Modeling of material discontinuity in gradient elasticity by using staggered mixed meshless collocation approach

  • Jalušić, Boris (University of Zagreb)
  • Jarak, Tomislav (University of Zagreb)
  • Sorić, Jurica (University of Zagreb)

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In recent years, various gradient elasticity theories have been derived and utilized in order to solve the physical phenomena in which the material microstructure has a significant impact. These theories take into account a lower scale of observation, elusive in classical material continuum theories, by including higher-order spatial gradients of state variables in constitutive equations. However, deriving efficient numerical methods for gradient elasticity theories is in general a difficult task because these theories are described by the fourth- or higher-order differential equations. In contrast to the finite element method formulations, in meshless methods shape functions of an arbitrary continuity order can be derived relatively straightforward. Nevertheless, the application of high-order derivatives of meshless approximants leads to increased computational effort and decreases the solution accuracy. The main focus of this contribution is related to modeling the interface of two different material zones within a heterogeneous material microstructure. In this contribution two different mixed meshless collocation approaches are considered for the gradient elasticity theories of the Helmholtz type, with only one unknown constitutive parameter. Hence, in these formulations the fourth-order equilibrium equations of gradient elasticity are solved by using the strain- and stress-based staggered procedures as uncoupled sequences of the two sets of second-order differential equations. Due to the employment of the two staggered procedures, different deformation responses around the material interface are expected. These responses will be analyzed along with the utilization of the appropriate natural boundary conditions in the mixed collocation methods. Since the collocation methods are used, the problems related to the numerical integration of meshless derivatives are completely circumvented, and some other problems associated with the meshless methods, such as high numerical costs or wide nodal connectivity, are alleviated to a certain extent. The application of the mixed approach results in the discretized equations, where only the first derivatives of meshless approximants appear, removing the problems associated with high-order meshless derivatives. The obtained results of all considered staggered solution strategies will be presented and discussed in the representative numerical examples often solved by usage of gradient elasticity theories