发展基于CB壳单元的扩展有限元模拟三维任意扩展裂纹

该文提出了一种新的基于连续体壳单元的扩展有限元格式,以用于对曲面上任意形状裂纹的扩展问题进行模拟。扩充形函数的构造和应力强度因子的计算都是基于三维实体单元进行,因此可以模拟复杂的三维断裂情况,壳体厚度的变化也可以得到考虑。三维应力强度因子的计算公式被引入到这种方法中。为模拟裂纹扩展,三维最大能量释放率准则被用作裂纹扩展准则。计算结果显示了曲面上的裂纹扩展路径可以与网格无关,并且由于在裂纹尖端的单元设置了具有奇异性的形函数,裂尖应力场被精确捕捉,从而证明了这种方法的优越性。     

Fracture in magnetoelectroelastic materials using the extended finite element method

Static fracture analyses in two‐dimensional linear magnetoelectroelastic (MEE) solids is studied by means of the extended finite element method (X‐FEM). In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and the crack‐tip asymptotic functions to the standard finite element approximation using the framework of partition of unity. In this study, media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. New enrichment functions for cracks in transversely isotropic MEE materials are derived, and the computation of fracture parameters using the domain form of the contour interaction integral is presented. The convergence rates in energy for topological and geometric enrichments are studied. Excellent accuracy of the proposed formulation is demonstrated on benchmark crack problems through comparisons with both analytical solutions and numerical results obtained by the dual boundary element method. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress-intensity factors for a cruciform crack with equal arm in an infinite elastic strip of finite thickness

The stress-intensity factors and the crack energy for a cruciform crack with equal arm, in an infinite elastic strip of finite thickness have been determined by solving the corresponding boundary value problem by using the Fourier Transform technique. The dual integral equations generated by using the necessary boundary conditions have been reduced to solving a pair of integral equations of Fredholm type involving two unknown functions. The integral equations have been solved by a numerical technique and the formulae for determining the quantities of physical interest have been expressed in terms of the solutions of the integral equations. Numerical results for these physical quantities have been furnished at the end for different values of the thickness of the strip.     

Strain energy release rate determination of prescribed cracks in adhesively-bonded single-lap composite joints with thick bondlines

An analytical model for determining the strain energy release rate due to a prescribed crack in an adhesively-bonded, single-lap composite joint with thick bondlines and subjected to axial tension is presented. An existing analytical model for determining the adhesive stresses within the joint is used as the foundation for the strain energy release rate calculation. In the stress model, the governing equations of displacements within the adherends are formulated using the first-order laminated plate theory. In order to simulate the thick bondlines, the field equations of the adhesive are formulated using the linear elastic theory to allow non-uniform stress distributions through the thickness. Based on the adhesive stress distributions, the equivalent crack tip forces are obtained and the strain energy release rate due to the crack extension is determined by using the virtual crack closure technique (VCCT). The specimen geometry of ASTM D3165 standard test is followed in the derivation. The system of second-order differential equations is solved to provide the adherend and adhesive stresses using the symbolic computational tool, Maple 7. Finite element analyses using J-integral as well as VCCT are performed to verify the developed analytical model. Finite element analyses are conducted using the commercial finite element analysis software ABAQUS™. The strain energy release rates determined using the analytical method correlate well with the results from the finite element analyses. It can be seen that the same prescribed crack has a higher strain energy release rate for the joints with thicker bondlines. This explains the reason that joints with thick bondlines tend to have a lower load carrying capacity.     

Calculation of the elastic strain energy of inclusions by the Green's function and finite element methods

Elastic strain energy of a free-standing body with many inclusions was calculated. In order to consider the effect of surrounding neighborhood inclusions, nonlocal terms of the Green's function were kept. The calculated strain energy by a derived analytic method was compared with that of the finite element method. The prediction of the new analytic method provided a correct tendency of the variation of the strain energy for various sample problems. Deviation of the analytic result from that of the finite element calculation became smaller when the value of the strain energy decreased. As the analytic method does not require solving the field equations, it can be useful for a situation demanding prompt computation of the strain energy.     

