Continuum Model Of Deformation Research Articles

A Continuum Deformation Model for Steel Coated with Nanolaminate Metallic Systems

This paper develops a continuum model of deformation of X70 steel coated with Cu/Nb–nanolaminate metallic systems (NMSs). The developed model is based on an elastic/viscoplastic framework that includes the effects of dislocation density, size effect, and strain hardening on the plastic deformation. The X70 steel is modeled according to a modified Taylor equation that considers the effects of statistically stored and geometrically necessary stored dislocation densities. NMSs obey two different mechanisms, the confined layer slip (CLS) and dislocation nucleation (Nu), during plastic deformation. NMSs are modeled according to the CLS mechanism when the individual lamina thickness is between 100 nm and 5 nm and the Nu mechanism when the individual lamina thickness is 5 nm or below. The model is implemented in a subroutine of a commercial finite element analysis code. Simulating a tensile test is the benchmark to generate flow curves of the model that agree with experimental data found in the literature.

Continuum Model of Deformation of Piezoelectric Materials with Cracks

The present paper addresses the continuum model describing deformation and accumulation of microdamages in electroelastic materials based on the generalized Eshelby principle. The microdamageability is considered as a process of appearance of flat elliptic or circular microcracks randomly dispersed over volume, the concentration of which increases with a load. The Eshelby method is based on the principle of equivalence of the deformation energy of fractured piezoelectric materials and the energy of medium, which is modeling these materials as a continuous medium. The key point of this approach is to determine the densities of the released elastic and electric energy.

A continuum model of deformation, transport and irreversible changes in atomic structure in amorphous lithium–silicon electrodes

Recent experiments and atomic scale computations indicate that the standard continuum models of diffusion in stressed solids do not accurately describe transport, deformation and stress in Li–Si alloys. We suggest that this is because classical models do not account for the irreversible changes in atomic structure of Si that are known to occur during a charge–discharge cycle. A more general model of diffusion in an amorphous solid is described, which permits unoccupied Si lattice sites to be created or destroyed. This may occur as a thermally activated process; or as a result of irreversible plastic deformation under stress. The model predicts a range of phenomena observed in experiment that cannot be captured using classical models, including irreversible changes in volume resulting from a charge–discharge cycle, asymmetry between the tensile and compressive yield stress, and a slow evolution in mechanical and electrochemical response over many charge–discharge cycles.

Model of fracture of metal melts and the strength of melts under dynamic conditions

The development of a continuum model of deformation and fracture of melts is needed for the description of the behavior of metals in extreme states, in particular, under high-current electron and ultrashort laser irradiation. The model proposed includes the equations of mechanics of a two-phase continuum and the equations of the kinetics of phase transitions. The change (exchange) of the volumes of dispersed and carrier phases and of the number of dispersed particles is described, and the energy and mass exchange between the phases due to phase transitions is taken into account. Molecular dynamic (MD) calculations are carried out with the use of the LAMMPS program. The continuum model is verified by MD, computational, and experimental data. The strength of aluminum, copper, and nickel is determined at various temperatures and strain rates. It is shown that an increase in the strain rate leads to an increase in the strength of a liquid metal, while an increase in temperature leads to a decrease in its strength.

Central summation in the quasicontinuum method

The quasicontinuum (QC) method [Tadmor, E.B., Phillips, R., Ortiz, M., 1996. Mixed atomistics and continuum models of deformation in solids. Langmuir 12, 4529–4534] is a multiscale methodology to significantly reduce the computational cost of atomistic simulations. The method ensures an accurate incorporation of small-scale atomistic effects in large-scale models. It essentially consists of an interpolation of the displacements of large numbers of atoms between representative atoms (repatoms) and an estimation of the total potential energy of the atomistic lattice by a so-called summation (or sampling) rule. In this paper a novel energy-based summation rule is presented for the QC method that allows for a seamless coupling between coarse domains and fully resolved domains. In the presented summation rule only the repatoms are used in combination with one extra sampling atom in the center of each interpolation triangle. The presented summation rule is therefore straightforward and computationally efficient. The performance of the proposed summation rule is evaluated for a number of two-dimensional and three-dimensional multiscale atomistic test problems.

A framework for coupled deformation–diffusion analysis with application to degradation/healing

SUMMARYThis paper deals with the formulation and numerical implementation of a fully coupled continuum model for deformation–diffusion in linearized elastic solids. The mathematical model takes into account the effect of the deformation on the diffusion process, and the affect of the transport of an inert chemical species on the deformation of the solid. We then present a robust computational framework for solving the proposed mathematical model, which consists of coupled non‐linear partial differential equations. It should be noted that many popular numerical formulations may produce unphysical negative values for the concentration, particularly, when the diffusion process is anisotropic. The violation of the non‐negative constraint by these numerical formulations is not mere numerical noise. In the proposed computational framework, we employ a novel numerical formulation that will ensure that the concentration of the diffusant be always snon‐negative, which is one of the main contributions of this paper. Representative numerical examples are presented to show the robustness, convergence, and performance of the proposed computational framework. Another contribution of this paper is to systematically study the affect of transport of the diffusant on the deformation of the solid and vice versa, and their implication in modeling degradation/healing of materials. We show that the coupled response is both qualitatively and quantitatively different from the uncoupled response. Copyright © 2011 John Wiley & Sons, Ltd.

