Singular Strain Fields Research Articles

Piecewise rigid displacement (PRD) method: a limit analysis-based approach to detect mechanisms and internal forces through two dual energy criteria

This paper proposes an extension of the Piecewise Rigid Displacement (PRD) method based on a new dual linear programming problem that minimises the complementary energy. Before, the PRD method had been applied to solve the kinematical problem for masonry-like structures composed of normal, rigid, no-tension (NRNT) material minimising the total potential energy. Specifically, the PRD method frames this minimum-energy search as a linear programming problem whose solutions are displacements and singular strain fields (cracks).Here, we show that the corresponding dual linear programming problem discretises the minimum of the complementary energy and returns, as solutions, admissible internal stress states compatible with the crack pattern obtained by solving the primal problem. Thus, these two minimum-energy criteria are dually connected, and their combined use allows coupling mechanisms and internal forces with settlements or homogeneous boundary displacements. This allows addressing different mechanical problems: equilibrium and stability of the reference configuration, effects of settlements, and mechanisms due to overloading (e.g. horizontal forces). Since the NRNT material represents the extension to continuum media of Heyman's material model, the PRD method offers an extremely fast, limit analysis-based, displacement approach that allows simultaneously finding mechanisms and compatible internal forces for any boundary condition, loads and geometry.

T-strain effects in kinked interfacial fracture of bonded composites

Strain-based criteria are employed to predict fracture initiation behavior in bi-material cracked components. Two versions of the maximum tangential strain criterion are developed theoretically for the case of a crack existing at the interface of two dissimilar isotropic elastic solids to predict kinking angle and the mixed-mode interfacial fracture toughness. The first version (MTSN) is developed based on the singular strain fields while the second version (EMTSN) takes into account the effects of the first non-singular strain term (T-strain) as well as the singular terms. The theoretical effect of T-strain on the calculation of the kinking angle and the mixed-mode interfacial fracture toughness is studied. Then, both MTSN and EMTSN criteria are examined by comparing theoretical predictions to the previously reported experimental results for the kinking angle and mixed-mode interfacial fracture toughness in different bonded composite specimens. It is found that considering T-strain can significantly improve the theoretical predictions of the experimental results provided by the maximum tangential strain criterion.

Stress and Strain of Cracked Orthotropic Sheet

The analytic function of complex variable including a material parameter is analyzed fully. The new stress function has been formed by the complex function method. Typical shear mode II crack states are considered to orthotropic materials. The boundary shear loading problem of cracked orthotropic sheet is studied and the formulae for stress fields are derived. The singular strain fields are also discussed. And the expressions of the singular stress and strain in the crack-tip are determined with inclusion of the material elasticity coefficients.

Mixed mode fracture analysis using extended maximum tangential strain criterion

Abstract The maximum tangential strain (MTSN) criterion is modified in this paper to predict fracture initiation in cracked specimens under mixed mode loading conditions. Theoretical investigations were performed to study the role of both first nonsingular strain term (T-strain) and Poisson's ratio on prediction of the crack propagation direction and fracture toughness. An extended version of the MTSN (EMTSN) criterion is proposed by taking into account the T-strain as well as the singular strain field. The EMTSN criterion is then examined using the experimental data reported for angled cracked plates made from polymethylmethacrylate (PMMA). It is shown that the EMTSN criterion provides more accurate estimation of the experimental results than conventional MTSN criterion.

Stability in the linearized problem of quantitative elastography

The goal of quantitative elastography is to identify biomechanical parameters from interior displacement data, which are provided by other modalities, such as ultrasound or magnetic resonance imaging. In this paper, we analyze the stability of several linearized problems in quantitative elastography. Our method is based on the theory of redundant systems of linear partial differential equations. We analyze the ellipticity properties of the corresponding PDE systems augmented with the interior displacement data; we explicitly characterize the kernel of the forward operators and show injectivity for particular linearizations. Stability criteria can then be deduced. While joint reconstruction of all parameters suffers from non-ellipticity even for more measurements, our results show stability of the separate reconstruction of shear modulus, pressure and density; they indicate that singular strain fields should be avoided, and show how additional measurements can help in ensuring stability of particular linearized problems.