Modeling arbitrary crack propagation in coupled shell/solid structures with X‐FEM

In this paper, the extended finite element method (X‐FEM) formulation for the modeling of arbitrary crack propagation in coupled shell/solid structures is developed based on the large deformation continuum‐based (CB) shell theory. The main features of the new method are as follows: (1) different kinematic equations are derived for different fibers in CB shell elements, including the fibers enriched by shifted jump function or crack tip functions and the fibers cut into two segments by the crack surface or connecting with solid elements. So the crack tip can locate inside the element, and the crack surface is not necessarily perpendicular to the middle surface. (2) The enhanced CB shell element is developed to realize the seamless transition of crack propagation between shell and solid structures. (3) A revised interaction integral is used to calculate the stress intensity factor (SIF) for shells, which avoids that the auxiliary fields for cracks in Mindlin–Reissner plates cannot satisfy exactly the equilibrium equations. Several numerical examples, including the calculation of SIF for the cracked plate under uniform bending and crack propagation between solid and shell structures are presented to demonstrate the performance of the developed method. Copyright © 2015 John Wiley & Sons, Ltd.     

基于虚拟裂纹闭合技术的应变能释放率分析

基于虚拟裂纹闭合技术(VCCT),建立了复合材料层合板层间裂纹尖端的应变能释放率(SERR)三维有限元计算模型。该模型考虑了裂纹尖端大转动和离散单元形状变化对应变能释放率计算的影响,修正了裂纹尖端应变能释放率的计算方法。利用该模型计算了裂纹长度为15 mm和35 mm时纯Ⅰ型和纯Ⅱ型的应变能释放率,纯Ⅰ型应变能释放率分别为 207 J/m2和 253 J/m2;纯Ⅱ型应变能释放率分别为 758 J / m 2和 1040 J / m2;计算值与试验值吻合得很好。同时,该模型计算了混合型不同比值 R=(G_Ⅱ/G_Ⅰ+G_Ⅱ)的长裂纹层合板层间断裂过程的应变能释放率,其中Ⅰ型和Ⅱ型应变能释放率计算值与试验平均值的最大误差为 11.4%,最小误差为 0.4%。该模型能有效计算裂纹尖端的应变能释放率。     

A coupled molecular dynamics and extended finite element method for dynamic crack propagation

A multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain. After recalling the basic equations of molecular dynamics and continuum mechanics, the discretization is discussed for the continuum subdomain where the partition‐of‐unity property of finite element shape functions is used, since in this fashion the crack in the wake of its tip is naturally modelled as a traction‐free discontinuity. Next, the zonal coupling method between the atomistic and continuum models is recapitulated. Finally, it is discussed how the stress has been computed in the atomic subdomain, and a two‐dimensional computation is presented of dynamic fracture using the coupled model. The result shows multiple branching, which is reminiscent of recent results from simulations on dynamic fracture using cohesive‐zone models. Copyright © 2009 John Wiley & Sons, Ltd.     

Stress–strain state of a piecewise homogeneous ferromagnetic body with an interfacial penny-shaped crack

This paper considers the stress–strain state of a body consisting of two bonded uniform ferromagnetic half-spaces, with an interfacial penny-shaped crack. The body is assumed to be acted on by a uniform magnetic field in a direction normal to the crack and interface. The problem is reduced to the solution of singular integral equations with respect to two unknown functions on a finite interval. These functions are defined analytically, allowing us to obtain formulae for the magnetoelastic stresses and coefficient of stress intensity. From these formulae, the influence of various physico-mechanical parameters and the magnetic field on the stress strain state of the body in the vicinity of crack is studied.     

