Mixed finite element formulation for the general anti-plane shear problem,including mode III crack computations,in the framework of dipolar linear gradient elasticity

A mixed formulation is developed and numerically validated for the general 2D anti-plane shear problem in micro-structured solids governed by dipolar strain gradient elasticity. The current mixed formulation employs the form II statement of the gradient elasticity theory and uses the double stress components and the displacement field as main variables. High order, C ⁰-continuous, conforming basis functions are employed in the finite element approximations (p-version). The results for the mode III crack problem reveal that, with proper mesh refinement at the areas of high solution gradients, the current approximation method captures the exact solution behaviour at different length scales, which depend on the size of material micro-structure. The latter is of vital importance because, near the crack tip, the nature of the exact solution, changes radically as we proceed from the macro- to micro-scale.     

Loss of specular reflection due to nonlinear crack-face interaction

The interaction between rough crack faces is modeled by nonlinear relations between the crack-face tractions and the crack-opening displacements. These relations account for crack closure and for the related resistance to crack-face sliding. The relations are used to investigate reflection and transmission of an incident pulse by an infinite flaw plane. The problem statement is reduced to a set of inhomogeneous nonlinear ordinary differential equations for the displacement discontinuities, [u] and [v], across the flaw plane. These equations have been solved numerically. The reflection and transmission of an incident pulse by a crack with interacting crack faces. Both incident longitudinal and transverse waves have been considered. The loss of specular reflection as compared to a perfect (traction-free) crack is exhibited by specific examples.     

Three dimensional hybrid‐Trefftz stress finite elements for plates and shells

Three‐dimensional hybrid‐Trefftz stress finite elements for plates and shells are proposed. Two independent fields are approximated: stresses within the element and displacement on their boundary. The required stress field derived from the Papkovitch‐Neuber solution of the Navier equation, which a priori satisfies the Trefftz constraint, is generated using homogeneous harmonic polynomials. Restriction on the polynomial degree in the coordinate measured along the thickness direction is imposed to reduce the number of independent terms. Explicit expressions of the corresponding independent polynomials are listed up to the fifth order. Illustrative applications to evaluate displacements and stresses are conducted by hexahedral hybrid‐Trefftz stress element models. The hierarchical p‐ and h‐refinement strategy are exploited in the numerical tests.     

Some results on crack propagation in media with spatially varying elastic moduli

Energy release rates are calculated for cracks propagating in media with spatially varying elastic moduli, this variation being in a direction perpendicular to the crack growth direction. Results are given for transient problems of semi-infinite cracks in infinite media for certain special forms of the variation in shear modulus. Steady state crack propagation in a displacement loaded strip is also considered and it is shown by the use of a certain path independent integral that a simple formula for the energy release rate can be obtained for quite general variations in clastic moduli provided these variations are in a direction perpendicular to the crack. Other path independent integrals are derived which may be of use for transient crack tip stress analysis in strips in a similar way to that used in [4] for the homogeneous elastic problem.     

Coupled Thermoelasticity in a Composite Half-Space

The problem of fully coupled thermoelasticity in a composite half-space is considered where the composite has variations in its physical properties in one direction only. The resulting one-dimensional problem thus depends on the so-called microscale of the composite. Homogenization of the fully coupled theory provides the leading-order system of coupled equations (independent of the microscale) together with the effective physical properties of the thermoelastic medium. In particular, the effective coupling parameter δ_* is found and it is shown to exhibit rather interesting properties; for a range of volume fractions in two-phase composites it is shown that δ_* lies below the corresponding coupling parameter for a homogeneous material made up of either phase. Transient boundary-value problems of the homogenized system are then solved and compared with the classical problem of a homogeneous half-space. The magnitude of resulting discontinuities in field variables and their derivatives are found and their dependence on the effective coupling parameter is exhibited.     

Conference diary

A variational higher-order theory for bending and stretching of linearly elastic orthotropic beams including the deformations due to transverse shearing and stretching of the transverse normal fibre is presented. The theory assumes a linear distribution for the longitudinal displacement and a parabolic variation of the transverse displacement across the thickness. Additionally, independent expansions are introduced for the through-thickness displacement gradients with the requirement of a least-square compatibility for the transverse strains and the satisfaction of exact stress boundary conditions at the top/bottom beam surfaces. The theory is shown to be well suited for finite element development requiring simple C⁰- and C^?1- continuous displacement interpolation fields. To demonstrate the computational utility of the theory, a simple two-node stretching-bending finite element is formulated. The analytic and finite element results are obtained for a simple bending problem for which an exact elasticity solution is available. It is shown that the inclusion of the transverse normal deformation in the present theory enables improved displacement, strain and stress predictions, particularly, in the analysis of deep beams.     

