An incremental stress-based constitutive modeling on anisotropic damaged materials

An incremental form of anisotropic damage constitutive equation is proposed both for brittle and ductile materials. Based on the concept of irreversible thermodynamics that damage processes are history independent coupled with irreversible energy dissipation, two types of definition for damage representation are established, known as damage tensor D and damage strain tensor ^d, to describe constitutive responses of damaged materials. A state variable coupled with damage and other observable state variables, i.e. ^d, is formulated separately from other internal variables and defined as an equivalent damage variable. A constitutive relation due to damage is finally formulated by introducing damage flow potential function employing the theory of irreducible integrity bases. A clear physical representation based on theoretical foundations and rigorous mathematical arguments of the conventional damage models defined in terms of damage effect tensor M(D) is also elucidated. Validity of the proposed model is verified by comparing with the formulations of conventional damage effect tensor. A plastic potential function coupled with damage is also introduced by employing the anisotropic plastic flow theory, so that the proposed damage model can be applied to characterize a wide range of damage problems of practical engineering interest.     

Damage zones based on Dugdale model for materials

In this paper, an infinite sheet with a crack is studied using continuum damage mechanics technique. The formulation is based on the hypothesis of incremental complementary energy equivalence model for damage evaluation. Damage distributions in the region of a macrocrack tip are calculated for an elastic-perfectly plastic material. The size of the damage zone is also derived via the Dugdale model with damage which considers the interactions between the macrocracks and microcracks. To assess the results, comparisons are made between proposed damage model, Dugdale plastic model and finite element solutions. Good agreement is observed.     

On the effects of microstress on macroscopic diffusion processes

Summary The effects of internal microstress fields are neglected in a usual simulation of the diffusion of a small solute trough solid heterogeneous media. However, when a heterogeneous material is used in a structure undergoing external mechanical loading, highly nonuniform stress fields can arise locally due to the variable microstructure. In this paper a simple quasi-Fickian model is studied which employs a spatially variable stress-dependent diffusivity,D . The structure ofD stems from an assumption that the stresses nonuniformly open and close the pores of the material microstructure, thus providing preferential sites for accumulation of the diffusing solute. When =0, the usual stress-free Fickian diffusivity,D ⁰, is recovered. Because of the highly oscillatory stress fields on the micro level, when employing numerical methods, such as the finite element or finite difference method, the distance between discretization nodes must be far smaller than the microstructural oscillations to obtain accurate simulations. This fact makes direct numerical simulations involvingD virtually impossible without computationally intensive, and complicated, special techniques. In this paper upper bounds are developed for the difference between solutions produced when usingD and alternativelyD ⁰ in the body under analysis. The general case, whenD ⁰, and consequentlyD , are spatially variable, is considered. The bounds are a function of onlyD ⁰ and and do not require any knowledge of the stress-dependent solution, and can thus be used as an a-priori check to determine whether potentially expensive computations are necessary.Dedicated to Prof. Dr. Peter Haupt on the occasion of his 60th birthday     

An assessment of the validity of J-controlled crack growth,and the stability of cracks in incremental-plastic/elastic solids

The validity of the J-integral as a characteristic parameter of the crack-tip plastic field is examined under conditions where stable crack growth precedes catastrophic fracture. It is shown that the widely accepted conditions for the validity of J-controlled crack growth [1] are rendered invalid by singular fan-shaped regions of severe non-proportional loading and elastic unloading, extending completely across the stress field of the dominant singularity, when both small-finite and infinitesimal crack extension occurs.In view of this result a new approach to the problem of crack stability, following finite stable crack growth, is developed from the classical Kelvin-Dirichlet condition for the stability of elastic/plastic solids. A sufficient condition for the stability of a crack in a plastic/elastic solid, under dead loading and rigid constraints on the surface, is obtained in terms of traction- and displacement-increments from the current equilibrium state. The stability condition is valid, within the limits of small deformation theory, for the physically realistic work-hardening incremental-plastic/elastic solid; the stability criterion can therefore be used to examine the conditions leading to the onset of catastrophic fracture following extensive stable crack growth.A tentative experimental procedure is presented as a basis for the practical application of the incremental crack-stability criterion to problems involving large stable crack growth.     

