Formulation Of Continuum Mechanics Research Articles

Incorporation of polymer diffusivity and migration into constitutive equations

Given a general one-particle constitutive equation for the stress tensor, we discuss how to incorporate the additional effects of polymer diffusivity and migration into that constitutive equation within the framework of continuum mechanics. For the example of an upper-convected Maxwell model representing the polymer contribution to the stress tensor of a dilute polymer solution, we describe i) how to modify the constitutive equation for the stress tensor to include diffusion and migration effects, ii) how to formulate a balance equation for the polymer mass density in order to describe the nonhomogeneous composition of the polymer solution resulting from migration, and iii) how to close the extended set of coupled equations by means of further constitutive equations for the migration velocity and the diffusion tensor. In order to guarantee the material objectivity for all equations, we formulate them in the body tensor formulation of continuum mechanics (and then translate them into Cartesian space). The proposed equations are compared to results of a recent kinetic-theory approach.

Force recovery procedures in nonlinear analysis

Conventionally, incremental-iterative schemes have been used in solving nonlinear problems. There are two phases involved in such analyses. The first or ‘predictor’ phase relates to solution of the structural displacement increments from the total incremental equations of equilibrium, while the second or ‘corrector’ phase is concerned with recovery of the element forces from the element displacement increments obtained in the first phase. In this paper, it will be demonstrated that accuracy of the numerical solutions depends primarily on the corrector or procedure for recovering the element forces. The expression used in the predictor can only affect the number of iterations required at each incremental step, but not the final shape of the load-deflection curves. A key factor in selecting the procedure for force recovery is that higher order nonlinear effects must be included, based on rigorous continuum mechanics formulations.

Finite-element models of continental extension

Numerical models of the initial deformation of extending continental lithosphere have been computed to investigate the control of pre-existing thermal and mechanical heterogeneities on the style of deformation. The finite element technique was used to calculate deformation with a viscoelastic-plastic model for the lithosphere. Initial heterogeneities tested included a variety of thermal anomalies, anomalous fracturing in the upper crust, and anomalous crustal thickening. Deformation is strongly localized by thermal, mechanical, or compositional heterogeneities. The style of initial deformation is controlled by the depth and width of the heterogeneities. Shallow weakening of the lithosphere causes broad uparching of the lithosphere in the early stages of deformation due to elastic bending stresses caused by non-uniform extensional deformation. Deeper weakening of the lithosphere has the opposite effect. The width of the zone of necking is closely controlled by the width of the initial heterogeneity, with a minimum width controlled by the layer structure of the model. The continuum-mechanics formulation of the models prevent the reproduction of fault details in the models, but general features of extensional style are indicated. Thermal heterogeneities are most effective in controlling deformational style: broad zones of high strain are predicted by wide thermal anomalies penetrating up to the Moho or shallower; brittle, low-strain extension is predicted by models with deeper, narrower thermal anomalies. Deep-penetrating low-angle faults are not required to reproduce the broad lithospheric deformational styles observed in lithospheric extension.

Design sensitivity analysis of critical load factor for nonlinear structural systems

A continuum formulation of sizing design sensitivity analysis of the critical load factor is developed for nonlinear structural systems that are subject to conservative static loadings. Both geometric and material nonlinear effects are considered. Sizing design variables, such as thickness and cross-sectional areas of structural components, are treated. The total Lagrangian formulation for an incremental equilibrium equation and one- and two-point linearized eigenvalue problems are utilized to estimate the critical load factor. A distributed parameter approach and the adjoint variable method are used to obtain design sensitivity expressions in continuum mechanics formulation. A numerical method is presented to evaluate design sensitivity expressions using established finite element codes. The proposed method is tested using several examples. The numerical results show that the proposed method for design sensitivity of the critical load factor is accurate. It is found that, as the applied load approaches the critical load, the design sensitivity of the critical load factor does not converge to the design sensitivity of the actual critical load, even though the estimated critical load converges to the actual critical load and the design sensitivity of the critical load factor is correct.