The displacement field due to an interface crack along an elastic inclusion in a differing elastic matrix

Summary A two dimensional mathematical model of an interface crack which lies along an elastic inclusion embedded in an elastic matrix with different elastic constants is considered. In contrast to a previous study by Toya, which determined only the displacement of the crack faces for a far-field biaxial load, a formula for theEntire displacement throughout the matrix and inclusion is obtained for a far-field biaxial load. Herrmann [16], which considered a fixed rigid inclusion, and this paper are the first solutions for the entire displacement of an interface crack problem with in plane far-field loading. From this expression for the displacement, a natural decomposition of the problem is identified and the extent of the predicted interpenetration of the crack faces is discussed for each case. It is seen, analogous to a Griffith crack, that interpenetration regions always occur and are large for most mixed far-field loads. This confirms the statement in England [10] that it might be expected... that a similar wrinkling and crossover phenomena will be observed near the ends of the crack. This elegant closed-form expression for the displacements throughout both the matrix and the inclusion is of interest either for use as a benchmark for numerical studies of interface problems or to determine a domain of influence for interface cracks in fiber-reinforced and particular composites.     

Finite elements with increased freedom in choosing shape functions

Starting with a mathematical statement of the convergence requirements for an element stiffness matrix, the paper discusses displacement shape functions that may be used in connection with the potential energy principle. In short, these functions must be force orthogonal and energy orthogonal, but they need not be conforming (satisfy interelement compatibility). It is shown that the requirements to the displacement functions may be greatly relaxed through slight modifications of the coupling stiffness between fundamental and higher order displacement modes. Several alternative formulations are examined. In particular, a new ‘free formulation’ is suggested. Using this form, which is very simple, the only requirement to the displacement patterns used is that they should contain the fundamental deformation modes and be linearly independent. Applications of the theory to triangular and rectangular plate bending elements are shown; the simple stiffness matrix for the latter is given explicitly. The numerical results compare favourably with other types of finite elements.     

GCPOF: a novel optical flow estimation algorithm leveraged by sparse ground control points

As a fundamental task in computer vision, optical flow estimation algorithms aim to establish dense pixel correspondences between image frames. This paper presents a novel optical flow estimation framework called GCPOF to handle large displacement and scale variations of scene objects, which appear frequently and pose great challenges in practice. Within the framework of GCPOF, large displacement and scale variations are captured by a new problem formulation leveraged by sparse ground control points. We present detailed theoretical derivation of the solution to the problem based on iterative reweighted least squares. Both qualitative and quantitative evaluations on synthetic and real images demonstrate that GCPOF is able to handle optical flow fields with large displacement and scale variations properly, and it runs significantly faster than relevant optical flow estimation methods.     

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.     

Radio characteristics evaluation tests of an experimental 3G WCDMA system in Taiwan suburban environments

Abstract

An experimental study on the radio characteristics of a 3G WCDMA system in a Taiwan suburban environment was conducted. This paper presents the results of handover behaviors. The examinations and performance evaluation are based on the measured E_b /I ₀, mobile system transmitting power, bit‐error‐rate and frame‐error‐rate statistic distributions with respect to the time variations, which corresponds to different measurement locations, under individual and independent but appropriate design of experiment procedures. The corresponding radio characteristics are analyzed and evaluated according to the measured data.     

Optimum design of cylindrical pressure vessels taking account of autofretting

Two statements are given for the problem of optimum design of cylindrical pressure vessels taking account of autofretting. The first statement assumes successive solution of two optimization problems. First the optimum residual stress distribution is determined with which strength restrictions are fulfilled for a cylinder of minimum thickness. Then the optimum autofretting pressure is found with which residual stress curves are close to the optimum according to a certain standard. The second statement does not require an iteration procedure and its makes it possible immediately to determine the optimum autofretting pressure corresponding to the minimum permissible cylinder thickness. The results obtained are compared. Possibilities are demonstrated for strengthening cylindrical pressure vessels by means of the autofretting process.Translated from Problemy Prochnosti, No. 2, pp. 78–82, February, 1992.     

Mixed finite element formulation for geometrically non-linear analysis of shells of revolution

A Finite Element formulation for the axisymmetrical analysis of rotational shells with geometric non-linearity is developed, using a mixed finite element model of a curved rotational shell type, the two global components of displacement (u_r, u_z) and the two stress couples (M_r, M_tH) being the free and independent unknowns of the problem. The prescribed geometry at every nodal circle comprises the co-ordinates and meridional slope angle and curvature. Newton-Raphson Iteration is used in solving the non-linear system of equations. Circular plates and a spherical cap are used as examples to test the formulation; good results were achieved which are presented graphically in comparison to the analytical solution.     

