Governing Field Equations Research Articles

Shear banding and material instability in finitely deformed compressible materials: An example in thin sheets

A finite elastoplasticity model for compressible materials is reviewed in order to emphasize some special features of the model. The criterion for material instability has an invariant form and is embodied exclusively in the constitutive relation. On the other hand, the condition for the onset of shear banding is derived based upon the full set of governing field equations. The condition is both necessary and sufficinet. The derivation accounts for both non-coaxial boundary loading and non-coaxial strain path. Through the analysis, the influence of both the plastic shear modulus and the stress state over the shear band behavior is identified. As an illustrative example, the case of proportional loading is discussed. The results are in good agreement with experimental data from the open literature and thus demonstrate the efficacy of the model.

Scaling thermal effects in radial flow

To adequately evaluate the environmental impact of siting nuclear waste repositories in basalt aquicludes, it is essential to know the effects on parameter identification algorithms of thermal gradients that exist in these basaltic aquicludes. Temperatures of approximately 60°C and pressures of approximately 150 atm can be expected at potential repository sites located at depths of approximately 1000 m. The phenomenon of over-recovery has been observed in some pumping tests conducted at the Hanford Nuclear Reservation located in the Pasco Basin adjacent to the Columbia River in the state of Washington, USA. This over-recovery phenomenon may possibly be due to variations in the fluid density caused by thermal gradients. To assess the potential effects of these thermal gradients on indirect parameter identification algorithms, a systematic scaling of the governing field equations is required in order to obtain dimensionless equations based on the principle of similarity. The constitutive relationships for the specific weight of the fluid and for the porosity of the aquiclude are shown to be exponentially dependent on the pressure gradient. The dynamic pressure is converted to the piezometric head and the flow equation for the piezometric head is then scaled in radial coordinates. Order-of-magnitude estimates are made for all variables in unsteady flow for a typical well test in a basaltic aquiclude. Retaining all nonlinear terms, the parametric dependency of the flow equation on the classical dimensionless thermal and hydraulic parameters is demonstrated. These classical parameters include the Batchelor, Fourier, Froude, Grashof, and Reynolds Numbers associated with thermal flows. The flow equation is linearized from order-of-magnitude estimates based on these classical parameters for application in parameter identification algorithms.

Numerical modelling of bifurcation and localisation in cohesive-frictional materials

Methods are reviewed for analysing highly localised failure and bifurcation modes in discretised mechanical systems as typically arise in numerical simulations of failure in soils, rocks, metals and concrete. By the example of a plane-strain biaxial test it is shown that strain softening and lack of normality in elasto-plastic constitutive equations and the ensuing loss of ellipticity of the governing field equations cause a pathological mesh dependence of numerical solutions for such problems, thus rendering the results effectively meaningless. The need for introduction of higher-order continuum models is emphasised to remedy this shortcoming of the conventional approach. For one such a continuum model, namely the unconstrained Cosserat continuum, it is demonstrated that meaningful and convergent solutions (in the sense that a finite width of the localisation zone is computed upon mesh refinement) can be obtained.

On the Lagrangian residual current and residual transport in a multiple time scale system of shallow seas

A multiple time scale perturbation method is used to discuss the Lagrangian residual current and residual transport on the basis of a weakly nonlinear dynamic model of shallow seas. The governing equations for the long-term variation of zero order “apparent concentration” (which is a linear combination of salinity, temperature of seawater and the concentration of any tracer which is conservative and passive) and its mean value over tidal cycles are obtained for the system with single tidal constituent, and for the one with multi-constituents, winds and thermohaline. The equations for the two cases are in the same form and show this long-term variation resulted from the cumulative effect of residual convection and turbulent diffusion. The multiple time scale variation of current is caused by tides, winds, and the thermohaline and the nonlinear effects of the system. The derived set of governing field equations of the Lagrangian current for this multiple time scale system is also in the same form as that for a single time scale system.

SIMULATION OF STRAIN LOCALIZATION: A REAPPRAISAL OF THE COSSERAT CONTINUUM

Classical continuum models, i.e. continuum models that do not incorporate an internal length scale, suffer from pathological mesh‐dependence when strain‐softening models are employed in failure analyses. In this contribution the governing field equations are regularized by adding rotational degrees‐of‐freedom to the conventional translational degrees‐of‐freedom. This so‐called elasto‐plastic Cosserat continuum model, for which an efficient and accurate integration algorithm and a consistent tangent operator are also derived in this contribution, warrants convergence of the load—deflection curve to a unique solution upon mesh refinement and a finite width of the localization zone. This is demonstrated for an infinitely long shear layer and a biaxial test of a strain‐softening elasto‐plastic von Mises material.

