MODIFIED CRACK CLOSURE INTEGRAL (MCCI) FOR 3-d PROBLEMS USING 20-NODED BRICK ELEMENTS

Abstract— The Modified Crack Closure Integral (MCCI) technique based on Irwin's crack closure integral concept is very effective for estimation of strain energy release rates G in individual as well as mixed-mode configurations in linear elastic fracture mechanics problems. In a finite element approach, MCCI can be evaluated in the post-processing stage in terms of nodal forces and displacements near the crack tip. The MCCI expressions are however, element dependent and require a systematic derivation using stress and displacement distributions in the crack tip elements.
Earlier a general procedure was proposed by the present authors for the derivation of MCCI expressions for 3-dimensional (3-d) crack problems modelled with 8-noded brick elements. A concept of sub-area integration was proposed to estimate strain energy release rates at a large number of points along the crack front. In the present paper a similar procedure is adopted for the derivation of MCCI expressions for 3-d cracks modelled with 20-noded brick elements. Numerical results are presented for center crack tension and edge crack shear specimens in thick slabs, showing a comparison between present results and those available in the literature.     

Reduced minimization of product and non-product type in Mindlin plate elements

This paper presents concepts of two-dimensional reduced minimization of product type and non-product type: the analytical reduced minimization theory in one dimension when extended to that of Mindlin plate elements reveals that the two-dimensional reduced minimization for Lagrangian plate elements can be obtained by the successive use of the one-dimensional reduced minimization. But this relation does not hold for serendipity elements due to an incompleteness of displacement functions. This paper examines effects of various integrations of shear strain energy by applying the one-dimensional successive reduced minimization to Lagrangian and serendipity plate elements. The results show why the selective 2×3 and 3×2 rule for the transverse shear strain energies of an 8-noded plate element cannot eliminate locking and why a conventional 12-noded plate element of serendipity family does not have one-order-lower, optimal Gauss strain locations. © 1997 John Wiley & Sons, Ltd.     

Reduced integration and the shear-flexible beam element

A clearer insight into the ‘shear locking’ phenomenon, which appears in the development of C₀ continuous element using shear-flexible or penalty type formulations, is obtained by a careful study of the Timoshenko beam element. When a penalty type argument is used to degenerate thick elements to thin elements, the various approximations of the shear related energy terms act as different types of constraints and, depending on the formulation, two types of constraints which are classified as true or spurious may emerge. The spurious constraints, where they exist, are responsible for the ‘shear locking’ phenomenon, and its manifestation and elimination is demonstrated in a very simple example. The source of difficulty is shown to be the mathematical operations involved in the various shape function definitions and subsequent integration of functionals. It is seen that formulations that ensure only true constraints in the extreme penalty limit cases display far superior performance in the thick element situation as well, and thus guidelines for the development of efficient elements are drawn. A similar type of behaviour is observed in a shallow curved beam element and here ‘inplane locking’ can be eliminated by selective integration to obtain an improved curved beam element. However, ‘inplane locking’ does not cause a spurious constraint as the error quickly vanishes with the reduction of element size for a reasonable radius of curvature conforming with shallow shell theory.     

Field- and edge-consistency synthesis of A 4-noded quadrilateral plate bending element

In this paper, we demonstrate the use of two conceptual principles, the field-consistency requirement and the edge-consistency requirement, as the basis for deriving a 4-noded quadrilateral plate bending element based on Mindlin plate theory using Jacobean transformations only. The derivation is now free of the use of such devices as strain-interpolation points and Hrennik off strain reference lines, etc., which have been the basis for many recent formulations of this element. The shear strain constraints are now consistently defined within the element domain, and ‘tangential’ shear strains are consistently matched at element boundaries so that there is no locking even under extreme distortion—e.g. even when two nodes are collapsed so that the quadrilateral becomes a triangle. Numerical experiments show that this synthesis produces an element that should be identical to other recent formulations of this element based on tensorial transformations or on shear constraint condensation on the edges, but now given a more complete and formal logical basis.     

An isoparametric quadratic thick curved beam element

A three-noded curved beam element with transverse shear deformation, based on independent isoparametric quadratic interpolations, is designed from field-consistency principles. It is shown that a quadratic element that is field-inconsistent in membrane strain suffers from ‘membrane locking’—i.e. an error of the second kind propagates indefinitely as the element length to thickness ratio and/or the element length to radius of curvature ratio increase, in nearly inextensional bending. However, field-inconsistency in shear strain does not lead to ‘shear locking’ but degrades its performance to exactly that of a field-consistent linear element. It is also seen that field-inconsistency leads to severe axial force and shear force oscillations. Error estimates for locking are derived, wherever possible, and confirmed by numerical experiments. The field-consistent element offered here is the most efficient quadratic curved beam element possible.     

