Finite deformation of a hollow sphere of linear elastic perfectly plastic material

Summary The total deformation is decomposed into its plastic and its elastic part. This means the final configuration of the plastic deformation is the stress-free reference configuration for the elastic deformation. However the intermediate state is not a continously differentiable vector field and the plastic and elastic deformation tensors are not gradients. Their incompatibilities are determined. The plastic work and the flow rule are considered. Finally the equations for the unloading process are given.     

Representation for stress from a stressed reference configuration

There are several problems wherein it would be convenient to have representations for the stress from a stressed configuration as reference. We assume that the body undergoes finite elastic deformations (i.e., non-dissipative processes) from a stressed reference configuration but it is possible that a dissipative (inelastic) process could be the cause for the origin of the stresses in the reference configuration. We restrict ourselves to determining the representation for the stress in a body, from a stressed reference configuration, whose symmetry group in the stress free configuration (that is accessible from the stressed reference configuration) coincides with the full orthogonal group. We also find restrictions on the constitutive relation so that we have a model that exhibits physically acceptable response characteristics.     

Static response of elastic inflated wrinkled membranes

In this paper we present an effective numerical algorithm for determining the equilibrium shapes of inflated elastic membranes susceptible to wrinkling. The use of a two-state constitutive law and the introduction of a suitable criterion allow for accounting for wrinkling of the membrane, although in an approximated way. In the active state, the material is able to transmit only tensile stresses; vice versa, in the passive state it is stress-free and can contract freely. Equilibrium of the membrane in the current inflated configuration is enforced by recourse to the minimum total potential energy principle, whereas the Lagrange multipliers method is used to solve the minimum problem by accounting for the aforesaid nonlinear constitutive law. We use an expressly developed iterative-incremental numerical algorithm, consistent with the established governing set of equations, for accurately monitoring the evolution of the stress field in the membrane during the inflation process. Specifically, we suppose that the membrane reaches its final shape at the end of a four-stage loading process corresponding to the temporary enforcement and the subsequent removal of a fictitious antagonist plane traction acting uniformly along its entire boundary. By this way it is possible to solve with great accuracy the set of governing equilibrium equations by means of a numerical procedure in which the membrane’s tangent stiffness is always kept different from zero. The soundness of the proposed algorithm is verified by comparing the results with well-known solutions available in the literature. In particular, for each specific value of pressure, the current configuration of the inflated membrane found by assuming that compressions are allowed is compared in details to the corresponding pseudo-deformed surface, obtained by assuming a tension-only response.     

Issues on the constitutive formulation at large elastoplastic deformations,part 2: Kinetics

Summary The coupling between kinematics and kinetics and the invariance requirements under superposed rigid body rotations, determine unambiguously the proper general form of the elastoplastic rate constitutive equations, in terms of the values of the state variables and their rates in reference to the current and any choice of the unstressed configuration. Topics such as the effect of changing the stress rate, small elastic deformations with or without large volumetric elastic strains, rate effects and viscoplasticity, an example on single slip, the effect of the plastic spin constitutive relations and the concept of an effectively unstressed configuration, are addressed in detail. Issues and different approaches debated in the past are discussed, compared and clarified.     

CALCULATION OF HYPERELASTIC RESPONSE OF FINITELY DEFORMED ELASTIC-VISCOPLASTIC MATERIALS

The objective of this paper is to present a numerical algorithm for calculating hyperelastic constitutive equations characterizing the thermomechanical response of elastically isotropic elastic-viscoplastic materials. The algorithm is developed within the context of an alternative formulation of plasticity in which elastic distortional deformation is determined directly by integrating an evolution equation which includes the current velocity gradient and quantities that depend only on the present state of the material. Consequently, the formulation is independent of the particular choice of a measure of plastic deformation, the reference configuration, and the total deformation gradient from the reference configuration. These features allow the constitutive equations to be easily implemented into computer codes which currently use a hypoelastic formulation for calculating plasticity.     