Analysis of finite elastoplastic deformation using a new finite element method

Abstract A direct finite element scheme based on a variational method of the least squares type is used to analyze large deformation of dilatant solids. The approach is incremental in nature. The scheme treats both Cauchy stress rates and velocities as dependent variables, and provides the requisite computational basis for mixed problems, that is, problems with combined elliptic and hyperbolic character. There are two unique features of the method, which distinguish the scheme from that based on the conventional displacement and the mixed formulations. One is that the method is insensitive to the type of the governing field equations, that is, the computational algorithm is the same in both elliptic and hyperbolic regions. The other is that the method is insensitive to the absence of work hardening, that is, the resulting matrix system is always symmetric and positive definite. Numerical results for a number of fundamental deformation problems demonstrate that the onset of progressive damage of the deforming material, either due to dilational cracking or the formation of shear bands, can emerge as the natural outcome of the analysis. The analysis predicts various failure modes as they are embedded in the governing field equations and does not employ any additional failure criterion. The results also demonstrate the significance of the infinitesimal assumption of the elastic component of the deformation rate in choosing time-step sizes during numerical implementation.

Model of Finite Strain Creep of Metals

The finite strain tensile behavior of polycrystalline metal bars at elevated temperatures is investigated. The starting point for the investigation of flow and rupture is a constitutive model which is capable of reproducing, at infinitesimal strain, the basic tensile behavior of elastic, anelastic, and viscous creep response and creep recovery. A finite strain version of the model, together with the governing field equations, is then used to study the response of a perfect cylindrical tensile bar subjected to constant load. Elongation and area histories are obtained, and the governing equations are examined with the purpose of extracting a rupture criterion. In order to study the necking behavior, a linear eigenvalue problem is formulated for both uniform and nonuniform bifurcation modes. The analysis indicates that a uniform mode is possible; however, it depends solely on the structure of the elastic law and formally coincides with rupture or the termination of the flow process. A nonuniform mode is show...

Conditions for shear banding and material instability in finite elastoplastic deformation

A necessary and sufficient condition for shear banding in two-dimensional space is derived for finite elastoplastic deformation of compressible materials. The criterion for the initiation of shear bands, a geometric instability, is that the governing field equations become hyperbolic. The results indicate that shear bands may occur before the material response reaches its peak load, irrespective of compressibility, and that the critical work-hardening rate, which represents the material factor governing shear banding, has the order of the ratio of yield stress to shear modulus. Moreover, the characteristics, or the orientations, of shear bands do not in general coincide with the direction of maximum shear stress, and so their directions may not be orthogonal. Also in the analysis, the critical stress ratio, which is a deformation factor controlling shear banding, is derived explicitly in terms of Poisson's ratio, the relative density of the aggregate, and the transverse stress ratio. The prediction of the orientations of shear bands explains well, at least qualitatively, experimental observations in the open literature. The criterion for material instability, unlike the onset of shear banding, is defined at a point when the stress-strain relation of the aggregate has no inverse. The criterion for material instability due to shear banding is derived. This provides a tool to distinguish shear band localization from shear fracture in general.

On the convergence of block constrained nonlinear equation solvers∗

The shortcomings associated with successive substitution nonlinear solvers are circumvented by introducing a so-called block constraint methodology. Specifically, by partitioning the governing field equations into various subsets, constraints can be applied individually to each separate block. To illustrate the formal properties of the scheme, a series of lemmas and a theorem will be developed. These define its convergence properties. Due to the generality of the approach taken, virtually any form of functional constraint may be employed.

Symbolic derivation of the electrodynamic Green’s tensor in an anisotropic medium

A symbolic inverse operator technique is used to derive the electromagnetic Green’s tensor of an anisotropic (uniaxial) medium. By using algebraic matrix identities, the tensor differential operator of the governing field equation is inverted in a straightforward way. Thus the correct result is obtained, which has been found elsewhere with the help of a much more complicated analysis.