各向异性损伤理论与复合材料层板损伤的实验研究(一)-各向异性损伤理论

本文讨论了各向异性损伤力学的基本概念并介绍了以应变等效性假设为基础的Lemaitre模型和以能量等效性假设为基础的Cordebois模型.在此基础上作者导出了两种损伤模型的损伤能释放率的表达式.文中作者定义了一维损伤时弹性材料板单元的一些损伤定义,并导出了两种模型在一维损伤时损伤能释放率Y₂的简单表达式,从而为复合材料层板损伤测量的实验研究提供了理论基础,     

Linear response and instability of vortex lattice in type-II superconductors

The wave vector dependent linear response of the vortex lattice in type-II superconductors is studied by calculating the superfluid density and is shown to have a singular behavior with respect to the long wave length perturbation of vector potential. The calculations are carried out in terms of the nonlinear elastic theory of the vortex lattice, which is, in principle, a low temperature expansion, and the terms up to the first order in temperature are taken into account, which is beyond the usual Gaussian elastic theory. The superfluid density is found to behave as ~ k_BT ¦k¦² log ¦k¦,for the small wave vector,k,perpendicular to the external magnetic field, where Tand k_B are the temperature and the Boltzmann constant, respectively. This behavior causes the divergence of the magnetic susceptibility at a critical wave vector, k_c.We show that this behavior is associated with an instability of vortex lattice to the externally applied perturbation.     

Structural,Elastic, Thermodynamic,Electronic, and Magnetic Investigations of Full-Heusler Compound Ag<Subscript>2</Subscript>CeAl: FP-LAPW Method

Structural, elastic, thermodynamic, electronic, and magnetic properties of the full-Heusler compound Ag₂CeAl were determined using generalized gradient approximation with exchange-correlation functional GGA (PBEsol) with spin-orbit coupling (SOC) correction. The elastic modulus and their pressure dependence are calculated. From the elastic parameter behavior, it is inferred that this compound is elastically stable and ductile in nature. Through the quasi-harmonic Debye model, in which the phononic effect is considered the effect of pressure P (0 to 50) and temperature T (0 to 1000) on the lattice constant, the elastic parameters, bulk modulus B, heat capacity and thermal expansion α, internal energy U, entropy S, Debye temperature ??_D, Helmholtz free energy A, and Gibbs free energy G are investigated. The thermodynamic properties show that the compound Ag₂CeAl is a heavy fermion material. The density of state (DOS), magnetic momentum, and band structure are computed, to investigate the magnetic and metallic characteristics. The calculated polarization of the compound is 77.34%. The obtained results are the first predictions of the physical properties for the rare-earth-based (Ce) full-Heusler Ag₂CeAl.     

Elastic energy of the vortex state in type II superconductors. II. Low inductions

The elastic energy of a distorted lattice of flux lines interacting by an arbitrary two-body potential is calculated. At low inductions, where the London model applies, the elastic response to localized forces appears to be larger than expected from local elasticity theory. This effect is most pronounced in hard superconductors, where the effective moduli for compression and bending waves may decrease by a factor 2² B/H _c2 . Such a behavior agrees with that obtained previously from the Ginzburg-Landau theory at large inductions. Those results can be interpreted by an effective interaction with range /(1-B/H _c2 )^1/2 and amplitude (H _{c2 - B)} .     

Harmonic waves in a prestressed thin elastic tube filled with a viscous fluid

In this work a theoretical analysis is presented for wave propagation ina thin-walled prestressed elastic tube filled with a viscous fluid. Thefluid is assumed to be incompressible and Newtonian, whereas the tubematerial is considered to be incompressible, isotropic and elastic.Considering the physiological conditions that the arteries experience, sucha tube is initially subjected to a mean pressure P_i and anaxial stretch _z. If it is assumed that in the course ofblood flow small incremental disturbances are superimposed on this initialfield, then the governing equations of this incremental motion are obtainedfor the fluid and the elastic tube. A harmonic-wave type of solution issought for these field equations and the dispersion relation is obtained.Some special cases, as well as the general case, are discussed and thepresent formulation is compared with some previous works on the samesubject.     

