Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “surface tension effects”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 469 records · Page 26
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Simulation insights into wetting properties of hydrogen-brine-clay for hydrogen geo-storage

Hydrogen geo-storage is attracting substantial interdisciplinary interest as a cost-effective and sustainable option for medium- and long-term storage. Hydrogen can be stored underground in diverse formations, including aquifers, salt caverns, and depleted oil and gas reservoirs. The wetting dynamics of the hydrogen-brine-rock system are critical for assessing both structural and residual storage capacities, and ensuring containment safety. Through molecular dynamics simulations, we explore how varying concentrations of cushion gases (CO 2 or CH 4 ) influence the wetting properties of hydrogen-brine-clay systems under geological conditions (15 MPa and 333 K). We employed models of talc and the hydroxylated basal face of kaolinite (kaoOH) as clay substrates. Our findings reveal that the effect of cushion gases on hydrogen-brine-clay wettability is strongly dependent on the clay-brine interactions. Notably, CO 2 and CH 4 reduce the water wettability of talc in hydrogen-brine-talc systems, while exerting no influence on the wettability of hydrogen-brine-kaoOH systems. Detailed analysis of free energy of cavity formation near clay surfaces, clay-brine interfacial tensions, and the Willard-Chandler surface for gas-brine interfaces elucidate the molecular mechanisms underlying wettability changes. Our simulations identify empirical correlations between wetting properties and the average free energy required to perturb a flat interface when clay-brine interactions are less dominant. Here, our thorough thermodynamic analysis of rock-fluid and fluid-fluid interactions, aligning with key experimental observations, underscores the utility of simulated interfacial properties in refining contact angle measurements and predicting experimentally relevant properties. These insights significantly enhance the assessment of gas geo-storage potential. Prospectively, the approaches and findings obtained from this study could form a basis for more advanced multiscale simulations that consider a range of geological and operational variables, potentially guiding the development and improvement of geo-storage systems in general, with a particular focus on hydrogen storage.

25 ENERGY STORAGE↗
NASA NTRS1984

Evaluation of the effect of crack closure on fatigue crack growth of simulated short cracks

A test program was performed to determine the influence of crack closure on fatigue crack growth (FCG) rates of short cracks. By use of the standard compact tension specimen, test procedures were devised to evaluate closure loads in the wake of the crack behind its tip. The first procedure determined the magnitude of crack closure as a function of the fatigued crack wave by incrementally removing the contacting wake surfaces and measuring closure load at each increment. The second procedure used a low-high loading sequence to simulate short crack behavior. Based on the results, it was concluded that crack closure is not the major reason for the more rapid growth of short cracks as compared to long crack growth.

Telesman, J.↗
NASA NTRS1987

Characteristics of fluid flow in the combustion synthesis of TiC from the elements

The results of a numerical investigation of finite reservoir effects on capillary spreading at small reservoir dimensions are presently related to wave propagation phenomena in the combustion synthesis of TiC from its two elemental constituents. It is noted that gravitational forces can affect bubble coalescence by nonbuoyant means under the suitable conditions, although these conditions are expected to be rare in combustion synthesis. Finite-curved reservoirs can drive capillary flow due to surface tension and wall contact forces; these cause the wall and the metal to be completely reconfigured during combustion synthesis.

Valone, S. M.↗
NASA NTRS1997

Microgravity Fluid Separation Physics: Experimental and Analytical Results

Effective, low power, two-phase separation systems are vital for the cost-effective study and utilization of two-phase flow systems and flow physics of two-phase flows. The study of microgravity flows have the potential to reveal significant insight into the controlling mechanisms for the behavior of flows in both normal and reduced gravity environments. The microgravity environment results in a reduction in gravity induced buoyancy forces acting on the discrete phases. Thus, surface tension, viscous, and inertial forces exert an increased influence on the behavior of the flow as demonstrated by the axisymmetric flow patterns. Several space technology and operations groups have studied the flow behavior in reduced gravity since gas-liquid flows are encountered in several systems such as cabin humidity control, wastewater treatment, thermal management, and Rankine power systems.