On the duality principle of conservation laws in finite elastodynamics

The duality principle of conservation laws which holds in finite elastodynamics is studied using the two-point tensor method. Based on the general Noether's theorem, two basic equations of variational invariance are first derived, which correspond to the action integrals given, respectively, in Lagrangian and Eulerian representations for a finite motion of an elastic body. The dual relations between the conservation laws in both representations are given. The procedure for constructing these dual relations is to apply simultaneously the same infinitesimal transformation of either time or position coordinates as well as field variables to the dual equations of variational invariance, where the position coordinates could be taken either from the reference configuration or from the deformed configuration of the material body. Based on these dual relations it is shown that the conservation equations of material momentum and moment of material momentum possess the same structure as those of physical momentum and physical moment of momentum. Furthermore, three pairs of dual relations between stress tensors and material momentum tensors of various kinds are derived based on the duality principle by using the two-point tensor method. Finally, using the dual integral forms of conservation laws the concepts of dynamic material force and moment acting on defects are introduced and analyzed. The force and moment can be decomposed into a pure kinetic part and a pure deformation part, the latter corresponding to the path-independent integral as suggested in elastostatics.     

The effect of pre-stress on the vibration and stability of elastic plates

Plane incremental vibrations superimposed on the pure homogeneous deformation of a rectangular block of incompressible isotropic elastic material are studied in detail. Frequency equations, which determine the frequencies of symmetric and antisymmetric modes of vibration in terms of the underlying deformation and stress and of the in-plane aspect ratio of the block, are obtained in respect of a general form of strain-energy function. Different cases, dependent on the properties of the strain-energy function and on the deformation and stress, are enumerated; results analogous to those found in the (compressible) linear theory are recovered for comparison.

The case of zero frequency is of special interest since the frequency equations then become bifurcation equations. Each bifurcation equation determines a set of values of the deformation and stress at which, on a path from the undeformed stress-free configuration, stability of the underlying deformed configuration is lost and bifurcation into a (quasi-static) non-homogeneous incremental mode of deformation becomes possible. Results of Sawyers and Rivlin (1974) and Ogden (1984) emerge as special cases of those obtained here.

For particular forms of strain-energy function calculations have been carried out to illustrate how the frequency depends on the deformation, stress and aspect ratio.     

A continuum approach to magnon-phonon couplings—I: General equations,background solution

It is proposed to study the magnon-phonon couplings, exhibited in elastic ferromagnetic bodies, on the basis of a purely continuum approach and its related mathematical methods. The present part of the work is mostly devoted to establishing the equations that govern weak-field dynamical disturbances by the method of superimposition. More precisely, having defined an ideal, mechanically unloaded and field-free reference configuration, with respect to which the material behaves isotropical]y, we built up an initial configuration which corresponds to a stationary rigid-body, one-ferromagnetic domain state by applying an intense magnetic field and appropriate mechanical loadings on the boundary of the body. This initial state induces hexagonal symmetry for the dynamical properties to be studied in Part II in a neighborhood of this state. Viscosity and spin-lattice relaxation processes are accounted for, and the case of weakly elastically anisotropic bodies is commented upon.     

固体力学问题数值解的一种验证方法

给出了固体力学几何非线性动态问题数值解的一种精度验证方法。通过假定初始构形和现时构形之间的映射关系,利用固体力学的控制方程即可求得产生这种构形的体积力,则假定构形和所得到的体积力就构成了问题的解析解。在这些解析解的基础上,提出了一种检验数值方法精度的标准试验,可用于二维和三维问题、隐式算法和显式算法、小变形和大变形分析、弹性材料和超弹性材料。在线性位移场的情况下本文方法是和传统的分片检验(patchtest)一致的。文中给出了无网格迦辽金法(EFGM)精度检验的几个算例。     

Strain formulation in elasto-plasticity of materials with relaxed configuration

The paper deals with finite elasto-plasticity of materials having internal state variables and relaxed configurations. Using the properties inferred on the basis of adopted definitions and hypotheses we emphasize the alternative strain formulations. Consequently, we deduce the necessary and sufficient conditions for elasto-plastic materials with relaxed configurations to be hypoelastic in Hill's sense. We prove that the acoustic plastic tensor of an elasto-plastic material with relaxed configuration is symmetric for all directions of propagation of Hadamard-type acceleration waves if and only if we deal with elasto-plastic materials which are hypoelastic in Hill's sence.     

