Coefficients Of Higher Order Terms Research Articles

Experimental evaluation of stress field around the sharp notches using photoelasticity

A higher order representation of the stress field around the sharp notch has been utilized for calculating notch stress intensity factors (NSIFs) as well as the coefficients of higher order terms by the technique of photoelasticity. Adding the higher order terms to the singular term makes it possible to collect the data points from a larger zone, which helps to simplify the data collection from experiments. Moreover, the effects of higher order terms in the region near the notch tip are taken into account. To utilize the advantages of whole-field photoelasticity and minimize the experimental errors, a large number of data points have been employed and the overdeterministic least squares method combined with the Newton–Raphson method have been used to solve the resulting system of nonlinear equations. The NSIFs for a laboratory specimen called V-notched Brazilian disk (V-BD) were calculated for various notch angles under pure mode I, pure mode II and mixed mode I/II loading conditions. In addition to NSIFs, the coefficient of the first non-singular term of 30° notch was calculated experimentally and the effects of this term on the stress distribution in the vicinity of notch tip were investigated. A good correlation was observed between the experimental results and the numerical results obtained from finite element analysis.

Direct determination of SIF and higher order terms of mixed mode cracks by a hybrid crack element

Recently, the authors (Karihaloo and Xiao, 2001a-c) extended the hybrid crack element (HCE) originally introduced by Tong et al. (1973) for evaluating the stress intensity factor (SIF) to calculate directly not only the SIF but also the coefficients of the higher order terms of the crack tip asymptotic field. Extensive studies have proved the versatility and accuracy of the element for pure mode I problems. This study is to show the versatility of the element for mode II and mixed mode cracks. Accuracy of the SIF and coefficients of higher order terms is validated by comparing with the available results in the literature, or results obtained by the boundary collocation method, which is powerful for relatively simple geometries and loading conditions.

Direct evaluation of accurate coefficients of the linear elastic crack tip asymptotic field

ABSTRACTAn improvement to the extended finite element method (XFEM) and generalised finite element method (GFEM) is introduced. It enriches the finite element approximation of the crack tip node as well as its surrounding nodes with not only the first term but also the higher order terms of the linear elastic crack tip asymptotic field using a partition of unity method (PUM). Numerical results show that together with a reduced quadrature rule to the enriched elements, this approach predicts accurate stress intensity factors (SIFs) directly (i.e. without extra post‐processing) after constraining the enriched nodes properly. However, it does not predict accurately the coefficients of the higher order terms. For that a hybrid crack element (HCE) is introduced which is powerful and convenient not only for directly determining the SIF but also the coefficients of higher order terms in the plane linear elastic crack tip asymptotic field. Finally, the general expressions for the coefficients of the second to fifth terms of the linear elastic crack tip asymptotic field of three‐point bend single edge notched beams (TPBs) with span to depth ratios widely used in testing are extended to very deep cracks with the use of the HCE.

Coefficients of the crack tip asymptotic field for wedge splitting specimens

The stress intensity factor (SIF) and the coefficients of higher order terms of the crack tip asymptotic field of typical wedge splitting specimens with two different loading arrangements are directly computed using a hybrid crack element. Accurate analytical expressions for the first five terms are obtained by fitting the computed data. Numerical results show that the coefficients of terms higher than three are negligibly small, this may explain that the wedge splitting specimen is more stable than other geometries. The first five terms are not sensitive to support conditions. However, for short cracks coefficients of terms, except the SIF, are quite sensitive to the loading arrangement even when the loads are statically equivalent.

Coefficients of the crack tip asymptotic field for a standard compact tension specimen

The coefficients of the crack tip asymptotic field of a standard compact tension (CT) specimen are computed using a hybrid crack element (HCE). It allows the direct calculation (without post-processing) of not only the stress intensity factor (SIF) but also the coefficients of higher order terms of the crack tip asymptotic field. Approximate closed-form expressions for the first five terms for the CT specimen that are accurate for shallow to very deep cracks are obtained by fitting the computed data. The SIF formula proposed by Brown and Srawley (1966) is shown to be accurate when the crack length to depth ratio (α) ranges from 0.35 to 0.75. The formula proposed by Newman (1974) and Srawley (1976) is accurate for α≥0.15. However, the accuracy of available formulas for the second T-term in the literature is quite disappointing. Numerical results also show that, unlike the notched three-point bend beam and the wedge splitting specimen, the second T-term of the CT specimen is always positive.

Accurate determination of the coefficients of elastic crack tip asymptotic field by a hybrid crack element with p-adaptivity

A hybrid crack element (HCE) originally introduced by Tong, Pian and Lasry for evaluating the stress intensity factor (SIF) is improved to have p-adaptivity and is applied to calculate directly not only the SIF but also the coefficients of the higher order terms of the elastic crack tip asymptotic field. The latter are of great relevance to describing the fracture behaviour of elastic–plastic materials and to interpreting the size effect of quasi-brittle materials. The ambiguities in the variational background, parameter matching condition and integration order of HCE, as well as the arguments on its applicability to crack problems are cleared. Numerical results show that the computed coefficients of the higher order terms as well as the SIF converge rapidly with p-refinement of the HCE and the h-refinement of the remaining regular elements, and that they are stable when the HCE size to the crack length ratio is larger than 0.25. The HCE is efficient down to very short cracks. As expected, the accurate determination of the coefficients of higher order terms is more difficult than that of the SIF; it requires a higher order HCE together with a finer subdivision of the remainder of the body by regular elements.

On the Jacobian Conjecture: Reduction of Coefficients

We prove that a polynomial map from Rn to itself with non-zero constant Jacobian determinant is a stably tame automorphism if its linear part is the identity and all the coefficients of its higher order terms are non-positive. We also prove that the Jacobian conjecture holds for any number of variables and any field of characteristic zero, if one can show that every polynomial map of Rn to itself is injective when it has a non-zero constant Jacobian determinant and has linear part the identity, and all the coefficients of higher order terms are non-negative. The proofs use special properties of matrices with non-positive off-diagonal elements and non-negative principal minors, and of matrices with vanishing principal minors. Furthermore we reduce the Jacobian conjecture to a polynomial matrix problem. Moreover, if the matrix has a positive answer, then every real polynomial automorphism is stably tame.

Effect of finite coil thickness in a magnetic deflection system

The effect of finite coil thickness on the magnetic field produced by a magnetic deflection coil in the shape of a long right circular cylinder is calculated. It is found that the higher ‘‘multipole’’ contents of the field (i.e., the coefficients of higher order terms in a Fourier expansion about the coil axis) decrease by progressively larger percent amounts compared to their values for a coil of zero thickness.

On the theory of plane stress for finite deformations

By means of parametric expansions in terms of thickness h0 of the undeformed plate, the equations of plane stress for finite deformations of homogeneous, isotropic, elastic materials are derived from the threedimensional theory of finite elasticity without making any usual geometrical or physical assumptions other than that deformation is symmetrical with respect to the middle plane of the plate. The equations of plane stress are obtained as the coefficients of zero power of h0 in the expansions. Coefficients of higher order terms provide interior corrections to the plane stress theory. The appropriate boundary conditions for these interior equations are derived by the variational formulation of the three-dimensional theory of finite elasticity.