Unified and mixed formulation of the 4-node quadrilateral elements by assumed strain method: Application to thermomechanical problems

In this paper, a class of ‘assumed strain’ mixed finite element methods based on the Hu–Washizu variational principle is presented. Special care is taken to avoid hourglass modes and shear locking as well as volumetric locking. An unified framework for the 4-node quadrilateral solid and thermal as well as thermomechanical coupling elements with uniform reduced integration (URI) and selective numerical integration (SI) schemes is developed. The approach is simply implemented by a small change of the standard technique and is applicable to arbitrary non-linear constitutive laws including isotropic and anisotropic material behaviours. The implementation of the proposed SI elements is straightforward, while the development of the proposed URI elements requires ‘anti-hourglass stresses’ which are evaluated by classical constitutive equations. Several numerical examples are given to demonstrate the performance of the suggested formulation, including static/dynamic mechanical problems, heat conduction and thermomechanical problems.     

Field-consistency and violent stress oscillations in the finite element method

The fact that finite element models can give rise to violent stress oscillations and that there are optimal locations where stresses can be correctly sampled in spite of the presence of these violent stress fluctuations has been known for some time. However, it is less well known that these oscillations arise in a specific class of problems—where there are multiple strainfields arising from one or more field-variables and where one or more of these strain-fields must be constrained in particular physical limits. In this paper, we show that unless the interpolations for these constrained strain-fields are ‘field-consistent’, violent oscillations would set in. These oscillations represent spurious self-equilibrating stress-fields generating spurious energy terms that lead to ‘locking’. The field-consistency interpretation offers a conceptual scheme to delineate these problems and an operational procedure called the functional reconstitution technique allows the errors resulting from field-inconsistency to be anticipated a priori. We demonstrate the power of this approach through an interesting example of a multi-strain-field problem—the inextensional/nearly inextensional deformation of a shear flexible curved beam.     

Adding transverse normal stresses to layered shell finite elements for the analysis of composite structures

This work presents a formulation developed to add capabilities for representing the through thickness distribution of the transverse normal stresses, σ_z, in first and higher order shear deformable shell elements within a finite element (FE) scheme. The formulation is developed within a displacement based shear deformation shell theory. Using the differential equilibrium equations for two-dimensional elasticity and the interlayer stress and strain continuity requirements, special treatment is developed for the transverse normal stresses, which are thus represented by a continuous piecewise cubic function. The implementation of this formulation requires only C⁰ continuity of the displacement functions regardless of whether it is added to a first or a higher order shell element. This makes the transverse normal stress treatment applicable to the most popular bilinear isoparametric 4-noded quadrilateral shell elements.

To assess the performance of the present approach it is included in the formulation of a recently developed third order shear deformable shell finite element. The element is added to the element library of the general nonlinear explicit dynamic FE code DYNA3D. Some illustrative problems are solved and results are presented and compared to other theoretical and numerical results.     

A new nine‐node solid‐shell finite element using complete 3D constitutive laws

The solid‐shell element presented in this paper has nine nodes: eight are classically located at the apexes and are fitted with three translational DOFs whereas the ninth is sited at the center and is endowed with only one DOF; a displacement along the ‘thickness’ direction. Indeed, to be used for modeling thin structures under bending effects, this kind of finite element has a favored direction where several integration points are distributed. Besides, there is solely one ‘in‐plane’ quadrature point to avoid locking phenomena and prohibitive CPU costs for large nonlinear computations. Because a reduced integration is not enough to completely prevent transverse shear locking, a shear–strain field is assumed. Compared with the other eight‐node ‘solid‐shell' bricks, the presence of a supplementary node has a main aim: getting a linear normal strain component which, along with a full three‐dimensional constitutive strain–stress behavior, allows to achieve similar results in bending cases as those obtained with the usual plane stress state hypothesis. For that, the ninth node DOF plays the role of an extra parameter essential for a quadratic interpolation of the displacement in the thickness direction. The advantage is that this DOF has a physical meaning and, for instance, a strength equivalent to a normal pressure can be prescribed. With a suitable nodal numbering, the band width is not significantly increased and meshes can easily be generated because the extra nodes are always located at element centers. To emphasize the peculiar features of such an element, a set of examples (linear and nonlinear) is carried out. Numerous comparisons with other elements show pretty good results in bending dominating problems while adding the event of a normal stress component in sheet metal forming simulations with double side contact. Copyright © 2012 John Wiley & Sons, Ltd.     

