Implementation of a cohesive zone model for the investigation of the dynamic behavior of a rotating shaft with a transverse crack

The influence of a transverse crack on the vibration of a rotating shaft has been at the focus of attention of many researchers. The knowledge of the dynamic behavior of cracked shaft has helped in predicting the presence of a crack in a rotor. Here, the changing stiffness of the cracked shaft is investigated based on a cohesive zone model. This model is developed for mode-I plane strain and accounts for triaxiality of the stress state explicitly by using basic elastic-plastic constitutive relations. Then, the proposed numerical solution is compared to the switching crack model, which is based on linear elastic fracture mechanics. The cohesive zone model is implemented in finite element techniques to predict and to analyse the dynamic behavior of cracked rotor system. Timoshenko beam theory is used to model the discrete shaft under the effect of gravity, unbalance force and gyroscopic effect. The analysis includes the cohesive function for describing the breathing crack and the reduction of the second moment of area of the element at the location of the crack. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cohesive zone model for a transverse breathing crack in a rotor

The breathing mechanism of a transversely cracked rotor and its influence on a rotor system that appears due to the shaft weight is studied. This breathing mechanism is based on experimental and simulation result for the crack shape reported in the literature. If the crack depth is small, the crack closure line is a straight line while for larger crack depths the crack closure becomes more curved. For both cases, a method is proposed for the evaluation of the stiffness losses in the cross section that contains the crack. This method is based on a cohesive zone model (CZM) instead of linear elastic fracture mechanics (LEFM) approach, because LEFM is valid only for the fully open crack and cannot be extended to other intermediate situations. As the crack is closed, the stress intensity factor (SIF) will not appear at the boundary between the closed cracked areas and the open cracked areas. The CZM is developed for mode-I plane strain conditions and accounts explicitly for triaxiality of the stress state by using constitutive relations. The proposed model gives more realistic results than models based on LEFM for the stiffness losses of the crack rotor system for a wide range of the crack depth. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A thermo-mechanical energetic cohesive zone model accounting for Kapitza interfaces

The objective of this contribution is to study computational aspects of modeling thermo-mechanical solids containing mechanically energetic, geometrically non-coherent Kapitza interfaces under cyclic loading. The interface is termed energetic in the sense that it possesses its own energy, entropy, constitutive relations and dissipation. To date, classical thermo-mechanical cohesive zone models do not account for elastic interfaces. Therefore we propose a novel interface model that couples the classical cohesive zone formulation to the interface elasticity theory under the Kapitza assumption within a thermo-mechanical framework. In other words, such an interface model allows for discontinuities in geometry, temperature and normal stress fields, while not permitting a jump in the normal heat flux across the interface. The equations governing a fully non-linear transient problem are given. In particular, a comparison is made between the results of the classical thermo-mechanical cohesive zone model and our novel (cohesive + energetic Kapitza) interface formulation. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of crack growth using cohesive crack method

We present an effective cohesive discrete crack method in the context of the Reproducing Kernel Particle Method (RKPM) in order to study fracture of concrete structures. The discrete crack approach is based on the visibility method and a simple node splitting scheme. We also present an effective implementation of the visibility method and an iteration free algorithm by including the cohesive force term directly into the stiffness equations. The crack is represented by straight-line segments and the cohesive zone model is employed to model the post-localization behavior of concrete. The method is applied to several examples involving mode I and mixed-mode fracture. These results are compared to experimental data and show good agreement.     

A finite element method for cohesive crack modelling

The present contribution is concerned with the computational modelling of cohesive cracks, whereby the discontinuity is not limited to interelement boundaries, but is allowed to propagate freely through the elements. Inelastic material behaviour is described by a discrete constitutive law, formulated in terms of tractions and displacements at the surface. Details on the implementation and numerical examples are given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A generalized cohesive element technique for arbitrary crack motion

A computational method for arbitrary crack motion through a finite element mesh, termed as the generalized cohesive element technique, is presented. In this method, an element with an internal discontinuity is replaced by two superimposed elements with a combination of original and imaginary nodes. Conventional cohesive zone modeling, limited to crack propagation along the edges of the elements, is extended to incorporate the intra-element mixed-mode crack propagation. Proposed numerical technique has been shown to be quite accurate, robust and mesh insensitive provided the cohesive zone ahead of the crack tip is resolved adequately. A series of numerical examples is presented to demonstrate the validity and applicability of the proposed method.     

