Studies of quality of geometrically nonlinear elasticity theory for small strains and arbitrary displacements

Earlier it was shown in [1, 2] that the equations of classical nonlinear elasticity constructed for the case of small strains and arbitrary displacements are ill posed, because their use in specific problems may result in the appearance of “spurious” bifurcation points. A detailed analysis of these equations and the construction, in their stead, of consistent equations of geometrically nonlinear theory of elasticity can be found in [3]. Certain steps in this direction were also made in [4, 5]. In [3], it was also stated that the methods and applied program packages (APPs) based on the use of the classical relations of nonlinear elasticity require some revision and correction. In the present paper, this conclusion is justified and confirmed by numerical finite-element solutions of several three-dimensional geometrically nonlinear deformation problems and linearized problems on the stability of equilibrium of a rectilinear beam. These solutions were obtained by using two APPs developed by the authors and the well-known APP “ANSYS.” It is shown that the classical equations of the geometrically nonlinear theory of elasticity, which underly the first of the developed APP and the well-known APP “ANSYS,” often lead to overestimated buckling loads for structural members as compared with the consistent equations proposed in [1–3].     

Relationships of the Timoshenko-type theory of thin shells with arbitrary displacements and strains

A new modified version of the Timoshenko theory of thin shells is proposed to describe the process of deformation of thin shells with arbitrary displacements and strains. The new version is based on introducing an unknown function in the form of a rotation vector whose components in the basis fitted to the deformed mid-surface of the shell are the components of the transverse shear vector and the extensibility in the transverse direction according to Chernykh. For the case with the shell mid-surface fitted to an arbitrary non-orthogonal system of curvilinear coordinates, relationships based on the use of true stresses and true strains in accordance with Novozhilov are obtained for internal forces and moments. Based on these relationships, a problem of static instability of an isotropic spherical shell experiencing internal pressure is solved. The shell is considered to be made either of a linear elastic material or of an elastomer (rubber), which is described by Chernykh’s relationships.     

A unified thermodynamic theory of elasticity and plasticity

A thermodynamics is developed for a unified theory of elasticity and plasticity in infinitesmal strain. The constitutive equations which relate stress and strain deviators are rate type differential equations. When they satisfy a Lipschitz condition, uniqueness for the initial value problem dictates that the stress and strain will be related through elastic relations. Failure of the Lipschitz condition occurs when a von Mises yield condition is achieved: Plastic yield then occurs and the deviator relations turn into the Prandtl-Reuss equations. The plastic yield solution is stable during loading and unstable during unloading. The requirement that the solution followed during unloading be stable dictates entry into an elastic regime. Appropriate thermodynamic functions are constructed. It then appears that stress deviator (not strain deviator) is a viable state variable, and the thermodynamic relations are constructed in terms of a Gibbs function. The energy balance leads to satisfaction of the Clausius-Duhem inequality (and thus the second law of thermodynamics) in an elastic regime because it is shown that in an elastic regime entropy production is caused only by heat flux. During yield, the proper method of differentiating yields entropy production terms in addition to those arising from heat flux. These terms are positive during loading, whence it is concluded that the requirement that a stable solution be followed leads to satisfaction of the Clausius-Duhem inequality during plastic as well as elastic behavior.     

A mixed system of equations of elasticity

A mixed system of six equations of elasticity is represented as a Hamiltonian (canonical) operator system in one of the spatial coordinates. It is shown that this system is the Euler equations for the Hellinger–Reissner principle with an appropriately modified integrand. One more functional with an operator integrand from which the canonical operator system can be derived is set up     

On the constitutive equations for cyclic plasticity under nonproportional loading

Experimental results on 316 stainless steel, at room temperature, under strain controled axial-torsion loadings, point out a large difference in strengthening between proportional and nonproportional loadings. The maximum difference is produced by 90° out of phase straining. In this paper we discuss the possibility of describing this additional hardening, without any new internal variable, by only modifying the expression for the yield criterion.     

A novel Volume-Compensated Particle method for 2D elasticity and plasticity analysis

A novel Volume-Compensated Particle model (VCPM) is proposed for the modeling of deformation and fracture in solids. In this proposed method, two potentials are introduced to model the interactions between material particles, i.e., a local pair-wise potential and a non-local multi-body potential. The local pair-wise potential is utilized to account for the constitutive relationship within the connecting bonds between particles while the non-local multi-body potential is employed for considering the volumetric effects under general mechanical loadings. The potential coefficients are determined by matching the potential energy stored in a discrete unit cell to the strain energy at the classical continuum level. A volume conservation scheme is proposed to model the plastic deformation. The validity of the proposed model is tested against the classical elasticity and elasto-plasticity benchmarks before its application to fracture problems. Several conclusions are drawn based on the proposed study.     

