A theory for the fracture of thin plates subjected to bending and twisting moments

Stress fields near the tip of a through crack in an elastic plate under bending and twisting moments are reviewed assuming both Kirchhoff and Reissner plate theories. The crack tip displacement and rotation fields based on the Reissner theory are calculated here for the first time. These results are used to calculate the J-integral (energy release rate) for both Kirchhoff and Reissner plate theories. Invoking Simmonds and Duva's [16] result that the value of the J-integral based on either theory is the same for thin plates, a universal relationship between the Kirchhoff theory stress intensity factors and the Reissner theory stress intensity factors is obtained for thin plates. Calculation of Kirchhoff theory stress intensity factors from finite elements based on energy release rate is illustrated. A small scale yielding like model of the crack tip fields is discussed, where the Kirchhoff theory fields are considered to be the far field conditions for the Reissner theory fields. It is proposed that, for thin plates, fracture toughness and crack growth rates be correlated with the Kirchhoff theory stress intensity factors.     

Computation of membrane and bending stress intensity factors for thin,cracked plates

The crack tip stress fields for plate bending and membrane loading problems are reviewed and the four stress intensity factors that determine these fields are defined. These four stress intensity factors arise from use of Kirchhoff plate theory to account for the bending loads and two dimensional plane stress elasticity to account for the membrane loads. The energy release rate is related to the stress intensity factors and to the stress resultants of plate theory. Virtual crack extension, nodal release and modified crack closure integral methods are discussed for computing components of the energy release rate from finite element analyses of cracked plates. Sample computations of stress intensity factors for single and mixed mode cases are presented for a crack in an infinite plate. Sample computations of stress intensity factors for a double edge notched tension-torsion test specimen are given as well.School of Civil and Environmental Engineering, Cornell University     

Benchmark computation and performance evaluation for a rhombic plate bending problem

The paper addresses the question of the accuracy and reliability of the computational analysis of the rhombic plate problem. The Kirchhoff model has a large error (measured in the energy norm) in comparison with the three dimensional solution with a soft simple support; also for very thin plates (t = 1/100a) and all angles (up to α = 90°). The Reissner–Mindlin model gives results 3–5 times better and is less sensitive to the change of the plate angle. The Kirchhoff model is a relatively good approximation of three dimensional setting for a hard simple support. The paradoxical (polygon) behaviour of the simply supported Kirchhoff plate extends to the Reissner–Mindlin model with a hard support. The finite element solution of the Kirchhoff model is addressed in detail. It is shown that higher degree methods are clearly preferable and that the skewness of the elements does not influence essentially the accuracy of the method; the singularity of the solution, which strongly depends on the skewness of the plate, is the primary cause of the deterioration of the performance of the FEM.     

Crack-tip fields in anisotropic shells

Asymptotic crack-tip fields including the effect of transverse shear deformation in an anisotropic shell are presented. The material anisotropy is defined here as a monoclinic material with a plane symmetry at x ₃=0. In general, the shell geometry near the local crack tip region can be considered as a shallow shell. Based on Reissner shallow shell theory, an asymptotic analysis is conducted in this local area. It can be verified that, up to the second order of the crack tip fields in anisotropic shells, the governing equations for bending, transverse shear and membrane deformation are mutually uncoupled. The forms of the solution for the first two terms are identical to those given by respectively the plane stress deformation and the antiplane deformation of anisotropic elasticity. Thus Stroh formalism can be used to characterize the crack tip fields in shells up to the second term and the energy release rate can be expressed in a very compact form in terms of stress intensity factors and Barnett–Lothe tensor L. The first two order terms of the crack-tip stress and displacement fields are derived. Several methods are proposed to determine the stress intensity factors and `T-stresses'. Three numerical examples of two circular cylindrical panels and a circular cylinder under symmetrical loading have demonstrated the validity of the approach.     

Some fundamental solutions for the Kirchhoff,Reissner and Mindlin plates and a unified BEM formulation

We derive analytical solutions for the deflection of thin circular plates, which are loaded by centrally located concentrated bending moments and transverse forces. Green's functions for clamped and simply supported plates are presented. Reduction of these Green's functions leads to the corresponding fundamental solution for the Kirchhoff plate bending model (K problem). This fundamental solution reduces to those obtained through the direct simplification of the fundamental solution for the sixth-order Reissner and Mindlin plate bending models (RM problem). This allows to decompose each fundamental tensor of the problem RM into the sum of the fundamental tensor of the problem K and a correction tensor (Sh problem), which contains the contribution of the shear strains, e.g. U_ij^RM(r)=U_ij^K(r)+U_ij^Sh(r). Within the boundary element analysis this enables the investigation of the contributions of the shear strains to the solutions of Reissner and Mindlin plate bending models, as corrections of the Kirchhoff values, all determined from the same BEM code. This opens up new possibilities for the analysis of plates by the BEM.     

