Nonlocal damage modelling using the element-free Galerkin method in the frame of finite strains

Prediction of size effects has been a challenging problem since some experiments found the size effect in material damage. Both material model and numerical algorithm have to be improved to consider the complex damage process. In the present paper we implement element-free Galerkin (EFG) method for a strain-gradient based nonlocal damage model and use it to analyze ductile material damage process. The EFG algorithm overcomes some drawbacks of the FEM in convergence of numerical iteration due to large deformations as well as evaluation of the higher-order gradients of the plastic strain. The numerical benchmarks show that the EFG method for the nonlocal damage model provides more stable numerical results. The size effect in notched specimens can be predicted in the computations. Both ductile fracture in tensile specimens as well as their size effects are investigated and the computational results agree very well with experiments.     

Computational algorithms and applications of element-free Galerkin methods for nonlocal damage models

In using complex material models, especially the strain-gradient-dependent damage models, the convergence of the finite element computation increasingly becomes a problem. Due to large strains in damaging elements the computation may often result in non-convergence. For the higher-order gradient plasticity the special element formulation would often be necessary, which causes additional difficulties in implementation and computations. In recent years, meshless methods have been developed as an alternative to the finite element method (FEM) and can overcome some known shortcomings of FEM. In the present paper an algorithm of element-free Galerkin (EFG) methods for strain-gradient based nonlocal damage models has been developed and used to simulate ductile material damage. The method provides a reliable and robust results for material failure with large damage zones. The strain gradient-dependent terms can be evaluated from the direct differentiation. The investigation confirms that the nonlocal damage model with element-free Galerkin method is suitable for computing the damage problems and predicting the size effects. With the help of the meshless method, material failure in specimens as well as the size effects are predicted accurately.     

Steady-state crack growth and fracture work based on the theory of mechanism-based strain gradient plasticity

Mode I steady-state crack growth is analyzed under plane strain conditions in small scale yielding. The elastic-plastic solid is characterized by the mechanism-based strain gradient (MSG) plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99]. The distributions of the normal separation stress and the effective stress along the plane ahead of the crack tip are computed using a special finite element method based on the steady-state fundamental relations and the MSG flow theory. The results show that during the steady-state crack growth, the normal separation stress on the plane ahead of the crack tip can achieve considerably high value within the MSG strain gradient sensitive zone. The results also show that the crack tip fields are insensitive to the cell size parameter in the MSG theory. Moreover, in the present research, the steady-state fracture toughness is computed by adopting the embedded process zone (EPZ) model. The results display that the steady-state fracture toughness strongly depends on the separation strength parameter of the EPZ model and the length scale parameter in the MSG theory. Furthermore, in order for the results of steady crack growth to be comparable, an approximate relation between the length scale parameters in the MSG theory and in the Fleck-Hutchinson strain gradient plasticity theory is obtained.     

无网格Galerkin法的理论进展及其应用研究

无网格Galerkin(Element-free Galerkin,EFG)法是无网格方法中应用比较广泛的一种,在介绍其基本特点和原理的基础上,对其移动最小二乘近似过程中涉及到的基函数、权函数的选择、影响域半径的确定等方面取得的新进展进行了介绍.并针对本征边界条件的满足,离散和积分方案的实施,自适应分析及误差分析的应用等一系列相关问题的研究现状及取得的成果进行了详细阐述.同时以受均布载荷的悬臂梁为例,编制了EFG平面弹性程序,验证了EFG法的可行性.最后针对EFG法存在的不足,提出了几个研究方向.     

Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method

 An efficient meshfree formulation based on the first-order shear deformation theory (FSDT) is presented for the static analysis of laminated composite beams and plates with integrated piezoelectric layers. This meshfree model is constructed based on the element-free Galerkin (EFG) method. The formulation is derived from the variational principle and the piezoelectric stiffness is taken into account in the model. In numerical test problems, bending control of piezoelectric bimorph beams was shown to have the efficiency and accuracy of the present EFG formulation for this class of problems. It is demonstrated that the different boundary conditions and applied actuate voltages affects the shape control of piezolaminated composite beams. The meshfree model is further extended to study the shape control of piezo-laminated composite plates. From the investigation, it is found that actuator patches bonded on high strain regions are significant in deflection control of laminated composite plates. Received: 23 October 2001 / Accepted: 29 July 2002     

RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates

A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the quasi-three-dimensional (3D) analysis of simply supported, multilayered composite and functionally graded material (FGM) plates. The strong and weak formulations of this 3D static problem are derived on the basis of the Reissner mixed variational theorem (RMVT) where the strong formulation consists of the Euler–Lagrange equations of the problem and its associated boundary conditions, and the weak formulation represents a weighted-residual integral in which the differentiation is equally distributed among the primary field variables and their variations. The early proposed DRK interpolation is used to construct the primary field variables where the Kronecker delta properties are satisfied, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. The system equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation. In the illustrative examples, it is shown that the solutions obtained from these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.     

RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates

A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the quasi-three-dimensional (3D) free vibration analysis of simply supported, multilayered composite and functionally graded material (FGM) plates. Based on the Reissner Mixed Variational Theorem (RMVT), the strong and weak formulations of this problem are derived, in which the material properties of each individual FGM layer, constituting the plate, are assumed to obey the power-law distributions of the volume fractions of the constituents. The system motion equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation, in which the shape functions of the unknown functions satisfy the Kronecker delta properties, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. In the illustrative examples, the natural frequencies and their corresponding modal field variables varying along the thickness coordinate of the plate are studied. It is shown that the solutions obtained using these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.     

A natural neighbour-based moving least-squares approach for the element-free Galerkin method

The element-free Galerkin method (EFG) and the natural element method (NEM) are two well known and widely used meshless methods. Whereas the EFG method can represent moving boundaries like cracks only by modifying the weighting functions the NEM requires an adaptation of the nodal set-up. But on the other hand the NEM is computationally more efficient than EFG. In this paper a new concept for the automatic adjustment of nodal influence domains in the EFG method is presented in order to obtain an efficiency similar to the NEM. This concept is based on the definition of natural neighbours for each meshless node which can be determined from a Voronoi diagram of the nodal set-up. In this approach adapted nodal influence domains are obtained by interpolating the distances to the natural neighbours depending on the direction. In the paper we show that this concept leads, especially for problems with grading node density, to a reduced number of influencing nodes at the interpolation points and consequently a significant reduction of the numerical effort. Copyright © 2006 John Wiley & Sons, Ltd.     

Computational analysis of thin coating layer failure using a cohesive model and gradient plasticity

Thermal barrier coatings (TBC) are widely used to prevent transient high temperature attack and allow components high durability. Due to strong inhomogeneous material properties the TBC failure often initiates near the interface between the brittle oxide layer and the ductile substrate. A reliable prediction of the TBC failure requires detailed information about the crack tip field and the consequent fracture criteria. In the present paper both cohesive model and gradient plasticity are used to simulate the failure process and to study interdependence of the interface stress distribution with the specific fracture energies. Computations confirm that combination of the two models is able to simulate different failure mechanisms in the TBC system. The computational model has the potential to give a realistic prediction of the crack propagation process.     

Temperature dependence of material length scale for strain gradient plasticity and its effect on near‐tip opening displacement

This paper investigates the temperature dependence of the material length scale in the conventional mechanism‐based strain gradient (CMSG) plasticity theory. The work reported here also examines the plastic strain gradient effect on the opening displacement near a sharp crack tip. The study examines the mechanical properties of two typical structural steels (S355 and S690) in onshore and offshore structures at two different temperatures (20 and 300 °C) through both the uniaxial tension test and the indentation test. The CMSG‐based finite element analysis then confirms a constant material length scale for these two steels at the two tested temperatures, despite the apparent temperature dependence of the macroscopic material parameters measured from the tension test. Using the calibrated material length scale, the subsequent numerical study demonstrates that the magnitude of the near‐tip crack opening displacement computed by the CMSG theory remains significantly lower than that computed from the classical plasticity.     

Free vibration analysis of double-bonded isotropic piezoelectric Timoshenko microbeam based on strain gradient and surface stress elasticity theories under initial stress using differential quadrature method

The size-dependent effect on free vibration of double-bonded isotropic piezoelectric Timoshenko microbeams using strain gradient and surface stress elasticity theories under initial stress is presented. This article is developed for isotropic piezoelectric material. Due to the high surface-to-volume ratio, surface stress has an important role with micro- and nanoscale materials. Thus, the Gurtin–Murdoch continuum mechanic approach is used. Governing equations of motion are derived by Hamilton's principle and solved by the differential quadrature method. The effects of pre-stress load, surface residual stress, surface mass density, surface piezoelectrics, Young's modulus of surface layers, three material length scale parameters, thickness to material length scale parameter ratios, various boundary conditions, and two elastic foundation coefficients are investigated. It is concluded that the effect of pre-stress load in greater modes is negligible for higher aspect ratios and this effect is similar to lower aspect ratios. Also, the size-dependent effect on the dimensionless natural frequency for strain gradient theory is higher than that for modified couple stress theory and classical theory, which is due to increasing stiffness of the Timoshenko microbeam model. Moreover, the results show that dimensionless natural frequency affects more by considering the material length scale parameters with respect to surface effect. The results are compared with the obtained results from the literature and show good agreement between them. It is concluded that the amplitude of the transverse displacements (w₀) for a microbeam (MB) is more than the transverse displacements (w₁) for a piezoelectric microbeam (PMB). On the other hand, using a piezoelectric layer for PMB, the amplitude of the transverse displacements (w₁) reduces considerably with respect to MB, in which this effect leads to increase the stiffness of the microbeam and stability of microstructures. With considering the piezoelectric layer, the obtained results can be used to control the amplitude and vibration of microstructures, prevent the resonance phenomenon, design smart structures, and can be employed for micro-electro-mechanical systems and nano-electro-mechanical systems.     

