Exponential Softening Law Research Articles

Analysis of 3D quasi-brittle solids failures by crack growth using the strong discontinuity approach with the boundary element method

The continuum strong discontinuity approach (CSDA) is suitable for the analysis of material failures of quasi-brittle solids when constitutive damage models are applied to kinematic equations with displacement discontinuities. Its association with the implicit version of the boundary element method (BEM), using cells with embedded strong discontinuity, allows the analysis of material failures with crack growth. Such cells are mandatory for the domain discretization, which are restricted to the region where the crack is supposed to occur. They are introduced in the model as the analysis progresses, without the need to re-evaluate the pre-existing terms of the matrices, but only through their appropriate expansion. Discontinuity in the displacement field is quantified, after internal equilibrium verification, assuming a uniform displacement jump inside each cell. This work uses this methodology for the analysis of three-dimensional solids under external loads. An elastic-degrading isotropic constitutive model with an exponential softening law is considered, with the strong discontinuity regime imposed directly after the end of the elastic regime. This is the first time that such cells are applied for full three-dimensional mixed mode fracture problems, in the context of the boundary element method. The results presented in the numerical examples are in good agreement with the references.

An improved mesoscale damage model for quasi-brittle fracture analysis of concrete with ordinary state-based peridynamics

Ordinary state-based peridynamics (OSPD) is a promising computational method to analyze damage and fracture process because its equation of motion is still satisfied at the presence of discontinuities. Though OSPD has been introduced to evaluate mesoscopic behavior of concrete, there are many unsolved issues to tackle quasi-brittle behavior of mesoscopic constituents. In this paper, an improved mesoscale damage model for two-dimensional fracture analysis of concrete is proposed. Instead of modeling the interfacial transition zone (ITZ) with material points, the mechanical behavior of ITZ is embedded in the corresponding bonds of mortar and aggregates to improve modeling and computational efficiency. During the failure process, the damage initiation criterion is established based on the stress computed from the scalar force density; while the softening is characterized by the fracture energy and an exponential softening law. The proposed model is verified by simulating cracking of ITZ in uniaxial tension as well as crack propagation of concrete in three-point bending test. The prediction results show good agreements with either analytical or experimental results. Then, the model is employed to study mesoscale crack initiation and propagation of concrete under uniaxial tension, considering the distributions of mortar, aggregates, ITZ, and voids. Influences of the mesoscopic fracture properties on the macro constitutive relationship are also investigated through parametric studies. The improved mesoscale damage model presented is a simple yet effective tool for fracture analysis of concrete.

Calibration of a combined XFEM and mode I cohesive zone model based on DIC measurements of cracks in structural scale wood composites

A combined experimental and numerical approach to characterize a fracture process in wood composites of Norway Spruce is presented. Processed results of crack growth during wedge split tests in TL (tangential - longitudinal), and RL (radial-longitudinal) planes for mode I (opening) are used as a basis for numerical fracture characterization of wood composites. A non-linear plane stress finite element model with extrinsic (monotonically decreasing) traction-separation law for fracture behaviour is used to model the experiment conditions. The data for the numerical model are validated against independent experimental measurements from two sources: displacement fields from DIC (Digital Image Correlation) and force-displacement logging history. The procedure of finding the optimal combination of model parameters resulting in the minimal error between experimental and numerical crack growth history is implemented. The model parameters are initial elastic compliance, damage initiation stress, crack separation at failure, shape coefficient for exponential softening law, and initial crack length for each experiment in both series. The space of the optimal sets of numerical parameters is established to be readily used in engineering applications. The mechanism of fracture process zone development in fibre composite is captured experimentally and numerically.

