Explicit expressions for the mode III asymptotic solutions of an unsteadily fast-propagating crack tip had not been derived before. In the present study, first, the governing equations for this problem were derived. Next, to obtain the solutions of the governing equations, a symbolic manipulation system was used. The explicit expressions for the mode-III transient stress and displacement fields were successfully derived for the first five terms (n=0∼4) in asymptotic series. It was found that the transient effects do not appear in the and constant-stress terms. but appear only in the higher-order terms greater than or equal to the third order (n≥3). These features of the asymptotic solutions agree with those of the in-plane mixed-mode asymptotic solutions obtained in a previous study.

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An analytical evaluation method of a domain integral in BEM inelastic analysis is described in this paper. A high order domain integral term originated from an inelastic strain rate is present in the internal stress rate formulation and, therefore, it is very difficult to calculate its principal value by numerical integration. However, an accurate evaluation of the internal stress rate is important to obtain a correct inelastic strain field at each load increment stage of nonlinear material BEM analysis. In the present paper, we derived a fully analytical form of the integral term for a plane problem by applying a triangular polar coordinate transformation which can reduce the order of singularity by one degree. Internal stresses for thermal strain problems and a plastic strain problem are calculated by this method. The accuracy and effectiveness of this method is discussed, compared with the conventional numerical integral method.

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An equation for investigating the intensity of singularity in the stress field near the apex in a three-phase bonded structure formed from isotropic homogeneous rectangular wedges was derived using the theory of elasticity, in a previous paper. In this paper, the distributions of the intensity K for free surfaces subjected to normal loadings in a stress field represented by the types Kr^P-1 and K log r near the apex in the three-phase bonded structure with arbitrary wedge angles, are investigated. Theoretical values of K and numerical ones of K analyzed by the finite element method (FEM) are shown and discussed for different combinations of materials and wedge angles.

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The stress fields near the apex in dissimilar materials composed of three isotropic homogeneous wedges with arbitrary angles subjected to surface tractions or thermal load are expressed by a linear combination of the solutions, no singularity ones and the particular ones. For the cases where the stresses near the apex of dissimilar materials subjected to thermal load represent the singularities of type γ^p-1, type log γ and no singularity, the stress distributions are examined theoretically and numerically. Moreover, the different influences of solutions with no singularity and the particular solutions on the stresses near the apex are theoretically clarified. It was examined how the characteristic of stress singularity clarified using FEM is changed by whether no singularity solutions and the particular solutions are taken into consideration in analyzing the stress singularity. Furthermore, the notable results on the stress singularity of type log γ in dissimilar materials formed from two rectangular wedges are shown.

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In the previous papers, characteristics of stress fields near the apex in dissimilar materials formed from several isotropic homogeneous wedges with arbitrary angles subjected to surface traction or thermal load were theoretically clarified. The stress fields near the apex are defined by a linear combination of the solutions K_γ^p-1 of typeγ^p-1 on root p of an eigenequation in 0 < Re(p) < 1 or K_g logγ of type logγ on a double root p=1, no singularity ones K_γ^p-1 on root p in Re(p) > 1 and the particular solution K_pa on root p=1. The singularity of type logγ appears at the boundary where characteristics of stress fields vary from no singularity solutions to solutions of typeγ^p-1 or its reverse (Re(p) > 1 &harr; 0 < Re(p) < 1). For variations Re(p) > 1 &harr; 0<Re(p) < 1 of root p, the distributions of stress intensity K_j on p =p_j and K_pa on P=1 are changed at the boundary. Moreover, it is shown that characteristics of stress fields cannot be determined by only the roots of the eigenequation.

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Explicit expressions for in-plane mixed-mode asymptotic solutions of an unsteadily fast-propagating crack tip had not been derived so far, due to seemingly insurmountable mathematical and algebraic difficulties To overcome these difficulties, a symbolic manipulation system was used in the present study. The explicit expressions for the mixed-mode transient stress and displacement fields up to the first five terms (n=0∼4) in asymptotic series were successfully derived. It was found that the transient effects do not appear in the first three terms (rigid body (n=O), (n=1) and constant-stress (n=2) terms), but appear only in the higher-order terms greater than or equal to the third order (n&ge;3). The transient effects in the near-tip stress distributions were also visualized using the symbolic manipulation system. It was found that both effects of the crack acceleration and of the changes in the asymptotic coefficients increase the stress distribution ahead of the propagating crack tip. These stress-increases in the unsteady part may clarify the problem of the crack-acceleration effects in the dynamic fracture toughness versus crack velocity curves, which is one of important unsolved problems in dynamic fracture mechanics.

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In this paper, a method of finding a local minimum is examined for LQ optimal control problem of constant output feedback. A necessary condition for optimality is given as matrix algebraic equations for a modified LQ performance index, and a method of tracing a solution path from a state feedback gain to an output feedback gain is proposed. In order to trace the solution curve that may have branch, a new homotopy method based on series expansion of approximate algebra is applied. Initial gains are obtained from not only a positive definite solution but also the other solutions of Riccati equation, then it is examined numerically whether more than one local minimum can be obtained by tracing the solutions starting from the initial gains. A numerical example which has two local minima shows that one local minimum is connected to the optimal state feedback gain, but the other is not connected to any gains given by the solutions of Riccati equation.

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In order to simulate three-dimensional dynamic crack propagation, a three-dimensional moving finite-element method was developed. A new expression was also derived for the dynamic J integral (J') in terms of the equivalent domain integral (EDI) method. The moving finite-element procedure together with the dynamic J integral evaluation procedure made it possible to evaluate the distribution of the dynamic energy release rate along the propagating crack front at each time step. Furthermore, a new formula for converting the dynamic J integral to the stress intensity factor along the crack front was derived by introducing a three-dimensionality parameter. Numerical simulation was carried out for several sample problems.

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Given a vector field, its simplified expression in terms of coordinate transformations is called a normal form of the dynamical system generated by the vector field. The purpose of this paper is to give a brief exposition of recent progress in the normal form theory for vector fields. More attention is paid to the algorithm for its computation as well as applications rather than the theoretical details. After explaining a brief idea about normal forms in §1, we start with the basic definition of normal forms and give a simple example in §2, and in §3 we explain a detailed algorithm for the computation of normal forms based on a simple general principle given in §2. §4 is devoted to a sketch of analysis of a laser equation as an application of normal forms to concrete differential equations.

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This paper is concerned with the problem of verifying an accuracy of a numerical solution of a linear system with an arbitrarily ill-conditioned coefficient matrix. In this paper, a method of obtaining an accurate numerical solution of such a linear system and its verified error bound is proposed. The proposed method is based on the accurate computation of dot product and IEEE standard 754 arithmetic. A verified and accurate numerical solution with a desired tolerance can be obtained by the proposed method with iterative refinement. Numerical results are presented for illustrating the effectiveness of the proposed method.

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