Field-consistency rules for a three-noded shear flexible beam element under non-uniform isoparametric mapping

Most studies on the three-noded quadratic Timoshenko beam element have been based on a 2-point Gaussian integration rule for the shear energy or on the use of a smoothed substitute shape function or assumed shear strain field to improve its performance. However, if the mid-node of the element is slightly displaced from the mid-point position, the consistency requirements for the constrained shear strain field are more complex. This paper investigates this problem and the findings throw more light on consistency procedures demanded for distorted quadrilateral plate and shell elements. The optimal element which emerges from this study has bending and shear strain fields which have a linear variation over the element under non-uniform mapping.     

New a posteriori error estimator and adaptive mesh refinement strategy for 2-D crack problems

A posteriori error estimation and adaptive refinement technique for fracture analysis of 2-D/3-D crack problems is the state-of-the-art. The objective of the present paper is to propose a new a posteriori error estimator based on strain energy release rate (SERR) or stress intensity factor (SIF) at the crack tip region and to use this along with the stress based error estimator available in the literature for the region away from the crack tip. The proposed a posteriori error estimator is called the K-S error estimator. Further, an adaptive mesh refinement (h-) strategy which can be used with K-S error estimator has been proposed for fracture analysis of 2-D crack problems. The performance of the proposed a posteriori error estimator and the h-adaptive refinement strategy have been demonstrated by employing the 4-noded, 8-noded and 9-noded plane stress finite elements. The proposed error estimator together with the h-adaptive refinement strategy will facilitate automation of fracture analysis process to provide reliable solutions.     

Incompressible and locking-free finite elements from Rayleigh mode vectors: quadratic polynomial displacement fields

Under pure bending, with an arbitrary patch of plane four-node finite elements, the exact analytical algebraic expressions of deformation, strain and stress fields are numerically captured by a computer algebra program for both compressible and incompressible continua. Linear combinations of Rayleigh displacement vectors yield the Ritz test functions. These coupled fields model pure bending of an Euler-Bernoulli beam with appropriate linearly varying axial strains devoid of shear. Such Courant admissible functions allow an undeformed straight side to curve in flexure. Since these displacement vectors satisfy equilibrium conditions, they are necessarily functions of the Poisson’s ratio. Applications in bio-, micro- and nano-mechanics motivated this formulation that blurs the frontier between the finite and the boundary element methods. Exact integration yields the element stiffness matrix of a compressible convex or concave quadrilateral, or a triangular element with a side node. For the generic energy density integral, the paper furnishes an analytical expression that can be incorporated in Fortran or C ⁺⁺. In isochoric plane strain problems, the Rayleigh kinematic mode of dilatation is replaced by a constant element pressure. The equivalent nodal loadings are calculated according to the Ritz variational statement. Subsequently, without assembling the global stiffness matrix, nodal compatibility and equilibrium equations are solved in terms of Rayleigh modal participation factors.     

A Cauchy‐stress based solution for a necking elastic constitutive model under large deformation

A finite element based method for solution of large‐deformation hyperelastic constitutive models is developed, which solves the Cauchy‐stress balance equation using a single rotation of stress from principal directions to a fixed co‐ordinate system. Features of the method include stress computation by central differencing of the hyperelastic energy function, mixed integration‐order incompressibility enforcement, and an iterative solution method that employs notional ‘small strain’ stiffness. The method is applied to an interesting and difficult elastic model that replicates polymer ‘necking’; the method is shown to give good agreement with published results from a well‐established finite element package, and with published experimental results. It is shown that details of the manner in which incompressibility is enforced affects whether key experimental phenomena are clearly resolved. Copyright © 2005 John Wiley & Sons, Ltd.     

