Being one of the most promising candidates for the modeling of localized failure in solids, so far the phase-field method has been applied only to brittle fracture with very few exceptions. In this ...work, a unified phase-field theory for the mechanics of damage and quasi-brittle failure is proposed within the framework of thermodynamics. Specifically, the crack phase-field and its gradient are introduced to regularize the sharp crack topology in a purely geometric context. The energy dissipation functional due to crack evolution and the stored energy functional of the bulk are characterized by a crack geometric function of polynomial type and an energetic degradation function of rational type, respectively. Standard arguments of thermodynamics then yield the macroscopic balance equation coupled with an extra evolution law of gradient type for the crack phase-field, governed by the aforesaid constitutive functions. The classical phase-field models for brittle fracture are recovered as particular examples. More importantly, the constitutive functions optimal for quasi-brittle failure are determined such that the proposed phase-field theory converges to a cohesive zone model for a vanishing length scale. Those general softening laws frequently adopted for quasi-brittle failure, e.g., linear, exponential, hyperbolic and Cornelissen et al. (1986) ones, etc., can be reproduced or fit with high precision. Except for the internal length scale, all the other model parameters can be determined from standard material properties (i.e., Young’s modulus, failure strength, fracture energy and the target softening law). Some representative numerical examples are presented for the validation. It is found that both the internal length scale and the mesh size have little influences on the overall global responses, so long as the former can be well resolved by sufficiently fine mesh. In particular, for the benchmark tests of concrete the numerical results of load versus displacement curve and crack paths both agree well with the experimental data, showing validity of the proposed phase-field theory for the modeling of damage and quasi-brittle failure in solids.
Recently several non-standard phase-field models different from the standard one for brittle fracture (Bourdin et al. 2000, 2008; Miehe et al. 2010a) have been proposed for the modeling of damage and ...fracture in solids. On the one hand, linear elastic behavior before the onset of damage and even cohesive cracking induced quasi-brittle failure can be considered by such non-standard phase-field models. On the other hand, they present great challenges to the numerical implementation since the damage boundedness is no longer automatically fulfilled. In this work, the numerical implementation of the unified phase-field damage model (Wu 2017, 2018) is addressed, though it also applies to the standard and other non-standard ones. In particular, an iterative alternate minimization (AM) algorithm enhanced with path-following strategies is presented. The phase-field or damage sub-problem is solved by the bound-constrained optimization solver in which the boundedness and irreversibility conditions of the crack phase-field are exactly enforced. Moreover, material softening induced snap-backs typically for localized failure in solids can also be effectively dealt with. The equilibrium paths in terms of the fracture surface and of the indirect displacement (e.g., the crack mouth opening or sliding displacement) are discussed in the context of the AM algorithm. The AM algorithm with the indirect displacement control is advocated, since both prescribed external forces and displacements can be naturally dealt with, which is more versatile than the one with the fracture surface control applicable only for prescribed displacements. Representative examples of benchmark tests show that the AM algorithm enhanced with the indirect displacement control is extremely robust even for large increment sizes. The insensitivity of the unified phase-field damage model to the incorporated length scale and mesh size is also confirmed.
•Robust numerical implementation of non-standard phase-field damage models is presented.•The crack irreversibility is exactly enforced by the bounded-constrained optimization solver.•The alternate minimization algorithm enhanced with the indirect displacement control is justified.•Robustness of the presented numerical algorithm is verified by examples of benchmark tests.
Aiming to bridge the gap between damage and fracture mechanics for modeling localized failure in solids, this paper addresses a novel geometrically regularized gradient-damage model with energetic ...equivalence for cracking evolution. With the free energy potential defined as usual in terms of the local strain and damage fields, the constitutive relations are derived consistently from the standard framework of thermodynamics. Upon the sharp crack topology geometrically regularized by the damage localization band with a length scale, the ensuing energetic equivalence naturally yields the damage evolution law of gradient-type and the associated boundary condition of Neumann-type. Compared to other gradient-damage models, no extra assumptions like the nonlocal energy residual and insulation condition are introduced. Moreover, the damage gradient, physically accounting for microscopic nonlocal interactions, is fully dissipative as expected. In line with the unified phase-field theory recently proposed by the author (Wu, 2017) during failure processes the material behavior is uniquely characterized by two constitutive functions, i.e., the degradation function defining the free energy potential of the bulk and the geometric function regularizing the sharp crack topology. In particular, optimal constitutive functions defining an equivalent cohesive zone model of general softening laws are postulated, with the involved parameters calibrated from standard material properties. The proposed model is numerically implemented into the multi-field finite element method and applied to several benchmark tests of concrete under mode-I and mixed-mode failure. It is found that the incorporated length scale can be regarded either a numerical parameter or a material property. For the former considered in this work, the length scale has negligible, if not no, effects on the global responses, so long as the sharp crack topology and the damage field of high gradients within the localization band are well resolved. Furthermore, the localization bandwidth does not exhibit spurious widening, but rather, it approaches to a finite value proportional to the length scale. Comparison between the numerical and experimental results, regarding the curve of load versus displacement and crack path, confirms validity of the proposed gradient-damage model for characterizing localized failure in quasi-brittle solids.
