Actual Boundary Conditions Research Articles

Response of stiffened and unstiffened plates subjected to blast loading

This paper describes the results of dynamic analyses carried out on both stiffened and unstiffened panels using both simplified and advanced analytical techniques. For unstiffened panels with in-plane restraint along their edges, the dynamic response of an imperfect panel was predicted using a large displacement elastic analysis based on Lagrange's equation, with the panel being treated as a shallow shell. For stiffened panels, the finite element (FE) technique was used to establish the validity of using the simplified technique to predict the inter-stiffener panel displacements for a simply supported panel. A parametric study has been carried out to analyse the effects of in-plane boundary conditions, local stiffener buckling and initial imperfections on the overall response. The significant effect of boundary conditions is demonstrated by including the actual boundary conditions of a test frame in the finite element modelling of a large-scale stiffened floorplate panel used in an experimental test series.

Anwendungen von Tensorfunktionen in der Kontinuumsmechanik anisotroper Materialien

During the last three decades much effort has been devoted to the elaboration of phenomenological theories describing the relation between force and deformation in bodies of materials which do not obey either the linear laws of the classical theories of elasticity or the hydrodynamics of viscous fluids. Such problems will play a central role for mathematicians, physicists, and engineers also in the future [1]. - Material laws and constitutive theories are the fundamental bases for describing the mechanical behaviour of materials under multi-axial states of stress involving actual boundary conditions. In solving such complex problems, the tensor function theory has become a powerful tool. This paper will provide a short survey of some recent advances in the mathematical modelling of materials behaviour including anisotropy and damage. The mechanical behaviour of anisotropic solids (materials with orientated internal structures, produced by forming processes and manufacturing procedures, or induced by permanent deformation) requires a suitable mathematical modelling. The properties of tensor functions with several argument tensors constitute a rational basis for a consistent mathematical modelling of complex material behaviour, This paper presents certain principles, methods, and recent successful applications of tensor functions in solid mechanics. The rules of specifying irreducible sets of tensor invariants, and tensor generators of material tensors of rank two and four are also discussed. Furthermore, it is very important to determine the scalar coefficients in constitutive and evolutional equations as functions of the integrity basis and experimental data. It is explained in detail that these coefficients can be determinded by using tensorial interpolation methods. Some examples for practical use are discussed. Finally, we have carried out our own experiments in order to examine the validity of the mathematical modelling. - Like applications in solid mechanics, tensor functions also play a significant role in mathematical modelling in fluid mechanics. This paper, however, is restricted to the mechanical behaviour of solids.

Effect of Mechanical Boundary Conditions on Phase Diagrams of Epitaxial Ferroelectric Thin Films

A phenomenological thermodynamic theory of ferroelectric thin films epitaxially grown on cubic substrates is developed using a new form of the thermodynamic potential, which corresponds to the actual mechanical boundary conditions of the problem. For single-domain ${\mathrm{BaTiO}}_{3}$ and ${\mathrm{PbTiO}}_{3}$ films, the ``misfit-temperature'' phase diagrams are constructed. It is found that the 2D clamping of the films, apart from a shift of the temperature of the ferroelectric transition, results in a change of its order. A change of the sequence of the phases and the appearance of phases forbidden in the bulk crystals are predicted.

Damage effect on the fracture toughness of nodular cast iron: Part-II. Damage zone characterization ahead of a crack tip

In order to understand the fracture toughness of nodular cast iron, the damage zone was studied by Scanning Electron Microscope (SEM) observations of the polished surface of a CT 25 specimen before and after ductile tearing. Damage is defined as decohesion at the graphite/matrix interface. It is shown that the damage zone is very large in nodular cast iron (almost throughout the whole remaining ligament ahead of the crack tip), so linear elastic fracture mechanics (LEFM) are not valid for small specimens. The size of the damage zone was calculated analytically by introducing a damage initiation criterion which was based both on observations of the debonding of the interface between matrix and graphite nodules and on measurements of the pressure sensitivity of cast iron. To take into account the actual boundary conditions, the damage zone was also calculated by numerical modeling using the modified Gurson’s model and by considering the nodular cast iron as a porous material. The calculated results led to good agreement with the damage zone observations. Plane stress and plane strain calculations yielded nearly the same size plastic zone. This result is opposite to those obtained for fully dense materials.