A model computation of moisture content in polythermal glaciers

Measurements in boreholes and internal radar horizons confirm the existence of polythermal structures in polar and subpolar glaciers and ice sheets. The occurrence of an intraglacial phase boundary makes it necessary to consider two layers and transition conditions between cold and temperate ice. The governing field equations and constitutive relations for a two‐layer model are stated with emphasis on boundary and transition conditions. A reduced plane flow model is introduced by using dimensionless variables and performing a scale analysis. According to the stated aim of this paper a parallel‐sided slab model is suitable for rough analyses of borehole measurements. Its governing equations are deduced, and their analytical and numerical treatment is discussed. Sensitivity tests reveal a strong dependence of moisture content profiles on the numerical value of the water diffusivity coefficient and especially on strain heating.

Plane universal streamline patterns of incompressible viscous fluids

Unlike the truly kinematical quantities such as speed, vorticity and acceleration, the streamlines and stream surfaces in a flow characterize purely geometrical aspects of the motion. While the class of dynamically admissible motions of a fluid is determined by the governing field equations, which in turn depend upon the constitutive functions, several familiar streamline patterns maintain their presence despite the variations in the material properties. For example, the concentric circular streamline pattern admits various Couette flows corresponding to different viscosity functions, and likewise the parallel rectilinear streamline pattern admits various Poiseuille flows. Most of the exact solutions of incompressible viscous fluids in the literature were indeed obtained by semi-inverse methods based on the assumption of isometric streamline p a t t e r n s i n particular, streamline patterns that are composed of parallel or radial lines, concentric circles, or logarithmic spirals*. This suggests a general question concerning the class of streamline patterns which admit different flows depending upon the material properties. For such flows the streamline pattern is unperturbed by certain variations in the constitutive functions, although other kinematical features are. It is clear that such streamline patterns are of particular interest in connection with the experimental determination of constitutive functions. There is an extensive literature on the related subject of the universal flows of incompressible viscous fluids. With appropriate conservative body forces, a universal flow satisfies the Navier-Stokes equations for arbitrary values of the viscosity. Generalizing the concept of a universal flow, one may consider a streamline pattern which admits some steady solution of the Navier-Stokes equation for each value of the viscosity. However, in the Navier-Stokes equations the effects of the viscous stress and of the inertial force combine in such a way that any particular steady solution Vo corresponding to a specific value of the kinematical

The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates

It is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation. For example, in the analysis of rotating structures, the coupling of the equations for geometrically inexact models, obtained by linearization or other approximations, with those for rotating rigid bodies can easily lead to misleading artificial “softening” effects that can significantly alter numerical results; see Simo and VuQuoc [1986c] (especially equations (3) and (5)). In this paper, we consider fully nonlinear geometrically exact models for rods, plates (and shells) which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models. These models can be obtained either from the three-dimensional theory by a systematic use of projection methods; see e.g., Antman [1972] and Naghdi [1972], or by a direct approach within the context of Cosserat continuum. Remarkably, the two approaches lead to essentially the same form of the governing field equations. In the present context, we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical Kirchhoff-Love model (see Love [1944]) to include shear deformations, as in

Self-stabilized nonlinear lateral modes of broad area lasers

The lateral modes of broad area lasers are investigated theoretically. The nonlinear interaction between optical field and effective refractive index leads to a saturable nonlinearity in the governing field equation, so that self-modulated solutions are found to be stable with increased current injection above saturation intensity. We derive approximate analytical solutions for traveling wave fields within the broad area laser. The field amplitude consists of a small ripple superimposed on a large dc value. Matching fields at the boundary determines the modulation depth and imparts an overall phase curvature to the traveling wave mode. There are multiple lateral modes for a given set of operating conditions, and modes with successively more lobes in the ripple have greater overall phase curvature. In contrast to the linear problem, several lateral modes can achieve the same modal gain, for a given injected current density, by saturating the gain to different extent. Thus, these modes would exhibit slightly different optical powers.

Stress assisted diffusion in elastic and viscoelastic materials

The governing field equations and boundary conditions for stress assisted diffusion are derived from basic principles of continuum mechanics and irreversible thermodynamics. Attention is confined to elastic and viscoelastic material response, with emphasis on isotropy and the case of small strains. It is shown that both the diffusion process and the saturation levels are stress dependent. In viscoelstic materials the time-dependence of the moisture boundary condition causes the saturation level to drift with time. In addition, the diffusion process becomes non-linear in stress, in spite of the linearity of the mechanical behavior.

