Some problems in basic metrology

The paper discusses some of the problems of modern metrology in the context of the past. These include those relating to physical quantities, uncertainties, the SI units, and the role of the fundamental physical constants.Paper presented at the European Metrology Conference, St. Petersburg, Russia, September 1–3, 1992.Translated from Izmeritel'naya Tekhnika, No. 10, pp. 66–70, October, 1993.     

Some problems in experimental fracture mechanics

A standard procedure for the determination of fracture toughness K_IC is discussed. The insufficiency of the existing K_ic determination confidence criteria is stressed and the following criteria are proposed instead: φ_max ? 1.5%; $\text{σ}{}_{\mathrm{fr}}{}^{\mathrm{net}}\text{σ}{}_{\mathrm{0.2\; ?\; 0.8}}$, in conjunction with the old criterion $\text{P}{}_{\max }\text{P}{}_{\mathrm{Q}}\text{? 1.1}$. Determination of K_IC from P_max should be used instead of from P_Q.A method for the determination of a point on the “force-displacement” diagram corresponding to crack growth initiation is set forth. The method is based on specimen compliance tests under repeated load-relief cycles. The crack growth initiation point is used to determine both the critical crack opening and plane strain fracture toughness. The indefinite effect of the growing crack (in the ease of crack opening or Cherepanov-Rice integral calculations) is thereby eliminated. Necessity is emphasized to determine the share of the J-integral which contributes to fracture process. A method for plotting the elastic displacement diagram is proposed which allows on the basis of preliminary estimates to determine fracture toughness of small-sized specimens without using special setups. The area ratio between the plastic and elastic strain diagrams is proposed to be adopted as fracture type criterion. Certain experiments to determine crack resistance of material specimens are described and discussed.     

Meshless techniques for convection dominated problems

In this paper, the stability problem in the analysis of the convection dominated problems using meshfree methods is first discussed through an example problem of steady state convection-diffusion. Several techniques are then developed to overcome the instability issues in convection dominated phenomenon simulated using meshfree collocation methods. These techniques include: the enlargement of the local support domain, the upwind support domain, the adaptive upwind support domain, the biased support domain, the nodal refinement, and the adaptive analysis. These techniques are then demonstrated in one- and two-dimensional problems. Numerical results for example problems demonstrate the techniques developed in this paper are effective to solve convection dominated problems, and in these techniques, using adaptive local support domain is the most effective method. Comparing with the conventional finite difference method (FDM) and the finite element method (FEM), the meshfree method has found some attractive advantages in solving the convection dominated problems, because it easily overcomes the instability issues.     

Two basic wave propagation problems for the non-uniform motion of displacement discontinuities in a half-plane

The splitting of elastic materials is often achieved by, effect, moving more rigid solids into the surface of the material or by causing a shear tearing characterized by the relative slip of adjacent material points. Therefore, both processes involve the motion of displacement discontinuities. This article examines basic wave propagation problems for the non-uniform motion of two arbitrary displacement discontinuities in an elastic half-plane along paths normal to the surface. The discontinuities impose relative displacements which are, respectively, normal to and along the path. Complete solutions are derived in forms which allow identification of each component of the reflection-complicated wave pattern, Moreover, the dependence of these components on specific aspects of the discontinuity behavior can be noted. Finally, two physical examples, the motion of a rigid wedge and the motion of edge dislocations in the half-plane, are studied.     

Some basic half-plane problems of the cohesive elasticity theory with surface energy

Summary We outline a procedure for obtaining the relevant influence functions for the cohesive plane strain half-plane with surface energy under any distribution of normal and tangential loads on its bounding surface, in terms of solutions of classical elasticity. This is achieved by modifying the classical isotropic elasticity theory, to account for the existence of higher order terms in the constitutive equations and additional boundary conditions. We also deal with the half-plane problem under concentrated edge forces and under a uniform distribution of shearing tractions, both of which involve load-induced concentrations of stress (or strain), and it is illustrated how the proposed cohesive elasticity theory can remove the strain singularities.     

Modelling of fracture process zones and singularity dominated zones

Models for fracture process zones and singularity characterizing zones are developed and their interrelation is studied on the basis of two-dimensional stress analysis concepts for brittle materials. Effects of specimen geometry and loading conditions on the stability of fracture initiation and progressive fracturing is investigated and compared with recent experimental results.     

