A thermodynamically consistent plastic-damage framework for localized failure in quasi-brittle solids: material model and strain localization analysis

Aiming for the modeling of localized failure in quasi-brittle solids, this paper addresses a thermodynamically consistent plastic-damage framework and the corresponding strain localization analysis. A unified elastoplastic damage model is first presented based on two alternative kinematic decompositions assuming infinitesimal deformations, with the evolution laws of involved internal variables characterized by a dissipative flow tensor. For the strong (or regularized) discontinuity to form in such inelastic quasi-brittle solids and to evolve eventually into a fully softened one, a novel strain localization analysis is then suggested. A kinematic constraint more demanding than the classical discontinuous bifurcation condition is derived by accounting for the traction continuity and the loading/unloading states consistent with the kinematics of a strong (or regularized) discontinuity. More specifically, the strain jumps characterized by Maxwell’s kinematic condition have to be completely inelastic (energy dissipative). Reproduction of this kinematics implies vanishing of the aforesaid dissipative flow tensorial components in the directions orthogonal to the discontinuity orientation. This property allows naturally developing a localized plastic-damage model for the discontinuity (band), with its orientation and the traction-based failure criterion consistently determined a posteriori from the given stress-based counterpart. The general results are then particularized to the 2D conditions of plane stress and plane strain. It is found that in the case of plane stress, strain localization into a strong (or regularized) discontinuity can occur at the onset of strain softening. Contrariwise, owing to an extra kinematic constraint, in the condition of plane strain some continuous inelastic deformations and substantial re-orientation of principal strain directions in general have to take place in the softening regime prior to strain localization. The classical Rankine, Mohr–Coulomb, von Mises (J2) and Drucker–Prager criteria are analyzed as illustrative examples. In particular, both the closed-form solutions for the discontinuity angles validated by numerical simulations and the corresponding traction-based failure criteria are obtained.Peer ReviewedPostprint (author's final draft

A novel positive/negative projection in energy norm for the damage modeling of quasi-brittle solids

The asymmetric tensile/compressive material behavior and microcracks closure-reopening (MCR) effects exhibited by quasi-brittle solids are of significant importance to the nonlinear responses of engineering structures under cyclic loading, e.g., earthquake excitations. Based on our previous work (Cervera et al., 1995; Faria et al., 1998; Wu et al., 2006) this work addresses a novel thermodynamically consistent unilateral damage model for concrete. In particular, the positive/negative projection (PNP) of the effective stress tensor and the additive bi-scalar damage constitutive relation are maintained owing to the conceptual simplicity and computational efficiency. It is found that the classical PNP widely adopted in the literature is not optimal for this damage model, since the resulting stiffness is not always of major symmetry. Consequently, a well-defined free energy potential does not exist in general cases and the model cannot be cast into the framework of thermodynamics with internal variables. Furthermore, the damage induced anisotropy cannot be captured, exhibiting excessive lateral deformationsunder uniaxial tension. To overcome the above issues, a novel PNP, variationally interpreted as the closest point projection of the effective stress in energy norm, is proposed with closed-form solution. With the novel PNP, the secant stiffness tensorof the proposed unilateral damage model always possesses major symmetry and exhibits orthotropic behavior under uniaxial tension and mixed tension/compression. The corresponding thermodynamics framework is then given, resulting in an energy release rate based rounded-Rankine type damage criterion appropriate for tensile failure in quasi-brittle solids. Several numerical examples of single-point verifications and benchmark tests are presented. It is demonstrated that the proposed model is capable of characterizing localized failureof concrete under proportional and non-proportional static loading, as well as the MCR effects under seismic cyclic loading

Strain localization and failure mechanics for elastoplastic damage models

This work investigates systematically strain localization and failure mechanics for elastoplastic damage solids. Two complementary methodologies, i.e., traction-based discontinuities localized in an elastic solid and strain localization of a stress-based inelastic softening solid, are addressed. In the former it is assumed a priori that the discontinuity (band) forms with a continuous stress field and along the known orientation. A traction-based failure criterion is introduced to characterize the discontinuity (band) and the orientation is determined from Mohr’s maximization postulate. If the (apparent) displacement jumps are retained as independent variables, the strong/regularized discontinuity approaches follow, requiring constitutive models for both the bulk and discontinuity (band). Elimination of the displacement jumps at the material point level results in the embedded/smeared discontinuity approaches in which an overall inelastic constitutive model fulfilling the static constraint suffices. The second methodology is then adopted to check whether the assumed strain localization can occur and identify its consequences on the resulting approaches. The kinematic constraint guaranteeing stress boundedness/continuity upon strain localization is established for general inelastic softening solids. Application to a unified elastoplastic damage model naturally yields all the ingredients of a localized model for the discontinuity (band), justifying the first methodology. Two dual but not necessarily equivalent approaches, i.e., the traction-based elastoplastic damage model and the stress-based projected discontinuity model, are identified. The former is equivalent to the embedded/smeared discontinuity approaches, whereas in the later the discontinuity orientation and associated failure criterion, not given a priori, are determined consistently from the kinematic constraint. The bi-directional connections and equivalence conditions between the traction- and stress-based approaches are classified. Closed-form 2D results under plane stress condition are also given, with the classical Rankine, Mohr-Coulomb, von Mises and Drucker-Prager criteria analyzed as the illustrative examples. A generic failure criterion of either elliptic, parabolic or hyperbolic type, is then considered in a unified manner, resulting in many failure criteria frequently employed in practice

