Showing papers on "Continuum mechanics published in 1997"

Fractals and fractional calculus in continuum mechanics

Abstract: A. Carpinteri: Self-Similarity and Fractality in Microcrack Coalescence and Solid Rupture.- B. Chiaia: Experimental Determination of the Fractal Dimension of Microcrack Patterns and Fracture Surfaces.- P.D. Panagiotopoulos, O.K. Panagouli: Fractal Geometry in Contact Mechanics and Numerical Applications.- R. Lenormand: Fractals and Porous Media: from Pore to Geological Scales.- R. Gorenflo, F. Mainardi: Fractional Calculus: Integral and Differential Equations of Fractional Order.- R. Gorenflo: Fractional Calculus: some Numerical Methods.- F. Mainardi: Fractional Calculus: some Basic Problems in Continuum and Statistical Mechanics.

An adhesion map for the contact of elastic spheres

Abstract: Several continuum mechanics models of the adhesion between elastic spheres have found application to compliant materials such as rubber and to fine particles in the air or in colloidal suspension. More recently they are being used in connection with experimental techniques such as the surface force apparatus and the atomic force microscope. The appropriate model to use depends on the conditions: the size and elasticity of the spheres and the load to which they are subjected. To guide this choice a map has been constructed with nondimensional coordinates m and P, where the elasticity parameter m can be interpreted as the ratio of elastic deformation resulting from adhesion to the effective range of surface forces and the load parameter P is the ratio of the applied load to the adhesive force. A closed form solution to the problem based on a Dugdale force-separation law is outlined and used to construct the map. The errors introduced by the Dugdale approximation are assessed by comparison with numerical solutions using the Lennard-Jones force law. Copyright 1997Academic Press

A continuum-mechanical formulation for shallow polythermal ice sheets

Abstract: This paper is concerned with a new theoretical approach to model grounded ice sheets in three dimensions. These are considered as polythermal, i.e. there will be regions with temperatures below the...

Diffraction elastic constants of a cubic polycrystal

Abstract: New expressions for the diffraction elastic constants are presented for a cubic polycrystal that is statistically homogeneous, isotropic and disordered. The results are an extension of the theory of statistical continuum mechanics and differ from a previous derivation given by Bollenrath, Hauk, & Muller [Z Metallkde (1967), 58, 76–82].

Continuum mechanics modelling and augmented Lagrangian formulation of large deformation frictional contact problems

Abstract: Reference LMAF-CONF-2000-002View record in Web of Science Record created on 2005-09-14, modified on 2017-09-24

On universal relations in continuum mechanics

Abstract: This paper is devoted to a systematic study of local universal relations in continuum mechanics. We show that it is possible to determine the complete set of independent universal relations whose characterization is obtained by linear universal rules. A historical review of the literature on the topic and various significant examples are given.

A comparison of a discrete dislocation model and a continuous description of cyclic crack tip plasticity

Abstract: Crack tip plasticity is simulated as motion of discrete dislocations. The results of the dislocation model are compared with the predictions of a continuous description of crack tip plasticity in the case of a stationary mode I fatigue crack. It is shown that both the dislocation model and the continuum mechanics lead to the same result at high loading levels but they differ significantly for small stress intensity ranges.

The role of heterogeneity on the flow and fracture of two-phase materials

Abstract: The mechanical behavior of two-phase alloys containing a high volume fraction of hard particles is sensitive to effects such as particle clustering and damage. Models for such behavior are necessarily complex. However, recent progress is such that global effects (e.g. the decrease of work hardening capacity due to continuum processes during compressive flow) can be predicted with reasonable accuracy. Parameters involving localized processes such as ductility are more difficult to predict. In this paper we review and extend models for the role of particle clustering and damage (in the form of particle cracking) on deformation. Further understanding of this problem will rely on a combination of critical experiments on materials with well controlled microstructures, and on models which synthesize the results of detailed FEM calculations and large scale continuum mechanics approaches.

Elastic waves in a hyperelastic solid near its plane-strain equibiaxial cohesive limit

Abstract: Propagation of elastic waves near the cohesive limit of a solid is of interest in understanding the speed at which strain energy is transported in front of a mode-I crack tip. It can be argued that the crack propagation velocity is limited by how fast the strain energy can be transported ahead of the crack tip to sustain the bond-breaking processes in the fracture process zone. From this point of view, the cohesive-state wave speed leads to the concept of local limiting fracture speed which provides a possible explanation for the 'mirror-mist-hackle' instabilities widely observed in experimental and numerical investigations of dynamic fracture. In this letter, wave speeds near the plane-strain equibiaxial cohesive stress sigmamax are studied using the hyperelasticity theory of continuum mechanics, with no specific assumptions on the atomic structure of the solid other than that it remains homogeneous and isotropic in the plane of analysis. It is found that the cohesivestate wave speed is equal to (σmax/ρ)...