Shoemaker, J. Michael↗
NASA NTRS2000

Remobilizing the Interfaces of Thermocapillary Driven Bubbles Retarded by the Adsorption of a Surfactant Impurity on the Bubble Surface

Thermocapillary migration is a method for moving bubbles in space in the absence of buoyancy. A temperature gradient is applied to the continuous phase in which a bubble is situated, and the applied gradient impressed on the bubble surface causes one pole of the drop to be cooler than the opposite pole. As the surface tension is a decreasing function of temperature, the cooler pole pulls at the warmer pole, creating a flow which propels the bubble in the direction of the warmer fluid. A major impediment to the practical use of thermocapillarity to direct the movement of bubbles in space is the fact that surfactant impurities which are unavoidably present in the continuous phase can significantly reduce the migration velocity. A surfactant impurity adsorbed onto the bubble interface is swept to the trailing end of the bubble. When bulk concentrations are low (which is the case with an impurity), diffusion of surfactant to the front end is slow relative to convection, and surfactant collects at the back end of the bubble. Collection at the back lowers the surface tension relative to the front end setting up a reverse tension gradient. For buoyancy driven bubble motions in the absence of a thermocapillarity, the tension gradient opposes the surface flow, and reduces the surface and terminal velocities (the interface becomes more solid-like). When thermocapillary forces are present, the reverse tension gradient set up by the surfactant accumulation reduces the temperature tension gradient, and decreases to near zero the thermocapillary velocity. The objective of our research is to develop a method for enhancing the thermocapillary migration of bubbles which have been retarded by the adsorption onto the bubble surface of a surfactant impurity, Our remobilization theory proposes to use surfactant molecules which kinetically rapidly exchange between the bulk and the surface and are at high bulk concentrations. Because the remobilizing surfactant is present at much higher concentrations than the impurity, it adsorbs to the bubble much faster than the impurity when the bubble is formed, and thereby prevents the impurity from adsorbing onto the surface. In addition the rapid kinetic exchange and high bulk concentration maintain a saturated surface with a uniform surface concentrations. This prevents retarding surface tension gradients and keeps the velocity high. In our first report last year, we detailed experimental results which verified the theory of remobilization in ground based experiments in which the steady velocity of rising bubbles was measured in a continuous phase consisting of a glycerol/water mixture containing a polyethylene glycol surfactant C12E6 (CH3(CH2)11(OCH2CH2)6OH). In our report this year, we detail our efforts to describe theoretically the remobilization observed. We construct a model in which a bubble rises steadily by buoyancy in a continuous (Newtonian) viscous fluid containing surfactant with a uniform far field bulk concentration. We account for the effects of inertia as well as viscosity in the flow in the continuous phase caused by the bubble motion (order one Reynolds number), and we assume that the bubble shape remains spherical (viscous and inertial forces are smaller than capillary forces, i e. small Weber and capillary numbers). The surfactant distribution is calculated by solving the mass transfer equations including convection and diffusion in the bulk, and finite kinetic exchange the bulk and the surface. Convective effects dominate diffusive mass transfer in the bulk of the liquid (high Peclet numbers) except in a thin boundary layer near the surface. A finite volume method is used to numerically solve the hydrodynamic and mass transfer equations on a staggered grid which accounts specifically for the thin boundary layer. We present the results of the nondimensional drag as a function of the bulk concentration of surfactant for different rates of kinetic exchange, from which we develop criteria for the concentration necessary to develop a prescribed degree of remobilization. The criteria compare favorably with the experimental results.

Palaparthi, Ravi↗
NASA NTRS1987

Plates and shells containing a surface crack under general loading conditions

Various through and part-through crack problems in plates and shells are considered. The line-spring model of Rice and Levy is generalized to the skew-symmetric case to solve surface crack problems involving mixed-mode, coplanar crack growth. Compliance functions are introduced which are valid for crack depth to thickness ratios at least up to .95. This includes expressions for tension and bending as well as expressions for in-plane shear, out-of-plane shear, and twisting. Transverse shear deformation is taken into account in the plate and shell theories and this effect is shown to be important in comparing stress intensity factors obtained from the plate theory with three-dimensional solutions. Stress intensity factors for cylinders obtained by the line-spring model also compare well with three-dimensional solution. By using the line-spring approach, stress intensity factors can be obtained for the through crack and for part-through crack of any crack front shape, without recalculation integrals that take up the bulk of the computer time. Therefore, parameter studies involving crack length, crack depth, shell type, and shell curvature are made in some detail. The results will be useful in brittle fracture and in fatigue crack propagation studies. All problems considered are of the mixed boundary value type and are reducted to strongly singular integral equations which make use of the finite-part integrals of Hadamard. The equations are solved numerically in a manner that is very efficient.

