Softening Law Research Articles

Progressive failure of bamboo-steel adhesive bonding interface subjected low-energy impact and tension in sequence

Bamboo–steel composite structure is a newly developed structure, composed of bamboo plywood and cold-formed thin-walled steel bonded by structural adhesive. This paper configured a three-dimensional (3D) numerical model to characterize the progressive failure of bamboo–steel adhesive bonding interface subjected low-energy impact and tension in sequence. A 3D cohesive zone model (CZM) with reloading trapezoid softening law was adopted to characterize the debonding behavior of the bamboo–steel interface. Investigations on the debonding damage propagation of the bamboo–steel interface subjected low-energy impact and tension after impact were completed, and the influence factors of the residual tensile strength were studied.

A geometrically regularized gradient-damage model with energetic equivalence

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.

Research on the progressive damage model and trigger geometry of composite waved beam to improve crashworthiness

A nonlinear progressive damage model is proposed to predict the failure response of composite waved beam subjected to quasi-static axial crushing. To predict the failure model, both of the intra-laminar failure and delamination are considered. Based on continuum damage mechanics(CDM), the in-plane damage initiation is modeled with the maximum-stress failure criterion. To simulate the failed material, a stiffness degradation method combined with exponential damage evolution laws is adopted. A fracture-energy approach in conjunction with the characteristic element length is chosen to predict the damage evolution. The delamination initiation is given by a quadratic nominal-stress criterion. Considering the progressive delamination failure process, a cohesive stiffness degradation method and an exponential softening law based on a mix-mode fracture criterion are presented to predict the delamination propagation. To achieve better crashworthiness of composite waved beam, the wedge-trigger, two types of W-triggers and bevel-triggers with hybrid angles are proposed. Additionally, effect of the angle of chamfer-trigger on crashworthiness is investigated. Numerical results show that predictions correlate well with experimental results. Moreover, the present damage model can accurately capture intra-laminar and inter-laminar failure mechanisms. The proposed bevel-triggers perform significant effect in decreasing the peak load, and angles of chamfer-trigger make little difference in the crashworthiness.

Numerical implementation of anisotropic damage mechanics

SummaryThis paper describes implementation of anisotropic damage mechanics in the material point method. The approach was based on previously proposed, fourth‐rank anisotropic damage tenors. For implementation, it was convenient to recast the stress update using a new damage strain partitioning tensor. This new tensor simplifies numerical implementation (a detailed algorithm is provided) and clarifies the connection between cracking strain and an implied physical crack with crack opening displacements. By using 2 softening laws and 3 damage parameters corresponding to 1 normal and 2 shear cracking strains, damage evolution can be directly connected to mixed tensile and shear fracture mechanics. Several examples illustrate interesting properties of robust anisotropic damage mechanics such as modeling of necking, multiple cracking in coatings, and compression failure. Direct comparisons between explicit crack modeling and damage mechanics in the same material point method code show that damage mechanics can quantitatively reproduce many features of explicit crack modeling. A caveat is that strengths and energies assigned to damage mechanics materials must be changed from measured material properties to apparent properties before damage mechanics can agree with fracture mechanics.

The Influence of Temperature on Time-Dependent Deformation and Failure in Granite: A Mesoscale Modeling Approach

An understanding of the influence of temperature on brittle creep in granite is important for the management and optimization of granitic nuclear waste repositories and geothermal resources. We propose here a two-dimensional, thermo-mechanical numerical model that describes the time-dependent brittle deformation (brittle creep) of low-porosity granite under different constant temperatures and confining pressures. The mesoscale model accounts for material heterogeneity through a stochastic local failure stress field, and local material degradation using an exponential material softening law. Importantly, the model introduces the concept of a mesoscopic renormalization to capture the co-operative interaction between microcracks in the transition from distributed to localized damage. The mesoscale physico-mechanical parameters for the model were first determined using a trial-and-error method (until the modeled output accurately captured mechanical data from constant strain rate experiments on low-porosity granite at three different confining pressures). The thermo-physical parameters required for the model, such as specific heat capacity, coefficient of linear thermal expansion, and thermal conductivity, were then determined from brittle creep experiments performed on the same low-porosity granite at temperatures of 23, 50, and 90 °C. The good agreement between the modeled output and the experimental data, using a unique set of thermo-physico-mechanical parameters, lends confidence to our numerical approach. Using these parameters, we then explore the influence of temperature, differential stress, confining pressure, and sample homogeneity on brittle creep in low-porosity granite. Our simulations show that increases in temperature and differential stress increase the creep strain rate and therefore reduce time-to-failure, while increases in confining pressure and sample homogeneity decrease creep strain rate and increase time-to-failure. We anticipate that the modeling presented herein will assist in the management and optimization of geotechnical engineering projects within granite.

