Theoretical and Applied Mechanics (TAM) Technical Reports
http://hdl.handle.net/2142/179
TAM technical reports include manuscripts intended for publication, theses judged to have general interest, notes prepared for short courses, symposia compiled from outstanding undergraduate projects, and reports prepared for research-sponsoring agencies.Buckling and post-buckling behavior of a cylindrical shell subjected to external pressure
http://hdl.handle.net/2142/18754
Buckling and post-buckling behavior of a cylindrical shell subjected to external pressure
Langhaar, Henry Louis, 1909-
Boresi, Arthur Peter, 1924-
In an earlier report (TAM Report No. 80), the authors considered the buckling and post-buckling behavior of an ideal elastic cylindrical shell loaded by uniform external pressure on its lateral surface, and by an axial compressive force. Assumptions were introduced which reduced the shell to a system with one degree of freedom. The present investigation is a generalization and a refinement of this theory. The shell is treated as a system with 21 degrees of freedom. By the imposition of constraints on the 21 generalized coordinates, various end conditions can be realized; for example, simply supported ends with flexible end
plates (no axial constraint), simply supported ends with rigid end plates, and
clamped ends. Also, effects of reinforcing rings have been incorporated in a more
general way than in TAM Report No. 80. The restrictive assumption that the centroidal axis of a ring coincides with the middle surface of the shell has been eliminated.
A pressure-deflection curve for an ideal cylindrical shell that is loaded by external pressure has the general form shown in Figure 1. The falling part of the curve (dotted in the figure) represents unstable equilibrium
configurations. Also, the continuation of line OE (dotted) represents unstable
unbuckled configurations. Actually, the shell snaps from some configuration A to another configuration B, as indicated by the dashed line in Figure 1. Theoretically, point A coincides with the maximum point E on lhe curve, but initial imperfections and accidental disturbances prevent the shell from reaching this point. Point E is the buckling pressure of the classical infinitesimal theory (called the "Euler crítical pressure", since Euler applied the
infinitesimal theory to columns). To some extent, point A is indeterminate, but it is presumably higher than the minimum point C unless the shell has excessive initial dents or lopsidedness. In TAM Report No. 80. a
hypothesis of Tsien was used to locate point A. In the present investigation, point A is not considered. Rather, attention is focused on the development of a theory that will determine the en tire load-deflection curve.
For short thick shells, such as the inter-ring bays of a submarine hull, the Euler critical pressures, determined by TAM Report No. 80, are too high, presumably because the assumption that the shell buckles without
incremental hoop strain is inadmissible in this range. The present report corrects this error. Numerical data on the Euler critical pressures of shells with simply supported ends and flexible end plates have been obtained with the aid of lhe Illiac, an electronic digital computer. The data are tabulated at the end of this reporto For short shells without rings, the buckling pressures are appreciably lower than those determined by von Mises' theory. The
numerical data for the Euler buckling pressures of sheUs with uniformly spaced
reinforcing rings are sufficiently extensive to permit interpolation to estimate effects of various ring sizes. Some exploratory numerical investigations of post-buckling behavior have been conducted with the Illiac. lt is not feasible, at the present time, to handle nonlinear equilibrium problems for systems with 18 degrees of freedom. Consequently, for the numerical work, some higher harmonics were discarded so that the system was reduaed to 7 degrees of freedom.
Even then, the numerical problem is formidable. The calculations were confined principally to the determination of the minimum point C on the post-buckling curve (Figure 1). The pressure at point C is the minimum pressure at which a buckled form can exist. It is found that the ordinate of point C, determined by TAM report No. 80, is somewhat too high. The two theories are compared by a table anq curves at the end of this report.
Buckling behavior
Sun, 01 Apr 1956 00:00:00 GMTThermoelastic properties of plain weave composites for multilayer circuit board applications
http://hdl.handle.net/2142/444
Thermoelastic properties of plain weave composites for multilayer circuit board applications
Brown, Eric N.
The thermoelastic properties of several common woven glass/epoxy substrate materials were characterized in both the warp and fill directions. Five common commercially pressed cores (1080, 2116, 2313, 3313, 7628) were obtained from Polyclad, Inc. The cores consisted of either one or two plies of C-staged woven glass epoxy substrate sandwiched between 1 ounce copper cladding. After the copper was removed from the cores, samples were cut for either mechanical property characterization or microstructural analysis with the test axis lying along either the warp or fill direction. The crimp angle and relative fiber volume fraction of each fabric was first determined from photomicrographs of polished cross-sections.
Next, Young's modulus was measured via standard tension tests at room temperature. The storage and loss moduli were then measured as functions of temperature using dynamic mechanical analysis (DMA). Finally, the coefficients of thermal expansion were determined using constant force thermal mechanical analysis (TMA) measurements.