Formulation of a membrane finite element with drilling degrees of freedom

A 4-noded membrane element with drilling degrees of freedom is developed. A functional for the linear problem, in which the drilling rotations are introduced as independent variables, is obtained from the functional for the corresponding nonlinear problem, initially derived by Atluri. A careful examination of the Euler equations shows that the Allman-type element fails to pass a one-element patch test with a minimum number of constraints. A 4-noded element, which passes the one-element patch test without constraining the drilling rotations, is proposed here with the use of the separate kinematics variables of displacement and rotation. Detailed numerical studies show the excellent performance of the element.Paper presented at the Workshop on Reliability of Finite Elements, TICOM, Texas, Nov. 1989     

Exact static element stiffness matrices of non‐symmetric thin‐walled curved beams

An evaluation procedure of exact static stiffness matrices for curved beams with non‐symmetric thin‐walled cross section are rigorously presented for the static analysis. Higher‐order differential equations for a uniform curved beam element are first transformed into a set of the first‐order simultaneous ordinary differential equations by introducing 14 displacement parameters where displacement modes corresponding to zero eigenvalues are suitably taken into account. This numerical technique is then accomplished via a generalized linear eigenvalue problem with non‐symmetric matrices. Next, the displacement functions of displacement parameters are exactly calculated by determining general solutions of simultaneous non‐homogeneous differential equations. Finally an exact stiffness matrix is evaluated using force–deformation relationships. In order to demonstrate the validity and effectiveness of this method, displacements and normal stresses of cantilever thin‐walled curved beams subjected to tip loads are evaluated and compared with those by thin‐walled curved beam elements as well as shell elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Stress intensity factors by enriched mixed finite elements

An enriched finite element model for linear elastic fracture mechanics is developed for a mixed variational statement. The independent approximations for the displacement and stress components are enriched by adding the near-field analytic expressions for a cracked body to the polynomial approximations of a conventional element. This allows for an accurate representation of the stress and displacement fields near the crack tip and also results in the direct calculation of the appropriate stress intensity factors. The accuracy of this formulation is demonstrated through several numerical examples.     

Effect of Loading Rate on Fracture Energy of High‐Strength Concrete

Abstract: This research deals with the sensitivity of several types of performance‐designed high‐strength concrete to the loading rate. Variations in the composition of the concrete produce the desired performance, for instance having null shrinkage or being able to be pumped at elevated heights without segregation, but they also produce variations in the fracture properties that are reported in this paper. We performed tests at five loading rates spanning six orders of magnitude in the displacement rate, from 1.74 × 10^?5 mm s^?1 to 17.4 mm s^?1. Load‐displacement curves show that their peak is higher as the displacement rate increases, whereas the corresponding displacement is almost constant. Fracture energy also increases, but only for loading rates higher than 0.01 mm s^?1. We use a formula based on a cohesive law with a viscous term [Anales de Mecánica de la Fractura 25 (2008) 793–797] to study the results. The correlation of the formula to the experimental results is good and it allows us to obtain the theoretical value for the fracture energy under strictly static conditions. In addition, both the fracture energy and the characteristic length of the concretes used in the study diminish as the compressive strength of their aggregates increases.     

Dynamic contact of bodies experiencing large deformations

Summary In the paper the statement ad basic features of the large displacement contact problem will be considered. Using the notion and formalism of a material singular surface in continua important physico-chemical phenomena of the contact interface may be described. Our attention is focused on details which follow from the geometrical nonlinearity and from the role of the interfacial layer.Dedicated to Prof. Dr. Dr. h. c. Franz Ziegler on the occasion of his 60th birthday     

Thermo-elastic analysis of a functionally graded piezoelectric rotating hollow cylindrical shell subjected to dynamic loads

In this study, dynamo-thermo-elastic analysis of a rotating piezoelectric hollow cylinder made of functionally material is presented. Coupled differential quadrature and finite difference methods are used to solve boundary/initial value equations of the problem. Material properties are assumed to be graded in radial direction and temperature independent. Numerical results obtained and convergence are studied, and then verified with reported results in literature. Effect of variations of the grading parameter, angular velocity, thermal gradient and ratio of the outer to inner radii on the stresses, radial displacement and electrical potential are presented.     

Mesh distortion immunity of finite elements and the best-fit paradigm

It has been known for some time that distorted finite elements produce relatively (and, sometimes, dramatically) poor results. This has been related to the completeness condition. In this paper, we investigate this issue and propose that the abstract mathematical viewpoint represented by the completeness condition is actually a statement of the physical need for a finite element computation to recover accurate stresses in the metric space. This follows from the projection theorem describing finite element analysis which shows that the stresses computed by the displacement finite element procedure are abest approximation of the true stresses at an element as well as global level. The simplest possible element is used to elucidate the principles.