Existence and uniqueness of the integrable-exactly hypoelastic equation\mathop \tau \limits^ \circ = \lambda (trD){\rm I} + 2\mu D and its significance to finite inelasticity

Summary In Eulerian rate type finite inelasticity models postulating the additive decomposition of the stretchingD, such as finite deformation elastoplasticity models, the simple rate equation indicated in the above title is widely used to characterize the elastic response withD replaced by its elastic part. In 1984 Simo and Pister (Compt. Meth. Appl. Mech. Engng.46, 201–215) proved that none of such rate equations with several commonly-known stress rates is exactly integrable to deliver an elastic relation, and thus any of them is incompatible with the notion of elasticity. Such incompatibility implies that Eulerian rate type inelasticity theory based on any commonly-known stress rate is self-inconsistent, and thus it is hardly surprising that some aberrant, spurious phenomena such as the so-called shear oscillatory response etc., may be resulted in. Then arises the questions: Whether or not is there a stress rate ? The answer for these questions is crucial to achieving rational, self-consistent Eulerian rate type formulations of finite inelasticity models. It seems that there has been no complete, natural and convincing treatment for the foregoing questions until now. It is the main goal of this article to prove the fact: among all possible (infinitely many) objective corotational stress rates and other well-known objective stress rates , there is one and only one such that the hypoelastic equation of grade zero with this stress rate is exactly integrable to define a hyperelastic relation, and this stress rate is just the newly discoveredlogarithmic stress rate by these authors and others. This result, which provides a complete answer for the aforementioned questions, indicates that in Eulerian rate type formulations of inelasticity models, the logarithmic stress rate is the only choice in the sense of compatibility of the hypoelastic equation of grade zero that is used to represent the elastic response with the notion of elasticity.     

Finite-size effects in surface tension: Thermodynamics and the Gaussian interface model

It has been suggested by Kayser that finite-size effects associated with capillary waves might play a significant role in some surface tension measurements; for capillary rise between plates a distance D apart, an effect varying as 1/D and apparently observable in measurements, was proposed. In reconsidering this problem, one must analyze the thermodynamics of finite-size corrections to surface tension. In particular, one sees that capillary rise between plates does not measure the interfacial free energy density but, rather, a derivative of the interfacial free energy with respect to a system dimension. The quantity needed to draw definite conclusions, the finite-size residual free energy, can be calculated within the harmonic or Gaussian capillary wave model in d spatial dimensions with the aid of Poisson summation techniques and should yield the correct leading asymptotic behavior. For d=3 and experimentally relevant parameter values, the results are independent of the short-wavelength cutoff needed in the model and can be checked against the theory of conformai covariance at two-dimensional critical points. It is found that the finite-size effects in capillary-rise measurements of surface tension vary as 1/D ² (with a universal coefficient) but are too small to be seen in current experiments.Invited paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.     

Correlation between fracture and damage for quasi-brittle bi-material interface cracks

Fracture at a bi-material interface is essentially mixed-mode, even when the geometry is symmetric with respect to the crack and loading is of pure Mode I, due to the differences in the elastic properties across an interface which disrupts the symmetry. The linear elastic solutions of the crack tip stress and displacement fields show an oscillatory type of singularity. This poses numerical difficulties while modeling discrete interface cracks. Alternatively, the discrete cracks may be modeled using a distributed band of micro-cracks or damage such that energy equivalence is maintained between the two systems. In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence concepts and to predict the damage scenario in quasi-brittle bi-material interface beams. The study is aimed at large size structures made of quasi-brittle materials failing at concrete-concrete interfaces. The objective is to smoothly move from fracture mechanics theory to damage mechanics theory or vice versa in order to characterize damage. It is concluded, that through the energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the same energy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such as the fracture energy can be obtained.     

Model for prediction of simultaneous time-dependent damage evolution in multiple plies of multidirectional polymer composite laminates and its influence on creep

A model to predict time-dependent evolution of simultaneous transverse cracking developed in multiple plies during creep loading and its effects on creep of multidirectional polymer matrix composite laminates is presented. The stress states in the intact regions of the plies are determined using the lamination theory during an incremental change in time. The stored elastic energy, determined using this stress state, is compared with a critical stored elastic energy value for damage to determine if a ply would fracture after the increment. If fracture is predicted, variational analysis is used to determine the perturbation in ply stresses due to cracking. This procedure is repeated to determine the crack evolution and creep strain. Model predictions compared well with experimental results for a [±θ_m/90_n]_s laminate.     

The equivalent elastic crack: 2. X-Y equivalences and asymptotic analysis

The use of crack growth resistance curves (R-a) to predict the behaviour of cracked specimens is a well-established practice for ceramics and materials. In Part 1 of this work, the authors showed that the use of R-a curves may imply a certain elastic equivalence between an actual specimen governed by a cohesive crack model and a virtual equivalent linear elastic specimen. Part 1 included the analysis of certain classes of equivalences (P-Y equivalences) where the loads acting on the actual and on the equivalent specimen were forced to be equal. This paper analyzes more general equivalences in which two arbitrary variables X and Y are forced to be identical in the actual and in the equivalent specimens. In particular, the J-CTOD equivalence and the size-effect-based equivalence put forward by Bazant are analyzed. The first part of the paper deals with the bases and general applicability of these equivalences. The second part presents the results of asymptotic analyses intended to assess the applicability of the equivalences to specimens of relatively large size.     