Surface and internal cracks in a residually stressed plate

The problem of a surface or an internal crack in a plate which contains residual stresses is examined. The line spring model, which reduces a three-dimensional elasticity problem into a two-dimensional problem in plate theory, is used to model the crack. The Reissner plate theory, which takes into account transverse shear deformations, is used to model the plate. The formulation is based on Fourier Transforms which lead to a pair of singular integral equations that are solved numerically. The line spring method requires the plane strain solution to both the edge and internally cracked strip with crack surface loads representative of tension, bending, and the given residual stress distribution. For general use, plane strain solutions are presented for polynomial loading through the thickness up to the fifth order. Comparisons are made between the results given by the line spring model for the Reissner plate theory and the finite element method.     

An elastic-plastic finite element analysis of crack tip fields under biaxial loading conditions

A square plate containing a central crack and subjected to biaxial stresses has been studied by a finite element analysis. An elastic analysis shows that the crack opening displacement and stress of separation ahead of the crack tip are not affected by the mode of biaxial loading and therefore the stress intensity factor adequately describes the crack tip states in an elastic continuum.

An elastic-plastic analysis involving more than localized yielding at the crack tip provides different solutions of crack tip stress fields and crack face displacements for the different modes of biaxial loading.

The equi-biaxial loading mode causes the greatest separation stress but the smallest plastic shear ear and crack displacement. The shear loading system induces the maximum size of shear ear and crack displacement but the smallest value of crack tip separation stress.

     

Transverse shear and pressurization effects in the bending of plates with reinforced cracks

The authors develop an eigth-order model for bending of transversally isotropic plates and use integral transforms and a collocation method to form a line-spring model for a cracked plate. The eigth-order model allows satisfaction of the three standard plate bending boundary conditions; the normal moment, twisting moment, and transverse shear force, and an additional shear stress resultant that allows analysis of transverse normal stresses near the crack tip. The line-spring model is used to develop geometry correction factors for bending of finite-thickness plates, accounting for transverse shear deformation and pressurization of the plate near the crack tip. The line-spring model is then applied to the problem of a plate with a reinforced crack, and the results are used to validate an interpolation solution based on an energy method. While not explicitly analysed, the models are applicable to many problems, including bending of bonded repairs, fracture and fatigue of composite and layered materials, surface cracks, crack tip plasticity and crack closure or crack face interaction.     

Mixed mode cracks in Reissner plates

Based on the sixth order Reissner plate theory, the generalized displacement functions for a cracked plate are derived by eigenfunction expansion method. The fractal two-level finite element method is employed to obtain the stress (moment and shear) intensity factors for the center cracked plate subjected to out-of-plane bending and twisting loads. The numerical results from the present method are checked with those available in literature. Highly accurate stress intensity factors are predicted for a wide range of thickness to crack length ratio and a full range of PoissonÆs ratio provided that the radius of fractal mesh to thickness ratio is not less than .     

Determination of K Id at or below NDTT using instrumented drop-weight testing

Two kinds of surface cracked specimens are numerically analyzed by the three dimensional elastic-plastic FEM. Near tip regions are divided into fine elements, and the stress and displacement fields at the crack tip are compared with HRR solutions.At first, surface cracked specimens subjected to bending are analyzed by changing the aspect ratio and depth/thickness ratio. The effect of the loading condition, crack shape and the crack depth on the stress and displacement fields are discussed. Then the pipe with surface crack subjected to bending is analyzed and the availability of the J-integral concept to the LBB analysis is discussed.In every specimen, it is shown that in the regions very near to the crack tip, the displacement field is similar to HRR solutions of plane strain. In the outer regions, however, the stress and displacement fields depend strongly on the shape, thickness, and loading conditions.     

Effect of mode mixity on K-dominance and three dimensionality in cracked plates

The method of Coherent Gradient Sensing (CGS) in transmission, in conjunction with two and three dimensional finite element methods, is used to study the effect of mode mixity on crack tip stress fields. Using a two dimensional finite element analysis the outer bounds of the region of K-dominance were determined. A three dimensional finite analysis was utilized to study the effect of mode mixity on the three dimensional nature of the stress field in the immediate vicinity of the crack tip and to obtain an inner bound of the region of K-dominance. It was noted that increasing mode mixity leads to an increased rotation of the three dimensional zone, keeping its shape and size unchanged. In contrast, the region of K-dominance is seen to dramatically depend on mode mixity, both in shape and size. In addition, an analysis of the CGS interferograms was conducted to obtain an estimate of the regions of K-dominance experimentally. A least squares fit data analysis technique was used to extract fracture parameters, namely the stress intensity factors K _I, K _II and subsequently the crack tip phase angle, . The data points used for the least square fitting were obtained from the determined regions of K-dominance. The same fracture parameters were also evaluated from the finite element analysis, and good agreement was found between experimental measurements and finite element predictions.     