Phenomenological theory of elastic semiconductors

A conceptually simple but nonetheless nonlinear, rotationally invariant and thermodynamically admissible, phenomenological theory of elastic semiconductors is presented. The primitive nonlinearity is essential in that it guarantees the obtaining of a consistent set of linearized equations about a bias state, a fact overlooked practically by all previous authors. Nonlinear representations are obtained for all effects of importance in elastic semiconductors (elasticity, electromechanical couplings, galvanomagnetic, thermoelectric and thermomagnetic effects, elastoresistance and magnetoresistance). In particular, generalized Ohm's and Fourier's laws are obtained in which gradients of the chemical potential associated with charge carriers and the conduction current play a crucial role. Upon linearizing the nonlinear field and constitutive equations about a rigid-body stationary solution (which is nonetheless endowed with initial electromagnetic fields and internal stresses) by means of the so-called Lagrangian variation, a complete set of bulk and surface electroacoustic equations is obtained for elastic piezoelectric semiconductors in which the influence of the initial electric and magnetic fields is duly recognized. In particular, these fields greatly complicate the picture (stiffening of material coefficients, breaking of material symmetry, allowance for further electromechanical couplings, etc.) although, for the sake of simplicity, they are assumed to vary much more slowly over space than the small dynamical fields superimposed on them. The equations thus obtained provide a general framework for the study of the coupling and excitation of both bulk and surface displacement and charge modes which can be controlled through the intensity and direction, as also through the frequency, of the bias fields.     

Delayed behavior of double shear perturbations of isotropic incompressible elastic-viscoplastic materials

A study of the instantaneous and delayed behavior of a double shear perturbation superimposed on an equilibrium state of an isotropic incompressible medium with internal variables is presented. Elastic media with general internal variables and true viscoplastic media where an intermediate configuration and particular internal parameters are chosen, are successively considered. In both cases, conditions on evolution laws and free energy are proposed, and proved to be sufficient to obtain a stable system of differential equations for the perturbations. As a consequence, the two delayed wave speeds are then real and less than or equal to the instantaneous elastic wave speeds. When the equilibrium state lies on a viscoplastic yield surface, delayed wave speeds and loading conditions may be identified with those we obtain in a plastically deforming (rate-independent) medium, with the same surface as a plastic yield surface. The signification of the relaxation time introduced is also discussed.     

Optimal design and optimal control of structures undergoing finite rotations and elastic deformations

In this work, we deal with the optimal design and optimal control of structures undergoing large rotations and large elastic deformations. In other words, we show how to find the corresponding initial configuration through optimal design or the corresponding set of multiple load parameters through optimal control, in order to recover a desired deformed configuration or some desirable features of the deformed configuration as specified more precisely by the objective or cost function. The model problem chosen to illustrate the proposed optimal design and optimal control methodologies is the one of geometrically exact beam. First, we present a non‐standard formulation of the optimal design and optimal control problems, relying on the method of Lagrange multipliers in order to make the mechanics state variables independent from either design or control variables and thus provide the most general basis for developing the best possible solution procedure. Two different solution procedures are then explored, one based on the diffuse approximation of response function and gradient method and the other one based on genetic algorithm. A number of numerical examples are given in order to illustrate both the advantages and potential drawbacks of each of the presented procedures. Copyright © 2004 John Wiley & Sons, Ltd.     

Nonlinear elastic equation of state of solids subjected to uniaxial homogeneous loading

Applying the finite deformation theory to a solid, which possesses either cubic or isotropic symmetry at stress-free natural state and is subsequently loaded homogeneously in uniaxial direction, one obtains a stress (or strain) dependence of the Young's modulus, Poisson's ratio, and a volume (or density) change, together with a nonlinear elastic relation between stress and strain. These are all expressed in terms of the second and third order elastic constants of the solid material. These expressions are illustrated with examples of cubic silicon crystal, isotropic carbon steel, Pyrex glass, and polystyrene at the relaxed state.     