Deformations in Transform Faults with Rotating Crustal Blocks

A continuum model of deformation of a transform fault is considered. The fault interior is modelled as a part of a plate fragmented into a large number of angular blocks. The blocks are not joined together by any binder, but are rather held together due to external compression such that they can, in principle, rotate independently. Significant numbers of blocks involved in the deformation process permit a continuum description of the deformation, which in view of the possibility of independent rotations, necessitates the application of the Cosserat theory. The crucial point in the described model is the interconnection between the rotations and the normal stresses associated with the angular shape of the blocks: the rotating blocks ‘elbow’ one another. Furthermore, elbowing produces compressive stresses independent of the direction of rotations. Consequently, the constitutive equations become non-linear involving absolute values of the components of the curvature tensor. The paper analyses a simple shear of a fault under constraining compression acting in the direction normal to the fault. An infinite layer subjected to opposite displacements and zero rotations at the edges is considered. It is shown that block rotations can lead to complex deformation patterns. There exists a displacement threshold proportional to the pressure: for imposed displacements below the threshold, the deformation pattern coincides with the conventional one as predicted for a classical elastic isotropic layer with uniform displacement gradient in the absence of block rotations. When the imposed displacement exceeds the threshold value, boundary zones of non-uniform rotations, displacement gradients and dilatation emerge. It is interesting to note that these features, which could be mistaken for indicators of non-elastic or localisation processes, occur in a situation where only elastic processes are acting at the scale of blocks and no friction or other energy dissipation processes take place.

Lateral Contraction in Theories of Multilayered Shells and Plates

A method is proposed to exactly satisfy all the constitutive equations for a layer material under conditions of their contact and conditions on the outside surfaces. In the well-known continuum models of deformation of multilayered plates and shells, the relationship between the transverse normal stresses and strains is integral only. This method increases considerably the order of differentiation of the working system of equations

Mixed Atomistic and Continuum Models of Deformation in Solids

The modeling of processes involving multiple length scales is an area of pressing concern, especially in problems such as nanoidentation and crack tip dislocation activity. In these cases, there is more than one characteristic dimension with the nanometer scale arising due to the presence of extended defects such as dislocations and a second length scale at least 2 orders of magnitude larger set by the scale of the indenter or the crack tip itself. To properly model such processes, both scales must be treated explicitly, which is normally beyond the scope of conventional atomistic and continuum analyses alike. This paper describes a quasicontinuum method which seizes upon the strengths of both atomistic and continuum techniques and allows for the simultaneous treatment of multiple scales. The method is based upon a continuum formulation of the problem of interest as a boundary value problem treated within the confines of the finite element method. We part company with traditional approaches by utilizing direct atomistic calculations as the source of the constitutive input used in the finite element analysis. The method is illustrated through application to the case of the structure and energetics of single dislocations. This case is a stringent test as it represents an extreme limit for the model since dislocation core structures are primarily dictated by lattice effects. It is then shown how the method may be applied to problems of tribological concern such as nanoindentation, where it is found that dislocations are initiated beneath the indenter.

Horizontal crustal deformation on the Hikurangi Margin

Abstract The first-order geodetic triangulation of New Zealand, completed between 1925 and 1949by the Department of Lands & Survey, has been reobserved in part between 1976 and 1984 along a series of narrow belts, as a contribution to to the national Earth Deformation Programme of the Royal Society of New Zealand. This has provided a set of data of uniformly high quality, which has been analysed by the method of simultaneous reduction, to yield a continuum model of horizontal deformation for the region between Christchurch and Auckland. This spans the Hikurangi Margin, the southern extension of the Tonga-Kermadec boundary of the subducting Pacific plate that terminates at the Chatham Rise. The continuum model of deformation comprises the 16 terms that specify the horizontal derivatives of the horizontal velocity field of orders 2–4, and their error covariance matrix. Between Hawke Bay and Chatham Rise the pattern of shear-strain rate shows a consistent azimuth of the axis of maximum relative shortening of 110/290°, with a maximum intensity (tensor) at the east coast of Wellington of about (0.20 ± 0.02) μrad/yr, decreasing rapidly westward. This direction is consistent with that that would be predicted for the axis of maximum horizontal compression in the overriding plate for the case of isotropic factional slip between the plates at the obliquely convergent subduction margin. North of Hawke Bay, the azimuth of the axis of maximum relative shortening swings to northeast-southwest. This confirms previous evidence of extension across the Central Volcanic Region, and also of trenchward extension in the East Cape peninsula; this effect may be due to gravity sliding of the superficial, and rapidly uplifted, Tertiary sediments (which carry the geodetic monuments). Analysis of bending in the East Cape peninsula shows axes of maximum bending rate concave to the SSE. In the northern South Island, strong bending, with concave normals to the axes of maximum bending lying in the northwest quadrant, marks the southern extremity of Pacific plate subduction, with its transition to the intense compressional mountain building of the central South Island. This pattern is not incompatible with the role of the Chatham Rise as a westward-thrusting rigid indenter. A fourth-rank invariant of the deformation model shows the inflexion of bending on the Hikurangi Margin to be of S-form.