scholarly journals Higher Order Weight Functions in Fracture Mechanics of Multimaterials

10.5772/55360 ◽  
2012 ◽  
Author(s):  
A. Yu.
Keyword(s):  
Fracture Mechanics ◽  
Higher Order ◽  
Weight Functions

Using the Multi-Parameter Fracture Mechanics for more Accurate Description of Stress/Displacement Crack Tip Fields

2013 ◽  
Vol 586 ◽  
pp. 237-240 ◽  
Author(s):  
Lucie Šestáková
Keyword(s):  
Stress Distribution ◽  
Fracture Mechanics ◽  
Boundary Conditions ◽  
Displacement Field ◽  
Singular Term ◽  
Crack Tip ◽  
Higher Order ◽  
Precise Description ◽  
Accurate Knowledge ◽  
Higher Order Terms

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.

Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials

2004 ◽  
Vol 57 (5) ◽  
pp. 325-343 ◽  
Author(s):  
Tian-You Fan ◽  
Yiu-Wing Mai
Keyword(s):  
Thermal Properties ◽  
Fracture Mechanics ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Higher Order ◽  
Classical Elasticity ◽  
Partial Differential ◽  
Crystalline Materials ◽  
Quasi Crystals

A review is given on the basic concepts and fundamental framework of the theory of elasticity for quasi-crystalline materials, including some 1D, 2D, and 3D quasi-crystals. The elasticity of quasi-crystals embodies some new concepts, field variables, and equations. It is much more complicated and beyond the scope of classical elasticity which holds only for conventional structural materials, including crystalline materials. Hence, some well-developed methods in classical elasticity cannot be directly applied to solve the problems of elasticity of quasi-crystalline materials. But the ideas of the classical theory of elasticity provide beneficial insight to treat this new subject. A decomposition and superposition procedure is suggested to simplify the elasticity problems of 1D and 2D quasi-crystals. Application of displacement and stress potentials further simplifies the problems. The large number of complicated equations involving elasticity is reduced to a single or a few partial differential equations of higher order by this technique. Also, efforts have been made to simplify the equations for 3D cubic quasi-crystals to a single partial differential equation of higher order. Simplification of the basic equations provides the possibility to solve boundary value or initial-boundary value problems of elasticity. For this purpose, some direct and systematic methods of mathematical physics and function theory are developed, and a series of analytic (classical) solutions, mainly for dislocations and cracks in materials, are derived. In addition, attention is drawn to those variational problems and generalized solutions (weak solutions) of boundary value problems and numerical implementation by the finite element method. The above may be seen as a development of the theory and methodology akin to those of classical elasticity. Based on the exact solutions of crack problems with different configurations under different motion states for different quasi-crystal systems, we put forward a framework of fracture mechanics of quasi-crystalline materials. This may be seen as an extension of the development of fracture mechanics for conventional structural materials. Also, some elastodynamic problems for some 1D and 2D quasi-crystals are studied, related results for dislocation and crack dynamics are found, and possible connections with certain thermal properties of quasi-crystalline materials, eg, specific heat and other thermo-dynamic functions, are discussed. There are 75 references cited in this review article.

scholarly journals On two kinds of the reverse half-discrete Mulholland-type inequalities involving higher-order derivative function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiang Chen ◽  
Bicheng Yang
Keyword(s):  
Type Inequality ◽  
Higher Order ◽  
Constant Factor ◽  
Order Derivative ◽  
Weight Functions ◽  
Real Analysis ◽  
Derivative Function ◽  
Hadamard’S Inequality ◽  
The Best Possible Constant ◽  
Higher Order Derivative

AbstractBy means of the weight functions, Hermite–Hadamard’s inequality, and the techniques of real analysis, a new more accurate reverse half-discrete Mulholland-type inequality involving one higher-order derivative function is given. The equivalent statements of the best possible constant factor related to a few parameters, the equivalent forms, and several particular inequalities are provided. Another kind of the reverses is also considered.

Sign in / Sign up

Export Citation Format

Share Document