All of the substrates showed significant differences in microstructure and material properties between the warp and fill directions. Most of the laminates had a much lower crimp angle in the warp direction which resulted in a higher modulus and lower coefficient of thermal expansion than the fill direction. Of the cores investigated, the properties of 3313 were the most balanced.
Thermoelasticity
Thu, 01 Jan 1998 00:00:00 GMTCyclic Deformation and Fatigue Behavior of Hardened Steels
http://hdl.handle.net/2142/441
Cyclic Deformation and Fatigue Behavior of Hardened Steels
Landgraf, R.W.
Changes in deformation resistance are studied during completely reversed strain cycling of steels hardened to yield strengths in excess of 200 ksi by quenching and tempering, quenching and deforming at elevated temperature, ausforming and maraging.
Untempered steel and ausformed steel exhibit cyclic hardening; slightly tempered steel is cyclically stable. Varying amounts of cyclic
softening occur in intermediate hardness quenched and tempered steel, quenched and deformed steel and maraging steel. Such cyclically induced changes can be predicted from a steel's monotonic strain hardening exponent and are characterized in terms of a cyclic stress -strain curve.
Log-log linear relations between elastic strain and fatigue life and plastic strain and fatigue life adequately describe the fatigue behavior of hardened steels. Monotonic true fracture strength and ductility approximate
intercept values in the relations thus providing reliable indications of a steel's
fatigue resistance. The optimum condition for maximum fatigue resistance shifts from high hardnesses at long lives, where strength is the determining factor, to lower hardnesses at shorter lives, where ductility becomes more
important.
Trends between structure and cyclic behavior are discussed along with approaches for attaining improved fatigue resistance in steel.
Mechanics
Metals
Steel
Fri, 01 Nov 1968 00:00:00 GMTSharp-interface nematic-isotropic phase transitions with flow
http://hdl.handle.net/2142/346
Sharp-interface nematic-isotropic phase transitions with flow
Fried, Eliot
We present a sharp-interface theory for phase transformations between the isotropic and uniaxial nematic phases of a flowing liquid crystal. These equations include a supplemental interface condition which expresses the balance of configurational momentum. As an application, we study the behavior a spherical isotropic drop surrounded by a radially-oriented nematic phase. For this problem, the bulk and interfacial equations collapse to a single nonlinear second-order ordinary differential equation for the radius of the droplet. We determine and study the stability of the corresponding equilibria. In addition, we find that motion of the interface generates a backflow in the nematic phase. Our analysis indicates that a backflow measurement can be used to determine independently the density difference between the isotropic and uniaxial nematic phases.
nematic-isotropic phase transitions
Thu, 01 Jun 2006 00:00:00 GMTStability of an evaporating thin liquid film
http://hdl.handle.net/2142/347
Stability of an evaporating thin liquid film
Shklyaev, Oleg E.
Fried, Eliot
We use a newly developed set of boundary conditions to revisit the problem of an evaporating thin liquid film. In particular, instead of the conventional Hertz–Knudsen–Langmuir equation for the evaporation mass flux we impose a more general equation expressing the balance of configurational momentum. This balance, which supplements the conventional conditions enforcing the balances of mass, momentum, and energy on the film surface, arises from a consideration of configurational forces within a thermodynamical framework. We study the influence of newly introduced terms such as the effective pressure, encompassing disjoining and capillary components, on the evolution of the liquid film. We demonstrate that this quantity affects a time-dependent base state of the evaporating film and is an important factor in applications involving liquid films with thicknesses of one or two monolayers. These factors lead to a revised understanding of the stability of an evaporating film. Dimensional considerations indicate that the most significant influence of these effects occurs for molten metals.
Boundary conditions
Thu, 01 Jun 2006 00:00:00 GMTStatistical and structural similarities between micro- and macro-scale wall turbulence
http://hdl.handle.net/2142/345
Statistical and structural similarities between micro- and macro-scale wall turbulence
Natrajan, Vinay K.
Yamaguchi, Eiichiro
Christensen, Kenneth T.
Microscopic particle-image velocimetry (micro-PIV) measurements are made in the streamwise–wall-normal plane of a 536-micron capillary at Re = 4500 to study the statistical and structural features of wall turbulence at the microscale. Single-point velocity statistics, including the mean velocity profile, the root-mean-square streamwise and wall-normal velocities, and the Reynolds shear stress profile, agree well with established direct numerical simulations of turbulence in the same geometry at Re = 5300. This consistency validates the efficacy of micro-PIV as an experimental tool for studying instantaneous, and even turbulent, flow behavior at the microscale. The instantaneous micro-PIV velocity fields reveal spanwise vortices that streamwise-align to form larger-scale interfaces that are inclined slightly away from the wall. These observations are entirely consistent with the signatures of hairpin vortices and hairpin vortex packets that are often noted in instantaneous PIV realizations of macroscale wall turbulence. Further, two-point velocity correlations and estimates of the conditionally-averaged velocity field given the presence of a spanwise vortex indicate that hairpin structures and their organization into larger-scale vortex packets are statistically-significant features of wall turbulence at the microscale.
wall turbulence
Sat, 01 Apr 2006 00:00:00 GMTSpatial signatures of retrograde spanwise vortices in wall turbulence
http://hdl.handle.net/2142/344
Spatial signatures of retrograde spanwise vortices in wall turbulence
Natrajan, Vinay K.