Hybrid displacement function element method: a simple hybrid‐Trefftz stress element method for analysis of Mindlin–Reissner plate

In order to develop robust finite element models for analysis of thin and moderately thick plates, a simple hybrid displacement function element method is presented. First, the variational functional of complementary energy for Mindlin–Reissner plates is modified to be expressed by a displacement function F, which can be used to derive displacement components satisfying all governing equations. Second, the assumed element resultant force fields, which can satisfy all related governing equations, are derived from the fundamental analytical solutions of F. Third, the displacements and shear strains along each element boundary are determined by the locking‐free formulae based on the Timoshenko's beam theory. Finally, by applying the principle of minimum complementary energy, the element stiffness matrix related to the conventional nodal displacement DOFs is obtained. Because the trial functions of the domain stress approximations a priori satisfy governing equations, this method is consistent with the hybrid‐Trefftz stress element method. As an example, a 4‐node, 12‐DOF quadrilateral plate bending element, HDF‐P4‐11 β, is formulated. Numerical benchmark examples have proved that the new model possesses excellent precision. It is also a shape‐free element that performs very well even when a severely distorted mesh containing concave quadrilateral and degenerated triangular elements is employed. Copyright © 2014 John Wiley & Sons, Ltd.     

XFEM simulation of stable crack growth using J–R curve under finite strain plasticity

In the present work, extended finite element method (XFEM) has been extended to simulate stable crack growth problems using J–R criterion under finite strain plasticity. In XFEM, a physical representation of crack is not required, and a crack is completely modeled by enrichment functions. The modeling of large deformation is performed using updated Lagrangian approach. The nonlinear equations obtained as a result of large deformation are solved by Newton–Raphson iterative method. Von-Mises yield criterion is used with isotropic hardening to model the finite strain plasticity. The elastic-predictor and plastic-corrector algorithm is employed for stress computation. Three problems i.e. crack growth in compact tension specimen; crack growth in triple point bend specimen and crack growth in bi-metallic triple point bend specimen are solved using J–R curve under plane stress condition to demonstrate the capability of XFEM in crack growth problems.     

Creep fracture parameter C* solutions for semi-elliptical surface cracks in plates under tensile and bending loads

In order to predict and assess creep life for plate structures with semi-elliptic surface cracks under high temperature condition, the accurate calculation of the creep fracture mechanics parameter C* is a critical step. In this paper, the effects of crack sizes, plate geometries, and material creep properties on the parameter C* have been investigated under tensile and bending loads by extensive finite element analyses. Based on the results, the creep influence functions Hc for calculating C* values were obtained and fitted into equations for surface cracks in plates under both loads. The equations have been verified by finite element calculations. The C* solutions were obtained through these equations which are suitable for wide ranges of crack sizes, plate geometries, and materials.     

Coupling finite elements and magnetic networks in magnetostatics

In this paper a new method for computing non-linear magnetostatic fields is introduced, which allows the simultaneous coupling of a finite element structure with a magnetic network. Combining the advantages of both methods while avoiding their drawbacks, this coupling yields both an accurate and time-efficient computation. The traditional method of the unknown mesh fluxes is applied for the solution of the magnetic network. The finite element solution, on the other hand, is based on a classical first-order interpolation of the unknown vector potential. The coupling is established by a proper organization of the unknowns on the boundary common to the finite element and network regions. In this way, a single system of non-linear equations is obtained. Moreover, it is shown that the coupled system of equations is equivalent to a single finite element system if generalized base functions are allowed. Consequently, various results from finite element theory may be applied. For instance, the matrix governing the iteratively linearized system of equations can directly be shown symmetrical and positive definite. Finally, the field inside a permanent magnet motor is calculated with the coupled method. Although the number of unknowns is dramatically reduced compared to a full FE calculation, the same level of accuracy is achieved. Hereby, the benefit of the coupled method is clearly proved.     

Physical shape functions in finite element analysis of moderately thick plates

A new rectangular finite element for moderately thick plates is presented. The element is based on the concept of physical shape functions, i.e. functions which contain in themselves physical and geometrical properties of the element. The analytic formulae of stiffness, geometric stiffness and mass matrices are presented for isotropic material. The element is free from locking and zero energy modes not corresponding to rigid body motions. Several examples are presented for static, initial stability and free vibration problems.     