Some benchmark problems and basic ideas on the accuracy of boundary element analysis

Taken the linear elasticity problems as examples, some benchmark problems are presented to investigate the influence of calculation error and discretization error on the accuracy of boundary element analysis. For the conventional boundary element analysis based on singular kernel function of fundamental solution and using Gaussian elimination method, the main calculation error comes from the integration of kernel and shape function product on each element. Based on some benchmark problems of “simple problem” without discretization error, it can be observed that sometimes a large number of integration points in Gaussian quadrature should be adopted. To guarantee the integration accuracy reliably, an improved adaptive Gaussian quadrature approach is presented and verified. The main error of boundary element analysis is the discretization error, provided the calculation error has been reduced effectively. Based on some benchmark problems, it can be observed that for the bending problems both the constant and linear element are not efficient, the quadratic element with a reasonable refined mesh is required to guarantee the accuracy of boundary element analysis. An error indicator to guide the mesh refinement in the convergence test towards the converged accurate results based on the distribution of boundary effective stress is presented and verified.     

Procedure to predict ultimate fracture failure history for plane stress plasticity problems

A finite element solution procedure is presented to predict the load-displacement history up to ultimate fracture failure for a structural system. Incremental plasticity theory for the von Mises yield criterion and isotropic strain hardening are used to march along the uniaxial stress-strain curve of the material up to fracture. When an element fractures its strain energy is distributed into the unfractured elements using an element nodal release method. If another element fractures during this redistribution process, then unstable crack growth is said to occur, and the total load at this stage is termed the ultimate fracture failure load of the structural system. The analysis steps to automate the solution procedure are described. Numerical results obtained for a center pre-cracked panel tension specimen are reported and compared with experimental results available in the literature.     

Some observations on stress wave emission as a measure of crack growth

A technique for measuring stress waves emitted during slow crack growth is briefly described. The number and size of the stress waves generated appears to bear a unique relationship to the amount of crack growth. From elasticity theory, a semi-empirical relationship is developed. Δ ≈ (Σg)² E/K², where ΔA is the incremental area swept out by the crack, Σg, is the sum of the stress-wave amplitudes associated with that increment of growth, E is the elastic modulus and K is the applied stress-intensity factor. This relationship is shown to be valid for about 100 data points obtained from crack growth in high-strength aluminum, titanium and steel alloys.     

Adaptive combined DE/FE algorithm for brittle fracture of plane stress problems

A novel adaptive combined DE/FE algorithm is proposed to simulate the fracture procedure of brittle materials of plane stress problems. The main concept of the approach is that a model is composed of the finite element completely at the initial stage without any discrete element generated until portion of the model grid becoming severely deformed; and then the model is fragmented into two subdomains, the finite element (FE) and the discrete element (DE) subdomains. The interface force between the two subdomains is calculated by using the penalty method. An extrinsic cohesive fracture model is employed to simulate the brittle fracture procedure only in the DE subdomain. The adaptive algorithm may allow for the use of the accurate and efficient FEs in the lower distorted region and the DEs which are automatically generated in the severely deformed FE region . The feasibility of the adaptive algorithm is validated by the impact fracture simulation of a glass beam. The comparison of calculation time consumption shows that the adaptive algorithm has a higher efficiency than the DEM. At last, the impact fracture behavior of a laminated glass beam is simulated, and the cracks propagation is compared with the experimental results showing that the adaptive algorithm can be implemented to capture some fracture characteristics of brittle materials.     

Two basic wave propagation problems for the non-uniform motion of displacement discontinuities across a bimaterial interface

The splitting of laminated materials is often achieved by, in effect, moving more rigid solids through the material or by causing a shear tearing characterized by the relative slip of adjacent material points. Both processes may take place across the bimaterial interfaces of the laminate. Moreover, both processes involve the motion of displacement discontinuities. This article examines basic wave propagation problems for the non-uniform motion of two arbitrary displacement discontinuities in perfectly-bonded half-planes along paths normal to the interface. The discontinuities impose relative displacements which are, respectively, normal to and along the path. Complete solutions are derived in forms which allow identification of each component of the reflection and transmission-complicated wave pattern. The dependence of these components on specific aspects of the discontinuity behavior can be noted. Finally, two physical examples, the motions of a rigid wedge and an edge dislocation across the interface are studied. It is found that the singularity order and direction of characteristic stresses can be affected by crossing the interface.     