A note on plasticity theory in matrix notation

AbstractIn the matrix formulation of continuum mechanics, stress and strain tensors are contracted to vectors, and this lower order formulation has some advantages. However, due to an inconvenient contraction, the matrix formulation cannot be fully utilized in the theory of plasticity. The traditional contraction with engineering strains is rather artificial, and in the present note it is shown how the natural √2 contraction for both stresses and strains makes a very clean formulation possible. The importance of projection matrices is pointed out.

On electromagnetic fields in the Hamiltonian description of continua

The group theoretical structure of an infinite dimensional Hamiltonian formulation of continuum mechanics is studied using as an example the Maxwell-Vlasov system. In contrast to earlier works, electromagnetism and charged matter are coupled via Poisson brackets without using the vector potential. The charged matter is described on the group of canonical transformations on R 6 and we show that its evolution arises from a symplectic structure, modified by the magnetic field. The configurations of the electromagnetic field must be constrained by the physical requirement of the Gauss law. With the energy-functional taken as a Hamiltonian this leads - even for relativistic particles - to the well-known equations of motion.

Forces and the existence of stresses in invariant continuum mechanics

In an invariant formulation of pth-grade continuum mechanics, forces are defined as elements of the cotangent bundle of the Banach manifold of Cp embeddings of the body in space. It is shown that forces can be represented by measures which generalize the stresses of continuum mechanics. The mathematical representation procedure makes the restriction of forces to subbodies possible. The local properties of the stress measures are examined. For the case where stresses are given in terms of smooth densities, it is shown that the structure of forces agrees with the form of forces one assumes in the traditional formulation, and the equilibrium differential equations are obtained.

Conservation laws and the material momentum tensor for the elastic dielectric

It is shown that a Langrangian formulation of continuum mechanics can provide not only the equations of motion, but the conservation laws related to the material symmetries in a perfect continuum interacting with an external electric field. These conservation laws in the presence of defects lead to the path-independent integrals widely used in fracture mechanics. They are basically related to the “material force” on a defect in a continuum. The quantity playing the role of the physical stress tensor in this formulation is the material momentum tensor. A material force in the form of a path-independent integral for the elastic dielectric is derived employing Toupin's [1] formulations.

Energy release rate calculations by the finite element method

The present paper outlines a finite element method for calculating the energy release rate. The method is based on a continuum mechanics formulation of the virtual crack extension principle and can be used with linear elastic materials as well as materials following the deformation theory of plasticity. The formulation is easily incorporated into a finite element program, but can also conveniently be used as part of a post-processing program, which uses stress and displacement data from a finite element analysis to calculate the energy release rate. The present formulation can be used as a unified approach for 2-D and 3-D fracture problems and includes the effect of body forces and traction loading on the crack faces.

On the Lagrangian formulation of continuum mechanics

Abstract The aim of this paper is to show that a Lagrangian formulation of continuum mechanics (in the spirit of field theory) can deliver, not only equations of motion, but the conservation laws related to the material symmetries in the perfect continuum. Those conservation laws in the presence of defects lead to the path-independent integrals broadly used in fracture mechanics. They are basically related to the (material) forces on a defect in a continuum and can be interpreted as the equations of motion for a defect with respect to the material. The quantity playing the role of the stress tensor in this formulation is the material momentum tensor. A simple example of a material force in the form of a path-independent integral in the fluid is considered.

Kinetic theory of dense fluids. IV. Entropy balance equation and irreversible thermodynamics

By making use of the kinetic equation and the global entropy formula reported in the first paper of this series, we have derived the entropy balance equation and the accompanying molecular formulas for local entropy density, entropy flux and entropy production for real fluids at an arbitrary density. We have thereby obtained an extension to real fluids of the Boltzmannian irreversible thermodynamics which is based on the Boltzmann equation for ideal fluids. The theory presented provides us with a molecular basis for the continuum mechanics formulation of irreversible thermodynamics by Coleman, Noll, Gurtin and their coworkers: The macroscopic constitutive functionals in their theory can now be calculated from the various molecular formulas for the relevant quantities in our theory.

