Line-spring Model Research Articles

Line-spring finite element for fully plastic crack growth—II. Surface-cracked plates and pipes

Crack-tip opening angle (CTOA) is generally considered as the most operable fracture descriptor for fully plastic, quasi-steady, extensive crack extension. In Part I, based on the CTOA crack growth criterion, a line-spring finite element model was presented to resolve through-thickness crack growth in plane strain single-edge-cracked specimens and surface-cracked plate/shell structures under fully plastic loading. With constraint-dependent CTOA and some supplemental kinematic relations given, the line-spring model ultimately monitors crack extension from the history of generalized displacements. The model was implemented in the implicit ABAQUS finite element code (1993b) in a user-defined element form. Following the plane strain parametric studies in Part I, we further apply the line-spring model to the problems of surface-cracked plates and pipes in Part II here. Effects of material hardening, configuration and location of the surface crack on the histories of penetration-displacement/pressure are examined in a remotely-stretched plate and a pressurized cylindrical vessel. As the most plausible exercise of a stable crack propagation leading to leak-before-break failure, a circumferentially cracked pipe subject to pure bending is selected. Evolution of CTOA along the crack-front and surface crack enlargement pattern are examined for each case. Experimentally-observed CTOA values for the remotely-stretched plate are interpreted in light of the model prediction.

Size Effect in Penetration of Sea Ice Plate with Part-Through Cracks. I: Theory

The paper analyzes the vertical penetration of a small object through a floating sea ice plate. The analysis takes into account the fact that the bending cracks reach only through part of the ice plate thickness and have a variable depth profile. The cracks are modeled according to the Rice-Levy nonlinear softening line spring model. The plate-crack interaction is characterized in terms of the compliance functions for the bending moments and normal forces in the crack plane, which are computed by an energy-based variational finite-difference method. The radial crack is divided into vertical strips, and a numerical algorithm with step-by-step loading is developed to calculate the vertical growth of the crack in each strip for a prescribed radial crack length increment. The initiation of crack strips from the surface of the plate is decided on the basis of a yield strength criterion with a fracture based flow rule. Systems of up to 300 nonlinear equations are solved by the Levenberg-Marquardt optimization algorithm. The maximum load is reached when the circumferential cracks begin to form. Numerical calculations, comparison of the results with test data, and a study of scaling laws are relegated to the companion paper, which follows in this issue. Numerical calculations show a typical quasi brittle size effect such that the plot of log σN versus log h (where σN= nominal stress at maximum load and h= plate thickness) is a descending curve whose slope is negligible only for h< 0.2 m and then gets gradually steeper, asymptotically approaching −1/2. The calculated size effect agrees with the existing test data, and contradicts previous plasticity solutions.

Line-spring finite element for fully plastic crack growth—I. Formulation and one-dimensional results

A line-spring finite element model is developed to resolve fully plastic, quasi-steady, through-thickness crack growth in plane strain single-edge-cracked (SEC) specimens and surface-cracked plate/shell structures. The plane strain sliding-off and cracking model of McClintock et al. (1995) is adopted to obtain the instantaneous crack-tip opening angle (CTOA) in terms of material parameters and the instantaneous slip-line angle and stress triaxiality at the crack-tip. The slip-line angle and crack-tip stress triaxiality are calculated approximately using the least upper bound method of Kim et al. (1996a). Utilizing these approaches, the generalized forces transmitted by the line-spring finite element provide the constraint-dependent CTOA. The increment of crack extension is then determined from the kinematic relation with incremental crack-tip opening displacement through CTOA. Detailed description of the model, as incorporated into the ABAQUS (1993) finite element code in the form of a user-defined element, is given in Part I. Parametric studies in plane strain SEC specimens are carried out to examine the effects of loading type, ductility and strain hardening on plane strain crack extension behavior, and the effects of elastic surroundings on structural stability. Applications of the line-spring model to problems of surface-cracked plate and pipe are presented in Part II (Lee and Parks, 1998).