Minimum energy based method to predict the multiple cracking pattern in quasi-brittle beam

A minimum energy based method is proposed to predict the evolution of multiple cracking pattern in a pure bending beam made of quasi-brittle material. The minimum energy is found from the energy landscape of the cracked beam calculated by the Finite Element (FE) Method, and the crack pattern corresponding to the minimum energy is taken as the crack pattern most likely to occur. The fracture behavior of the quasi-brittle material is simulated by cohesive zone model (CZM) using the bilinear tension softening law. After appropriate normalization based on dimensional analysis, the multiple cracking process is found to be governed by only three parameters and their effects are systematically studied. Physically, the ratio between the “crack-tip toughness” and the “bridging toughness” is important for the crack pattern evolution. As a validation, the proposed method is used to predict the crack pattern of fiber reinforced concrete beam and good agreement with experiment is achieved. The goal of this paper is twofold. First, with a systematic parametric study focusing on the constitutive law of quasi-brittle material, this investigation would facilitate material design or selection for crack control; Second, by quantitatively determining the crack pattern development during the whole multiple cracking process based on a sound physical principle, i.e., the commonly adopted minimum energy criterion, the result can serve as a benchmark for checking the reliability of other methods for multiple cracking analysis.

Multiple solutions in cohesive zone models of fracture

The use of cohesive zone models (CZM) for studying crack initiation and propagation can lead to non-unique deformed configurations. This situation can lead to solution branches that may be non-physical, leading to difficulty in interpreting computed results. These aspects are studied in this paper using the double cantilever beam (DCB) specimen first and next in the context of a mode I center crack in a thin sheet of infinite extent subjected to remote tensile loading. Analytical solutions to the CZM that employ constant stress and linear softening laws are presented, and it is shown that when the linear softening cohesive law is used, two cohesive zone sizes are valid for the same external loads, leading to two equally possible deformed configurations, satisfying all the field equations and boundary conditions. The smaller of the two cohesive zones has a lower energy which proposed as a criterion to render a unique solution.

A unified phase-field theory for the mechanics of damage and quasi-brittle failure

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.

Strategies for numerical modeling the behavior of RC beams strengthened in shear using the ETS technique

The strategy to simulate the crack shear stress transfer in a fixed smeared crack model for concrete is crucial to correctly predict the deformational and cracking behavior of RC elements that exhibit shear failure. This work presents a shear softening law able to correctly simulate the shear stiffness degradation and stress transfer during the crack opening. In cracked concrete, the element of the matrix corresponding to the fracture mode II is represented here by a τtcr-γtcr diagram, describing the shear behavior of the ith crack. The predictive performance is evaluated simulating RC beams failing in shear strengthened using the Embedded Through-Section Technique. According to this technique holes are opened through the cross section, with the desired inclination, and steel or FRP bars are introduced into these holes and bonded to the concrete substrate with adhesive materials. The parameters influencing the shear softening law are analyzed through a parametric study, furthermore a simple rule to estimate these parameters is provided and its predictive performance is assessed. The presented approach represents a valid alternative to the shear retention function.

Bone fracture characterization under mixed-mode I + II loading using the MMB test

Fracture characterization of bovine cortical bone under mixed-mode I+II loading was performed in this work. With this aim, a miniaturized version of the mixed-mode bending testing setup was constructed. Three different mode ratios were analysed to cover a wide range of mode mixities, thus leading to an appropriate definition of fracture criterion of bovine cortical bone under mixed-mode I+II loading. A suitable data reduction scheme based on crack equivalent concept was used to get the Resistance-curves of the strain energy release rate components. The proposed procedure was validated by means of a mixed-mode I+II cohesive zone model using a bilinear softening law. In general, it was verified that the numerical load-displacement as well as the Resistance-curves are in agreement with the global trend obtained experimentally, thus validating the proposed procedure concerning mixed-mode I+II fracture characterization.

Peak Load Response of a Concrete Beam in Mixed-Mode Fracture

A major difficulty in simulating load response of a concrete structure in mixed-mode fracture lies in the fact that crack path is not known a priori. Predicting both the crack path and the associated load response involves advanced simulation techniques and novel numerical methodologies. Here, an intrinsic cohesive crack model is employed to study mixed-mode fracture in a concrete beam. The present approach requires neither preliminary results from linear elastic fracture mechanics simulations nor a re-meshing procedure or special implementation to prevent crack locking. Simulations with regular meshes illustrate that this concise approach can provide a reasonable estimation of peak load of the pre-cracked concrete beams in mixed-mode fracture. This study shows that the energy ratio in the bilinear softening law has larger effects than the stress ratio.