Stress and Hyperstress as Fundamental Concepts in Continuum Mechanics and in Relativistic Field Theory

Abstract: The notions of stress and hyperstress are anchored in 3-dimensional continuum mechanics. Within the framework of the 4-dimensional spacetime continuum, stress and hyperstress translate into the energy-momentum and the hypermomentum current, respectively. These currents describe the inertial properties of classical matter fields in relativistic field theory. The hypermomentum current can be split into spin, dilation, and shear current. We discuss the conservation laws of momentum and hypermomentum and point out under which conditions the momentum current becomes symmetric.

A new class of particle methods

Abstract: Particle methods are numerical methods designed to solve problems in fluid mechanics and related problems in continuum mechanics. A general approach to the construction of such particle methods is presented in this article. The particles are no mass points but possess a finite extension. They can rotate in space and have a spin. The conservation of mass is automatically guaranteed by the ansatz. The forces of interaction between the particles are derived in a canonical way from the force laws of continuum mechanics and are directly based on a regularized stress tensor. In the absence of external forces and of heat sources and sinks, momentum, angular momentum, and energy are conserved as in the continuum case.

Transition from laminar to rotational motion in plasticity

Abstract: On the basis of observations made by electron microscopy of the distribution of dislocations around undeformable particles embedded in a matrix of plastic crystalline metal, it is possible to see a clear transition from ‘laminar plastic flow’ to ‘rotational flow’ at a critical strain which depends mostly upon the particle size, larger particles displaying smaller critical strains. These patterns are produced by plastic relaxation of the internal stress caused by deformation. Although there seems to be no secure way of predicting the patterns on the basis of continuum mechanics, nor on an atomistic basis of dislocation mechanics, it is possible by a combination of dislocation theory and continuum patterns of flow to devise simple models which go some way to explaining the transition and to predicting the distribution of misorientation in the rotational structures. The results have important consequences for understanding recrystallization phenomena, as well as overall work–hardening behaviour. They suggest that indentation plasticity might behave similarly.

A method of seamlessly combining a crack tip molecular dynamics enclave with a linear elastic outer domain in simulating elastic–plastic crack advance

Abstract: Molecular dynamics is applicable only to an extremely small region of simulation. In order to simulate a large region, it is necessary to combine molecular dynamics with continuum mechanics. Therefore, we propose a new model where molecular dynamics is combined with micromechanics. In this model, we apply molecular dynamics to the crack tip region and apply micromechanics to the surrounding region. Serious problems exist at the boundary between the two regions. In this study, we manage to solve these problems, and make possible the simulation of the process of crack propagation at the atomic level. In order to examine the validity of this model, we use α-iron for simulation. If the present model is valid, stress and displacement should vary continuously across the boundary between the molecular dynamics region and the micromechanics region. Our model exhibits just such behavior.

Configurational Forces and the Dynamics of Planar Cracks in Three-Dimensional Bodies

Abstract: This paper develops a three-dimensional framework for the evolution of planar cracks, concentrating on the derivation of balances and constitutive equations that describe the motion of the crack tip. The theory is based on the notion of configurational forces in conjunction with a mechanical version of the second law.

Finite Sized Atomistic Simulations of Screw Dislocations

Abstract: The interaction of screw dislocations with an applied stress is studied using atomistic simulations in conjunction with a continuum treatment of the role played by the far-field boundary condition. A finite cell of atoms is used to consider the response of dislocations to an applied stress and this introduces an additional force on the dislocation due to the presence of the boundary. Continuum mechanics is used to calculate the boundary force which is subsequently accounted for in the equilibrium condition for the dislocation. Using this formulation, the lattice resistance curve and the associated Peierls stress are calculated for screw dislocations in several close-packed metals. As a concrete example of the boundary force method, we compute the bow-out of a pinned screw dislocation; the line tension of the dislocation is calculated from the results of the atomistic simulations using a variational principle that explicitly accounts for the boundary force.

Forty Years of Non-Linear Continuum Mechanics

Abstract: An account is given of the author’s involvement with the development of nonlinear continuum mechanics. The manner in which the formulation of a phenomenological theory for finite elastic deformations of rubber-like materials led naturally to the development of phenomenological theories for the deformation of viscoelastic solids and fluids is described. This in turn led to the development of a general theory for. the formulation of non-linear constitutive equations.

Elastodynamics of Multibody Systems Using Generalized Inertial Coordinates and Structural Damping

Abstract: A new formulation to describe the elastodynamics of flexible multibody systems using efficient generalized inertial coordinates is presented and discussed in this paper. The finite element method is initially applied to the continuum mechanics equations of the system components, leading to equations of motion for flexible bodies in which the linear elastodynamics is effectively coupled with the body gross motion by a time variant mass matrix. However, the coefficients of the mass matrix must be derived for each panicular type of finite element used in the description of the flexible body. Applying a lumped mass formulation and referring nodal displacements to the inertial frame, rather than to the body-fixed coordinate frame, yields a constant diagonal mass matrix for a flexible body. Coupling between the large rigid body motion and the small elastic deformations is still preserved. Kinematic constraints are introduced in the multibody system equations, using the new coordinates. Efficiencies and...