Snow Pressure on Rigid Obstacles

Abstract In this paper a continuum mechanical formulation of snow pressure is given. The snow pressure against retaining structures on slopes is considered in two separate parts: (1) the static component due to the regression of transverse expansion as discussed by Haefeli, and (2) the dynamic component due to the interruption of the creep (internal deformation) and glide (slip of the entire snow cover over the ground). Snow-pressure calculations made using the non-linear viscous constitutive equations given by McClung (in press) are given for the plane strain-rate problem of a rigid barrier on a snow-covered slope. These calculations are compared with the previous formulation of snow pressure given by Haefeli for the expected range of boundary conditions for the structure and for the snow-earth interface. The results show that the original formulation by Haefeli gives a dynamic component of similar magnitude to the present calculations. Substantial differences are apparent when the up-slope distances for interruption of the creep and glide processes are compared.

Snow Pressure on Rigid Obstacles

Abstract In this paper a continuum mechanical formulation of snow pressure is given. The snow pressure against retaining structures on slopes is considered in two separate parts: (1) the static component due to the regression of transverse expansion as discussed by Haefeli, and (2) the dynamic component due to the interruption of the creep (internal deformation) and glide (slip of the entire snow cover over the ground). Snow-pressure calculations made using the non-linear viscous constitutive equations given by McClung (in press) are given for the plane strain-rate problem of a rigid barrier on a snow-covered slope. These calculations are compared with the previous formulation of snow pressure given by Haefeli for the expected range of boundary conditions for the structure and for the snow-earth interface. The results show that the original formulation by Haefeli gives a dynamic component of similar magnitude to the present calculations. Substantial differences are apparent when the up-slope distances for interruption of the creep and glide processes are compared.

Stable and Unstable Crack Growth in Viscoelastic Media

The failure of load-bearing structures by fracture is generally important in all phases of our society. It may concern small household items as well as expensive structures of civil or space applications and accordingly may cause varying degrees of economic distress. While the state of failure is usually easily determined as either failed or completely failed, the estimation of how close to either state a structure is, poses a much more difficult problem. It is important to recognize, however, that from an engineering point of view, the latter problem is the important one because it would allow, in principle, the prediction of the conditions leading to fracture and thus to a close estimate of the service life of a structure. Inasmuch as failures by fracture involve the growth of cracks it appears that keeping track of the size of a crack in a particular structure provides a means of assessing *quantitatively* the strength prior to complete failure. If one agrees that the description of structural strength is rationalized in terms of the size of the defects, it foll0ws that one must attempt to understand the laws that govern the growth of such defects in order to predict complete failure. Fracture of materials is a complicated process which encompasses atomistic aspects, as well as microscopic and large-scale continuum mechanical considerations. Although one of these aspects should not be considered without the other we shall be concerned with the continuum-mechanical formulation of the problem of fracture growth in viscoelastic materials. From this viewpoint the prediction of failure comprises three phases: first an examination of the physical situation presented by a static or growing defect in a material, second the translation of this physically observable situation into a mathematical model which is amenable to analysis by currently available or extendable tools of mathematics, third the theoretical exploitation of the mathematical model in an attempt to predict the behavior of defects under load and the comparison of these results with experimentally observable phenomena to assess the validity of the modelling process as given from phase one and phase two. While there are many important details that have bearing on such a development we shall be concerned more with the principles of the analysis and show how the various considerations of the three phases enter into the overall structure of the crack propagation problem. In keeping analytic work as simple as possible it is intended to emphasize what type of results may be obtained with the aid of continuum mechanics and where continuum mechanics requires support by microscopic considerations.

The Reciprocal Theorem for Harmonic Oscillations of an Elastic, Isotropic Medium

Using the formulation of continuum mechanics, a dynamic reciprocal theorem is derived for sinusoidal motion of an elastic medium. It is found that, exactly as in the static case, the work of the forces of State 1 acting through the displacements of State 2, equals the work of the forces of State 2 acting through the displacements of State 1. A simple example is given, which illustrates the application of the reciprocal theorem to the analysis of forced vibrations of beams.