Wedge Forces Research Articles

Net Worth, Risk Taking and Investment

This paper analyzes how agency problems in financial contracting determine risk-taking and investment. In perfect capital markets a risk-neutral firm would invest until the expected marginal return equals the interest rate. However, as firms with little net-worth face agency cost in financial contracting the shadow cost of external funding are increased. It is often argued that this wedge forces firms to underinvest (the 'balance-sheet-effect'). However, this claim is premature. Firms with low net worth, hence high cost of external financing, might even overinvest. We establish a robust though more subtle balance sheet effect. Agency problems in financial contracting generate risk avoidance at the firm level even when all actors are risk neutral. This may explain why during major economic downturns firms and banks shun risks that could be accepted in better times.

Approximate Green's functions for singular and higher order terms of an edge crack in a finite plate

An edge crack in a finite plate (FSECP) subjected to wedge forces is solved by the superposition of the analytical solution of a semi-infinite crack, and the numerical solution of a FSECP with free crack faces, which is solved by the Williams expansion. The unknown coefficients in the expansion are determined by a continuous least squares method after comparing it with the direct boundary collocation and the point or discrete least squares methods. The results are then used to validate the stress intensity factor (SIF) formula provided by Tada et al. that interpolates the numerical results of Kaya and Erdogan, and an approximate crack face opening displacement formula obtained in this paper by Castigliano's theorem and the SIF formula of Tada et al. These approximate formulae are accurate except for point forces very close to the outer edge, and can be used as Green's functions in the crack-closure based crack growth analysis, as well as in interpreting the size effect of quasi-brittle materials. Green's functions for coefficients relevant to the second to the fifth terms in the crack tip asymptotic field are also provided. Finally, a FSECP with a uniform pressure over a part of the crack faces is solved to illustrate the application of the obtained Green's functions and to further assess their accuracy by comparing with a finite element analysis.

Mode I intra-laminar crack growth in composites — modelling of R-curves from measured bridging laws

In this paper R-curves for mode I crack growth in composites are modelled based on measured bridging laws. It is shown that simulated and measured R-curves are in good agreement. Simulations show that variations in the measured bridging law parameters can explain the scatter in overall R-curves. Finite element procedures for treating a generalised nonlinear law for intra-laminar fibre bridging (longitudinal splitting) in combination with R-curve modelling are demonstrated for mode I loading. The difference between calculating the crack growth resistance by linear elastic fracture mechanics and by the J integral for the double cantilever beam specimen loaded by wedge forces is elucidated. It is shown that calculating the crack growth resistance by linear elastic fracture mechanics results in overestimation of the steady-state crack growth resistance.

Evaluation of bimaterial stress intensity factors using a finite element-boundary element alternating method

The finite element-boundary element alternating method is extended to the analysis of a crack in a bimaterial test specimen. The crack is perpendicular to, and terminates at, the interface between two dissimilar. The finite method is used to model the finite-size specimen without the crack. The displacement discontinuity method (boundary element method) is used to find the stresses in the infinite bimaterial medium due to crack surface tractions. The stress intensity factors are extracted by using a closed form solution for a semi-infinite crack in an infinite bimaterial medium opened by a pair of wedge forces. The alternating method is verified by solving bimaterial crack problems for which solutions are known. A parametric study is performed for a specimen proposed for evaluating the fracture toughness of bimaterial interfaces.

Delamination R-curve phenomena due to damage

Resistance to delamination in composites can be enhanced by a variety of bridging mechanisms. The bridging zone size is usually several times the lamina thickness, so it is questionable to think of delamination resistance as a material property independent of specimen size and geometry. When measured with slender beams, the plateau resistance is found to be independent of the beam thickness. However, the steady-state bridging zone size increases with the beam thickness. Further implications of the large-scale bridging are studied using a family of steady-state, mixed-mode delamination beams, in conjunction with an idealized damage response. The complete solution is obtained for the model, which allows the R-curves to be constructed for given model parameters. The significance of the steady-state cracking, which is crucial in understanding delamination R-curves, is elucidated by contrasting double-cantilever beams loaded by moments and by wedge forces. As an inverse process, it is recommended that R-curves be used as an experimental probe to study localized damage response, such as polymer craze and interface separation.

抜歯時における歯根の形態差が歯根および歯槽骨に及ぼす力学的影響

The purpose of this study was to investigate mechanisms of tooth extraction, with special reference to unusual root shapes. Stress analysis with the use of finite element method was carried out to clarify the mechanical effects of tooth root morphology on the tooth and alveolar bone in tooth extraction, under the cantilever and wedge forces as extraction force. Finite element models of single root and double roots were developed as controls, and both models were modified from average shape to unusual shapes, according to the classification of oral pathology.The results obtained were as follows:1. The effects of the cantilever force on the root were lower than in the case of the wedge force.2. Stresses generated in the root tip area of various models in which the resistance to the loading incleased were higher than those of control model, according to the root morphology of the single root models.3. In the curved models of single root, stresses induced in the root tip area were lower under the loading from the outer side of root curvaturer than from the inner side.4. In the double roots model, stresses generated in the root tip area of the divergence model were higher than those of convergence model.From the above results mechanical effects of tooth root morphology on the tooth root and alveolar bone were clarified. These results also support the clinical procedures of tooth extraction based on clinical experience.