Joseph, Paul F.↗
NASA NTRS2008

Gimballed Shoulders for Friction Stir Welding

In a proposed improvement of tooling for friction stir welding, gimballed shoulders would supplant shoulders that, heretofore, have been fixedly aligned with pins. The proposal is especially relevant to self-reacting friction stir welding. Some definitions of terms, recapitulated from related prior NASA Tech Briefs articles, are prerequisite to a meaningful description of the proposed improvement. In friction stir welding, one uses a tool that includes (1) a rotating shoulder on top (or front) of the workpiece and (2) a pin that rotates with the shoulder and protrudes from the shoulder into the depth of the workpiece. In conventional friction stir welding, the main axial force exerted by the tool on the workpiece is reacted through a ridged backing anvil under (behind) the workpiece. When conventional friction stir welding is augmented with an auto-adjustable pin-tool (APT) capability, the depth of penetration of the pin into the workpiece is varied in real time by a position- or forcecontrol system that extends or retracts the pin as needed to obtain the desired effect. In self-reacting (also known as self-reacted) friction stir welding as practiced heretofore, there are two shoulders: one on top (or front) and one on the bottom (or back) of the workpiece. In this case, a threaded shaft protrudes from the tip of the pin to beyond the back surface of the workpiece. The back shoulder is held axially in place against tension by a nut on the threaded shaft. Both shoulders rotate with the pin and remain aligned coaxially with the pin. The main axial force exerted on the workpiece by the tool and front shoulder is reacted through the back shoulder and the threaded shaft into the friction-stir-welding machine head, so that a backing anvil is no longer needed. A key transmits torque between the bottom shoulder and the threaded shaft, so that the bottom shoulder rotates with the shaft. This concludes the prerequisite definitions of terms.

Carter, Robert↗
NASA NTRS1978

Liquid hydrogen tankage design

For this, the tankage experienced only minor thermal loadings and flight loads, hence, the predominate loading was the tank pressurization. Since the tanks were basically tension designed, the fatigue and damage tolerance criteria played the most important part in the overall design of the tank. Sensitivity factors were generated for the reference LH2-fueled airplane to provide a basis for evaluation of the effects of changes from the baseline design. For example, the various candidate insulation systems and tank structural concepts offer tradeoffs of the tankage offset distance (distance between exterior surface to tank) and weight. As the offset varies, the aircraft fuselage length must change to provide the required fuel volume within the fixed fuselage cross-section and hence the sensitivity on the aircraft DOC could be assessed.

Davis, G. W.↗
NASA NTRS1988

Liquid/vapour phase separation in He-4 using electric fields

In space, a replacement must be found for gravity to physically control and, in certain instances, contain cryogenic liquids. A program has been started at the Jet Propulsion Laboratory to study the use of electric field generated forces to establish the required orienting effects. Measurements which show that it is possible to apply strong enough electric fields to a liquid/vapor interface of He-4 to obtain an orienting force comparable to gravity are presented. Our measurements span the temperature range 1.7-4.2 K and demonstrate the applicability of Pashen's law for maximum attainable field before breakdown occurs. Some advantages of the electric field separator as opposed to passive surface tension devices are identified.

Israelsson, U. E.↗
NASA NTRS1993

Containerless measurements on liquids at high temperatures

The application of containerless techniques for measurements of the thermophysical properties of high temperature liquids is reviewed. Recent results obtained in the materials research laboratories at Intersonics are also presented. Work to measure high temperature liquid properties is motivated by both the need for reliable property data for modeling of industrial processes involving molten materials and generation of data form basic modeling of materials behavior. The motivation for this work and examples of variations in thermophysical property values from the literature are presented. The variations may be attributed to changes in the specimen properties caused by chemical changes in the specimen and/or to measurement errors. The two methods used to achieve containerless conditions were aeroacoustic levitation and electromagnetic levitation. Their qualities are presented. The accompanying slides show the layout of levitation equipment and present examples of levitated metallic and ceramic specimens. Containerless techniques provide a high degree of control over specimen chemistry, nucleation and allow precise control of liquid composition to be achieved. Effects of minor additions can thus be measured in a systematic way. Operation in reduced gravity enables enhanced control of liquid motion which can allow measurement of liquid transport properties. Examples of nucleation control, the thermodynamics of oxide contamination removal, and control of the chromium content of liquid aluminum oxide by high temperature containerless processes are presented. The feasibility of measuring temperature, emissivity, liquidus temperature, enthalpy, surface tension, density, viscosity, and thermal diffusivity are discussed in the final section of the paper.