Finite multiplicative plasticity for small elastic strains with linear balance equations and grain boundary relaxation

This paper is concerned with the formulation of a phenomenological model of finite elasto-plasticity valid for small elastic strains for initially isotropic polycrystalline material. As a basic we assume the multiplicative split of the deformation gradient into elastic and plastic part. A key feature of the model is the introduction of an independent field of 'elastic' rotations which eliminate the remaining geometrical nonlinearities coming from finite elasticity in the presence of small elastic strains. In contrast to micro-polar theories an evolution equation for is presented which relates to making use of a new device found by the author to perform the polar decomposition asymptotically. The model is shown to be invariant under both change of frame and rotation of the so called intermediate configuration. The corresponding equilibrium equations at frozen plastic and viscoelastic configuration constitute then a linear, elliptic system with nonconstant coefficients which makes this model amenable to a rigorous mathematical analysis. The introduced hysteresis effects within the elastic region are related to viscous elastic rotations of the grains of the polycrystal due to internal friction at the grain boundaries and constitute as such a rate dependent transient texture effect. The inclusion of work hardening will be addressed in future work. Received March 07, 2002 / Published online February 17, 2003 RID="*" ID="*"Communicated by Kolumban Hutter, Darmstadt     

薄板在冲击载荷下线弹性理想塑性响应的相似性研究

对于比例模型和原型使用不同弹塑性材料的冲击相似性问题,由于弹性和塑性阶段材料特性的差别及其在不同变形阶段的弹性塑性共存现象,将导致原有的结构冲击相似性理论失效。基于薄板冲击问题的理论模型,采用方程分析方法重新推导了材料线弹性以及理想刚塑性共存时的冲击响应的相似律。提出了一种能够同时考虑弹性变形和塑性变形的结构缩放响应相似性分析的厚度补偿方法,并利用数值分析验证了提出方法的适用性。分析结果表明,使用厚度补偿方法得到的比例模型结构响应能够准确地预测原型结构的冲击响应。

     

谈谈弹塑性力学中的简化问题

讨论了弹塑性力学中的基本假设、本构模型和简化分析方法. 指出在这一领域中, 简化是非常重要的和必需的.     

Diagonalization of a three-dimensional system of equations in terms of displacements of the linear theory of elasticity of transversely isotropic media

A dynamic three-dimensional system of linear equations in terms of displacements of the theory of elasticity of transversely isotropic media is given explicit expressions for phase velocities and polarization vectors of plane waves. All the longitudinal normals are found. For some values of the elasticity moduli, the system of equations is reduced to a diagonal shape. For static equations, all the conditions of the system ellipticity are determined. Two new representations of displacements through potential functions that satisfy three independent quasi-harmonic equations are given. Constraints on elasticity moludi, at which the corresponding coefficients in these representations are real, different, equal, or complex, are determined. It is shown that these representations are general and complete. Each representation corresponds to a recursion (symmetry) operator, i.e., a formula of production of new solutions.     

The longitudinal-torsional displacements of space structural elements with variable section and elasticity

The general exact solution to a boundary-value problem is constructed in the form of a differential operator for partial differential equations with variable cross sections and elasticity. The equations characterize the longitudinal-torsional displacements of the elements of large space structures with conditions as a time function specified at the ends of the girder. International University of Civil Aviation, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 99–105, July, 2000.     

Invariant recording of elasticity theory equations

An invariant (with respect to rotations) formalization of equations of linear and nonlinear elasticity theory is proposed. An equation of state (in the form of a convex generating potential) for various crystallographic systems is written. An algebraic approach is used, which does not require any geometric constructions related to the analysis of symmetry in crystals. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 127–142, September–October, 2008.     

Crack-opening displacements and normal strains in centrally notched plates

Experimentally determined crack-opening displacements of stationary and running cracks and of inclined stationary cracks in centrally notched plates are reported in this paper. The moiré-fringe technique was used for the determination of displacement fields in test specimens of magnesium, 7075-T6 and 7178-T6 aluminum alloys. Experimentally determined crack-opening displacements were compared with corresponding re ults based on theoretical models of Westergaard, Dugdale, Craggs and Craggs-Dugdale. In addition, normal-strain fields derived from the moiré-fringe data were compared with static or dynamic strain fields of these theoretical models. The results of this investigation indicate that while the Dugdale crack is a fair model of a stationary crack in ductile materials, the Craggs crack appears to be a better representation of a running crack in the ductile materials investigated.     

Generalized variational principles on nonlinear theory of elasticity with finite displacements

Two generalized variational principles on nonlinear theory of elasticity with finitedisplacements in which the σ_(ij),e_(i j)and u_i are all three kinds of independent functionsare suggested in this paper.It isproved that these two generalized variational principles areequivalent to each other if the stress-strain relation is satisfied as constraint.Some specialcases,i.e.generalized variational principles on nonlinear theory of elasticity with smalldeformation,on linear theory with finite deformation and on linear theory with smalldeformation together with the corresponding equivalent theorems are also obtained.All ofthem are related to the three kinds of independent variables.     

A class of exact solutions of the ideal plasticity equations

A class of exact solutions is formulated for the ideal plasticity equations in the case of plane deformation. The solutions describe the plastic state of various wedges, notched half-planes, and domains in the form of funnels produced by detonation, the plastic state of domains having a horn configuration, etc. Many of the solutions have natural boundaries comprising envelopes of slip lines. The equations for the boundaries, slip lines, and stresses are presented in explicit form.     

Finite gradient elasticity and plasticity: a constitutive thermodynamical framework

In Bertram (Continuum Mech Thermodyn. doi: 10.1007/s00161-014-0387-0, 2015), a mechanical framework for finite gradient elasticity and plasticity has been given. In the present paper, this is extended to thermodynamics. The mechanical theory is only briefly repeated here. A format for a rather general constitutive theory including all thermodynamic fields is given in a Euclidian invariant setting. The plasticity theory is rate-independent and unconstrained. The Clausius–Duhem inequality is exploited to find necessary and sufficient conditions for thermodynamic consistency. The residual dissipation inequality restricts the flow and hardening rules in combination with the yield criterion.