Derivation of geometrically nonlinear finite shell elements via tensor notation

To develop geometrically nonlinear, doubly curved finite shell elements the basic equations of nonlinear shell theories have to be transferred into the finite element model. As these equations in general are written in tensor notation, their implementation into the finite element matrix formulation requires considerable effort. The present paper will demonstrate how to derive the nonlinear element matrices directly from the incrementally formulated nonlinear shell equations using a tensor-oriented procedure. This enables the numerical realization of all structural responses, e.g. the calculation of pre- and post-buckling branches in snap-through analysis and especially in bifurcation analysis, including the detection of critical points and the consideration of geometric imperfections. To avoid loss of accuracy care is taken for a realistic computation of the geometric properties as well as of the external loads. Finally, the developed family of shell elements will be presented and its efficiency will be demonstrated by some applications to linear and geometrically nonlinear structural phenomena.     

Evaluation of energy release rate for a planar crack in heterogeneous media

A modified version of domain integral method is developed for evaluation of energy release rate with finite element solutions for problems with a 2-D crack located in a heterogeneous elastic field. The heterogeneous field considered in this work generally contains various materials, with discontinuous mechanical moduli across the interfaces. The formulation is proved to be patch-independent, in a generalized sense, and valid for problems under both small and large deformations. The results of calculation appear to be very insensitive to the crack tip finite element models when the tip is away from the material interface. However, strong dependency on the local modeling is observed in case the tip is located at the interface. Alternative studies on this particular case are thus required.     

Extended finite element method and fast marching method for three-dimensional fatigue crack propagation

A numerical technique for planar three-dimensional fatigue crack growth simulations is proposed. The new technique couples the extended finite element method (X-FEM) to the fast marching method (FMM). In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. Stress intensity factors for planar three-dimensional cracks are computed, and fatigue crack growth simulations for planar cracks are presented. Good agreement between the numerical results and theory is realized.     

Simulation of crack growth in composite material shell structures

This paper implements a domain integral energy method for modelling crack growth in composite material shell structures using the finite element method. Volume integral expressions to evaluate the dynamic energy release rate in a through‐thickness three‐dimensional crack are derived. Using the domain integral, the energy release rate computation is implemented in the DYNA3D explicit non‐linear dynamic finite element analysis program wherein crack propagation is modelled by releasing the constraints between initially constrained node pairs. The implementation enables the program to either determine the energy resistance response for the material (provided experimental data is available) or predict the rate of crack propagation in shell structures. The numerical implementation was verified by simulating mode I and mode III slow crack growth problems in semi‐infinite transversely isotropic media, for which analytic solutions are available. Oscillations of energy following the release of nodal constraints as the crack propagates in discrete increments were suppressed using light mass proportional damping and a moving averaging scheme. Copyright © 2004 John Wiley & Sons, Ltd.     

混凝土夹芯板滑移与变形的理论计算及分析

对均布荷载作用下简支混凝土夹芯板的滑移及其对混凝土夹芯板变形挠度影响的理论计算进行了研究和分析,推导出了简支混凝土夹芯板的滑移和变形挠度的理论计算公式,该公式既能描述混凝土夹芯板的滑移规律,也能体现滑移对混凝土夹芯板变形挠度的影响。同时推导出了开裂弯矩的计算公式,确定了滑移和变形挠度计算公式的适用范围。通过与试验和有限元分析结果的比较表明:理论公式计算的结果与试验和有限元分析结果较吻合,可以在实际工程中用来计算混凝土夹芯板的滑移和弯曲变形。     

Stable 3D XFEM/vector level sets for non‐planar 3D crack propagation and comparison of enrichment schemes

We present a three‐dimensional vector level set method coupled to a recently developed stable extended finite element method (XFEM). We further investigate a new enrichment approach for XFEM adopting discontinuous linear enrichment functions in place of the asymptotic near‐tip functions. Through the vector level set method, level set values for propagating cracks are obtained via simple geometrical operations, eliminating the need for solution of differential evolution equations. The first XFEM variant ensures optimal convergence rates by means of geometrical enrichment, ie, the use of enriched elements in a fixed volume around the crack front, without giving rise to conditioning problems. The linear enrichment approach, significantly simplifies implementation and reduces the computational cost associated with numerical integration, while providing nonoptimal convergence rates similar to standard finite elements. The 2 dicretization schemes are tested for different benchmark problems, and their combination to the vector level set method is verified for nonplanar crack propagation problems.