FLOW MODELING OF A POWER‐LAW FLUID IN A FULLY‐WIPED, CO‐ROTATING TWIN‐SCREW EXTRUDER1

ABSTRACTThe two‐dimensional flow theory of a power law fluid in a fully‐wiped co‐rotating twin‐screw extruder was investigated under steady‐state, isothermal conditions. The important distinction between this approach and previously published theories (Denson and Hwang 1980; Booy 1980; Wang and White 1989; Lai‐Fook et al. 1989, 1991) is that actual boundary conditions were used instead of boundary conditions based on the parallel plate model developed for single screw extruders. The generalized screw curves constructed for both double‐lead and single‐lead screw elements were similar in shape to those published for single screw extruders with rectangular channels. Noticeable increases in the closed discharge pressure and the open discharge flow rate were observed, which are the result of the screw geometry. Experimental results with corn meal were in agreement with theoretical predictions for single‐lead screw elements. Theoretical predictions for double‐lead screw elements using the new solution also showed reasonable agreement with experimental data from published literature after the incorporation of the intermeshing effect.

Confined compression of articular cartilage: Linearity in ramp and sinusoidal tests and the importance of interdigitation and incomplete confinement

Experimental and theoretical methods were used to investigate the linearity of the stress response of articular cartilage to ramp and sinusoidal tests in confined compression, as well as the role of cartilage-porous platen and lateral confinement boundary conditions in determining material responses. With respect to linearity, we posed the question as to whether the elicited stress responses to ramp compression, ramp release and sinusoidal tests were similar. With respect to boundary conditions we inquired as to the necessity of specifying a detailed interdigitating contact with the porous filter and of specifying the level of confinement present at the lateral edge of the disk. We found that the stress responses to the three types of tests were dissimilar, with ramp compression the only test exhibiting linear behavior. Ramp release from a static compression offset was non-linear in a manner such that the cartilage maintained a compressive stress higher than expected by a linear theory. Sinusoidal compression also displayed a non-linear response consistent with the presence of a release phase in each cycle. The actual boundary conditions present at the cartilage/porous-filter interface were visualized histologically. Areas (tens of microns) of cartilage in contact with the metal of the filter were interspersed with areas expanded into the pores of the filter. Finite-element analysis incorporating this information suggested that precise specification of this interface and of the level of the extent of lateral confinement would be necessary for the estimation of material properties, such as the hydraulic permeability, from these tests. The trends of the linearity studies did not appear to be significantly affected by the problems posed by these difficult to quantitate boundary conditions. The non-linear cartilage response to release and sinusoidal displacements therefore appear to be physiologically interesting. The maintained, that is higher than would be linear, compressive stress observed during release may be a beneficial adaptation to repeated loading where temporal variations in tissue stresses would be minimized.

Closed form analytic solutions describing glow discharge plasma

On the basis of an analytic model developed previously [S. T. Pai, J. Appl. Phys. 71, 5820 (1992)], an improved version of the model for the description of dc glow discharge plasma was successfully developed. A set of closed form solutions was obtained from the governing equations. The two-dimensional, analytic solutions are functional and completely satisfy the governing equations, the actual boundary conditions, and Maxwell equations. They can be readily used to carry out numerical calculations without the necessity of employing any assumed boundary conditions. Results obtained from the model reveal that as the discharge gap spacing or pressure increases the maximum value in the electron density distribution moves toward the cathode. At a sufficiently large value of gap spacing, the positive column phenomenon begins to appear in the discharge region. The model has the capability of treating the positive column and negative glow as a continuous system without the necessity of studying them separately. The model also predicts a sharp rise of the positive ion density near the cathode and field reversal in the anode region. Variation of the electrode radius produces little effect on the axial spatial distribution of physical quantities studied.