An assumed-stress hybrid 4-node shell element with drilling degrees of freedom

An assumed-stress hybrid/mixed 4-node quadrilateral shell element is introduced that alleviates most of the deficiencies associated with such elements. The formulation of the element is based on the assumed-stress hybrid/mixed method using the Hellinger-Reissner variational principle. The membrane part of the element has 12 degrees of freedom including rotational or ‘drilling’ degrees of freedom at the nodes. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner-Mindlin plate theory which takes into account the transverse shear contributions. The element formulation is derived from an 8-node isoparametric element by expressing the midside displacement degrees of freedom in terms of displacement and rotational degrees of freedom at corner nodes. The element passes the patch test, is nearly insensitive to mesh distortion, does not ‘lock’, possesses the desirable invariance properties, has no hidden spurious modes, and for the majority of test cases used in this paper produces more accurate results than the other elements employed herein for comparison.     

A new 3-D finite element model to evaluate added mass for analysis of fluid-structure interaction problems

The paper presents the results of investigations conducted to evaluate the added mass to represent fluid-structure interaction effects in vibration/dynamic analysis of floating bodies such as ship hulls. While the structural plating is idealized by 9-noded plate/shell finite elements, the fluid domain is modelled by 20-noded/21-noded 3-D finite elements in the investigations conducted. A new 8-noded element has been developed to model the interface between the structure and the fluid. An efficient computational methodology has been used for computation of added mass. The finite element models are validated by comparing the results with those given by analytical solution for a submerged sphere. The efficacy of the finite element model is demonstrated through convergence of the results obtained for a floating barge problem. A better convergence rate and distribution of added mass in three orthogonal directions have been obtained.     

Quadratic NMS Mindlin‐plate‐bending element

The development of a robust and efficient quadratic Mindlin‐plate‐bending elements mainly by the use of non‐conforming displacement modes is presented in this paper. A brief review on the previous efforts to develop efficient non‐conforming Mindlin plate bending elements is also given. The behaviour of the newly proposed plate element is further improved by the combined use of nonconforming displacement modes, the selectively reduced integration scheme, and the assumed shear strain fields. Thus, the newly developed element has been designated as ‘NMS‐8P’. The improvement achieved may be attributable to the fact that the merits of these improvement techniques are merged in the formation of the new element in a complementary manner. The proposed 8‐node element passes the patch tests, does not show spurious mechanism, and does not produce shear locking phenomena even with distorted meshes. It is also shown that the element produces reliable solutions through numerical tests for standard benchmark problems. Copypright © 1999 John Wiley & Sons, Ltd.     

Qualitative errors in laminated composite plate models

The presence of ‘qualitative’ errors, i.e. errors that misrepresent the sign or direction of deformation in a problem, is identified in laminated composite plate elements. The errors are found to be due to the presence of parasitic shear terms introduced when an element is formed using incompatible polynomials. Although these modelling errors can be removed from four-node elements with reduced order Gauss quadrature integration, they cannot be removed from eight-node elements by this method. However, procedures based on a physically interpretable notation are available to remove these errors from individual elements with any number of nodes. The procedures for correcting the laminated composite plate elements can be used to remove parasitic shear and spurious zero energy modes from other types of elements. Three examples are used to demonstrate the developments.     

The curved beam/deep arch/finite ring element revisited

In this paper, an attempt is made to understand the errors arising in curved finite elements which undergo both flexural and membrane deformations. It is shown that with elements of finite size (i.e. a practical level of discretization at which reasonably accurate results can be expected), there can be errors of a special nature that arise because the membrane strain fields are not consistently interpolated with terms from the two independent field functions that characterize such a problem. These lead to errors, described here as of the ‘second kind’ and a physical phenomenon called ‘membrane locking’. The findings here emerge from recent research on the effect of reduced integration on shallow curved beam elements and on the use of coupled displacement fields in finite rings. The failures which have occurred in earlier attempts to use independent polynomial displacement fields for curved elements may not have been due to neglect of rigid body motions or failure to achieve constant strain states, but because of locking due to spurious constraints. These emerge in the penalty limits of extreme thinness (an inextensional regime), when exact integration of the energy functional of an element based on low order independent interpolations for the in-plane and normal displacements is used. It seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam/deep arch/finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints. The much maligned ‘cubic in w–lincar in u’ curved beam element is now reworked to show its excellent behaviour in all situations. What is emphasized is that the choice of shape functions, or subsequent operations to determine the discretized functionals, must consistently model the physical requirements the problem imposes on the field variables. In this manner, we can restore an old element to respectability and thereby indicate clearly the underlying principles. These are: the importance of ‘field consistency’ so that arch and shell problems can be modelled consistently by independent polynomial displacement fields, and the role that reduced integration or some equivalent construction can play to achieve this.