Computational method of inverse elastostatics for anisotropic hyperelastic solids

The paper presents a computational method for predicting the initial geometry of a finitely deforming anisotropic elastic body from a given deformed state. The method is imperative for a class of problem in stress analysis, particularly in biomechanical applications. While the basic idea has been established elsewhere Comput. Methods Appl. Mech. Eng. 1996; 136 :47–57; Int. J. Numer. Meth. Engng 1998; 43 : 821–838), the implementation in general anisotropic solids is not a trivial exercise, but comes after a systematic development of Eulerian representations of constitutive equations. In this paper, we discuss the general representation in the context of fibrous hyperelastic solids, and provide explicit stress functions for some commonly used soft tissue models including the Fung model and the Holzapfel model. A three‐field mixed formulation is introduced to enforce quasi‐incompressibility constraints. The practical utility of this method is demonstrated using an example of aneurysm stress analysis. Copyright © 2006 John Wiley & Sons, Ltd.     

Fluid-fluid and -solid interaction problems: Variational principles revisited

We systematically review some unified variational principles for a strong interaction problem in both a stratified fluid region and a fluid-solid region. The problem is described by a general Lagrangian formulation for an anisotropic elastic solid region, either surrounded by an incompressible non-Newtonian fluid region or surrounding the fluid region. In the first part, we express the fundamental equations of the regular fluid and solid regions in differential form. Then, we deduce the variational principles respectively from the principle of virtual power and the principle of virtual work for the fluid and solid regions. The physics principles are modified through an involutory transformation together with a dislocation potential. In the second part, we similarly establish some multi-field variational principles for a stratified fluid of two or more distinct fluid layers of different thicknesses and mass densities. In the third part, we derive the variational principles for the interior and exterior interaction problems in a fluid region with a surface piercing solid, within either a rigid or an elastic structure. The variational principles, which operate on all the field variables lead to the fundamental equations of the regions, including the interface conditions, as their Euler-Lagrange equations. Some special cases of the variational principles are given.     

Wave-crack interaction in finite elastic bodies

This paper continues studies in Lalegname et al. (Int J Fract 152:97–125, 2008) on crack propagation in a bounded linear elastic body under the influence of incident waves. In Lalegname et al. (2008) we have considered shear waves, whereas in this paper we discuss the influence of plane elastic waves to a running crack. Actually, the time dependent problem is formulated in a two-dimensional current cracked configuration by a system of linear elasto-dynamic equations. In order to describe the behaviour of the elastic fields near the straight crack tip, we transform these equations to a reference configuration and derive the dynamic stress singularities. Furthermore, we assume that an energy balance law is valid. Exploiting the knowledge on the singular behaviour of the crack fields, we derive from the energy balance law a dynamic energy release rate. Comparing this energy release rate with an experimentally given fracture toughness we get an ordinary differential equation for the crack tip motion. We present first numerical simulations for a Mode I crack propagation.     

Spherical-symmetric steady-state response of fluid-filled laminate piezoelectric spherical shell under external excitation

Summary Spherical-symmetric steady-state response problem of a fluid-filled piezoelectric spherical shell is discussed in the paper. In the absence of body force and free charges, the general solutions of mechanical displacement, stresses, strains, potential and electric displacement are derived from constitutive relations, geometric and motion equations under external excitation (i.e., applied surface traction and potential) in spherical coordinate system. As an application of the general solutions, the problem of a fluid-filled elastic spherical shell with piezoelectric actuator and sensor layers was solved. The results could provide theoretical basis for the spherical-symmetric dynamic control problem of fluid-structure coupling piezoelectric intelligent structure. Furthermore, the solutions can serve as reference for the research on the control of general fluid-structure dynamic problems.