A hybrid micro-macro BEM with micro-scale inclusion-crack interactions

Local analysis schemes capable of detailed representations of micro-features of a problem are integrated with a macro-scale BEM technique capable of handling finite geometries and realistic boundary conditions. This paper focuses on micro-scale interactions among cracks and inclusions as well as their ramifications on macro-scale damage evaluations. The micro-scale effects are introduced into the macro-scale BEM computations through an augmented fundamental solution obtained from an integral equation representation of the micro-scale features. The proposed hybrid micro-macro BEM formulation allows complete decomposition of the real problem into two sub-problems, one residing entirely at the micro-level while the other resides at the macro-level. This allows for investigations of the micro-structural attributes while retaining the macro-scale geometric features and actual boundary conditions for the structural component under consideration. As a first attempt, dilute inclusion densities with strong inclusion-crack and crack-crack interactions are considered. The numerical results obtained from the hybrid BEM analysis establish the accuracy and effectiveness of the proposed micro-macro computational scheme for this class of problems. The proposed micro-macro BEM formulation can be easily extended to investigate the effects of other micro-features (e.g., interfaces, short or continuous fibers, in the context of linear elasticity) on macro-scale failure modes observed in structural components.

Field distribution in cable terminations from a quasi-static approximation of the Maxwell equations

A new model for the evaluation of the electric field in a cable termination realized through a nonlinear stress control tube (SCT), is presented in this paper. It is based on the electro-quasistatic approximation of the Maxwell equations: the Laplace equation describes the field in the nonconducting regions whereas a diffusion-like equation gives the field dynamics in the stress control tube. A numerical model is devised by solving the Laplace equation by finite difference and diffusion equations by the Galerkin method. It is shown that even the well-known RC transmission line model can be derived from this general approach. The underlying approximations leading to the circuital model are discussed in detail. The proposed model, in contrast with the circuital one, allows us to take into account properly the nonlinear SCT characteristics and the actual boundary conditions: in this way both spatial and temporal effects of the nonlinearity are-considered. The numerical results obtained by considering the general field approach and by using the transmission line model are compared.

The modal density of anisotropic structural components

The modal density of a structural component is a key parameter in high-frequency vibration prediction techniques such as statistical energy analysis (SEA). A study of the literature reveals that the standard method of computing the modal density of a two-dimensional component, such as a plate or shell, is restricted to orthotropic components. This method (which is usually referred to as Courant’s method) is extended here to a generally anisotropic component by considering initially the case of periodic (or Born–von Kármán) boundary conditions. It is then shown that the resulting modal density is independent of the nature of the actual boundary conditions which act on the component. On a related issue, doubts have been expressed in the literature as to whether the results yielded by Courant’s method are applicable to general boundary conditions if the component exhibits degeneracy of the dynamic edge effect; the present analysis demonstrates that the method is in fact valid in such cases. The developed technique is applied to the bending vibrations of an anisotropic plate, and good agreement is found with empirical results for the modal density derived from natural frequency computations.

Convection naturelle en cavité carrée différentiellement chauffée: investigation expérimentale à Ra= 1,69 × 10 9

Experimental investigations were conducted in a cubic air filled cavity ( L = 1.04 m, H = 0.94 m, P = 0.70 m). The cavity was equipped with two opposing differentially heated vertical walls maintained at uniform temperatures, with two insulated horizontal walls and two adiabatic lateral vertical walls. The experimental apparatus, designed to obtain well controlled boundary conditions, is reported in details. An examination of temperature fields in the vicinity of the ceiling and the floor allowed the characterization of the actual boundary conditions on these walls. An analytic law was determined, giving the temperature distribution along each horizontal wall. Flow visualizations, using laser tomography and spectral analysis of time dependent signal of the temperature recorded at several characteristic points in the cavity, were performed. It pointed out the existence of unsteadiness and revealed a complex interaction between internal gravity waves, thermal instabilities along the floor and Tollmien-Schlichting waves in the hot vertical boundary layer. Heat transfer along the active walls was evaluated from temperature measurements taken very close to the walls.

Modeling of flow in a single screw extruder

A new analytical solution of an isothermal, Newtonian flow in a single screw extruder with a finite channel is developed with the actual boundary conditions encountered. Down channel velocity distributions are presented in three-dimensional plots. The boundary conditions, velocity distributions, and screw characteristics predicted by the new solution are tested using the experimental data from published literature (Choo et al., 1980, Polymer Engineering and Science, 20, 349-56; Griffith, 1962, Industrial and Engineering Chemistry Fundamentals, 1, 180-7). The results are found to be more accurate than existing theories (Rowell & Finlayson, 1922, Engineering, 126, 249-87; Tadmor & Gogos, 1979, Principles of Polymer Processing, Wiley; Rauwendaal, 1986, Polymer Extrusion, Hanser).