Weber, Richard↗
NASA NTRS2003

Liquid Propellant Manipulated Acoustically

Fluids are difficult to manage in the space environment. Without gravity, the liquid and gas do not always remain separated as they do in the 1g environment of Earth. Instead the liquid and gas volumes mix and migrate under the influence of surface tension, thermodynamic forces, and external disturbances. As a result, liquid propellants may not be in a useable location or may even form a chaotic mix of liquid and gas bubbles. In the past, mechanical pumps, baffles, and a variety of specialized passive devices have been used to control the liquid and gas volumes. These methods need to be carefully tuned to a specific configuration to be effective. With increasing emphasis on long-term human activity in space there is a trend toward liquid systems that are more flexible and provide greater control. We are exploring new methods of manipulating liquids by using the nonlinear acoustic effects achieved by using beams of highly directed high-intensity acoustic waves.

Oeftering, Richard C.↗
NASA NTRS2006

Tool for Two Types of Friction Stir Welding

A tool that would be useable in both conventional and self-reacting friction stir welding (FSW) has been proposed. The tool would embody both a prior tooling concept for self-reacting FSW and an auto-adjustable pin-tool (APT) capability developed previously as an augmentation for conventional FSW. Some definitions of terms are prerequisite to a meaningful description of the proposed tool. In conventional FSW, depicted in Figure 1, one uses a tool that includes (1) a rotating shoulder on top (or front) of the workpiece and (2) a rotating pin that protrudes from the shoulder into the depth of the workpiece. The main axial force exerted by the tool on the workpiece is reacted through a ridged backing anvil under (behind) the workpiece. When conventional FSW is augmented with an APT capability, the depth of penetration of the pin into the workpiece is varied in real time by a position- or force-control system that extends or retracts the pin as needed to obtain the desired effect. In self-reacting (also known as self-reacted) friction stir welding (SR-FSW), there are two rotating shoulders: one on top (or front) and one on the bottom (or back) of the workpiece. In this case, a threaded shaft protrudes from the tip of the pin to beyond the back surface of the workpiece. The back shoulder is held axially in place against tension by a nut on the threaded shaft. The main axial force exerted on the workpiece by the tool and front shoulder is reacted through the back shoulder and the threaded shaft, back into the FSW machine head, so that a backing anvil is no longer needed. A key transmits torque between the bottom shoulder and the threaded shaft, so that the bottom shoulder rotates with the shaft. A tool for SRFSW embodying this concept was reported in "Mechanism for Self-Reacted Friction Stir Welding" (MFS-31914), NASA Tech Briefs, Vol. 28, No. 10 (October 2004), page 53. In its outward appearance, the proposed tool (see Figure 2) would fit the above description of an SR-FSW tool. In this case, the FSW machine would have an APT capability and the pin would be modified to accept a bottom shoulder. The APT capability could be used to vary the distance between the front and back shoulders in real time to accommodate process and workpiece-thickness variations. The tool could readily be converted to a conventional FSW tool, with or without APT capability, by simply replacing the modified pin with a conventional FSW pin.