Fracture mechanics analysis using the wavelet Galerkin method and extended finite element method

This paper presents fracture mechanics analysis using the wavelet Galerkin method and extended finite element method. The wavelet Galerkin method is a new methodology to solve partial differential equations where scaling/wavelet functions are used as basis functions. In solid/structural analyses, the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain. To improve accuracy, wavelet functions are superposed on the scaling functions within a region having a high stress concentration, such as near a hole or notch. Thus, the method can be considered a refinement technique in fixed‐grid approaches. However, because the basis functions are assumed to be continuous in applications of the wavelet Galerkin method, there are difficulties in treating displacement discontinuities across the crack surface. In the present research, we introduce enrichment functions in the wavelet Galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method. This paper presents the mathematical formulation and numerical implementation of the proposed technique. As numerical examples, stress intensity factor evaluations and crack propagation analyses for two‐dimensional cracks are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

An element‐free Galerkin method for 3D crack propagation simulation under complicated stress conditions

This study develops an element‐free Galerkin method based on the moving least‐squares approximation to trace three‐dimensional crack propagation under complicated stress conditions. The crack surfaces are modelled by a collection of planar triangles that are added when cracks propagate. The visibility criterion is adopted to treat the screening effect of the cracks on the influenced domain of a Gaussian point. Cracks are assumed to propagate in the perpendicular planes at crack front points when the strain energy release rates reach the material fracture toughness. This method is unique in that it uses a nonlinear contact iterative algorithm to consider contributions of crack surface interaction to the global equilibrium equations, so that crack opening, sliding and closing under complicated stress states can be efficiently modelled. Two numerical examples of three‐dimensional quasi‐static crack propagation were modelled with satisfactory results. Copyright © 2012 John Wiley & Sons, Ltd.     

A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems

This paper presents a novel numerical method for effectively simulating the singular stress field for mode-I fracture problems based on the edge-based smoothed finite element method (ES-FEM). Using the unique feature of the ES-FEM formulation, we need only the assumed displacement values (not the derivatives) on the boundary of the smoothing domains, and hence a new technique to construct singular shape functions is devised for the crack tip elements. Some examples have demonstrated that results of the present singular ES-FEM in terms of strain energy, displacement and J-integral are much more accurate than the finite element method using the same mesh.     

Analytical solution for the singular stress distribution due to V-notch in an orthotropic plate material

On the basis of general solutions of two-dimensional linear elasticity, displacement and singular stress fields near the singular point in orthotropic materials are derived in closed form expressions. According to the presented expressions, analysis formulas of displacement and singular stress fields near the tip of a V-notch under the symmetric and the anti-symmetric modes are obtained subsequently. The open literatures devoted to developing stress singularity near the tip of the V-notch in anisotropic or orthotropic materials. In this study, however, not only direct eigenequations were derived, but also the explicit solutions of displacement and singular stress fields were obtained. At the end, the correctness of the formulas of the singular stress field near the tip of the V-notch has been verified by FEM analysis.     

A finite element analysis of the singular stress fields in anisotropic materials loaded in antiplane shear

A finite element formulation is developed to determine the order and angular variation of singular stress states at material and geometric discontinuities in anisotropic materials subject to antiplane shear loading. The displacement field of the sectorial element is quadratic in the angular co-ordinate direction and asymptotic in the radial direction measured from the singular point. The formulation of Yamada and Okumura¹⁴ for in-plane problems is adapted for this purpose. The simplicity and accuracy of the formulation are demonstrated by comparison to several analytical antiplane shear solutions for both isotropic and anisotropic multi-material wedges and junctions with and without disbonds. The nature and speed of convergence of the eigensolution suggests that the solution presented here could be used in developing enriched elements for accurate and computationally efficient evaluation of stress intensity factors in problems having complex global geometries.     

Fracture analysis in the conventional theory of mechanism-based strain gradient (CMSG) plasticity

In a remarkable series of experiments, Elssner et al. (1994) and Korn et al. (2002) observed cleavage cracking along a bimaterial interface between Nb and sapphire. The stress required for cleavage cracking is around the theoretical strength of the material. Classical plasticity models fall short to reach such a high stress level. We use the conventional theory of mechanism-based strain gradient plasticity (Huang et al., 2004) to investigate the stress field around the tip of an interface crack between Nb and sapphire. The tensile stress at a distance of 0.1 m to the interface crack tip reaches 13.3_Y, where _Y is the yield stress of Nb. This stress is nearly 4 times of that predicted by classical plasticity theory (3.6_Y) at the same distance to the crack tip, and is high enough to trigger cleavage cracking in materials and interfaces. This is consistent with Elssner et al.'s (1994) and Korn et al.'s (2002) experimental observations.