Long-term deformation and control structure of rheological tunnels based on numerical simulation and on-site monitoring

The time-varying and creep failure of rocks affect the long-term stability of support structures in deep rheological tunnels. Structure instability is the macroscopic result of damage evolution and strength degradation in microstructures. Hence, the calculation model of long-term deformation and failure of soft rock tunnels is established. The model explains the heterogeneity of rocks by a random local stress field, and explains the local degradation of rocks by an exponential softening law. The effect of stress field distribution on failure mode and long-term instability is revealed. The deformation characteristics of different structure sections are analyzed, and the damage evolution process, failure, and instability properties are further clarified in rock tunnels. The results indicate that the coefficient of lateral pressure determines the long-term creep and damage properties of the surrounding rock, and the rocks show a strong time effect from a large to small deformation rate. The stability of curve shapes is better than that of straight ones in deep tunnels. The shoulder and foot of the structure are always damaged firstly, and continue to evolve in an arc until fracture. A mechanical mechanism is applied to the design of a stable structure, and the control principle of soft rocks is proposed: stress–deformation applicability and space–time applicability. The failure mechanism, control principle, and stability structure “yielding bolt + W steel strip + shotcrete” are demonstrated with the soft rock tunnels of cross mining in Baijiao Coal Mine. Finally, results of this study promote the utilization of soft rock tunnels and provide a new control structure for engineering stability in deep-cross mining.

The Time-Dependent Failure Mechanism of Rocks and Associated Application in Slope Engineering: An Explanation Based on Numerical Investigation

In this study, a numerical model for long-term deformation and progressive failure of rock slope is presented. The model accounts for both rock heterogeneity and the initiation, activation, nucleation, and coalescence of cracks in rock slope through a stochastic local stress field and local rock degradation by using an exponential softening law. The time-dependent behaviour of rocks is taken as a macroscopic consequence of damage evolution and strength degradation in microstructure. A series of demonstrative slope cases containing preexisting joints are constructed and investigated. The slope instability occurs at a particular point in time when the rock strength is reduced to a certain value. The temporal and spatial evolution of joint linkage structures is numerically obtained, which clearly shows how the local stress field and damage evolution within the joint network contribute to the fracture pattern and the long-term instability. Then, a practical slope case in jointed and layered rock formations in Yunyang city is studied. The prevailing failure phenomena of the slope, including gradual surface scaling, sliding collapses, and block falling, are numerically reproduced, with an emphasis placed on the slope failure process and development tendency. There is a good agreement on the failure mode and instability time between the numerical simulations and the field observations.

Lattice modelling of gravity and stress-driven failures of rock tunnels

In this paper, an investigation is performed regarding the applicability of the Rigid Body Spring Network method (RBSN) to the analysis of hard rock tunnels in massive and fractured rock masses. For this objective, the method is enhanced with a new meshing scheme and a Cohesive Zone Model (CZM) formulation based on an exponential softening law. While the new meshing scheme provides for models containing complex Discrete Fracture Networks, the CZM affords realistic representations of fracturing phenomena. In order to demonstrate the robustness of the contributions, a verification model and three study cases are presented. The verification concerns the analysis of an elastic jointed medium with a circular hole, from which stresses and displacements are calculated and checked against Finite Element solution. The study cases are based on the real Monte-Seco and Mine-By tunnels and on a hypothetical tunnel. Simulations are performed, and the results are compared to either observed in situ behavior or Finite Element analyses. In the end, the cases show that the enhanced RBSN can realistically represent both gravity and stress-driven types of failures that are encountered in hard rock tunneling. Additional results highlight the method’s simple material calibration procedure and the practicality of the new meshing scheme.