Wu, Yanhua
Christensen, Kenneth T.
The spatial signatures of retrograde spanwise vortices in wall turbulence are assessed from particle-image velocimetry measurements in the streamwise–wall-normal plane of a zero-pressure-gradient turbulent boundary layer at Ret = 2350. The present results suggest that a portion of retrograde spanwise vortices have a well-defined spatial relationship with neighboring prograde vortices. Two-point cross-correlations and conditionally averaged velocity fields given a retrograde vortex reveal that such structures are typically oriented either upstream/below or downstream/above a prograde core. While the former preferred orientation is consistent with the typical-eddy patterns reported by Falco and co-workers, we offer an alternative interpretation for a portion of these retrograde/prograde pairs. In particular, the arrangement of a retrograde core upstream and below of a prograde core is also consistent with the spatial signature revealed if an omega-shaped hairpin structure were sliced through its shoulder region by a fixed streamwise–wall-normal measurement plane.
wall turbulence
Sat, 01 Apr 2006 00:00:00 GMTA numerical method for a second-gradient theory of incompressible fluid flow
http://hdl.handle.net/2142/343
A numerical method for a second-gradient theory of incompressible fluid flow
Kim, Tae-Yeon
Dolbow, John E.
Fried, Eliot
This work concerns the development of a finite-element method for discretizing a recent second-gradient theory for the flow of incompressible fluids. The new theory gives rise to a flow equation involving higher-order gradients of the velocity field and introduces an accompanying length scale and boundary conditions. Finite-element methods based on similar equations involving fourth-order differential operators typically rely on C1-continuous basis functions or a mixed approach, both of which entail certain implementational difficulties. Here, we examine the adaptation of a relatively inexpensive, nonconforming method based on C0-continuous basis functions. We first develop the variational form of the method and establish consistency. The method weakly enforces continuity of the vorticity, traction, and hypertraction across interelement boundaries. Stabilization is achieved via Nitsche’s method. Further, pressure stabilization scales with the higher-order moduli, so that a classical formulation is recovered as a particular limit. The numerical method is verified for the problem of steady, plane Poiseuille flow. We then provide several numerical examples illustrating the robustness of the method and contrasting the predictions to those provided by classical Navier–Stokes theory.
Sat, 01 Apr 2006 00:00:00 GMTSquire's theorem for the Rayleigh–Taylor problem with a phase boundary
http://hdl.handle.net/2142/342
Squire's theorem for the Rayleigh–Taylor problem with a phase boundary
Chen, Xuemei
Fried, Eliot
We consider the RayleighTaylor problem with a phase transformation. For simplicity, we restrict our attention to base states in which the interface convects with the phases, so that mass exchange between the phases occurs only in response to a disturbance. We find that every unstable three-dimensional disturbance (involving a pair of modes transverse to the interface) is associated with a more unstable two-dimensional disturbance (involving a single mode transverse to the interface) at lower values of the Weber, Froude, Reynolds, Voronkov, and Gurtin numbers. This constitutes an appropriate version of Squires theorem.
Wed, 01 Mar 2006 00:00:00 GMTInertial effects on rotating flow in a porous layer
http://hdl.handle.net/2142/340
Inertial effects on rotating flow in a porous layer
Riahi, Daniel N.
Inertial effects on flow instabilities in a horizontal reactive porous layer with deformed upper boundary are studied using a linear stability analysis and under the condition that the porous layer, which is also referred to as a dendrite or mushy layer, is rotating about an inclined axis during the solidification of a binary alloy. The linear stability analysis leads to new results about the effects of the inertial force on the existence and the number of the oscillatory modes and on the preference of either left- or right-traveling longitudinal rolls, which can depend on the angle of inclination ã of the rotation axis with respect to the vertical axis. For some 0° < ã < 90° and for the rotation rate beyond some particular value, the preferred flow solution in the form of left-traveling rolls can be replaced by the one in the form of right-traveling rolls or vice-versa. The preferred flow pattern, the period of oscillation of the flow solution, the critical Rayleigh number and the shape and structure of the deformed upper boundary of the layer are found to depend on the inertial effect.
Wed, 01 Feb 2006 00:00:00 GMT