STRESS ANALYSIS OF CONCRETE SUBJECTED TO SPLITTING BOND ACTION

In order to investigate the splitting crack patterns due to the wedge action and dowel action of deformed bars and estimate the splitting bond strength, the two-dimensional non-linear finite element analysis was carried out on reinforced concrete members. The wedge force or dowel force was replaced by uniform or non-uniform internal pressure, respectively, and the subject of analysis was limited to concrete only. Concrete cover and bar spacing were selected as analytical parameters.The analytical results show that three splitting crack patterns proposed by Jirsa and Morita: V-notch, side splitting and corner splitting type are also observed by this analysis, and that the dowel action aggravate especially the side splitting failure. On the basis of the analytical results, a tentative equation to estimate the splitting bond strength was proposed.

A new method for quantitative modelling of quasibrittle fracture

A slow-crack method for precise measurement of the stress intensity factor K I for cracks in bodies of complex shapes is proposed. It is based on registration of slow-crack growth under static loads in the model prepared of a polymer glass for which the fundamental dependence of K I on the crack rate is known. It is shown that PMMA glasses meet all the main requirements for model material. The comparison of experimental and theoretical values of K I for a central crack loaded by two wedge forces in rectangular and circular plates was carried out. It shows that the precision of K I measurements is no less than 2–3%. The proposed method may be effectively applied for solution of many other problems of the fracture mechanics.

An elastic strip with multiple cracks and applications to tapered specimens

In this paper an infinite elastic strip containing arbitrarily oriented cracks and subjected to uniform tension and a pair of concentrated forces is formulated in terms of a system of singular integral equations. Even though the technique is sufficiently general to solve any multiple crack problem, with the objective of applying the results to tapered specimens, only a certain symmetric crack geometry and loading conditions are considered. The stress intensity factors are calculated for edge cracks in the strip under uniform tension and for a “compact” and a “slender” tapered specimen (the latter simulating the double cantilever beam) under concentrated forces or crack surface wedge forces.

The Formation of Chips in the Penetration of Elastic-Brittle Materials (Rock)

An analysis is presented for the formation of chips in a two-dimensional, isotropic, elastic-brittle solid, due to the penetration of a rigid wedge. The analysis can be viewed in three parts. It is first necessary to obtain the static stress field due to a prescribed traction on the boundary. An integral method is used for this which is particularly convenient for the present problem. With the stresses known the field is searched to determine where fracture is initiated. Fracture is assumed to satisfy a Coulomb-Mohr fracture criterion. It is found that for certain realistic parameter ranges fracture initiation is somewhat insensitive to the details of traction distribution and wedge angle but is dependent on the material parameters. An approximate fracture path is proposed for stable fracture growth based on a certain fracture function. It is further proposed that the transition from stable to unstable fracture growth takes place at the inflection point of the fracture function versus fracture path length plot. Arguments are based on the Griffith theory of fracture and experimental data on crack velocity. Ratios of chip formation wedge forces to incipient fracture wedge forces are calculated from the theory and are of the order of 4. The results appear to be in reasonable quantitative agreement with some wedge penetration experiments on charcoal grey granite.

Environmental fatigue crack propagation of aluminum alloys at low stress intensity levels

Environmental fatigue crack propagation in 2024-T3, 7075-T6, and 7178-T6 has been studied at low levels of cyclic amplitude of stress intensity, ΔK. Both wedge force loading and remote loading techniques were employed to achieve the desired ΔK levels, and preliminary experiments were designed to test their compatibility. Testing was carried out in humid air, distilled water, and 3.5 pct sodium chloride solution, and the observed crack growth rates compared with those in desiccated air. Later studies were also conducted in an inert reference environment with a total water content of less than 2 ppm. When the data are plotted as log ΔK vs log d2a /dN, alloy 2024-T3 exhibits a marked slope transition, alloy 7075-T6 a slight slope transition, and alloy 7178-T6 a rectilinear behavior throughout the whole range of ΔK studied. The basic shape of these curves is discussed in terms of state-of-stress conditions at the crack tip, frequency effects, environmental effects, strain rate sensitivity, and metallurgical structure. An attempt is also made to correlate the rate of fatigue crack propagation in a particular environment and at a particular ΔK level with the fracture topography.