Crack propagation in dynamics by embedded strong discontinuity approach: Enhanced solid versus discrete lattice model

In this work we propose and compare the two models for crack propagation in dynamics. Both models are based on embedded strong discontinuities for localized cohesive type crack description and both provide the advantage to not to require tracking algorithms. The first one is based on discrete lattice approach, where the domain is discretized with Voronoi cells held together prior to crack occurrence by cohesive links represented in terms of Timoshenko beams. The second one is based on constant strain triangular solid element. In both models, propagation of cracks activates enhancements in the displacement field leading to embedded strong discontinuities. The latter remain localized inside the element, regulated by the localized traction separation behavior defined through exponential softening law. Thus, the both models provide the result that remain mesh-independent, with fracture energy as the model parameter. We show that implementation in dynamics framework can be obtained by adding inertial effects without modifying the existing quasi-statics models. In order to provide reliable results, classical implicit Newmark algorithm can be used for time integration. The two presented models are subjected to dynamic crack propagation benchmarks, where detailed analysis on strain, kinetic, plastic free and dissipated energy during simulation is verified by comparison to the amount of total work which is introduced into the system. The main strength of the proposed approach is that cracks initiation, propagation, their coalescence, merging and branching are automatically obtained without any tracking algorithms. In addition, since the discontinuities remain localized inside elements, accurate results can be obtained even with coarser grids, leading to efficient methodology capable of capturing complex crack patterns in dynamics.

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.

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.

Unloading Behavior Study on FRP-Concrete Interface under an Exponential Softening Law

Loading behavior studies on FRP-concrete interface are refined enough. Because of the complexity of the problem, there exist relatively few unloading behavior studies. Occurrence mechanism of unloading needs to be clarified. Through a combination of theory and finite element method, this paper studied the unloading behavior involved in interfacial failure process under an exponential softening law, clarified when and where unloading occurred and estimated the effect of unloading.

Studies on FRP-concrete interface with hardening and softening bond-slip law

This latest experimental study proposes a theory that the bond-slip law for a FRP-concrete interface contains linear hardening and exponential softening. On the basis of this law, the paper studies the mechanic behavior and debonding process of a FRP-concrete interface. Firstly, through nonlinear fracture mechanics, the analytical solutions of the interface shear stress, the axial normal stress of FRP and the load–displacement relationship at the loaded end with the single shear test model of FRP-concrete are acquired. The shear stress propagation as well as the debonding process of the whole interface for different bond lengths could be predicted. Secondly, a simplified interface bond-slip law is used by changing the exponential softening law into a linear softening law. In addition, the analytical solutions for the simplified interface bond-slip law could also be obtained. Finally, based on the analytical solutions of the two bond-slip laws, the influences of the FRP bond length and stiffness on load–displacement curve and the ultimate load, as well as stiffness on effective bond length were discussed, with the similarities and differences between the two bond-slip laws also being studied.

Modelling the FRP-concrete delamination by means of an exponential softening law

Among rehabilitation strategies, bonding of Fibre Reinforced Polymers (FRP) plates is becoming more and more popular, especially for what concerns concrete structures. The performance of the interface between FRP and concrete is one of the key factors affecting the behaviour of the strengthened structure. Up to now, closed-form analytical solutions exist only for the local bond–slip law with linear softening. The aim of the present paper is to show that analytical solutions can be achieved also assuming an exponential decaying softening law. Accordingly, the expressions for the interfacial shear stress distribution and the load–displacement response are derived for the different loading stages. A full parametric analysis of the problem has been performed, highlighting the size effect on the structural behaviour as well as the effects of the bond length, of the FRP stiffness and of the interface cohesive law. A comparison with other analytical models as well as with experimental data available in the literature concludes the paper.

Numerical simulation of quasi-brittle fracture using damaging cohesive surfaces

The cohesive surface methodology is used in a numerical study of fracture of concrete. The traction vs. separation response is governed by an isotropic damage law in which damage evolves according to a prescribed one-dimensional linear or exponential softening law. Cohesive surfaces are immersed in the continuum to allow for a maximum freedom of crack path selection. The single edge notched four point shear beam and the double edge notched tensile bar are used to study: (i) the influence of the tangential cohesive response on the development of the fracture path and (ii) the mesh alignment sensitivity. It is shown that in the present formulation, the tangential cohesive response has a minor influence on both crack path and global characteristics. Mesh alignment does have a significant influence on the outcome of the numerical analysis.