A hybrid BEM formulation for multiple cracks in orthotropic elastic components

Local analysis schemes capable of detailed representations of micro-features (e.g. micro-crack interactions) of a problem are integrated with a macro-scale BEM technique capable of handling orthotropic elastic components with complex finite geometries and realistic boundary conditions. The micro-scale effects are introduced into the macro-scale BEM analysis through an augmented fundamental solution obtained from an integral equation representation of the micro-scale features. The proposed hybrid micro-macro BEM formulation allows decomposition of the complete problem into two sub-problems, one residing entirely at the micro-level and the other at the macro-level. This allows for investigations of the effects of the micro-structural attributes while retaining the macro-scale geometric features and actual boundary conditions for the component or structure under consideration. As a first attempt, elastic fracture mechanics problems with interacting cracks at close spacings are considered. The proposed micro-macro BEM formulation can easily be extended to investigate the effects of other micro-features (e.g. interfaces, short or continuous fiber reinforcements, voids, and inclusions, in the context of linear elasticity) on macroscopic failure modes observed in structural components.

A hybrid micro–macro BEM formulation for micro‐crack clusters in elastic components

AbstractLocal analysis schemes capable of detailed representations of the micro‐features of a problem are integrated with a macro‐scale BEM technique capable of handling complex finite geometries and realistic boundary conditions. The micro‐scale effects are introduced into the macro‐scale BEM analysis through an augmented fundamental solution obtained from an integral equation representation of the micro‐scale features. The proposed hybrid micro‐macro BEM formulation allows decomposition of the complete problem into two sub‐problems, one residing entirely at the micro‐level and the other at the macro‐level. This allows for investigations of the effects of the micro‐structural attributes while retaining the macro‐scale geometric features and actual boundary conditions for the component or structure under consideration. As a first attempt, elastic fracture mechanics problems with interacting cracks at close spacings are considered. The numerical results obtained from the hybrid BEM analysis establish the accuracy and effectiveness of the proposed micro–macro computational scheme for this class of problems. The proposed micro–macro BEM formulation can easily be extended to investigate the effects of other micro‐features (e.g. interfaces, short or continuous fibre reinforcements, voids and inclusions, in the context of linear elasticity) on macroscopic failure modes observed in structural components.

Selection of 3D Geostatistical Reservoir Representations Based on Dynamic Production Criteria: ABSTRACT

Geostatistical models allow to take into account uncertainties in reservoir description. In this way, several possible reservoir images are generated, and must be considered when evaluating production forecasts. Nevertheless, flow simulations are too expensive to be performed on a high number of geostatistical representations; in addition, one can expect that some images have the same production behavior. Hence, reservoir images representative of the possible production behaviors of the reservoir must be selected. In this paper, we propose a selection method based on dynamic criteria: simplified and much more less expensive 3D flow simulations are performed on many geostatistical representations. Production results are obtained and representative reservoir images are selected by considering these results. Simplified flow simulations consist in computing only one constant pressure field corresponding to a steady-state flow within the reservoir. Actual boundary conditions are kept unchanged. Saturations calculations take into account multiphase parameters. Production results obtained by simplified flow simulations are wrong, but allow to obtain a right sorting of the images for cases belonging to the validity domain of the method. The selection method has been applied on reservoir cases corresponding to di- or triphasic fluid flow. These tests have shown that the proposed selection method ismore » valid for diphasic water-oil fluid flow with no significant capillary and gravity effects compared to viscous effects. The proposed selection method based on low-cost simplified fluid flow simulations is one efficient solution to the problem of dealing with multiple geostatistical representations. And this approach could be advantageously mixed with homogenization methods.« less

Observation of dihedral transverse patterning of Gaussian beams in nonlinear optics.

We present experimental evidence that the spontaneous transverse filamentation of a laser beam back reflected onto a thin Kerr optical medium can be strongly affected by the actual boundary conditions, namely the Gaussian shape of the beam. As the filament spacing is made to increase by moving the feedback mirror, well-known hexagonal patterns no longer grow up but rather the laser spot breaks up into lower dihedral symmetry patterns, such as pentagonal, rhombic, or triangular ones, in very good agreement with previous numerical predictions. Further increasing the laser intensity after the symmetry breaking leads to secondary bifurcations. The multistability, itineracy, and turbulence of those structures were also observed.