•A geometrically regularized gradient-damage model with energetic equivalence is proposed.•The damage evolution law emerges from the geometric regularization and energetic equivalence.•The crack is implicitly represented, and cumbersome crack tracking strategies are not required.•The length scale can be regarded either as a small numerical parameter or as a material property.•The localization bandwidth approaches to a finite limit value, not exhibiting spurious damage growth.
This paper presents a novel fractal defected ground structure (FDGS) to reduce mutual coupling between coplanar spaced microstrip antenna elements. The structure of the proposed FDGS is studied. ...Bandgap characteristic of second and third iterative FDGS is achieved. The second and third iterative FDGSs are used to reduce mutual coupling between microstrip antenna elements. Mutual coupling reduction performance of third iterative FDGS is better than that of second iterative FDGS. And dimension of FDGS can be decreased by using higher level iterative FDGS. The third iterative FDGS is fabricated and measured. Simulated and measured results both show that more than 35-dB mutual coupling reduction is obtained by using the third iterative FDGS. Moreover, the envelope correlation of antenna elements with FDGS is quite smaller than that of antenna elements without FDGS.
A highly sensitive optical fiber surface plasmon resonance (SPR) biosensor based on graphene oxide (GO) and staphylococcal protein A (SPA) co-modified tilted fiber Bragg grating (TFBG) is proposed ...and demonstrated for the detection of human immunoglobulin G (IgG) for the first time. The gold film on the surface of the sensor was first fixed with GO and then modified with an SPA to improve the sensitivity of the sensor. Large specific surface area and abundant functional groups of GO can adsorb more antibodies. The combination of SPA and the antibody molecule Fc region makes the Fab area with antigen-binding sites extend outward, resulting in highly oriented antibody immobilization on the sensor surface and high antigen-antibody binding efficiency. The experimental results show that the sensitivity as well as the limit of detection of GO-SPA-modified TFBG-SPR biosensor is around 0.096 dB/(<inline-formula> <tex-math notation="LaTeX">\mu \text{g} </tex-math></inline-formula>/mL) and <inline-formula> <tex-math notation="LaTeX">0.5~\mu \text{g} </tex-math></inline-formula>/mL, showing better responses to human IgG solutions with a concentration range of 30-<inline-formula> <tex-math notation="LaTeX">100~\mu \text{g} </tex-math></inline-formula>/mL compared with the TFBG-SPR biosensors modified singly with GO or SPA. The biosensor exhibits the advantages of small size, ease of fabrication, high sensitivity, label-free, and rapid response, and provides a new solution for detecting low concentration of biological solution, presenting great application potential in the biochemistry field.
•A phase-field regularized cohesive zone model for brittle fracture is proposed.•The relations to other phase-field models and Barenblatt’s CZM are clarified.•Neither explicit crack ...representation/tracking nor penalty stiffness is needed.•The numerical result is insensitive to the incorporated length scale parameter.
Being able to model complex nucleation, propagation, branching and merging of cracks in solids within a unified framework, the classical phase-field models for brittle fracture fail in predicting length scale independent global responses for a solid lacking elastic singularities (e.g., corners, notches, etc.). Motivated from Barenblatt’s approximation of Griffith’s brittle fracture with a vanishing Irwin’s internal length, this paper extends our recent work in quasi-brittle failure (Wu, 2017, 2018a) and presents for the first time a length scale insensitive phase-field damage model for brittle fracture. More specifically, with a set of optimal characteristic functions, a phase-field regularized cohesive zone model (CZM) with linear softening law is addressed and applied to brittle fracture. Both the failure strength and the traction – separation law are independent of the incorporated length scale parameter. Compared to other phase-field models and CZM based discontinuous approaches for brittle fracture, the proposed phase-field regularized CZM is of several merits. On the one hand, being theoretically equivalent to Barenblatt’s CZM (at least in the 1-D case), it needs neither the explicit crack representation/tracking nor the elastic penalty stiffness which both are necessary but cumbersome for discontinuous approaches. On the other hand, it gives length scale independent global responses for problems with or without elastic singularities while preserving the expected Γ-convergence property of phase-field models. Representative numerical examples of several well-known benchmark tests support the above conclusions, validating its capability of modeling both mode-I and mixed-mode brittle fracture.
