Luqun Ni

A constitutive model for fcc crystals with application to polycrystalline OFHC copper

Based on the results of a series of experiments on commercially pure OFHC copper (an fcc polycrystal), a physically based, rate- and temperature-dependent constitutive model is proposed for fcc single crystals. Using this constitutive model and the Taylor averaging method, numerical calculations are performed to simulate the experimental results for polycrystalline OFHC copper. The model calculation is based on a new eAcient algorithm which has been successfully used to simulate the flow stress of polycrystalline tantalum over broad ranges of temperature, strain rate, and strain (Nemat-Nasser, S., Okinaka, T., Ni, L., 1998. J. Mech. Phys. Solids 46, 1009). The model eAectively simulates a large body of experimental data, over a broad range of strain rates (0.001‐8000 s ˇ1 ), and temperatures (77‐1096 K), with strains close to 100%. Few adjustable constitutive parameters of the model are fixed at the outset for a given material. All other involved constitutive parameters are estimated based on the crystal structure and the physics of the plastic flow. ” 1998 Published by Elsevier Science Ltd. All rights reserved.

A physically-based constitutive model for bcc crystals with application to polycrystalline tantalum

Abstract Based on the results of an extensive series of systematic experiments on commercially pure tantalum (bcc crystals), a physically-based, rate- and temperature-dependent constitutive model is proposed for bcc single crystals and is applied to simulate the experimental results, using the Taylor averaging method. The model calculation is based on a new efficient algorithm for the numerical solution of the finite deformation of bcc single crystals, involving up to 48 potentially active slip systems. The accuracy and efficiency of the proposed algorithm are checked through comparison with the results of the conventional explicit Euler time-integration scheme, using a very large number of timesteps. The model effectively simulates a large body of experimental data, over a broad range of strain rates (10−3 − 4 × 104/s), and temperatures (77 to 1300 K), with strains exceeding 100%, using very few adjustable parameters whose values are fixed at the outset for a given material. All other involved constitutive parameters are estimated based on the crystal structure and the physics of plastic flow.

Interface cracks in anisotropic dissimilar materials : An analytic solution

Abstract Based on linear elastic fracture mechanics, analytic solutions are given for displacement and stress fields of in-plane deformation of two anisotropic half-planes, forming a composite bimaterial, with an interface crack, assuming strictly two-dimensional problems; this requires suitable orientation of the material symmetry axes to ensure decoupling of the anti-plane fracture mode from the in-plane ones. It is shown that the field equations are fully characterized in terms of four dimensionless parameters, and these parameters are expressed in terms of the twelve involved elastic constants, six for each half-plane. Analytic solutions are given for two models: (1) the fully open-crack model, involving oscillatory square-root singularities at crack tips ; and (2) the Comninou model which allows possible small contact zones close to the crack tips. Analytic expressions are obtained for the crack opening displacement, the size of the contact zone, the total force transmitted across the contact zone, and the stress field. The results are discussed and related to those for isotropic bimaterials, given by Gautesen and Dundurs.

Shear bands as surfaces of discontinuity

A theoretical characterization of a shear band as a surface of discontinuity is developed. For the one-dimensional problem of unidirectional shearing of a slab, expressions for the jump discontinuities across a possible shear band are derived. This provides a simplified formulation of the problem, which yields a pair of equations for the evolution of stress and temperature along the surface representing the shear band. Two illustrative examples are examined.