Surface flaws behavior under tension,bending and biaxial cyclic loading

Fatigue surface crack growth and in-plane and out-of-plane constraint effects are studied through experiments and computations for the aluminum alloy D16T. A tension/bending central notched plate and cruciform specimens under different biaxial loadings with external semi-elliptical surface cracks are studied. The variation of the fatigue crack growth rate and surface crack paths is studied under cyclic tension, bending and biaxial tension–compression loading. For the experimental surface crack paths in the tested specimens, the T-stress, out-of-plane T_z factor, local triaxiality parameter h and the governing parameter for the 3D-fields of the stresses and strains at the crack tip in the form of the I_n-integral are calculated as a function of the aspect ratio by finite element analysis to characterize the constraint effects along the semi-elliptical crack front. The plastic stress intensity factor approach is applied to the fatigue crack growth on the free surface, as well as at the deepest point of the semi-elliptical surface crack front, of the tested tension/bending plate and cruciform specimens. From the results, characteristics of the fatigue surface crack growth rate as a function of the loading conditions are established.     

Adaptive finite element analysis of geometrically non-linear plates and shells,especially buckling

Based on both moderate and finite rotation bending theories of thin elastic shells including shear deformation, adaptive non-linear static finite element analysis is treated within a displacement approach and h-adaptivity. The a posteriori error indicator given by Rheinboldt, gained by linearization, is investigated in order to decide whether the deformations influence the indicator explicitely and how parameter dependent problems (like the Reissner–Mindlin model) behave in the process of adaptation. In order to achieve overall consistency, dimensional adaptivity (to 3-D elasticity) is implemented within disturbed subdomains, especially at supports. Results are that Rheinboldt's error indicator is valid under certain restrictions but not directly at bifurcation points and that robustness is not improved by adaptation. Nested quadrilateral finite elements are used for studying pre- and post-buckling states of plates and shells.     

An improved discrete Kirchhoff quadrilateral element based on third‐order zigzag theory for static analysis of composite and sandwich plates

A new improved discrete Kirchhoff quadrilateral element based on the third‐order zigzag theory is developed for the static analysis of composite and sandwich plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid‐surface. The usual requirement of C¹ continuity of interpolation functions of the deflection in the third‐order zigzag theory is circumvented by employing the improved discrete Kirchhoff constraint technique. The element is free from the shear locking. The finite element formulation and the computer program are validated by comparing the results for simply supported plate with the analytical Navier solution of the zigzag theory. Comparison of the present results with those using other available elements based on zigzag theories for composite and sandwich plates establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed by comparing the finite element results of the square all‐round clamped composite plates with the converged three‐dimensional finite element solution obtained using ABAQUS. The comparisons also establish the superiority of the zigzag theory over the smeared third‐order theory having the same number of degrees of freedom. Copyright © 2006 John Wiley & Sons, Ltd.     

On the bending stress distribution at the tip of a stationary crack from Reissner's theory

A general solution is developed for the symmetric bending stress distribution at the tip of a crack in a plate taking shear deformation into account through Reissner's theory. The solution is obtained in terms of polar coordinates at the crack tip and includes the complete class of solutions satisfying all the three boundary conditions along the crack. The solution has arbitrary multiplicative constants and in specific problems, these constants can be determined from conditions on the exterior boundary by well-known numerical techniques such as collocation, successive integration. Results of a numerical solution for a square plate with a central crack subject to uniaxial bending are presented along with a critical discussion of the sensitivity of the numerical solution which is associated with the exponential character of Bessel terms in this higher order analytical solution.

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Résumé On développe une solution générale à la distribution symétrique de contraintes de flexion à l'extrémité d'une entaille dans une plaque soumise à une déformation de cisaillement, en tenant compte de la théorie de Reissner. La solution est obtenue sous forme de coordonnées polaires à l'extrémité de la fissure et comporte l'ensemble des solutions satisfaisant à toutes les conditions de frontières le long de la fissure. La solution comporte des constantes multiplicatives arbitraires et, dans des problèmes spécifiques, ces constantes peuvent être déterminées à partir des conditions de confinement extérieur à l'aide de techniques numériques bien connues telles que la collocation ou l'intégration successive. Les résultats d'une solution numérique dans le cas d'une plaque carrée comportant une fissure centrale soumise à flexion uniaxiale sont présentés en même temps qu'une discussion critique de la sensibilité de la solution numérique associée au caractère exponentiel des termes de Bessel dans cette solution analytique d'une ordre supérieur.