Variational principles of nonlinear dynamical fluid-solid interaction systems

Abstract: Based on the fundamental equations of continuum mechanics, the concept of Hamilton's principle and the adoption of Eulerian and Lagrangian descriptions of fluid and solid, respectively,variational principles admitting variable boundary conditions are developed to model mathematically the nonlinear dynamical behaviour of the responses and interactions between fluid and solid. The nonlinearity of the fluid is introduced through nonlinear field equations and nonlinear boundary conditions on the free surface and fluid–solid interaction interface. The structure is treated as a nonlinear elastic body. This model assumes the fluid inviscid, incompressible or compressible and the fluid motion irrotational or rotational but isentropic along the flow path of each fluid particle. The stationary conditions of the variational principles include the governing equations of nonlinear elastic dynamics, fluid dynamics and those relating to the fluid-structure interaction interface as well as the imposed boundary conditions. A family of variational principles are obtained depending on the assumptions introduced into the mathematical model (i.e. fluid incompressible, motion irrotational, etc.) and these provide a foundation to construct numerical schemes of study to assess the dynamical behaviour of nonlinear fluid–solid interaction systems. Two simple illustrative examples are presented demonstrating the applicability of the proposed theoretical approach.

On a Model of Nonlocal Continuum Mechanics Part II: Structure, Asymptotics, and Computations

Abstract: We consider the asymptotic behavior and local structure of solutions to the nonlocal variational problem developed in the companion article to this work, On a Model of Nonlocal Continuum Mechanics Part I: Existence and Regularity. After a brief review of the basic setup and results of Part I, we conduct a thorough analysis of the phase plane related to an integro-differential Euler--Lagrange equation and classify all admissible structures that arise as energy minimizing strain states. We find that for highly elastic materials with relatively weak particle-particle interactions, the maximum number of internal phase boundaries is two. Moreover, we also develop explicit bounds for the number of internal phase boundaries supported by any material and show that this bound is essentially inversely related to the particle size. To understand the question of asymptotics, we utilize the Young measure and show that in the sense of energetics and averages, minimizers of the full nonlocal problem converge to minimizers of two limiting problems corresponding to both the large and small particle limits. In fact, in the small particle limit, we find that the minimizing fields converge, up to a subsequence in strong-Lp, for 1 ≤ p < ∞, to fields that support either a single internal phase boundary, or two internal phase boundaries that are distributed symmetrically about the body midpoint. We close this work with some computations that illustrate these asymptotic limits and provide insight into the notion of nonlocal metastability.

A continuum theory for the thermomechanics of solidification

Abstract: A general theory for the phenomenon of solidification is presented in which the coupling between the thermal and kinematical fields is fully taken into account. The resulting model descsribes the liquid region as an ordinary Newtonian liquid and the solid phase as an elastic material. In the case of multi-component solidification it allows for the existence of a mixed region separating the pure phases whose behavior is modeled as a non-linear viscoelastic material. After a preliminary analysis of the jump conditions across the singular surface separating the two phases, the strong interdependence between the thermomechanical field, the geometry of the singular surface and the freezing temperature θf is examined in detail. A simple one-dimensional problem (Boussinesq problem) has been discussed to show how only a dynamical theory can predict with reasonable accuracy the final shape of the solid.

Basic Equations in Eulerian Continuum Mechanics

Abstract: We present a general framework applicable to several mathematical models in continuum mechanics. Particularly, we demonstrate that mass, momentum and energy balancing along with appropriate constitutive laws, form the basis of a broad class of boundary value problems for macroscopic motion of a solid, fluid, gas and mixture of several media. We also point out the similarities in the final governing equations and the resulting advantage of this when constructing numerical solution methods.

Combining molecular and continuum mechanics concepts for constitutive equations of polymer melt flow

Abstract: Molecular-level modelling of linear viscoelastic properties of a polymer melt is combined with a continuum-based model for the recoverable deformation in domains of fluid in a flow field to predict nonlinear viscoelastic properties. A nonlinear differential constitutive equation is derived, whose parameters are elemental linear viscoelastic properties which can be calculated as a function of polymer composition and environmental conditions. The model is demonstrated by means of examples of important nonlinear effects: recoverable strain and Non-Newtonian shear viscosity, extrudate swell, nonlinear stress relaxation, the onset of shear, and elongational viscosity.

Viscoelastic Model of Finitely Deforming Rubber and Its Finite Element Analysis

Abstract: A viscoelastic model of finitely deforming rubber is proposed and its nonlinear finite element approximation and numerical simulation are carried out. This viscoelastic model based on continuum mechanics is an extended model of Johnson and Quigley's one-dimensional model. In the extended model, the kinematic configurations and measures based on continuum mechanics are rigorously defined and by using these kinematic measures, constitutive relations are introduced. The obtained highly nonlinear equations are approximated by the nonlinear finite element method, where a mixture of the total and updated Lagrangian descriptions is used. To verify the theory and the computer code, uniaxial stretch tests are simulated for various stretch rates and compared with actual experiments. As a practical example, an axisymmetric rubber plate under various time-dependent pressure loading conditions is analyzed.