A general duality principle in elasticity

Abstract Anisotropic elasticity problems are considered, where the field variables depend on only two spatial coordinate variables, say, the rectangular Cartesian coordinates, x1 and x1. The equilibrium equations are then identically satisfied (no body forces), if the stress components, σij, are such that σj1 = ∂Φj/∂x2 and σj2 = −∂Φj/∂x1, j = 1, 2, 3, where Φ with the component Φj is the stress vector potential. In terms of the six-dimensional vector field η = [u, Φ], the basic elasticity equations reduce to {∂/∂x2}[u, Φ]T = N{∂/∂x1}[u, Φ]T, where u is the displacement field and N is called the fundamental elasticity matrix, given by the elastic moduli of the solid. This formulation and the corresponding duality remove the distinction between the displacement vector field u, and the stress vector-potential field Φ. Indeed, these two vector fields are, in many respects, each others dual, in the sense that, for a given solid with given geometry and elasticity, the solution to a class of prescribed mixed displacement-traction boundary-value problems also provides the solution of the corresponding dual mixed traction-displacement boundary-value problems, without a change in geometry or elasticity of the solid. This duality principle is first proved rigorously. Then the results are applied to a general mixed boundary-value problem of a multiply connected heterogeneous finite solid with arbitrary piecewise constant anisotropic elasticity. A number of interesting illustrations involving dislocations and concentrated forces, cracks and anticracks (rigid line inclusions), interface problems, and the half-space Green functions, are also presented as special cases of the general duality principle. In addition to the general duality principle which provides a two-way relation between u and Φ, a number of other one-way relations between the displacement field u and the stress vector potential Φ of the same problem, as well as u of one problem and u of its dual, and Φ of one problem and Φ of s dual, are also presented.

The transient motion of a dislocation with a ramp-like core

The general solution of the radiated field from a nonuniformly moving ramp dislocation with general time-dependent ramp displacement function is obtained. Special cases of it, for arctan displacement with certain constant or time-dependent width and step function displacement, and for motions starting from rest at time t=0 and moving thereafter nonuniformly, are presented. The solution is in terms of quadratures which will be evaluated when the time-dependent functions become known. A special case, the ramp dislocation starting from rest and moving with a constant velocity thereafter, is analyzed with particular emphasis on the wave-front and near-field behaviour.

The Debonded Interface Anticrack

The debonded interface anticrack is treated analytically and the singularity at the tip is found to vary between 1/2 and I, with the dependence on the material constants combination explicitly obtained. While the case of uniform tension and shear loading at infinity has been solved, the method of solution, which consists of distributing dislocation and line load densities, readily lends itself to solution for other point loadings, such as concentrated forces or dislocations.

Bridged interface cracks in anisotropic bimaterials

Abstract The bonding strength between dissimilar materials in composites is often relatively weak. Interface debonding and cracking are of common occurrence. As an effective method to minimize such failures, fibre stitching is used to enhance the bonding strength when joining layers in polymer composites. With the presence of the reinforcing fibre, the overall elastic response for each component of the composite is, in general, anisotropic. This paper investigates the elastic behaviour of bridged interface cracks in anisotropic bimaterials. From the equilibrium of the interface crack, a system of Cauchy integral equations for the required distributed dislocation density is obtained. When the bridging force depends linearly on the crack-opening displacement, explicit solutions are given in terms of a series of Jacobi polynomials. For illustration, a bridged interface crack in isotropic bimaterials is examined in detail. Results suggest that bridging fibres effectively enhance the toughness of the compositive. When the bridging force increases, the effect of interface mismatch becomes insignificant, being overshadowed by the effect of the bridging forces.

The singular nature of the stress field near an arbitrarily moving dislocation loop

Abstract It is proved that, for the most singular 1/ϵ terms in the stress field at a point near the current position of a moving loop, the loop may be approximated by its tangent translating with the same instantaneous velocity v and acceleration a as the loop has at that point, while, as far as the coefficients of the In ϵ terms are concerned, the loop is approximated by its tangent at that point translating and rotating as the loop, and by its osculating circle expanding with the same instantaneous velocity as the loop.