Analysis of concrete fracture using a novel cohesive crack method

Numerical analysis of fracture in concrete is studied with a simplified discrete crack method. The discrete crack method is a meshless method in which the crack is modeled by discrete cohesive crack segments passing through the nodes. The cohesive crack segments govern the non-linear response of concrete in tension softening and introduce anisotropy in the material model. The advantage of the presented discrete crack method over other discrete crack method is its simplicity and applicability to many cracks. In contrast to most other discrete crack methods, no representation of the crack surface is needed. On the other hand, the accuracy of discrete crack methods is maintained. This is demonstrated through several examples.     

Thermodynamically and variationally consistent cohesive zone models for large deformation

The thermodynamical and variational consistency of cohesive zone models is analyzed in the present contribution. Based on a naive application of the classical Coleman & Noll procedure, it is shown that the second law of thermodynamics is not fulfilled in general. This can even be seen, in case of hyperelastic interfaces. For guaranteeing thermomechanical consistency, additional surface stresses acting at the interface have to be introduced. Based on such findings, a thermomechanically consistent model including dissipative effects is proposed. This model possesses a natural variational structure. More precisely, all state variables can naturally and jointly be computed by minimizing an incrementally defined potential. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of a cohesive zone element for the prediction of damage in nano/micro coating

Many tools in production technology are nowadays coated to obtain a satisfactory lifetime and degradation resistance. Therefore, the main goal of this study is to investigate antiadhesive and wear resistant coatings made of ceramics, plastics and metals produced by High Power Pulsed Magnetron Sputtering (HPPMS) technique [1]. A cohesive zone element technique (CZ) is applied to model the interactions of the coatings and the substrate surfaces (see [2]). This goes along with the investigations of the delamination and failure behavior of the involved surfaces. To illustrate the applicability of the model, several structural simulations are performed. The developed CZ element model is capable of modeling the separation, the contact and also the irreversible reloading conditions in both normal and tangential directions [3]. The model is further developed to be applicable for different structures including different bonding behaviors, with a higher stability. The talk concludes with a detailed discussion of the numerical results of different material and interface properties. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eigenvalue analysis for a crack in a power-law material

A nonlinear eigenvalue problem related to determining the stress and strain fields near the tip of a transverse crack in a power-law material is studied. The eigenvalues are found by a perturbation method based on representations of an eigenvalue, the corresponding eigenfunction, and the material nonlinearity parameter in the form of series expansions in powers of a small parameter equal to the difference between the eigenvalues in the linear and nonlinear problems. The resulting eigenvalues are compared with the accurate numerical solution of the nonlinear eigenvalue problem.     

Investigation of the local behavior of different cohesive zone models

Cohesive interface elements are well suited for three-dimensional crack propagation analyses as long as the crack path is known. This is the case e.g. in delamination analyses of laminated composite structures or failure analyses of adhesively bonded joints. Actually, they are widely used in such applications for both brittle and ductile systems. As long as the strength and fracture toughness of the material are accurately captured it is generally accepted that the shape of the cohesive law has little to no influence on the mechanical behavior of the investigated structures. However, when having a look on the local behavior of different cohesive zone models, such as stress distribution in the fracture process zone, the results exhibit certain differences. These will be studied in the present contribution. Especially the local stress distribution will be investigated and the effect on the computational efficiency will be pointed out. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类具有保护区域的Leslie-Gower捕食-食饵模型的分歧分析

研究了一类Neumann边界条件下带有保护区域的Leslie-Gower捕食-食饵模型,分析稳态系统从半平凡解处发生分歧的条件,得到了分歧方向及分歧值的唯一性,得到了在确定参数范围内,从半平凡解出发的分支解曲线的稳定性.     

Gas-powder flow in blast furnace with different shapes of cohesive zone

With high PCI rate operations, a large quantity of unburned coal/char fines will flow together with the gas into the blast furnace. Under some operating conditions, the holdup of fines results in deterioration of furnace permeability and lower production efficiency. Therefore, it is important to understand the behaviour of powder (unburnt coal/char) inside the blast furnace when operating with different cohesive zone (CZ) shapes. This work is mainly concerned with the effect of cohesive zone shape on the powder flow and accumulation in a blast furnace. A model is presented which is capable of simulating a clear and stable accumulation region in the lower central region of the furnace. The results indicate that powder is likely to accumulate at the lower part of W-shaped CZs and the upper part of V- and inverse V-shaped CZs. For the same CZ shape, a thick cohesive layer can result in a large pressure drop while the resistance of narrow cohesive layers to gas-powder flow is found to be relatively small. Implications of the findings to blast furnace operation are also discussed.     

