Formulation Of Continuum Mechanics Research Articles

Relativistic anelasticity

A formulation of continuum mechanics within the context of general relativity is presented that allows for the incorporation of certain types of anelastic material behaviour, such as viscoelasticity and plasticity. The approach is based on the concept of a four-dimensional body-time complex structured as a principal bundle. The anelastic response is regarded as the result of a continuous distribution of inhomogeneities, whose evolution is dictated by a suggested relativistic version of the Eshelby tensor. The role played by various groups is emphasized throughout the presentation and illustrated by means of the example of an anelastic fluid.

A Framework for Structured Modeling of Skeletal Muscle

The aim of this study is to present a detailed continuum mechanics formulation, and the corresponding algorithms, to predict the deformation of skeletal muscle at different structural levels, starting from the muscle fiber level. The model is used to investigate force production and structural changes during isometric and dynamic contractions of the cat medial gastrocnemius. From a comparison with experimental data obtained in our own laboratories, we conclude that the model faithfully predicts all of the observations pertaining to force production, fascicle length and angle of pennation under various test conditions.

Surface displacement characteristics of line crack trespassing the Rayleigh wave speed barrier as influenced by velocity and applied stress dependent local zone of restrain

Surface displacement characteristics of line crack trespassing the Rayleigh wave speed barrier as influenced by velocity and applied stress dependent local zone of restrain

Onset of failure in finitely strained layered composites subjected to combined normal and shear loading

A limiting factor in the design of fiber-reinforced composites is their failure under axial compression along the fiber direction. These critical axial stresses are significantly reduced in the presence of shear stresses. This investigation is motivated by the desire to study the onset of failure in fiber-reinforced composites under arbitrary multi-axial loading and in the absence of the experimentally inevitable imperfections and finite boundaries. By using a finite strain continuum mechanics formulation for the bifurcation (buckling) problem of a rate-independent, perfectly periodic (layered) solid of infinite extent, we are able to study the influence of load orientation, material properties and fiber volume fraction on the onset of instability in fiber-reinforced composites. Two applications of the general theory are presented in detail, one for a finitely strained elastic rubber composite and another for a graphite–epoxy composite, whose constitutive properties have been determined experimentally. For the latter case, extensive comparisons are made between the predictions of our general theory and the available experimental results as well as to the existing approximate structural theories. It is found that the load orientation, material properties and fiber volume fraction have substantial effects on the onset of failure stresses as well as on the type of the corresponding mode (local or global).

Simulation of meso–macro dynamic behavior using steel as an example

Presented in this paper is a model, where strain hardening and strain rate sensitivity are taken into account. The cellular automata rules are included into a continuum mechanics formulation to describe shear band propagation at the mesoscale. We apply the model in simulating deformation of the 20MnMoNi55 steel as an example. The macro stress–strain behavior and plastic strain patterns at the mesolevel are analyzed in the course of a computational tension test.

Iper-ideal kinetic constraints in continuum mechanics

A geometrical approach to Lagrangian formulation of continuum mechanics in the language of jet-bundles theory is proposed. The resulting environment allows one to deal with a particular class of kinetic constraints, through a suitable implementation of Gauss’ principle. A variational formulation of systems subject to such constraints is also outlined. Two explicit examples are given.

Some issues about anisotropic elastic–plastic models at finite strain

Abstract The description of anisotropic elastic–plastic behaviour at large strain is difficult. In the rigid plastic case, anisotropic plasticity is now reasonably well understood and is based on formulating anisotropic constitutive equations in a moving frame which, correctly chosen, will ensure objectivity and which can be postulated: (i) a priori to follow in some sense the material rotations (kinematical rotating frame); (ii) a posteriori by an appropriate constitutive evolution equation (plastic spin equation). With respect to the classical continuum models this is in fact an alternative for the formulation of frame indifferent constitutive equations. This rotating frame formulation of continuum mechanics is first formally presented and applied to anisotropic elasticity to show that in this case the proper rotating frame must be used to get a true elasticity. The remaining part of the paper is then devoted to the elastic–plastic case starting from a direct extension of the formalism with derivation of the elastic constitutive equation and plastic flow rule which is best written in the so-called isoclinic configuration. Finally the kinematic plastic rotating frames will be introduced and exemplified as the appropriate choice for a correct description of an anisotropic elastic–plastic behaviour.