Use of strain gages for determining generalized stress intensity factors of sharp V-notched plates under transverse bending

A strain gage method has been developed for determining the generalized stress intensity factors (GSIFs) of sharp V-notched plates subjected to transverse bending. The study is based on Mindlin’s theory of plate bending. The bisector of the notch opening angle is inclined to the edge of the plate, and the mixed mode loading is created simultaneously. The singular strain fields around the notch tip including the non-singular terms are derived in terms of the GSIFs. These strain fields of dominant terms do not coincide with those of the plane problem. The crack strain fields, however, are coincident with those of the plane problem, if we take a limiting value in the notch strain fields obtained. In order to separate the mixed-mode GSIFs, two strain gages with five element grids are severally positioned along the two lines extending from the bisector of the notch, and a simple procedure to determine the GSIFs for each mode is presented. Experiments on the specimens in strip form with various notch opening angles and inclination angles as well as an inclined crack in rectangular plates are carried out to compare with the results of finite element analyses according to Mindlin’s plate theory. It is shown that the reasonable agreement is obtained between them.

Numerically determined enrichment functions for the extended finite element method and applications to bi‐material anisotropic fracture and polycrystals

AbstractStrain singularities appear in many linear elasticity problems. A very fine mesh has to be used in the vicinity of the singularity in order to obtain acceptable numerical solutions with the finite element method (FEM). Special enrichment functions describing this singular behavior can be used in the extended finite element method (X‐FEM) to circumvent this problem. These functions have to be known in advance, but their analytical form is unknown in many cases. Li et al. described a method to calculate singular strain fields at the tip of a notch numerically. A slight modification of this approach makes it possible to calculate singular fields also in the interior of the structural domain. We will show in numerical experiments that convergence rates can be significantly enhanced by using these approximations in the X‐FEM. The convergence rates have been compared with the ones obtained by the FEM. This was done for a series of problems including a polycrystalline structure. Copyright © 2010 John Wiley & Sons, Ltd.

Hybrid displacement FE formulations including a hole

The paper deals with the problem related to the modelling of riveted assemblies for crashworthiness analysis of full-scale complete aircraft structures. Comparisons between experiments and standard FE computations on high-energy accidental situations onto aluminium riveted panels show that macroscopic plastic strains are not sufficiently localised in the FE shells connected to rivet elements. The main reason is related to the structural embrittlement caused by holes, which are currently not modelled. Consequently, standard displacement FE models do not succeed in initialising and propagating the rupture in sheet metal plates and along rivet rows as observed in the experiments. However, the literature survey show that it is possible to formulate super-elements featuring defects that both give accurate singular strain fields and are compatible with standard displacement finite elements. These super-elements can be related to the displacement model of the hybrid-Trefftz principle of the finite element method, which is a kind of domain decomposition method. A feature of hybrid-Trefftz finite elements is that they are mainly used for elastic computations. It is thus proposed to investigate the possibility of formulating a hybrid displacement finite element, including the effects of a hole, dedicated to crashworthiness analysis of full-scale aeronautic structures.

Finite element methods for the multi-scale modeling of softening hinge lines in plates at failure

This paper presents a multi-scale framework for the modeling of softening hinge lines in inelastic plates and its finite element implementation. The proposed framework considers the Reissner–Mindlin problem of thick plates as the large-scale problem with the hinge introduced locally in small neighborhoods modeling the small scales of the material response. The hinges are defined as strong discontinuities of the generalized displacements describing the plate deformations. The consideration of the large-scale limit of vanishing small scales leads to a large-scale model of these localized failures of plates, involving the solution of a problem in terms of those large-scale generalized displacements with the standard regularity conditions. Enhanced finite elements define the natural framework for the numerical implementation of these considerations. The new finite elements are enhanced with the singular strain fields associated with the discontinuities or hinge lines, which given their local character are statically condensed at the element level. We present a general strategy for the definition of these local strain fields so the kinematics of the hinge lines are captured correctly, allowing in particular their fully softened state with no stresses and hence avoiding the so-called stress locking. We present several particular finite elements formulated in this framework, including triangular and quadrilateral finite elements. Several numerical simulations are presented illustrating their performance.

Effect of External Sources on Finite Time Singularity

A local model of a singular strain field for a finite time singular vortex in both Euler and Navier–Stokes equations is employed to investigate the effect of an external line source of matter. This line source is assumed to be nested at the core of the singularity which can handle a forced singularity with infinite energy. Two different geometries of flows are considered with some solutions for each of them. Some of these solutions are shown to prevent the singularity to collapse on the source which is a new effect only due to the source. A finite area of singularity around the source can be created. Some other solutions of Burgurs type turned out to possess a structure of maxima and minima for the vorticity.

