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The finite cover method (FCM), which is a coverbased generalized finite element method, is extended for analyses of progressive failure processes involving cohesive zone fracture, starting from an interface debonding and evolving toward one of the constituents of heterogeneous solids and structures. Assuming that the constituents fail according to the maximum principal stress, we are able to represent the evolution of the resulting failure surfaces of discontinuity independent of mesh alignment owing to the distinctive features of the FCM. Also, interface elements with Lagrange multipliers are introduced to impose compatibility conditions on the material interface so that debonding is judged by the multipliers. Representative numerical examples demonstrate the capability of the proposed method in tracing the smooth transition of crack paths from interfacial to internal failure, and vice versa.

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We have studied the electrical resistivity of PrB 6 under pressure up to P = 0.8 GPa, with magnetic fields applied along the ͓001͔ and ͓111͔ directions. The commensurate ͑C͒ magnetic ordered phase is significantly stabilized by pressure, and another extra phase ͑denoted phase A͒ is observed under a low pressure of ϳ0.3 GPa. This phase appears above the transition temperature T IC1 = 7 K at which the incommensurate magnetic order sets in at the ambient pressure. The magnetic phase diagram for H ʈ ͓111͔ changes drastically under pressure. Short-range interactions, which are strongly dependent on the Pr-Pr distances, are likely to play a dominant role in the relative stability of the different phases.Multipole order is one of the recent topics in the f electron systems. RB 6 ͑R = rare-earth element͒ is a good system to study the multipole interactions because of the cubic simple crystal structure. The crystalline electric-field ͑CEF͒ level schemes of CeB 6 , PrB 6 , and NdB 6 are known and the CEF ground state of these compounds contains the multipole degrees of freedom. 1 It is well known that CeB 6 exhibits the unusual antiferroquadrupole ͑AFQ͒ order dominated by the O xy -type AFQ and T xyz -antiferro-octupole interactions. [2][3][4] Here, O xy = ͱ 3 / 2 J x J y and T xyz = ͱ 15/ 6 J x J y J z where the bars denote the sums of all possible permutations of indices. In CeB 6 , there exist three phases called as I, II, and III. Phase I is paramagnetic. Phase II below T Q = 3.2 K is the O xy -type AFQ ordered phase with the wave vector Q = ͓1 / 2,1/ 2,1/ 2͔. Phase III below T N = 2.3 K is the noncollinear magnetic ordered phase with the wave vectors k ͑1,2͒ = ͓1 / 4, Ϯ 1 / 4,1/ 2͔ and kЈ ͑1,2͒ = ͓1 / 4, Ϯ 1 / 4,0͔, which is dominated by the O xy -type AFQ order. 2 In PrB 6 and NdB 6 , while the dipole interaction is dominant, the magnetic structures are affected by the higher order multipole interactions.In PrB 6 , there exist four magnetic ordered phases of C, IC1, IC2, and C H as are shown in Fig. 1͑a͒. [5][6][7][8][9] We define the transition temperature from the paramagnetic to IC1 phase as T IC1 and that from the IC1 to C phase as T N . The C phase is the commensurate ͑C͒, double-k magnetic ordered phase with the wave vectors k C ͑1,2͒ = ͓1 / 4, Ϯ 1 / 4,1/ 2͔, where the nonclolinear alignment of the magnetic moment with the easy axis along the twofold one is realized. 5 The magnetic structure in the C phase is similar to that in phase III of CeB 6 . This similarity strongly suggests the importance of the O xy -type AFQ interaction also in PrB 6 . The IC1 phase is the incommensurate ͑IC͒, double-k magnetic ordered phase with the wave vectors k IC1 ͑1,2͒ = ͓1 / 4−␦ , Ϯ 1 / 4,1/ 2͔, where ␦ = 0.05. 5 At H = 0, there exist the planar-type three domains of K xy , K yz , and K zx . The K xy domain is dominated by the O xy -AFQ order, etc. In the IC1 phase, there exist K xy IC1 , K yz IC1 , and K zx IC1 . The magnetic phase diagram for H ʈ ͓001͔ is normal as is observed in a usual antiferromagnetic ͑AF͒ magnet where T ...

A Chua-type magnetization model is derived by means of a Fourier series while the magnetic flux density is sinusoidally varying with time. It is shown that this Chua-type model is well suited for practical computations of magnetodynamic field. A geometrical duality between the Delaunay triangles and associated Voronoi polygons is utilized to implement a dual energy finite element approach. As an example, the magnetodynamic fields in a toroidal reactor including the effects of transients, eddy currents, and hysteresis are computed.