Fracture analyses of surface-cracked pipes and elbows using the line-spring/shell model

The line-spring/shell model proposed by Rice and Levy (1972) and further modified by Parks and co-workers (1981, 1982, 1995) provides an attractive and inexpensive alternative to analyzing surface-cracked structures. This model idealizes the complex three-dimensional surface crack problem in a plate- or shell-like structure as a more tractable two-dimensional problem. Though used in some previous studies, this model has largely been ignored in the study of surface-cracked nuclear piping components. While it is true that the line-spring method has some restrictions, it can provide reasonably accurate predictions of important fracture parameters of cracked components within the domain of its applicability. Through some detailed studies of surface-cracked pipes and elbows, the predictive nature of the line-spring model is demonstrated by comparing its predictions with that of a three-dimensional finite element model as well as with experimental measurements. Excellent comparisons between the predictions of the line-spring/shell and three-dimensional finite element models are shown not only using the three-dimensional finite element calculations performed by the author, but also using the three-dimensional calculations performed by two other independent investigators. It is hoped that the results shown in this paper invalidate the criticisms and reestablish the usefulness and reasonable accuracy of the line-spring mode, within its domain of applicability, in the analyses of surface-cracked structures.

Vibration analysis of edge-cracked beams using a line-spring model

The vibration characteristics of a uniform Bernoulli-Euler beam with a single edge crack are investigated using a modified line-spring model. The governing matrix equation for free vibrations of the cracked beam is constructed from the basic standard beam elements combined with the modified line-spring model. The natural frequencies and the corresponding mode shapes are determined for uniform beams having edge cracks of different depths at different positions. The numerical results for the natural frequencies of cracked hinged-hinged and cantilever beam are shown to be in excellent agreement with available numerical and experimental data for crack depth ratios a/ W of less than about 0.5. It is also shown that the abrupt changes in the higher mode shapes of vibration for the cracked beam make it possible to predict and monitor the crack location in the beam.

An approximate method of response analysis of vibrations for cracked beams

In this paper a method based on a line-spring model was proposed to analyze approximately vibration responses of cracked beams. The method in conjunction with the Euler-Bernoulli beam theory, modal analysis and fracture mechanics principle was applied to derive a characteristic equation for the cracked beam vibration. As application examples, natural frequency responses for a cracked hinged-hinged beam and a cracked cantilever beam were examined. It was shown that the present solutions obtained are quite in agreement with the solutions or experimental results in available references.

Ligament-collapse load of plates and cylinders with an axial semi-elliptical flaw

The loads for collapse of plates with semi-elliptical surface flaws have been evaluated by a weight-function method incorporating stress intensity factors using results from a line-spring model. A simplified equation has been developed for the collapse load, and incorporated in a modified strip yield model including toughness dependence, for estimating failure stresses of cylinders with external axial surface defects. The predicted results are in good agreement with the experimental ones.

The perturbed stress state of a closed cylindrical shell with longitudinal cracks on its inner surface

Using the Rice-Levy line spring model we find an approximate solution of the problem of elastic balance of a closed infinite circular cylindrical shell weakened by a periodic system of longitudinal interior cracks of identical length. The stress intensity factors are found at the center of a crack for various numbers of cracks with various parameters.

Dynamic stress intensity factors analysis of interface crack using line-spring model

In this study, a new method for calculating the dynamic stress intensity factors of a bimaterial bending specimen with an interface crack is proposed by making use of a line-spring model. A pre-cracked bending specimen is modeled by one-dimensional beam finite elements and a line-spring representing the stiffness or compliance of a cracked part. The proposed method enables the one-dimensional analysis of a two-dimensional crack problem; thus the time variations of the dynamic stress intensity factors of a bimaterial bending specimen with an interface crack can be obtained by making use of a personal computer within a few minutes. The results obtained by the proposed method agree reasonably well with those obtained by the two-dimensional finite element method, although a slight difference in period can be found. The proposed method enables rapid evaluation of dynamic stress intensity factors. So a rapid evaluation system of the dynamic fracture toughness of a bimaterial with an interface crack can be achieved by combining an instrumented impact test apparatus with a computer program based on the proposed method which runs on a personal computer.