Nonlinear Deformation And Crack Interactions In Semiconductor Packages

In this study, three-dimensional enriched crack tip elements were employed to analyze interracial cracking problems encountered in electronic packaging. These elements are located at the crack tip region and contain the correct 3-D crack tip displacement and singular strain fields with unknown stress intensity factors, Using enriched elements, the elastic-plastic behavior of solder balls and interaction effects with nearby cracks in a flip-chip semiconductor package were investigated. Applying periodicity bounda~ conditions, a slice near the central region of the device was considered, Results from two differently shaped interracial cracks near the fillet region are presented, Including plasticity effects on the solder balls, the interaction between plastic deformation and fracture parameters was investigated (elastic crack tip behavior). The two interracial crack configurations are; a vertical crack between the fillet and the Si-chip, and an “L” shaped crack between the undertlll material and the chip, The results from these calculations showed that plastic deformation, especially on the outermost solder ball, is affected by the presence of a crack in its vicinity and vice versa. It was observed that the plastic strains increase as the crack tip approaches the outermost solder ball. Depending on the type of crack under consideration, the fracture parameters can also vary significantly along the crack front even for the straight through cracks presented in this paper. In studying the “L” shaped crack, it was concluded that inclusion of crack surface contact is critical for accurate determination of plastic strains and fracture parameters.

Asymptotic analysis of the nonlinear Boussinesq problem for a kind of incompressible rubber material (compression case)

A half rubber space compressed by a concentrated force normal to the surface is asymptotically analyzed based on large strain elastic theory. The material domain near the singular point is divided into an expanding domain and a shrinking domain. The asymptotic equations are derived and solved individually for both domains. The solutions to expanding and shrinking domains are further assembled together so that there is only one free parameter left which can indicate the amplitude of the singular stress and strain field. Finally, the amplitude parameter of the field is determined by the given concentrated force.

Application of special finite elements for determination of the parameters of fracture mechanics of nonlinear elastomers with cracks

To simulate singular stress and strain fields in elastomers with cracks, we use spatial square finite elements with intermediate nodes shifted toward the crack front taking into account the weak compressibility of the material. We construct the shape functions reflecting the discontinuity of the distribution of displacements of order\(\sqrt r \). Since, in the region of the crack, the elastomer is subjected to significant deformations, the analysis of the structure is performed according to a nonlinear algorithm based on the modified Newton-Kantorovich method. We present the results of numerical computation of the stress-strain state of rectangular plates with through and surface semielliptical cracks for tensile strains of up to 40%. The obtained solutions are in good agreement with the data available from the literature. The character of distribution of nonlinear stresses enables us to conclude that, in the investigated loading range, the stress intensity factors can be applied for the solution of problems of the fracture mechanics of elastomers.

Notch-tip fields in rubber-like materials under tension and shear mixed load

This paper introduces a new method to solve the singular stress strain field near a notch-tip in rubber-like materials. The constitutive relation used herein is valid even when the strain tends to infinity. The singular field is divided into shrinking sectors and expanding sectors for which the asymptotic equations are derived separately. The solution to the expanding sector is obtained analytically while that for the shrinking sectors are solved numerically. The singular behavior of the notch tip field is revealed clearly.

Finite element analysis of creep fracture initiation in a model superalloy material

Detailed finite element analyses were performed for a single edge-cracked specimen geometry under both plane stress and plane strain constraint for a superalloy material that obeys a power-law creep relationship. The objectives of these analyses were to elucidate the stationary creep crack-tip fields and to provide guidance for the experimental measurement of crack-tip deformations. New results demonstrate that, for both plane stress and plane strain, the angular variations in the creep strain fields do not agree with HRR-type predictions, although the radial variations are in agreement with HRR-type creep strain field predictions in a zone very near the crack tip. Thus, the use of experimental measurement of surface displacement and/or strain data for the location of HRR-type fields may not be possible, unless modifications to the existing HRR-type theory are made. It is also noted that the size of the stress-based HRR-dominance zone is only a fraction of the creep zone size in plane stress, and is very small (especially along θ=0°) compared to the creep zone size in plane strain. Furthermore, the dominance of the singular strain fields are at least two orders of magnitude smaller than the corresponding stress dominance zones. As such, unless the microstructural features of the material are smaller than the dimensions of the dominance zones, the basis for using stress or strain-based fracture parameters derived from the HRR-type fields for prediction of creep fracture initiation is unclear.