Prediction of fracture in grain boundaries of nano-coatings using cohesive zone elements

A cohesive zone element technique (CZ) is applied to study grain boundary fracture in nano coating layers (see [1]). This goes along with the investigations of the delamination and fracture behavior of the coatings and the substrate interface. The main motivation is to investigate antiadhesive and wear resistant properties of coatings made of ceramics produced by the High Power Pulsed Magnetron Sputtering (HPPMS) technique [2]. Different physical conditions in HPPMS result into different grain morphologies with different mechanical properties. Therefore prediction of fracture and damage in such systems can lead to the optimum choice of process parameters in order to gain the best fracture resistance properties for the coatings. To illustrate the applicability of the model, several simulations with different mechanical and structural properties are performed. The developed CZ element model is capable of modeling the separation, the contact and also the irreversible reloading conditions in different directions [3]. The model is further developed to be applicable for geometrically complex interfaces including different bonding behaviors, with a high robustness. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The end zone of a shear crack propagating at an intersonic velocity

An asymptotic solution of the problem for a shear crack propagating at an intersonic velocity is given that determines the size of the end zone, the distribution of the displacement jump in it and the dependence of the propagation velocity on an effective stress intensity factor, introduced in this paper. Numerical data are presented for the case of linear softening in the end zone and a comparison is made with the results for the Leonov–Panasyuk–Dugdale model. It is established using an effective stress intensity factor that the propagation is unstable at velocities close to the velocity of transverse waves; it becomes stable when approaching the velocity of longitudinal waves.     

An experimental-analytical hybrid model for the elasto-plastic stresses at a crack

The primary obstacle preventing the analytical determination of physically sensible stresses at a crack tip is the presence of a mathematical singularity there. This singularity is best known in its elastic form; however it persists even in elasto-plastic crack-tip stresses. To overcome the difficulty we adopt the following strategy: we attempt to capture initial elastic stresses experimentally, than track subsequent elasto-plastic stress distributions analytically.We infer a finite stress at a crack tip from the experimental behaviour of cracked specimens at fracture when the specimens are made of a truly brittle material. Given a size-independent result, we argue that the crack-tip stress at fracture must equal the ultimate stress for such a material; thus dividing by the applied stress at the same point gives a measure of the stress concentration factor, K_T. The approach is checked for size independence and against hole configurations with known theoretical, yet physically reasonable, K_T. Then the effective experimental KT are taken as inputs for the second phase of the study in which we model the crack as being a smooth notch having the same stress concentration factor as found experimentally. In this way our configuration initially shares the same stresses at the crack tip as we inferred physically. Next we track effects of incremental plastic flow on a set of finite element grids. Satisfactory resolution in return for modest computational effort is obtained by employing a substructuring method. The accuracy in both the elastic and the elasto-plastic regime is checked against trial problems with exact solutions. Thereafter, physically interpretable stress distributions ahead of the crack are determined for a range of materials and for varying load levels.     

Gradient bounds for minimizers of free discontinuity problems related to cohesive zone models in fracture mechanics

In this note we consider a free discontinuity problem for a scalar function, whose energy depends also on the size of the jump. We prove that the gradient of every smooth local minimizer never exceeds a constant, determined only by the data of the problem.     

On the dynamic model and motion planning for a spherical rolling robot actuated by orthogonal internal rotors

The paper deals with the dynamics of a spherical rolling robot actuated by internal rotors that are placed on orthogonal axes. The driving principle for such a robot exploits nonholonomic constraints to propel the rolling carrier. A full mathematical model as well as its reduced version are derived, and the inverse dynamics are addressed. It is shown that if the rotors are mounted on three orthogonal axes, any feasible kinematic trajectory of the rolling robot is dynamically realizable. For the case of only two rotors the conditions of controllability and dynamic realizability are established. It is shown that in moving the robot by tracing straight lines and circles in the contact plane the dynamically realizable trajectories are not represented by the circles on the sphere, which is a feature of the kinematic model of pure rolling. The implication of this fact to motion planning is explored under a case study. It is shown there that in maneuvering the robot by tracing circles on the sphere the dynamically realizable trajectories are essentially different from those resulted from kinematic models. The dynamic motion planning problem is then formulated in the optimal control settings, and properties of the optimal trajectories are illustrated under simulation.