Statistical continuum theory for large plastic deformation of polycrystalline materials

This paper focuses on the application of statistical continuum mechanics to the prediction of mechanical response of polycrystalline materials and microstructure evolution under large plastic deformations. A statistical continuum mechanics formulation is developed by applying a Green's function solution to the equations of stress equilibrium in an infinite domain. The distribution and morphology of grains (crystals) in polycrystalline materials is represented by a set of correlation functions that are described by the corresponding probability functions. The elastic deformation is neglected and a viscoplastic power law is employed for crystallographic slip in single crystals. In this formulation, two- and three-point probability functions are used. A secant modulus-based formulation is used. The statistical analysis is applied to simulate homogeneous deformation processes under uniaxial tension, uniaxial compression and plane strain compression of an FCC polycrystal. The results are compared to the well-known Taylor upper bound model and discussed in comparison to experimental observations.

A finite element procedure for the analysis of thermo-mechanical solids in contact

We present a finite element procedure for the analysis of fully coupled thermo-elasto-plastic response of solids including contact conditions. The continuum mechanics formulation for the solid and contact conditions is summarized and effective finite element techniques for solution are given. The constraint function method is employed to impose the contact conditions at the Gauss points of the contact surface. Other procedures widely used in finite element analysis can be considered as particular cases of the constraint function method discussed herein.

Progress of phase boundaries and walls in thermo-deformable solids : A canonical approach

Four approaches at different scales or levels of understanding are proposed to study the propagation of phase boundaries and domain walls in thermo-deformable solids: (i) a microscopic one relying on a lattice model that yields the representation of the thin transition zone as a solitonic structure; (ii) a thermodynamic macroscopic one that is based on the use ofjump relations associated with the canonical formulation of continuum mechanics on the material manifold, (iii) a mesoscopic continuum approach using directly a mixed viscosity-strain-gradient modelling, and (iv) a global quasi-particle viewpoint which, although accounting for the structure of the transition zones, is also canonical and allows for the study of transient motions of these zones. The common points and respective interests of the approaches are discussed.

Statistical continuum theory for inelastic behavior of a two-phase medium

Abstract A statistical continuum mechanics formulation is presented to predict the inelastic behavior of a medium consisting of two isotropic phases. The phase distribution and morphology are represented by a two-point probability function. The isotropic behavior of the single phase medium is represented by a power law relationship between the strain rate and the resolved local shear stress. It is assumed that the elastic contribution to deformation is negligible. A Green’s function solution to the equations of stress equilibrium is used to obtain the constitutive law for the heterogeneous medium. This relationship links the local velocity gradient to the macroscopic velocity gradient and local viscoplastic modulus. The statistical continuum theory is introduced into the localization relation to obtain a closed form solution. Using a Taylor series expansion an approximate solution is obtained and compared to the Taylor’s upper-bound for the inelastic effective modulus. The model is applied for the two classical cases of spherical and unidirectional discontinuous fiber-reinforced two-phase media with varying size and orientation.

Virtual crack extension in geometries under large displacements/rotations

The present paper deals with a development of the virtual crack extension (VCE) method to arbitrary three-dimensional cracks under large displacements and/or large rotations. For this purpose, continuum mechanics formulations are introduced and the material is assumed to be nonlinear elastic. Illustrative examples are given for two crack problems under either large rotations or large displacements as well.

A new theory for free-surface formation in solid continua

In essence, fracture mechanics consist of the conventional boundary-value problem formulation of continuum mechanics, along with a variety of fracture criteria that govern the advance of crack fronts. The assumption that the material response is governed by a local constitutive model everywhere in the body, even at points that are arbitrarily close to the crack front, generally leads to an unbounded stress field as the crack front is approached. Also, processes such as metal-cutting and penetration share with fracture the essential feature of new-free-surface formation, but do not fit easily within the conventional fracture theory. For these and other reasons, it seems worthwhile to seek a broader theoretical construct which encompasses surface separation in a more general setting. With this motivation, a new theory is proposed which applies generally to surface separation in solid continua, but which nonetheless yields fracture-mechanics-type predictions in appropriate special cases. The proposed exclusion region theory involves identification of a small material neighborhood that contains the separation front. A generalized constitutive description that derives directly from the local constitutive model is constructed for the exclusion region. A separation criterion is formulated with reference to tractions on, and/or distortion of, the exclusion region. The direction-of-advance of the separation front is determined as a natural consequence of the separation criterion. The material parameters appearing in the separation criterion can generally be determined from conventional fracture tests. The theory has been implemented in a finite element code. Two example problems illustrating certain important aspects of the theory are presented.

