This article is prompted by the existing confusion about correctness of responses of beams and plates produced by middle surface (MS) and neutral surface (NS) formulations. This study mathematically analyzes both formulations in the context of the bending of bi-directional functionally graded (BDFG) plates and discusses where the misconceptions are. The relation between in-plane displacement field variables on NS and on MS are derived. These relations are utilized to define a modified set of boundary conditions (BCs) for immovable simply supported plates that enables either formulation to apply fixation conditions on the refence plane of the other formulation. A four-variable higher order shear deformation theory is adopted to present the displacement fields of BDFG plates. A 2D plane stress constitution is used to govern stress–strain relations. Based on MS and NS, Hamilton’s principles are exploited to derive the equilibrium equations which are described by variable coefficient partial differential equations. The governing equations in terms of stress resultants are discretized by the differential quadrature method (DQM). In addition, analytical expressions that relate rigidity terms and stress resultants associated with the two formulations are proved. Both the theoretical analysis and the numerical results demonstrate that the responses of BDFG plates based on MS and NS formulations are identical in the cases of clamped BCs and movable simply supported BCs. However, the difference in responses of immovable simply supported BCs is expected since each formulation assumes plate fixation at different planes. Further, numerical results show that the responses of immovable simply supported BDFG plates obtained using the NS formulation are identical to those obtained by the MS formulation if the transferred boundary condition (from NS- to MS-planes) are applied. Theoretical and numerical results demonstrate also that both MS and NS formulations are correct even for immovable simply supported BCs if fixation constraints at different planes are treated properly.

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Mathematical and Physical Analyses of Middle/Neutral Surfaces Formulations for Static Response of Bi-Directional FG Plates with Movable/Immovable Boundary Conditions

A pure complex variable enrichment method for modeling progressive fracture of orthotropic functionally gradient materials

In this paper, dynamically propagating cracks in anisotropic plates exposed to a generalised thermal shock are investigated. Here, the governing equations are considered according to the Lord-Shulman (LS) model as a fully coupled generalised thermoelasticity theory. The crack is modelled in the context of the eXtended Finite Element Method (XFEM). To evaluate Stress Intensity Factors (SIFs), a technique based on the J integral and crack tip opening and sliding displacements, previously applied for stationary cracks, is developed for a dynamically propagating crack in orthotropic solids. Besides, a variety of functions for enriching the displacement field are derived relying on the behaviour near the tip of a dynamically propagating crack in anisotropic materials. The maximum hoop stress criterion is also modified by using the asymptotic stress field for a dynamically propagating crack. In addition, the effect of the angular variation of fracture toughness in orthotropic materials is accounted in computing the crack propagation speed. In some examples, dynamic crack growth in complex cracked orthotropic structures subjected to LS thermal shock is studied in detail. The impacts of fibre orientations and LS relaxation time are also investigated in these examples. In conclusion, applying non-Fourier thermal shock for modelling dynamically propagating cracks in orthotropic structures is recommended to obtain more realistic results. In addition, by increasing the LS relaxation time, the crack grows less oscillatory and is more aligned along the material fibres. 

Dynamically propagating cracks in anisotropic plates subjected to hyperbolic thermal shock

The fracture properties of concretes, which are often characterized by the tensile fracture strength, compressive fracture strength, fracture energy and characteristic length are closely related to temperature. However, there are still few systematic theoretical models of temperature-dependent fracture properties of concretes. In this work, a series of novel temperature-dependent theoretical models are developed to determine the fracture properties of concretes based on the force-heat equivalence energy density principle and classical concrete fracture theories. The models establish the quantitative relationship between the high-temperature tensile fracture strength, compressive fracture strength and characteristic length, and the basic material parameters including the Young's modulus and melting point. It is worth noting that this work quantitatively characterizes the fracture properties of concretes subjected to high temperatures using the simple material parameters, without the need to carry out any data fitting. The fracture properties of concretes with different aggregate types, different fiber-reinforced concretes, and ordinary Portland cement-based concretes are predicted and systematically analyzed using the proposed models. The model-predicted tensile fracture strength, compressive fracture strength and characteristic length of materials up to 800 °C agree well with the experimental measurements without using any fitting parameters. The coincidence rate at many temperature points could reach up to and above 90%, even approaching about 100%. The fracture properties of concretes and their main mechanisms at various temperatures thus can be depicted by using the developed models. Additionally, this work offers a new and simple testing method to determine the characteristic length of concretes and its change with temperature. 

Non-fitting theoretical models for the fracture properties of concretes subjected to high temperature

A novel voxel-particle energy approach to predict 3D microscopic fracture surface of porous geomaterials and fracture permeability modeling

https://doi.org/10.1016/j.cma.2022.115402

On tracking arbitrary crack path with complex variable meshless methods

An enriched improved complex variable element-free Galerkin method for efficient fracture analysis of orthotropic materials

Buckling analysis of variable stiffness composite plates with elliptical cutouts using an efficient RPIM based on naturally stabilized nodal integration scheme

D.M. Li

Papers

On tracking arbitrary crack path with complex variable meshless methods

An enriched improved complex variable element-free Galerkin method for efficient fracture analysis of orthotropic materials

Buckling analysis of variable stiffness composite plates with elliptical cutouts using an efficient RPIM based on naturally stabilized nodal integration scheme

On compacting pattern control of finite-size 2D soft periodic structures through combined loading

A pure complex variable enrichment method for modeling progressive fracture of orthotropic functionally gradient materials