A combination of the weight function method and the line spring model: A surface-cracked cylindrical shell subjected to stress gradients

In this paper, the problem of a part-through crack with arbitrary loads on the crack faces is considered. By introducing the solution of the weight function for a two-dimensional edge crack, combined with the line spring model, the extra loads on crack faces can be represented as two equivalent quantities in terms of two-dimensional stress intensity factors. The problem for a cylindrical shell containing an axial inner surface crack subjected to stress gradients can then be reduced to two integral equations. The model is used to compute the stress intensity factors for four stress distributions—uniform, linear, quadratic and cubic—and the results give very good agreement with those obtained by the finite element method. The numerical results of the calculations for axial inner surface crack initiation in the shell subjected to thermal shock are also presented and discussed.

Surface cracks in toroidal shells

Toroidal shells containing a surface crack and subjected to various symmetric and antisymmetric loading conditions are considered. A singular integral formulation of a through crack located along a principal line of curvature of a shell is used along with the line spring model to treat the surface crack problem. Transverse shear deformations are taken into account in the shell theory. Locating the crack along a line or principal curvature uncouples the symmetric mode I deformations from the antisymmetric mode II and III deformations, which are inherently coupled. Mode I loadings considered include membrane loading and bending, while the mode II and III cases include in plane shear, twisting and transverse shear. In addition to these loadings, residual stresses resulting in mode I deformations are considered. Stress intensity factors for the various loadings are given for shell geometries representative of spherical caps, cylindrical pipes and pipe elbows for a number of crack locations.

Enhanced elastic-plastic line-spring finite element

The elastic-plastic line-spring finite element of Parks and White (1982), as implemented in the ABAQUS finite element code (1988b), is modified in several ways for a better evaluation of crack-tip deformation parameters such as K1, J, crack-tip opening displacement and crack-tip stress triaxiality, the last of which is inferred or measured by the so-called T-stress and Q-stress parameters. An effective crack length formulation is introduced to capture the nonlinearity due to contained plane strain crack-tip plastic deformation, the size and orientation of which are known to be sensitive to the sign and magnitude (relative to yield strength) of the T-stress. More accurate yield surfaces, obtained from finite element limit analyses of plane strain single-edge cracked specimens, are employed to more sharply define the behavior of the line-spring model in the fully plastic region. More accurate relationships between incremental plastic crack-tip opening displacement and incremental load point displacement/rotation are also incorporated into the model for shallow crack configurations under fully plastic loading. The modified line-spring model is then applied to surface cracked plate problems having a wide range of crack-tip constraints, and the feasibility of implementing J-T(Q) based two-parameter characterizations of elastic-plastic crack front fields via simplified line-spring elements is examined.

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.

Dynamic Stress Intensity Factor Analysis of Interface Crack Using Line-Spring Model.

In this study, a method for calculating the dynamic stress intensity factor of a bimaterial bending specimen with an interface crack is proposed for the first time by making use of a line-spring model. A precracked bending specimen is modeled by one-dimensional beam finite elements and a line-spring representing the stiffness or compliance of a cracked part. The present method enables the one-dimensional analysis of a two -dimensional crack problem ; thus the time variation of dynamic stress intensity factors of a bimaterial bending specimen with an interface crack can be obtained by making use of a personal computer within a few minutes. The results obtained from the present method agree reasonably well with those obtained from the two-dimensional finite element method, although a slight difference in period can be found. The present method enables rapid evaluation of dynamic stress intensity factors. Thus a rapid evaluation system of dynamic fracture toughness of a bimaterial with an interface crack can be achieved by combining an impact test apparatus with a computer program based on the present method which runs on a personal computer.