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Nomenclature 2a crack length - h plate thickness - k ² - E Young's Modulus - v Poisson's ratio - D bending rigidity of plate,Eh ³/12(1-v ²) - M ₀ reference bending moment - ₀ extreme fibre stress due toM ₀, 6M ₀/h ² - r, polar coordinates with crack tip as the origin (Fig. 1), r being nondimensionalized with respect toa - W normal displacement, nondimensionalized with respect toM ₀ a ²/D - M _r ,M ,M _r bending and twisting moments per unit length of plate element (Fig. 1), nondimensionalized with respect toM ₀ - _r , transverse shear forces per unit length of plate element (Fig. I), nondimensionalized with respect toM ₀/a - _{r r} stresses on that surface where a positive moment produces tension, nondimensionalized with respect to ₀ - a stress function in Reissner's theory, nondimensionalized with respect toM ₀ - K _(b) bending stress intensity factor - K_(b) Modified Bessel Functions of first and second kinds respectively - (,m) ( + 1)( + 2)...( +m)form5 0 = 1 form = 0 - Gamma function Structures DivisionMaterials Division     

Bending modified J–Q theory and crack-tip constraint quantification

It is well known that the J–Q theory can characterize the crack-tip fields and quantify constraint levels for various geometry and loading configurations in elastic–plastic materials, but it fails at bending-dominant large deformation. This drawback seriously restricts its applications to fracture constraint analysis. A modification of J–Q theory is developed as a three-term solution with an additional term to address the global bending stress to offset this restriction. The nonlinear bending stress is approximately linearized in the region of interest under large-scale yielding (LSY), with the linearization factor determined using a two-point matching method at each loading for a specific cracked geometry in bending. To validate the proposed solution, detailed elastic–plastic finite element analysis (FEA) is conducted under plane strain conditions for three conventional bending specimens with different crack lengths for X80 pipeline steel. These include single edge notched bend (SENB), single edge notched tension (SENT) and compact tension (CT) specimens from small-scale yielding (SSY) to LSY. Results show that the bending modified J–Q solution can well match FEA results of crack-tip stress fields for all bending specimens at all deformation levels from SSY to LSY, with the modified Q being a load- and distance-independent constraint parameter under LSY. Therefore, the modified parameter Q can be effectively used to quantify crack-tip constraint for bending geometries. Its application to fracture constraint analysis is demonstrated by determining constraint corrected J–R curves.     

Study on cracked plates, shells and three dimensional bodies

This paper presents a summary of the authors' recent work in following areas: (1) The stress-strain fields at crack tip in Reissner's plate. (2) The calculations of the stress intensity factors in finite size plates. (3) The stress-strain fields at crack tip in Reissner's shell. (4) The calculations of the stress intensity factors and bulging coefficients in finite size spherical shells. (5) The stress-strain fields along crack tip in three dimensional body with surface crack. (6) The calculation of stress intensity factors in a plate with surface crack.     

基于Kirchhoff板中裂纹尖端辛本征解的有限元应力恢复方法

对于含穿透裂纹的板结构,裂纹尖端应力场及应力强度因子的计算精度对评估板的安全性具有非常重要的影响。基于含裂纹Kirchhoff板弯曲问题中裂纹尖端场的辛本征解析解,该文提出了一个提高裂纹尖端应力场计算精度的有限元应力恢复方法。首先利用常规有限元程序对含裂纹板弯曲问题进行分析,得到裂纹尖端附近的单元节点位移;然后根据节点位移确定辛本征解中的待定系数,得到裂纹尖端附近应力场的显式表达式。数值结果表明,该方法给出的应力分析精度得到较大提高,并具有良好的数值稳定性。     

Automatically inf − sup compliant diamond‐mixed finite elements for Kirchhoff plates

We develop a mixed finite‐element approximation scheme for Kirchhoff plate theory based on the reformulation of Kirchhoff plate theory of Ortiz and Morris [1]. In this reformulation the moment‐equilibrium problem for the rotations is in direct analogy to the problem of incompressible two‐dimensional elasticity. This analogy in turn opens the way for the application of diamond approximation schemes (Hauret et al. [2]) to Kirchhoff plate theory. We show that a special class of meshes derived from an arbitrary triangulation of the domain, the diamond meshes, results in the automatic satisfaction of the corresponding inf ? sup condition for Kirchhoff plate theory. The attendant optimal convergence properties of the diamond approximation scheme are demonstrated by means of the several standard benchmark tests. We also provide a reinterpretation of the diamond approximation scheme for Kirchhoff plate theory within the framework of discrete mechanics. In this interpretation, the discrete moment‐equilibrium problem is formally identical to the classical continuous problem, and the two differ only in the choice of differential structures. It also follows that the satisfaction of the inf ? sup condition is a property of the cohomology of a certain discrete transverse differential complex. This close connection between the classical inf ? sup condition and cohomology evinces the important role that the topology of the discretization plays in determining convergence in mixed problems. Copyright © 2013 John Wiley & Sons, Ltd.