On the conformity of strong, regularized, embedded and smeared discontinuity approaches for the modeling of localized failure in solids

Once strain localization occurs in softening solids, inelastic loading behavior is restricted within a narrow band while the bulk unloads elastically. Accordingly, localized failure in solids can be approached by embedding or smearing a traction-based inelastic discontinuity (band) within an (equivalent) elastic matrix along a specific orientation. In this context, the conformity of the strong/regularized and embedded/smeared discontinuity approaches are investigated, regarding the strategies dealing with the kinematics and statics. On one hand, the traction continuity condition imposed in weak form results in the strong and regularized discontinuity approaches, with respect to the approximation of displacement and strain discontinuities. In addition to the elastic bulk, consistent plastic-damage cohesive models for the discontinuities are established. The conformity between the strong discontinuity approach and its regularized counterpart is shown through the fracture energy analysis. On the other hand, the traction continuity condition can also be enforced point-wisely in strong form so that the standard principle of virtual work applies. In this case, the static constraint resulting from traction continuity can be used to eliminate the kinematic variable associated with the discontinuity (band) at the material level. This strategy leads to embedded and smeared discontinuity models for the overall weakened solid which can also be cast into the elastoplastic degradation framework with a different kinematic decomposition. Being equivalent to the kinematic constraint guaranteeing stress continuity upon strain localization, Mohr’s maximization postulate is adopted for the determination of the discontinuity orientation. Closed-form results are presented in plane stress conditions, with the classical Rankine, Mohr–Coulomb, von Mises and Drucker–Prager criteria as illustrative examples. The orientation of the discontinuity (band) and the stress-based failure criteria consistent with the given traction-based counterparts are derived. Finally, a generic failure criterion of either elliptic, parabolic or hyperbolic type, appropriate for the modeling of mixed-mode failure, is analyzed in a unified manner. Furthermore, a novel method is proposed to calibrate the involved mesoscopic parameters from available macroscopic test data, which is then validated against Willam’s numerical test

Recursions in Calogero-Sutherland Model Based on Virasoro Singular Vectors

The present work is much motivated by finding an explicit way in the construction of the Jack symmetric function, which is the spectrum generating function for the Calogero-Sutherland(CS) model. To accomplish this work, the hidden Virasoro structure in the CS model is much explored. In particular, we found that the Virasoro singular vectors form a skew hierarchy in the CS model. Literally, skew is analogous to coset, but here specifically refer to the operation on the Young tableaux. In fact, based on the construction of the Virasoro singular vectors, this hierarchical structure can be used to give a complete construction of the CS states, i.e. the Jack symmetric functions, recursively. The construction is given both in operator formalism as well as in integral representation. This new integral representation for the Jack symmetric functions may shed some insights on the spectrum constructions for the other integrable systems.Comment: Latex, 32pages, 4 figure

Strain localization of elastic-damaging frictional-cohesive materials: analytical results and numerical verification

Damage-induced strain softening is of vital importance for the modeling of localized failure in frictional-cohesive materials. This paper addresses strain localization of damaging solids and the resulting consistent frictional-cohesive crack models. As a supplement to the framework recently established for stress-based continuum material models in rate form (Wu and Cervera 2015, 2016), several classical strain-based damage models, expressed usually in total and secant format, are considered. Upon strain localization of such damaging solids, Maxwell's kinematics of a strong (or regularized) discontinuity has to be reproduced by the inelastic damage strains, which are defined by a bounded characteristic tensor and an unbounded scalar related to the damage variable. This kinematic constraint yields a set of nonlinear equations from which the discontinuity orientation and damage-type localized cohesive relations can be derived. It is found that for thePeer ReviewedPostprint (published version

Reconsideration on the elastic damage/degradation theory for the modeling of microcrack closure-reopening (MCR) effects

AbstractDespite the substantial and noteworthy contributions, the modeling of damage induced anisotropy remains an unsolved issue, especially when the microcrack closure-reopening (MCR) effects are accounted for. Theoretically speaking, the most challenging problem might be the lack of energy conservation for all the existing models employing the spectral decomposition of the stress or strain tensor and the associated positive/negative projection operators (Carol and Willam, 1996). In this paper this crucial problem is solved by introducing a new definition of thermodynamically consistent projection operators into the classical elastic damage/degradation model. The orthogonality between the rates of the new projection operators and the stress (or strain) automatically guarantees the fulfillment of energy conservation under any arbitrary (proportional or non-proportional) load history. Moreover, with this extra orthogonal property the thermodynamically consistent projection operators can be exclusively determined in a unique form. The above advantages lend to their promising use in the modeling of damage induced anisotropy and MCR effects simultaneously. With the aid of the thermodynamically consistent projection operators, the existing elastic damage/degradation models can be reformulated so that the energy conservation is restored only with minor modifications. As a prototype example, a stress-based elastic damage/degradation model with multiple loading surfaces is developed analogously to the classical multisurface plasticity model. Finally, a simple closed-loop load path accompanied with rotation of principal stress directions is constructed to verify its thermodynamically consistency