Complete solutions of a coupled system of partial differential equations arising in thermoelasticity

Three general, complete solutions of a coupled hyperbolic or hyperbolic-parabolic system of two second-order linear partial differential equations are presented. The system includes among its particular cases the governing field equations of the conventional as well as generalized thermoelasticity theories. The solutions obtained are analogous to the Lamé, Papkovitch, and Galerkin solutions in classical elasticity. The interrelationships among the solutions are also exhibited. Some solutions obtained in earlier works are deduced as special cases of the unified solutions obtained here.

Predicted axial temperature gradient in a viscoplastic uniaxial bar due to thermomechanical coupling

AbstractThe thermomechanical response of a uniaxial bar with thermoviscoplastic constitution is predicted herein using the finite element method. After a brief review of the governing field equations, variational principles are constructed for the one‐dimensional conservation of momentum and energy equations. These equations are coupled in that the temperature field affects the displacements and vice versa. Due to the differing physical nature of the temperature and displacements, first‐order and second‐order elements are utilized for these variables, respectively. The resulting semi‐discretized equations are then discretized in time using finite differencing. This is accomplished by Euler's method, which is utilized due to the stiff nature of the constitutive equations. The model is utilized in conjunction with stress‐strain relations developed by Bodner and Partom to predict the axial temperature field in a bar subjected to cyclic mechanical end displacements and temperature boundary conditions. It is found that spacial and time variation of the temperature field is significantly affected by the boundary conditions. The nomenclature used is given in an Appendix.

Nonlinear spectral characteristics of large deformation elasticity theory

This paper investigates the spectral properties of general structural systems undergoing large elastic deformations. The governing field equations used in the analysis consist of large deformation elasticity theory wherein the Hookes type constitutive law is defined in terms of the second Piola-Kirchhoff stress and Lagrangian strain tensor combination. Due to the generality of the approach taken, the effects of external loading fields containing a perfusion of harmonics can be handled. In this context, special emphasis is given to ascertaining the various forms of sub/super and combination type harmonics. Overall a wide range of combination behavior is considered. This includes mixed interactions between external harmonics and the various system spectra. Special emphasis is given to dealing with cross talk which may occur between all the various participating system spectral branches.

Hydraulic fracturing of a saturated porous medium-II: Special cases

This analysis investigates the steady-state propagation of a hydraulic fracture in an infinite isotropic fluid-saturated elastic porous medium in the limiting cases of slow and rapid crack growth. In the limiting case of slow crack propagation, it is found that the governing field equations become decoupled, and that a closed-form solution for the stress and pore pressure components can be obtained by solving these equations subject to the appropriate boundary conditions. In the limiting case of rapid crack propagation, it is found that the field equations reduce to a degenerate form, and that the stress and pore pressure components given by the solution of the equations do not satisfy the boundary conditions imposed at the crack faces and at the crack tip. This suggests the presence of stress and pore pressure boundary layers along the surfaces and at the tip of the propagating crack. Closed-form solutions for stress and pore pressure components in the boundary layer are then obtained by formulating a singular perturbation problem which is subsequently investigated by the method of matched asymptotic expansions.

Operator-valued connections, Lie connections, and Gauge field theory

Noether's first theorem tells us that the global symmetry groupGr of an action integral is a Lie group of point transformations that acts on the Cartesian product of the space-time manifold with the space of states and their derivatives. Gauge theory constructs are thus required for symmetry groups that act indiscriminately on the independent and dependent variables where the group structure can not necessarily be realized as a subgroup of the general linear group. Noting that the Lie algebra of a general symmetry groupGr can be realized as a Lie algebragr of Lie derivatives on an appropriately structured manifold,Gr-covariant derivatives are introduced through study of connection 1-forms that take their values in the Lie algebragr of Lie derivatives (operator-valued connections). This leads to a general theory of operator-valued curvature 2-forms and to the important special class of Lie connections. The latter are naturally associated with the minimal replacement and minimal coupling constructs of gauge theory when the symmetry groupGr is allowed to act locally. Lie connections give rise to the gauge fields that compensate for the local action ofGr in a natural way. All governing field equations and their integrability conditions are derived for an arbitrary finite dimensional Lie group of symmetries. The case whereGr contains the ten-parameter Poincare group on a flat space-timeM4 is considered. The Lorentz structure ofM4 is shown to give a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of the Poincare group. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of the Poincare group.