A New Systematic Approach to Derive Functionals and Nonclassical Conservation Laws

It is shown in detail that Noether's theorem represents a mathematical identity which always exists for any nondegenerate functional and includes two parts: one is Euler–Lagrange expression, and the other is a possible conservation law. If and only if any physical equations are the same as Euler–Lagrange expressions, the following three points follow: (i) there always exist conservation laws or, at least, balance laws; (ii) the symmetry transformations between inertial frames can be used first to check the absolute invariance; (iii) the field variables included in spacetime transformations can be canceled for absolute invariance, and space and time transformations are irrelevant. A theorem for getting nonclassical conservation laws from the general solution of a physical system is presented. The necessary conditions in a theorem's form are also presented to construct Lagrangians for second- and fourth-order partial differential equations (PDEs). Based on these theorems, the Lagrangians and conservation laws of nonlinear heat and KdV equations are constructed and given. For Boussinesq's solution, a nonclassical conservation integral is also found for application. Moreover, it is shown that there exists only the same conservation laws derived from the generalized variational principle as those of the principle of minimum potential energy in elasticity.

J-Integral Computation in Mixed Mode Ι and ΙΙ

Computation of a path independent contour integration but it yields directly the independent values of the opening mode and shear mode stress intensity factors. The calculations are not sensitive to the choice of contour. To the J-integral calculation, the present scheme calculates separately the factors KI and KII. A simple method to determine the SIFs for the three-dimensional crack is proposed based on the conservation integral.

A New Interaction Integral Method for Nonhomogeneous Materials

A new interaction integral technique is derived for computation of mixed-mode stress intensity factors (SIFs) in nonhomogeneous materials with continuous or discontinuous properties. This method is based on a conservation integral that relies on two admissible mechanical states (actual and auxiliary fields). In the equivalent domain formulation, the integrand does not involve any derivatives of material properties. Moreover, the formulation is proved to be still valid when the integral domain contains material interfaces. Therefore, its applicable range is greatly enlarged. The method is combined with the extended finite element method (XFEM) to calculate the SIFs for different integral domains. Numerical results show that the interaction integral has excellent convergence for material nonhomogeneity and discontinuity.

Effect of Impurities on Σ3 (111) Grain-Boundary Fracture in Tungsten—Atomistic Simulation

ABSTRACTThe effect of various impurities and micro-alloying additions (B, N, C, O, Al, Si, S and P) on the intrinsic resistance of the Σ3 (111) GB in Tungsten has been investigated using Molecular Dynamics simulation. The atomic interactions have been accounted for through the use of the Finnis-Sinclair interatomic potentials. The fracture resistance of the GB has been characterized by computing, in each case, the ideal work of GB separation, the Mode I stress intensity factor and the Eshelby F - conservation integral at the onset of crack propagation. The results obtained suggest that pure Tungsten is relatively resistant toward GB decohesion and that this resistance is further enhanced by the presence of B, C and N. Elements such as O, Al and Si on the other hand, have a relatively minor effect on the cohesion strength of the Σ3 (111) GB. In sharp contrast, S and P greatly reduce this strength making Tungsten quite brittle. These findings have been correlated with the effect of the impurity atoms on material evolution at the crack tip.