“Driving forces” and radiated fields for expanding/shrinking half-space and strip inclusions with general eigenstrain

A half-space constrained Eshelby inclusion (in an infinite elastic matrix) with general uniform eigenstrain (or transformation strain) is analyzed when the plane boundary is moving in general subsonic motion starting from rest. The radiated fields are calculated based on the Willis expression for constrained time-dependent inclusions, which involves the three-dimensional dynamic Green’s function in an infinite tractionfree body, and they constitute the unique elastodynamic solution, with initial condition the Eshelby static fields obtained as the unique minimum energy solutions by a limiting process from the spherical inclusion. The mechanical energy-release rate and associated “driving force” to create dynamically an incremental region of eigenstrain (due to any physical process) is calculated for general uniform eigenstrain. For dilatational eigenstrain the solution coincides with the one obtained by a limiting process from a spherically expanding inclusion, while for shear eigenstrain the fields are due to the propagation of the rotation. The “driving force” has the same expression both for expanding and shrinking motions, resulting in expenditure of the energy rate for motion of the boundary in both cases. By superposition from the half-space inclusions, the fields and “driving force” for a strip inclusion with both boundaries moving are obtained. The “driving force” consists also of a contribution from the other boundary when it has time to arrive. The presence of applied loading contributes the counterpart of the Peach-Koehler force of dislocations, in addition to the self-force. Introduction. In a recent publication, Markenscoff and Ni (2010) obtained the energy-release rate required to create dynamically an incremental region of dilatational Received February 1, 2010. 2000 Mathematics Subject Classification. Primary 74N20, 74H05, 74B99. Partial support by NSF (grant # CMS 0555280) is gratefully acknowledged. The first author wishes to thank Professors Rodney Clifton and Lev Truskinovsky for suggesting the problem of the moving plane boundary and initial comments. c ©2011 Brown University 529 License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf 530 X. MARKENSCOFF AND L. NI uniform eigenstrain by an expanding spherical inclusion, as well as by an expanding plane boundary, through a limiting process from the sphere. Here, the energy release rate to create an incremental region of general uniform eigenstrain εij by a moving plane boundary of a constrained inclusion is obtained. Markenscoff and Ni (2010) obtained the radiated fields from a spherical inclusion with dilatational eigenstrain (constrained in an infinite linearly elastic matrix) expanding in a general subsonic motion based on the analysis of Willis (1965) for inclusions with time-dependent eigenstrain (transformation strain) constrained in an elastic matrix that is traction-free at the boundary at infinity. It results in an expression for the displacement in terms of the dynamic Green’s function, analogous to the Eshelby one (1957) for static inclusions. The static Eshelby solution for a spherical inclusion (1957) was obtained from this elastodynamic expression when evaluated from t = −∞ to t = 0, and the Hadamard jump conditions were shown to be satisfied. Using these fields, the energy-release rate required to create an incremental volume of eigenstrain as the spherical inclusion expands was computed. It may be noted here that the energy-release rate expression of Atkinson and Eshelby (1968), Rice (1968), and Freund (1972), derived initially for moving cracks when evaluated for a singularity that is a jump discontinuity (Stolz, 2003), gives an expression which coincides with that of the associated “driving force” in the thermodynamic literature (Truskinovky, 1982) for a system that is purely mechanical. The energy-release rate is equivalent to the pathindependent dynamic J integral derived on the basis of Noether’s theorem (Freund (1990), Maugin (1990), Gupta and Markenscoff (in preparation)) for an “elastic singularity” for which the integrals involved exist (as Cauchy Principal Values). The radiated fields and energy-release rate to move a plane boundary with dilatational eigenstrain were obtained by Markenscoff and Ni (2010) by a limiting process from the spherically expanding inclusion, as the radius of the sphere tends to infinity, and that solution, radiated fields and self-force is recovered here as a special case of eigenstrain. The energy-release rate, and associated “driving force”, or “self-force” of the moving plane boundary, has a static part coinciding with the one based on the expression given by Eshelby (1970, 1977) and independently calculated by Gavazza (1977) for a spherical inclusion. The dynamic part of the self-force for a plane boundary depends only on the current value of the velocity, and not