasma is obtained with the fundamental emission of a pulsed Nd:YAG laser. Simultaneously, plasma shape and spectrally resolved data are taken with a temporal resolution down to 2 ns. First, the temporal evolutions of a high-pressure nitrogen plasma are analyzed as function of spark energy. It is shown that plasma changes orientation from an elongated shape parallel to the laser line to a perpendicular one in a very short time. Results are reported for both spatial and spectral variations. Afterward, the effects of increased carbon concentration are discussed in both shape and spectra. It is seen that strong intensity due to the atomic carbon emissions appear for the high-pressure case. From those experiments, calibration strategies are proposed to get equivalence ratio under high-pressure conditions with a ratio of carbon versus nitrogen and oxygen. The delay between plasma and measurements is set to 2,000 ns and the signal is integrated for 5,000 ns, so as to yield a good signal to noise ratio and a good sensitivity of the technique to changes in mixture fraction. Calibration curves are reported for equivalence ratio up to 1.00 and for pressure from 1.0 to 5.0 MPa. It is shown that typical uncertainties are limited to 7.5% regardless the equivalence ratio in a single shot approach using a spectral fit procedure, whereas it accounts to two times more in a more classical peak ratio approach. Increasing the pressure tends to increase the precision as lower pressure had higher uncertainties.     

Load imperfections disrupting symmetry in the spatial loss of stability with buckling

Budapest, Hungary. Translated from Prikladnaya Mekhanika, Vol. 27, No. 8, pp. 101–105, August, 1991.     

On the structure of the group g1 appearing in hyperelasticity

In the representation theorem for the stored energy function of a hyperelastic material, it is necessary to determine a subgroup g^*₁ of the symmetry group g of the stress tensor. It is the purpose of this note to study the construction of g^*₁ and its relation to the other subgroups ot g, viz. the commutator subgroup g_c and the orthogonal subgroup g₀.     

在无单元法里直接准确施加位移边界条件和材料不连续条件

在无单元伽辽金法(EFG)里,由于其滑动最小二乘近似位移函数不满足Kronecker条件,使得它不能准确地施加本质边界条件和材料不连续条件,从而极大地限制了EFG法的发展和进一步应用。本文在位移边界和不同材料交界面的离散结点上采用实际的结点位移值,提出了一种准确施加位移边界和材料不连续条件的方法,该方法实施简单、稳定、求解精度高,而且其推导得出的整体刚度矩阵具有正定、对称和带状分布的特点,可以和有限单元法一样,直接利用各种成熟、高效的线性方程组解法求解系统平衡方程。数值算例结果表明了文中理论和方法的正确性和可靠性。     

摩擦力对两个恢复系数公式等价性条件的影响

文章讨论两个作一般运动的刚体在考虑摩擦力的情况下碰撞(不受外力作用)．首先由欧勒动力学方程和质心运动定理导出在碰撞的压缩阶段和恢复阶段二碰撞点沿公法线的相对速度的变化量, 然后给出两个恢复系数公式的等价性条件与刚体之间摩擦力的关系．     

Strain-gradient elastic-plastic material models and assessment of the higher order boundary conditions

A gradient elastic material model exhibiting gradient kinematic and isotropic hardening is addressed within a thermodynamic framework suitable to cope with nonlocal-type continua. The Clausius–Duhem inequality is used, in conjunction with the concepts of energy residual, insulation condition and locality recovery condition, to derive all the pertinent restrictions upon the constitutive equations, including the PDEs and the related higher order (HO) boundary conditions that govern the gradient material behaviour. Through a suitable limiting procedure, the HO boundary conditions are shown to interpret the action, upon the body's boundary surface, of idealized extra HO constraints capable to impede the onset of strain as a nonlocality source and to react with a double traction (of dimension moment/area), work-conjugate of the impeded strain. The HO boundary conditions for the internal moving elastic/plastic boundary are also provided. A number of variational principles are proved. A few simple illustrative numerical examples are worked out.