Failure Prediction in Fiber Metal Laminates for Next Generation Aero Materials

In aerospace industry, there is huge demand for low density and low cost materials with better mechanical properties. In this view, there are many researchers developed new materials interms of composites. Similar manner, the present paper also aimed to produce a new approach for cost effective materials of 3D weaved glass fiber metal laminates (FML) with different compositions using a numerical study. A method for the simulation of progressive delamination based on de-cohesion elements has been presented. De-cohesion elements are placed between layers of solid elements that open and shear in response to the loading situation. The onset of damage and the growth of delamination are simulated without previous knowledge about the location, the size, or the direction of propagation of the de-laminations. A softening law for mixed-mode delamination that can be applied to any interaction criterion is also proposed. The constitutive equation proposed uses a single variable, the maximum relative displacement, to track the damage at the interface under general loading conditions. The material properties required to define the element constitutive equation are the inter-laminar fracture toughness's, the penalty stiffness, and the strengths.

A progressive intra- and inter-laminar damage model to predict the effect of out-of-plane confinement on pin-bearing behaviour of laminated composites

In this paper, a progressive intra- and inter-laminar damage model is proposed to study the effect of out-of-plane confinement on the pin-bearing behaviour of laminated composite. The constitutive law for fibre direction is formulated at the ply scale and is related to classical models dealing with matrix and inter-laminar damage. The influence of shear and transverse stresses on fibre damage is represented through a modified damage evolution law. Specific issues resulting from the use of a softening law in a structural analysis are discussed from a physical point of view in order to select a relevant modelling strategy whose limitations are well understood. The experimental procedure is based on an extensive set of tests in order to identify separately most of the material constants. Pin-bearing tests on configurations with and without confinement are used to evaluate the model performances. Results show the model’s ability to predict the damage scenario and the loss of stiffness implied by first damage stages.

Numerical analysis of single particle impact in the context of Cold Spray: a new adhesion model

A new adhesion model for numerical simulation of single particle impact in the context of Cold Spray is introduced. As in other studies, cohesive forces are put between the particle and substrate to account for adhesion. In this study however, the forces are put only when a local physical criterion is met. The physical phenomenon most often attributed to Cold Spray adhesion is a shear stress instability. The Johnson-Cook material law is used with a shear damage softening law to enable strong localization at the interface without the need for an extremely fine mesh. This localization is then detected as a drop in local yield stress value by the algorithm, which then implements a local cohesive force. The evolution of this cohesive force is defined by an energy dissipative cohesive model, using a surface adhesion energy as a material parameter. Each cohesive link is broken once all its associated surface energy is dissipated. A criterion on the damage value is also used to break a cohesive bond prematurely, to account for the effect of erosion at higher speeds. This model is found to reproduce the Cold Spray-like adhesion behavior with observed critical and maximum speeds.

Nonlinear analysis of masonry panels strengthened with textile reinforced mortar

The paper proposes a simplified modelling approach for the analysis of the in-plane shear behavior of tuff stone masonry panels strengthened with a surface bonded composite Basalt Textile Reinforced Mortar (BTRM) system subjected to diagonal compression. The nonlinear behavior of the unreinforced and reinforced panels is reproduced by means of a macroscopic smeared crack approach. The masonry is modeled as an isotropic continuum material characterized by different nonlinear softening laws in tension and compression. The BTRM composite is modeled with two layers representing the basalt textile and the matrix with different constitutive relationships. Sensitivity numerical analyses are developed in order to investigate the influence of each parameter in the global nonlinear response of the panel. Then, representative numerical models for the unreinforced and the reinforced panel are presented. Two BTRM layouts are investigated: single-side and double-side reinforcement. The models aim to capture the main aspects of the observed experimental nonlinear response, such as stiffness, strength-deformation shear capacity and failure mechanism. The numerical results are compared with the experimental outcomes of diagonal compression tests to investigate the feasibility of the models in the description of the in-plane shear behavior of tuff stone panels strengthened with the BTRM system.