the acceleration, and thus the plane phase boundary has no effective mass, in contrast to the dislocation (Ni and Markenscoff, 2008). However, for a spherical inclusion the furthermost point of the back of the inclusion, where a discontinuity occurs, also contributes to the “driving force” on the front boundary. In the present treatment, the radiated fields from a constrained (in an elastic matrix) three-dimensional linearly elastic inclusion occupying x1 ≤ R0 for t ≤ 0, and expanding/shrinking in a general subsonic motion of the plane inclusion boundary according to x1 = R0 + (t), are calculated based on Willis (1965, equation (26)) for inclusions with time-dependent boundaries constrained in an elastic matrix that is traction-free on the boundary. The Willis expression involves the three-dimensional dynamic Green’s function for a point force in an infinite elastic body, and is the exact dynamic analog to the static Eshelby expression (1957). The eigenstrain is general, but due to antisymmetries in some terms of the dynamic Green’s function, the evaluation of the integrals is simplified. The solution for the displacement is obtained (modulo rigid body motion), License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf “DRIVING FORCES” AND RADIATED FIELDS 531 from which the strains, rotations, jumps thereof, and “driving force” are obtained for general uniform eigenstrain. In the dynamic case, here as well as in Markenscoff and Ni (2010), for the same reason as in the static half-plane inclusion (Dundurs and Markenscoff, 2009), the obtained solution is unique, since it is derived by the elasticity solution for a constrained inclusion in an infinite medium with zero tractions on the boundary at infinity, having as initial condition the Eshelby static fields. No superposed compatible externally applied fields at infinity are allowed (which would increase the energy, e.g. Mura (1982), and, which were called by Dundurs and Markenscoff, 2009, “rogue states”). The “driving force” has the same expression both for expanding and shrinking motion, resulting in expenditure of energy for motion of the boundary both cases. The case of shear eigenstrain ε12, which is frequently of interest in phase transformations (e.g. Mura, 1982), is part of the solution. By superposition of the half-space fields, the radiated fields for a strip inclusion with shear eigenstrain, expanding and shrinking in either direction, are obtained, and the “driving force” computed. The “driving force” has a contribution also from the jump discontinuity at the other boundary, when it has the time to arrive, similar to the contribution to the front boundary from the back of the spherically expanding inclusion (Markenscoff and Ni, 2010). In the present treatment, the radiated fields from a constrained (in an elastic matrix) three-dimensional linearly elastic inclusion occupying x1 ≤ R0 for t ≤ 0, and expanding/shrinking in a general subsonic motion of the plane inclusion boundary according to x1 = R0 + (t), are calculated based on Willis (1965, equation (26)) for inclusions with time-dependent boundaries constrained in an elastic matrix that is traction-free on the boundary. The Willis expression involves the three-dimensional dynamic Green’s function for a point force in an infinite elastic body, and is the exact dynamic analog to the static Eshelby expression (1957). The eigenstrain is general, but due to antisymmetries in some terms of the dynamic Green’s function, the evaluation of the integrals is simplified. The solution for the displacement is obtained (modulo rigid body motion), from which the strains, rotations, jumps thereof, and “driving force” are obtained for general uniform eigenstrain. In the dynamic case, here as well as in Markenscoff and Ni (2010), for the same reason as in the static half-plane inclusion (Dundurs and Markenscoff, 2009), the obtained solution is unique, since it is derived by the elasticity solution for a constrained inclusion in an infinite medium with zero tractions on the boundary at infinity, having as initial condition the Eshelby static fields. The static Eshelby fields for the half-space inclusion are unique minimum energy ones, as derived from the minimum energy solution of the spherical inclusion by a limiting process. No superposed compatible externally applied fields at infinity are allowed (which would increase the energy, e.g. Mura (1982), and which were called by Dundurs and Markenscoff, 2009, “rogue states”). The “driving force” has the same expression both for expanding and shrinking motion, resulting in expenditure of energy for motion of the boundary in both cases. The case of shear eigenstrain ε12, which is frequently of interest in phase transformations (e.g. Mura, 1982), is part of the solution. By superposition of the half-space fields, the radiated fields for a strip inclusion with shear eigenstrain, expanding and shrinking in either direction, are obtained, and the “driving force” computed. The