A formulation of continuum mechanics as a dimensional reduction of a finite-dimensional dynamical system

IN THE PAPER a generalized formulation of the continuum mechanics is suggested. The generalization consists in the assumption that the energy balance equation is not satisfied for all subbodies of a body but only for their chosen family. This formulation leads to fields in the continuum which create a finite-dimensional space. With the help of the chosen family of subbodies, a volume of averaging related to the continuum model is defined. This volume is connected with a more elementary dynamical system which takes part in determination of the form of constitutive equations. In general, the mechanical model of the continuum is seen as a dimensional reduction of the more elementary dynamical system related to another continuum or to a discrete set of material points.

Theoretical and numerical formulation of a molecular based constitutive tube-model of rubber elasticity

The main physical features of the tube-model of rubber elasticity are reviewed. Based on an affine-coupling-hypothesis of network strands between successive entanglements a simple scaling derivation of the non-affine deformation of the lateral tube dimension is presented. The mathematical form of the tube deformation essentially determines the structure of the high elastic potential in a continuum mechanical formulation that is suitable for an efficient implementation into a finite element method. Experimental data from van den Bogert and de Borst are taken for the comparison with numerical results. The material parameters are identified in a uniaxial experiment. Subsequently, a non-homogeneous complex shear experiment is modelled for the comparison in order to assess the predictive character of the finite element simulation. From the numerical part we can conclude that the tube-model can be used to carry out realistic simulations. However, results are sensitive to the material parameters used and parameter identification is of great importance. The tube-model when applied to FEM provides the possibility to relate molecular parameters of the vulcanizate to the properties and behaviour of rubbers under operating conditions.

Transformation-Induced Plasticity (TRIP)

This review article attempts to explain TRIP generally and its appearance in different materials. Continuum mechanics formulations are mainly used to explain this phase change phenomenon. An overview is given of the published literature which is often not very easily accessible. Technological aspects are presented, such as constitutive equations of materials showing TRIP. Aspects of material selection and future material design are also treated. This article contains 315 references.

The theory of fuzzy structures—A statistical continuum mechanics interpretation

The theory of fuzzy structures is demonstrated to follow from a renormalization of a more fundamental continuum mechanics formulation. The derivation accepts an explicitly statistical interpretation of the attached fuzzy and the achieved renormalization applies for estimating the ensemble averaged response of the master structure. An alternative interpretation of the attached fuzzy acting on a much smaller length scale than that required for describing the master structure response is briefly discussed. Several issues are identified that suggest the theory cannot be rigorously justified as a general mathematical framework. These issues relate to the uniqueness of predictions, to the independence of the effective impedance operator that reflects the presence of the fuzzy structure attachments to the master structure, and to the well-posedness of the inverse problem for determining this effective impedance from experiments of the response of the master structure. The case of point-connected fuzzy structures is investigated in detail and a distinction between an added inertia force and an added stiffness is emphasized. The simplification of the renormalization required for the added inertia forces is shown to reproduce a result of a statistical independence assumption.

The updated co-moving coordinate formulation of continuum mechanics based on the SR decomposition theorem

This paper presents a new updated co-moving coordinate formulation for nonlinear analysis of continuum mechanics. The incremental variational equation is derived by using the energy principle of power form. Then, the updated co-moving coordinate formulation is developed. Numerical examples are presented.

The principle of virtual work and a symplectic reduction of non-local continuum mechanics

We study a symplectic formulation of continuum mechanics on the Fréchet manifold of (smooth) embeddings J : M → N. From a virtual work principle we derive a generalized Hamiltonian dynamics, which allows to include the description non-hyperelastic media as well as a non-linear and non-local material behaviour. Dividing out the (rigid) translation group, the deformation gradient and the stress tensor appear—via Marsden-Weinstein reduction—as natural geometric objects. The balance law on the reduced phase space becomes a weak equation for exact differential forms, which relates the dynamics of the deformation gradient to the stress. This equation is subjected to an interesting gauge freedom and yields the classical continuum dynamics as well as boundary conditions for the stress.

On the application of G( Θ) method and its comparison with De Lorenzi's approach

Based on a continuum mechanics formulation, Destuynder [1] proposed recently a Lagrangian method, called G( Θ) method, for calculating the energy release rate G. The present paper outlines a detailed finite element application of this method for use with the 3-noded DKT shell elements [2] and the 20-noded isoparametric 3-D elements. The representation in this paper allows one to calculate G either as an integral part of a finite element analysis or separately in a post-processing program using the stress and strain as input for calculations. Example computations are given and compared with the Lorenzi's approach [3] for several elastic as well as elasto-plastic crack problems.