On the dynamic susceptibility of the bulging domain wall model of polycrystalline magnetic oxides

The static initial magnetic susceptibility due to domain wall motion in a polycrystalline magnetic oxide has been explained by Globuset al. [3] using a model of a bulging domain wall inside grains of uniform diameter,D. The present work deals with the dynamic response of this model by solving the equation of motion of such a wall. The resultant solution reproduces Globus' relation for the static case and further shows that the dispersion frequency is D ^–1 for small grainsizes and D ^–2 for large grain sizes.     

Large deformation theory of coupled thermoplasticity including kinematic hardening

Summary Thermoplasticity is a topic central to important applications such as metalforming, ballistics and welding. The current investigation introduces a thermoplastic constitutive model accommodating the difficult issues of finite strain and kinematic hardening. Two potential functions are used. One is interpreted as the Helmholtz free energy. Its reversible portion describes elastic behavior, while its irreversible portion describes kinematic hardening. The second potential function describes dissipative effects and arises directly from the entropy production inequality. It is shown that the dissipation potential can be interpreted as a yield function. With two simplifying assumptions, the formulation leads to a simple energy equation, which is used to derive a rate variational principle. Together with the Principle of Virtual Work in rate form, finite element equations governing coupled thermal and mechanical effects are presented. Using a uniqueness argument, an inequality is derived which is interpreted as a finite strain thermoplastic counterpart to the classical inequality for stability in the small. A simple example is introduced using a von Mises yield function with linear kinematic hardening, linear isotropic hardening and linear thermal softening.Symbols D rate of deformation tensor - d VEC(D) - F deformation gradient tensor - h heat generation per unit mass - L velocity gradient tensor - q heat flux vector - workless internal variable - Lagrangian strain - e VEC() - E quasi-Eulerian strain - entropy - internal energy per unit mass - Helmholtz free energy - Cauchy stress tensor - Truesdell stress flux tensor - t VEC() - yield function - First Piola Kirchhoff stress - Second Piola Kirchhoff stress - s VEC() - s ^* backstress, center of the yield surface - Kronecker product symbol - VEC vectorization operator - tr(.) trace - DEV deviator of a tensor - TEN22 Kronecker tensor operator     

The geometry of nonlinear elasticity

Elasticity is the prototype of constitutive models in Continuum Mechanics. In the nonlinear range, the elastic model claims for a geometrically consistent physico-mathematical formulation providing also the logical premise for linearized approximations. A theoretic framework is envisaged here with the aim of contributing a conceptually clear, physically consistent, and computationally convenient formulation. A reasoning about the physics of the model, from a geometric point of view, leads to conceive constitutive relations as instantaneous incremental responses to a finite set of tensorial state variables and to their time rates along the space-time motion. Integrability of the tangent elastic compliance, existence of an elastic stress potential, and conservativeness of the elastic response, under the conservation of mass, are given a brand new treatment. Finite elastic strains have no physical interpretation in the new rate theory, and referential local placements are appealed to, just as loci for operations of linear calculus. Frame invariance is assessed with a consistent geometric treatment, and the clear distinction between the new notion and the property of isotropy is pointed out, thus overcoming the improper statement of material frame indifference. Extension of the theory to elasto-visco-plastic constitutive models is briefly addressed. Basic computational steps are described to illustrate feasibility and convenience of calculations according to the new theory of elasticity.     

Microscopic Theory of Organic Superconductors

A microscopic theory of organic superconductors based on the concept of partial electron dielectrization is developed from first principles. Self-consistent equations for the superconducting order parameter () and insulating order parameter (D) are derived using a Green's function technique and equation of motion method. The theory is applied to explain the experimental results in the two-dimensional organic superconductor k-(BEDT-TTF)₂ Cu(NCS)₂. The present model explains coexistence of spin density wave (SDW) state and superconductivity state in the system. The behavior of superconducting order parameter (), insulating order parameter (D), specific heat, density of state, free energy, and critical field is also studied for the system k-(BEDT-TTF)₂ Cu(NCS)₂.