The effect of crack closure on the stress intensity factors for two and three dimensional cracked bending problems

ABSTRACT The influence of crack closure on the stress intensity factors of cracked bending plates is investigated by the combination of the hybrid mongrel singular finite elements and the mixed method of contact problems. Embedding the asymptotic singular stress field in the singular element by the concepts of hybrid mongrel displacement model and a singular strain field by the jacobian determinant of the isoparametric element, the stress intensity factors are directly obtained by the stress unknown parameters or indirectly by the nodal displacement values. For the contact problems, the local flexibility matrix involving only the contact nodes is modified at each incremental step instead of the global stiffness matrix. The effects of crack closure on the stress intensity factors are analyzed in two cases. 1) Line contact: employing a two dimensional model for a single edge cracked plate loaded by tensile stress and bending moment. 2) Area contact: employing a three dimensional model for a plate subjected ...

Application of the hybrid mongrel displacement finite method to the computation of stress intensity factors in anisotropic material

A conformable and convergent method is proposed for the calculation of stress intensity factors in anisotropic plates. This approach is an extension to anisotropic materials of the one developed by the authors for 2-D isotropic material which is based on the formulation of the hybrid mongrel displacement finite element concept. Using singular six-node triangular hybrid mongrel isoparametric elements which contain the anisotropic singular stress field and the singular strain field, the stress intensity factors are directly obtained by the stress unknown parameters or indirectly by newly developed formulae. Very good agreement is shown for the crack problem in an orthotropic plate whose solutions are known.

A mode III crack subjected to high strain rates

T he displacement rate is given for a mode III crack growing in a steady state under dynamic conditions. The material is assumed to be visco-plastic and high strain rates are considered. A regular perturbation expansion is performed. First-order perturbing strains are, with one exception, square root singular near the crack-tip. For overstress exponents less, or equal to 2, a reasonably large region around the crack tip, i.e. > 10 −3 of the linear extent of the plastic zone, is dominated by the square root singular strain field. For an overstress exponent of 5/2 the region extends to about 10 −5 of the plastic zone size, which significantly reduces the possibilities of using the strength of the square root singular field as a fracture parameter. For the overstress exponent 3 the region vanishes.

Brittle Fracture Reliability by Probabilistic Finite Elements

The fusion of the probabilistic finite element method (PFEM) and reliability analysis for probabilistic fracture mechanics (PFM) is presented. The PFEM is extended to PFM using an enriched element that has the near crack‐tip singular strain field embedded. Static condensation is used to solve for modes I and II stress intensity factors, and the adjoint approach to PFEM is employed for evaluating the derivatives of the stress intensity factors with respect to the random variables. Statistical moments (e.g., expectation, covariance, and correlation) of stress intensity factors are calculated for uncertainties in load, material properties including fracture toughness, component geometry, and crack geometry (i.e., crack length, orientation, and position). In addition, the first‐order probability of brittle fracture is calculated. In order to calculate the probability of fracture, an optimization procedure is employed to determine the reliability index. The methodology is demonstrated on two mode I fracture examples. The fusion of PFEM and reliability for fracture mechanics is computationally quite efficient and provides a powerful tool for the design engineer.

Some basic issues in dynamic crack growth in elastic-plastic solids

Differential equations for the asymptotic solution of the near-tip fields in the problem of steady rectilinear crack growth in elastic—plastic solids, are examined in a systematic manner. Two classes of elastoplastic materials are considered and the corresponding solutions are obtained and compared. The first class is the usual J 2-plasticity, with a smooth yield surface and plastic strain rate normal to the yield surface and, hence, coaxial with the stress deviator. For the second class of materials we still use the J 2 or the Mises yield surface, but allow for deviation from normality, in order to model the vertex structure of the yield surface common in ductile crystalline solids. It is shown that the basic differential equations which govern the near-tip strain and stress fields are essentially the same whether or not plastic normality is imposed. This fact is discussed and its consequences are examined in some detail. It is shown that the near-field solution when the normality rule is imposed at the outset, is an isolated solution which cannot be obtained as a limiting case of the solutions with the yield surface tangential component of the plastic strain rate tending to zero. These latter solutions involve logarithmically singular strain fields and stress (but not strain) jump discontinuities, whereas the isolated solution does not have any singularities or discontinuities.