Some considerations on dynamic biem—II: Applications to two-dimensional fracture mechanics problems

Part I of this report, published in a detached part, the theoretical and numerical fundamentals of BIEM techniques in dynamic were presented.In Part II, numerical results for input problems obtained by BIEM, through a dynamic program of general applications, are shown.     

Some viscoelastic wave equations

The authors study the problem of existence, uniqueness and asymptotic behavior for t of (weak or strong) solutions of equations in the form% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% abaeqabaGaamyDamaaBaaaleaacaWG0bGaamiDaaqabaGccqGHsisl% cqaH7oaBcqqHuoarcaWG1bWaaSbaaSqaaiaadshaaeqaaOGaeyOeI0% YaaabCaeaacqGHciITcaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWG% Pbaabeaakiabeo8aZnaaBaaaleaacaWGPbaabeaakiaacIcacaWG1b% WaaSbaaSqaaiaadIhadaWgaaadbaGaamyAaaqabaaaleqaaOGaaiyk% aiabgUcaRaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGobaaniabgg% HiLdGccaWGMbGaaiikaiaadwhacaGGSaGaamyDamaaBaaaleaacaWG% 0baabeaakiaacMcacaqGGaGaeyypa0Jaaeiiaiaaicdaaeaaaeaaae% aaaeaacqaH7oaBcaqGGaGaeyyzImRaaeiiaiaaicdacaqGGaGaaiik% aiaadIhacaGGSaGaamiDaiaacMcacqGHiiIZcqGHPoWvcaqGGaGaae% iEaiaabccacaGGOaGaaGimaiaacYcacaWGubGaaiykaaqaaaqaaaqa% aaqaaiabgM6axjaabccacaqG9aGaaeiiaiaabggacaqGGaGaaeizai% aab+gacaqGTbGaaeyyaiaabMgacaqGUbGaaeiiaiaabMgacaqGUbGa% aeiiaiabl2riHoaaCaaaleqabaGaamOBaaaakiaacYcaaaaa!879A!\[\begin{array}{l}u_{tt} - \lambda \Delta u_t - \sum\limits_{i = 1}^N {\partial /\partial x_i \sigma _i (u_{x_i } ) + } f(u,u_t ){\rm{ }} = {\rm{ }}0 \\\\\\\\\lambda {\rm{ }} \ge {\rm{ }}0{\rm{ }}(x,t) \in \Omega {\rm{ x }}(0,T) \\\\\\\\\Omega {\rm{ = a domain in }}^n , \\\end{array}\]with various boundary and initial conditions on u(x, t). The case >0 corresponds to a nonlinear Voigt model (for nonlinear). The case =0, N=1 and f(u, u ₁ )=|u ₁ | sgn u ₁ , 0<<1 with nonhomogeneous boundary conditions corresponds to the motion of a linearly elastic rod in a nonlinearly viscous medium. The method followed is the Galerkin method.

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Résumé En suivant la méthode de Gaberkin, les auteurs ont étudié le problème de l'existence, de l'unicité et du comportement asymptotique lorsque t des solutions des équations d'état des ondes visco-élastiques, pour diverses conditions initiales et aux limites de U (x, t). On analyse les cas auxquels correspondent des valeurs positives (modèle non linéaire de Voigt) ou nulle du paramètre, ce dernier cas étant représentatif du mouvement d'un barreau élastique linéaire dans un milieu visqueux non linéaire, monyennant l'adoption de diverses conditions aux limites.

     

Some comments on the effect of biaxial stress on fatigue crack growth and fracture

The influence of biaxial loading on plastic zone size and crack opening displacement has been examined. Loading parallel to the crack plane was accounted for in satisfying the yield criteria, proposed by Von Mises, along the crack plane. Comparing plastic zone sizes for biaxial and uniaxial loading, when the applied stress normal to the crack plane is the same in both cases, shows a reduction of up to 65 per cent in plastic zone size for the biaxial loading case. The crack opening displacement is also reduced up to 33 per cent for biaxial loading. Based on critical crack opening displacement, fatigue crack propagation rates would be lower than for uniaxial loading, and an apparent increase in fracture toughness would also occur. The theoretical predictions are supported by a limited amount of experimental data.