Three‐Dimensional Model for Wood‐Pole‐Strength Predictions

A three‐dimensional finite element model was developed to predict the strength and failure location of nine wood transmission poles made from three commonly used North American species. All poles were 1,524–1,829 cm (50–60 ft) long, and were tested to failure as cantilever beams. The analysis methodology involved considering several 45.7‐cm‐ (18‐in.‐) long segments, located along the pole length, that contained the most severe inherent characteristics. Each segment was analyzed recognizing actual material properties and appropriate boundary conditions. Important material properties for each segment were determined by measuring clear‐wood elastic and strength parameters in undamaged sections taken from tested poles. The flow‐grain model was used to describe the grain deviation around knots. For the nine poles studied, predicted and experimental strength values differed on average by 7%. The correct location was predicted in six of the nine poles studied. Where failure location was not predicted correctly, ...

Experimental investigation of capillarity effects on surface gravity waves: non-wetting boundary conditions

Damping and eigenfrequencies of surface capillary—gravity waves greatly depend on the boundary conditions. To the best of our knowledge, so far no direct measurement has been made of the dynamic behaviour of the contact angle at the three-phase interface (fluid—vapour—solid walls) in the presence of surface oscillation. Therefore, theoretical models of surface gravity–capillary waves involve ad hoc phenomenological assumptions as far as the behavior of the contact angle is concerned. In this paper we report a systematic experimental investigation of the static and dynamic properties of surface waves in a cylindrical container where the free surface makes a static contact angle $\theta_{\rm c} = 62^{\circ}$ with the vertical walls. The actual boundary condition relating the contact angle to the velocity of the contact line is obtained using a new stroboscopic optical method. The experimental results are compared with the theoretical expressions to be found in the literature. Two different regimes are observed: (i) a low-amplitude regime, where the contact line always remains at rest and the contact angle oscillates during the oscillation of the free surface; (ii) a higher-amplitude regime, where the contact line slides on the vertical walls. The profile, the eigenfrequency and the damping rate of the first non-axisymmetric mode of the surface gravity waves are investigated. The eigenfrequency and damping rate in regime (i) are in satisfactory agreement with the predictions of the Graham-Eagle theory (1983) of pinned-end edge conditions. The eigenfrequency and damping rate in regime (ii) show a strongly nonlinear dependence on the oscillation amplitude of the free surface. All the experimental results concerning regime (ii) can be explained in terms of the Hocking (1987 a) and Miles (1967, 1991) models of capillary damping by introducing an ‘effective’ capillary coefficient $\lambda_{\rm eft}$. This coefficient is directly obtained for the first time in our experiment from dynamic measurements on the contact line. A satisfactory agreement is found to exist between theory and experiment.

FRACTURE OF A PLATE UNDER TRANSIENT THERMAL STRESSES

The cracking in a plate of finite thickness due to sudden thermal transient stresses is considered for an edge crack. The elastic strip is assumed to be insulated on one face and cooled suddenly on the face containing the edge crack. Even though the assumption of a step change in temperature on the boundary gives conservative results, it is not a realistic representation of the actual boundary condition. A ramp cooling function is assumed at the boundary, which is more realistic than the assumption of a step function. The inertia effects and the temperature dependence of the elastic constants are negligible. The thermal stresses for the uncracked problem are computed as a function of time. Then, these stresses are used as the crack surface tractions with opposite sign to formulate the mixed boundary value problem. This leads to a singular integral equation of Cauchy-type, which can be solved numerically. The numerical results for the stress intensity factor are computed as a function of the nondimensional time, the crack size, and the duration of cooling rate. Also, the temperature and the thermal stress distributions for the uncracked problem are included.

Numerical modeling of nonsteady transient processes in a sphere with cavity subjected to internal dynamic load

The article suggests a numerical algorithm for investigating shock-wave processes in a hollow sphere with limited thickness subjected to internal dynamic surface loads. On the example of actual boundary conditions the article evaluates the contribution of all groups of elastic waves to the formation of an overall structure of rapidly occurring wave processes within the medium under consideration. The effect of the boundary conditions on the distribution field of the components of displacements, speeds, strains and stresses across the hollow sphere upon multiple passage of stress waves was discovered, and some aspects of the realized numerical algorithm connected with the stability of the scheme were investigated.