Impact dynamics and puncture failure of pressurized tank cars with fluid–structure interaction: A multiphase modeling approach

This paper presents a computational framework that analyzes the effect of fluid–structure interaction (FSI) on the impact dynamics and puncture failure of pressurized commodity tank cars carrying hazardous materials. Shell (side) impact tests have been conducted on full scale tank cars resulting in deformed or punctured tank cars. A finite element (FE) modeling method is applied that explicitly simulates the three distinct phases in a tank car loaded with a liquefied substance: pressurized gas, pressurized liquid and solid structure. Furthermore, an equivalent plastic strain based fracture initiation criterion expressed as a function of stress triaxiality is adopted to depict the fracture behavior of the tank car steel material. The fracture initiation is implemented for ductile, shear and mixed fracture modes and followed by further material deterioration governed by a strain softening law. The force, displacement and impact energy results obtained from the FE analysis show good agreement with the corresponding shell impact test data. The simulations demonstrate that FSI plays a critical role in predicting the correct dynamics of tank car impact. The puncture resistance of a tank car, characterized as limit impact conditions in terms of puncture energy or puncture velocity, is further analyzed in shell impact scenarios. The puncture energy is shown to increase as the initial fluid pressure decreases, the tank car thickness increases or the effective impactor size increases. Quantitative correlations between puncture energy/velocity and each of these factors are obtained using the FE analysis method developed in this paper.

A stable X‐FEM in cohesive transition from closed to open crack

SummaryIgnoring crack tip effects, the stability of the X‐FEM discretizations is trivial for open cracks but remains a challenge if we constrain the crack to be closed (i.e., the bi‐material problem). Here, we develop a formulation for general cohesive interactions between crack faces within the X‐FEM framework. The stability of the new formulation is demonstrated for any cohesive crack stiffness (including the closed crack) and illustrated for a nonlinear cohesive softening law. A benchmark of the new model is carried out with simpler approaches for a closed crack (i.e., Lagrange multipliers) and for a cohesive crack (i.e., penalty approach). Due to the analogies between stable cohesive X‐FEM and Nitsche's methods, the new method simplifies the implementation and is attractive in dynamic explicit codes. Copyright © 2014 John Wiley & Sons, Ltd.

Three-dimensional heterogeneous fracture simulation of asphalt mixture under uniaxial tension with cohesive crack model

A three-dimensional (3D) heterogeneous fracture modeling technology is presented to simulate complex crack evolution in quasi-brittle asphalt mixture. In this technology, the random aggregate generation and packing algorithm is employed to create 3D heterogeneous numerical model of asphalt mixture, and the cohesive elements with the tension/shear softening laws are inserted into both the mastic matrix and the aggregate–mastic interfaces as potential cracks. The nucleation and coalescence of microcracks, and inception and propagation of main macrocracks are carefully studied under uniaxial tension and temperature of −10°C. The effects of the averaged coarse aggregate size and the cohesive fracture parameters on performance of asphalt mixture are also evaluated.

Rate-Dependent Embedded Discontinuity Approach Incorporating Heterogeneity for Numerical Modeling of Rock Fracture

In this paper, the embedded discontinuity approach is applied in finite element modeling of rock in compression and tension. For this end, a rate-dependent constitutive model based on (strong) embedded displacement discontinuity model is developed to describe the mode I, mode II and mixed mode fracture of rock. The constitutive model describes the bulk material as linear elastic until reaching the elastic limit. Beyond the elastic limit, the rate-dependent exponential softening law governs the evolution of the displacement jump. Rock heterogeneity is incorporated in the present approach by random description of the mineral texture of rock. Moreover, initial microcrack population always present in natural rocks is accounted for as randomly-oriented embedded discontinuities. In the numerical examples, the model properties are extensively studied in uniaxial compression. The effect of loading rate and confining pressure is also tested in the 2D (plane strain) numerical simulations. These simulations demonstrate that the model captures the salient features of rock in confined compression and uniaxial tension. The developed method has the computational efficiency of continuum plasticity models. However, it also has the advantage, over these models, of accounting for the orientation of introduced microcracks. This feature is crucial with respect to the fracture behavior of rock in compression as shown in this paper.

Buckling and post-buckling of gradient and nonlocal plasticity columns experiencing softening

The buckling and the post-buckling behaviors of a perfect axially loaded column are analytically investigated through a global bilinear moment–curvature elastoplastic constitutive law. Three plasticity cases are studied, namely the linear hardening plasticity law, the perfect elastoplastic case and the softening case. The applications of such a study can be found in various structural engineering problems, including reinforced concrete, steel, timber or composite structures. It is analytically shown that for all kinds of elastoplastic behaviors, the plasticity phenomena lead to a global softening branch in the load–deflection diagram. The propagation of the plasticity zone during the post-buckling process is analytically characterized in case of linear hardening or softening plasticity laws. However, it is shown that the unphysical elastic unloading solution necessarily occurs in presence of local softening moment–curvature constitutive law. A nonlocal plasticity moment–curvature softening law is then used to control the localization branch in the post-buckling stage. This nonlocal plasticity law includes the explicit and the implicit gradient plasticity law. Higher-order plasticity boundary conditions are derived from an extended variational principle. Some parametric studies finally illustrate the main findings of this paper, including the plasticity modulus effect on the post-buckling behavior of these plasticity structural systems.