•The phase-field regularized CZM is applied to fitting test data of size and boundary effects in concrete.•The size and boundary effects in notched and unnotched concrete beams under mode-I failure ...are captured.•The size effect of notched concrete beams transited from mode-I failure to mixed-mode one is predicted.•Not only the predicted peak loads but also the softening regimes agree fairly well with experimental results.
A phase-field regularized cohesive zone model (CZM) was recently proposed for both brittle fracture and cohesive failure within the framework of the unified phase-field damage theory. Motivated from the fact that this model gives length scale and mesh independent global responses for problems with or without elastic singularities, we further apply it in this work against the size and boundary effects of concrete under both mode-I and mixed-mode failure. More specifically, for the two independent experimental campaigns of three-point bending notched and unnotched concrete beams under mode-I failure, the quality of data-fitting is, at least, comparable to the best results reported in the literature. For another series of eccentrically notched concrete beam tests, the size effect with transition from mode-I fracture to mixed-mode failure is also predicted. In all numerical examples, not only the peak loads but also the softening regimes agree well with the experimental results using a single set of material parameters for a specific series of tests. Being accompanied with other merits, e.g., generic softening laws, no lateral widening, no need of extrinsic crack tracking nor the penalty stiffness, etc., the presented phase-field regularized CZM can be used as a promising approach in the modeling of damage and failure in solids and structures.
Despite the popularity in modeling complex and arbitrary crack configurations in solids, phase-field damage models suffer from burdensome computational cost. This issue arises largely due to the ...robust but inefficient alternating minimization (AM) or staggered algorithm usually employed to solve the coupled damage–displacement governing equations. Aiming to tackle this difficulty, we propose in this work, for the first time, to use the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm to solve in a monolithic manner the system of coupled governing equations, rather than the standard Newton one which is notoriously poor for problems involving non-convex energy functional. It is found that, the BFGS algorithm yields identical results to the AM/staggered solver, and is also robust for both brittle fracture and quasi-brittle failure with a single or multiple cracks. However, much less iterations are needed to achieve convergence. Furthermore, as the system matrix is less reformed per increment, the quasi-Newton monolithic algorithm is much more efficient than the AM/staggered solver. Representative numerical examples show that the saving in CPU time is about factor 3∼7, and the larger the problem is, the more saving it gains. As the BFGS monolithic algorithm has been incorporated in many commercial software packages, it can be easily implemented and is thus attractive in the phase-field damage modeling of localized failure in solids.
•A robust and efficient quasi-Newton monolithic algorithm is proposed for phase-field damage models.•The robustness comes from the BFGS method with a symmetric and positive-definite stiffness matrix.•The proposed quasi-Newton monolithic algorithm is about 3∼7 times faster than the staggered solver.•The phase-field regularized cohesive zone model is not numerically affected by the solution algorithm.
•Three different implementation strategies of phase-field damage models in Abaqus are addressed.•The sources codes and input files of all these three implementation schemes can be open ...accessed.•Representative examples validating the implementation strategies and source codes are presented.•The UEL subroutine together with the BFGS monolithic solver is of best numerical performances.•The UMAT subroutine is viable if the BFGS solver is available in coupled thermal–stress analyses.
Despite the versatility in modeling complex crack configurations, phase-field damage models for fracture usually count only on in-house codes, greatly restricting their potential applications. It is thus of vital importance to implement them in those commonly used commercial software packages like Abaqus. However, so far only the less robust Newton’s monolithic algorithm or the inefficient staggered solver has been considered. In this work, taking the unified phase-field damage theory (Wu, 2017) as the particular example, we present three distinct strategies of implementing phase-field damage models into Abaqus: (i) UMAT-Newton-M: a thermo-mechanically coupled user-defined material (UMAT) implementation of the modified Newton monolithic solver; (ii) UEL-Staggered: a novel user-defined element (UEL) implementation of the iterative staggered (alternate minimization) algorithm with dummy dofs; (iii) UEL-BFGS: a UEL implementation of the recently advocated BFGS quasi-Newton monolithic algorithm. The aforesaid implementation strategies are then validated against several representative benchmark problems of brittle fracture and quasi-brittle failure. It is found that, the UMAT-Newton-M implementation is the simplest but not robust enough, while the UEL-Staggered implementation is robust but extremely inefficient. Comparatively, in all cases the UEL-BFGS scheme is of the best numerical performance with lest iterations and sufficient robustness. For the sake of reproducibility of the presented numerical results and the promotion of phase-field damage models, the source codes (programmed in the free format syntax of FORTRAN90) are also provided and interested users can download them at https://github.com/jianyingwu/pfczm-abaqus.
Prevention and management of lung cancer in China Hong, Qun‐Ying; Wu, Guo-Ming; Qian, Gui‐Sheng ...
Cancer,
September 1, 2015, 2015-Sep-01, 2015-09-00, 20150901, Letnik:
121, Številka:
S17
Journal Article