Carter, Robert↗
DOE OSTI2023

INVESTIGATION OF ROCK-FLUID INTERACTIONS USING GEOMATERIAL MICROFLUIDICS

Illite and Illite-Smectite clays are reportedly the Caney Shale's most prevalent clay minerals. The swelling and fines migration of clays may occur when water-based fluids contact the clay minerals. This may result in the reduction of pore space and permeability, and wettability alteration. Therefore, we quantified the swelling potentials, fines migration and wettability alteration of Illite and Illite-Smectite clays upon exposure to model brines and a produced water, and a model oil and a crude oil. We measured density, viscosity, pH, TDS, and interfacial tension as they are important in multiphase flow and displacement characteristics. Geomaterial microfluidics is recently being used to study physicochemical interactions of solid-fluid systems. It facilitates visualizing the surface behavior upon exposure to various fluids. This study includes preparation and characterization of Illite-Smectite clay-coated glass capillary tubes. Illite and Illite-Smectite clay-coated geomaterial microfluidic surfaces were prepared to investigate the effects of first contact fluid, brine salinity, and aging on wettability. In this work, advancing and receding contact angles of model brine-air, model oil-air, produced water-air, crude oil-air, model brine-model oil systems, and produced water-crude oil systems have been measured in the untreated, and clay-coated microfluidic channels of 1000 µm width and 50 µm depth. The aging effect was studied by retaining the non-aqueous and aqueous solutions in the corresponding microfluidic channels for approximately 24 h in the case of channels first contacted by aqueous and non-aqueous solutions, respectively. The advancing and receding contact angles of Caney shale-crude oil-produced water systems were also measured. The experimental findings indicate the swelling potential, fines migration, and wettability alteration of Illite and Illite-Smectite clays. The results signify the influence of first contact fluid, brine salinity, and aging on wettability of the untreated and clay-coated microfluidic surfaces. The experiments also demonstrate the wetting nature of the Caney shale samples from reservoir and non-reservoir zones.

02 PETROLEUM↗
NASA NTRS1993

Marangoni effect induced convection in material processing under microgravity (M-18)

On the ground the upper layer of the liquid in a vessel heated from below becomes hotter than the lower layer. This well-known phenomenon results from buoyancy-induced convection. Though this means that the buoyancy-induced convection is not dominant under microgravity conditions in space, another convection called Marangoni convection may possibly occur. This convection results from an intermolecular force which acts on a free surface, that is surface tension. Since it is stronger at lower temperatures, the liquid surface near the heated wall is pulled to the cooled side as shown. This surface movement causes the inner convection. Marangoni convection, however, may be negligible on the Earth, for the molecular force is generally smaller than buoyancy. Various tests on material processing were recently conducted in space and some high quality crystals free from buoyancy convection were obtained. But, at the same time, others proved to be less uniform than expected in the components' distribution. This nonuniformity seems to be mainly caused by Marangoni convection. It is, therefore, very important to know how to control the convection by studying its characteristics, but the problem is that on the ground it is impossible to carry out the experiment without gravity. That is what a space experiment aboard the space shuttle is planned as the First Material Processing Tests (FMPT) of Japan. The experiment on Marangoni flow visualization is being performed in order to investigate the characteristics of convection in uni-dimensional melt growth under microgravity conditions.

Enya, S.↗
NASA NTRS1994

Small-crack effects in high-strength aluminum alloys

The National Aeronautics and Space Administration and the Chinese Aeronautical Establishment participated in a Fatigue and Fracture Mechanics Cooperative Program. The program objectives were to identify and characterize crack initiation and growth of small cracks (10 microns to 2 mm long) in commonly used US and PRC aluminum alloys, to improve fracture mechanics analyses of surface- and corner-crack configurations, and to develop improved life-prediction methods. Fatigue and small-crack tests were performed on single-edgenotch tension (SENT) specimens and large-crack tests were conducted on center-crack tension specimens for constant-amplitude (stress ratios of -1, 0, and 0.5) and Mini-TWIST spectrum loading. The plastic replica method was used to monitor the initiation and growth of small fatigue cracks at the semicircular notch. Crack growth results from each laboratory on 7075-T6 bare and LC9cs clad aluminum alloys agreed well and showed that fatigue life was mostly crack propagation from a material defect (inclusion particles or void) or from the cladding layer. Finite-element and weight-function methods were used to determine stress intensity factors for surface and corner cracks in the SENT specimens. Equations were then developed and used in a crack growth and crack-closure model to correlate small- and large-crack data and to make life predictions for various load histories. The cooperative program produced useful experimental data and efficient analysis methods for improving life predictions. The results should ultimately improve aircraft structural reliability and safety.