Evaluation of the J-integral for surface cracks using the line-spring model

The line-spring model for the analysis of surface cracks is discussed in this paper. The complementary energy method [J. Press. Vess. Technol.108, 305 (1986)] is extended to derive the fully plastic solutions of a single edge cracked plate in combinations of tension and bending. A simple method is proposed to specify the undetermined factors in complementary energy by making use of Shih's [J. appl. Mech.51, 48 (1984)] finite element results. The obtained fully plastic solutions are in a quasi-analytic form, and suitable for arbitrary combinations of tension and bending. Finally, the present line-spring model is incorporated into a plate-shell finite element code, and the semi-elliptical surface cracked plates in tension are analyzed. The estimated stress intensity factors and J-integral are in keeping with those of three-dimensional finite element analysis.

On large scale sliding in fiber-reinforced composites

A critical e xamination is made of the use of the line-spring model to represent fiber bridging of matrix cracks in the analysis of failure phenomena in fiber-reinforced brittle matrix composites. Attention is focused on composite systems designed to undergo fiber debonding and sliding when matrix cracking occurs. For most composites of this class, it is found that the distance along the fiber within which sliding occurs is often too large to justify use of the line-spring representation. A model which allows for large scale sliding (the LSS model) is proposed and applied to three problems : a matrix crack emerging from a semi-infinite unbridged through-crack in a uni-directional fiber-reinforced composite, the same problem for the finite length unbridged through-crack, and matrix cracking of a cross-ply composite. Primary emphasis is placed on the stress concentration in the bridging fibers. Predictions from the LSS model are compared with those from the line-spring model. In general, the line-spring model is found to overestimate the stress concentration in critically located fibers. A discussion of the significance of the lower estimates of the stress concentration factors is given for several composite systems.

Crack surface contact of surface and internal cracks in a plate with residual stresses

The problem of a mode I surface or internal crack in a plate with residual stresses is examined. The emphasis in this study is in accounting for crack surface contact. The importance of such problems is realized in both crack detection and in the stress intensity factor calculations. The line spring model is used to iteratively determine the border of the closed portion of the crack and the stress intensity factors along the open portion. It is demonstrated that the model is very versatile for solving contact problems. Examples for both surface and internal cracks are given. It is shown that if crack contact is ignored, the stress intensity factor may be significantly underestimated.

A new method of analyzing surface cracks

The authors have developed a new line-spring boundary element method in the present paper, which combines the advantage of the line-spring model with that of the boundary element method. This method reduces the three-dimension problem of the surface cracks into a quasi-one-dimension problem and can be used to analyze the surface cracked plate under various loading conditions. In this paper theoretical analyses and numerical verifications are carried out. The calculated results are reported, which indicate that the present method is efficient and can be used to analyze the surface crack problem on a personal computer.

Analysis of surface cracks using the line-spring boundary element method and the virtual crack extension technique

The authors have developed a new line-spring boundary element method which couples the line-spring model with the boundary element method to deal with the problem of a surface cracked plate. However, the drawback of the line-spring model is that a reliable stress intensity factor could not be directly obtained near the free surface intersection. Therefore, the virtual crack extension technique is employed in a post-processor of the line-spring boundary element method to obtain the stress intensity factor at the crack front-free surface intersection. Theoretical analysis is described. Stress intensity factors for surface cracks are calculated to verify the proposed method. The interaction of two surface cracks is also investigated. The solutions obtained by the line-spring boundary element method show that the method proposed is efficient and reasonably accurate.

Fracture mechanics of orthotropic laminated plates—III. The surface crack problem

In this paper the line spring model originally developed to treat the surface crack problems in homogeneous plates is extended to laminates which may consist of bonded isotropic or orthotropic dissimilar elastic layers. By using the two fundamental solutions for a laminate with a through crack and the corresponding plane strain problem for an edge crack, the surface crack problem in laminates is reduced to a system of singular integral equations. Sample results showing the stress intensity factors are given for bonded orthotropic and isotropic layers which correspond to fiber-reinforced structural composites and ceramic-coated metal substrates, respectively.