Newman, J. C., Jr.↗
NASA NTRS1994

Rarefied solids

One important limit to creating low density materials is the objects' own weight. As a solid or colloidal matrix becomes more rarefied, gravity acts destructively to compress its suporting skeleton. We describe experimental results and propose a model which matches the low gravity behavior of rarefied or fractal solids. On parabolic airplane flights, we sought to demonstrate a key component of producing higher surface area fractals. Flight paths were selected to give a range of gravity levels: 0.01 g/g(sub 0) (low), 0.16 g(sub 0) (Lunar), 0.33 g/g(sub 0) (Martian), 1 g/g(sub 0) (Earth) and 1.8 g/g(sub 0) (high) (where g(sub 0) = 980 cm/sq s). Results using the model material of hydrophobic silica indicated that stable agglomeration of such tenuous objects can increase markedly in reduced gravity. Optical characterization revealed that fractal dimension changed directly with varying gravity. As measured by fractal dimension, effective surface area and roughness increased by 40% in low gravity. This finding supports the conclusion that relieving internal weight stresses on delicate aggregates can enhance their overall size (by two orders of magnitude) and internal surface area. We conclude that gravitational restructuring limits the overall size and void content of low-density solids. These sparse colloidal regimes may present new and technologically attractive physics, ranging from improved insulators, liquid-like tension in a 'solid' matrix, and characteristically low conductivities for sound and (8 to 14 micrometers wavelength) infrared radiation.

Noever, D. A.↗
NASA NTRS2015

Space Biophysics: Accomplishments, Trends, Challenges

Physics and biology are inextricably linked. All the chemical and biological processes of life are dutifully bound to follow the rules and laws of physics. In space, these physical laws seem to turn on their head and biological systems, from microbes to humans, adapt and evolve in myriad ways to cope with the changed physical influences of the space environment. Gravity is the most prominent change in space that influences biology. In microgravity, the physical processes of sedimentation, density-driven convective flow, influence of surface tension and fluid pressure profoundly influence biology at the molecular and cellular level as well as at the whole-body level. Gravity sensing mechanisms are altered, structural and functional components of biology (such as bone and muscle) are reduced and changes in the way fluids and gasses behave also drive the way microbial systems and biofilms grow as well as the way plants and animals adapt. The radiation environment also effects life in space. Solar particle events and high energy cosmic radiation can cause serious damage to DNA and other biomolecules. The results can cause mutation, cellular damage or death, leading to health consequences of acute radiation damage or long-term health consequences such as increased cancer risk. Space Biophysics is the study and utilization of physical changes in space that cause changes in biological systems. The unique physical environment in space has been used successfully to grow high-quality protein crystals and 3D tissue cultures that could not be grown in the presence of unidirectional gravitational acceleration here on Earth. All biological processes that change in space have their root in a biophysical alteration due to microgravity and/or the radiation environment of space. In order to fully-understand the risks to human health in space and to fully-understand how humans, plants, animals and microbes can safely and effectively travel and eventually live for long periods beyond the protective environment of Earth, the biophysical properties underlying these changes must be studied, characterized and understood. This lecture reviews the current state of NASA biophysics research accomplishments and identifies future trends and challenges for biophysics research on the International Space Station and beyond.

Space Biophysics: Accomplishments↗
NASA NTRS1949

Some New Problems on Shells and Thin Structures

Cylindrical shells of arbitrary section, reinforced by longitudinal and transverse members (stringers and ribs) are considered by us, for a sufficiently close spacing of the ribs, as in our previously published papers (references 1 end 2), as thin-walled orthotropic spatial systems at the cross-sections of which only axial (normal and shearing) forces can arise. The longitudinal bending and twisting moments, due to their weak effect on the stress state of the shell, are taken equal to zero. Along the longitudinal sections of the shell there may arise transverse forces in addition to the normal ~d shearing forces. Under the so-called static assumptions there is taken for the computation model of the shell a thin-walled spatial system consisting along its length (along a generator) of an infinite number of elementary strips capable of bending. Each of these strips is likened to a curved rod operating in each of its sections not only in tension (compression)but also in transverse bending and shear. The interaction between two adjoining transverse strips in the shell expresses itself in the transmission from one strip to the other of only the normal and shearing stresses. The static structure of the computation model here described is shown in figure 1, where the connections through which the normal and shearing stresses transmitted from one transverse strip to smother are indicated schematically by the rods located in the middle surface of the shell. In addition to the static hypothesis we introduce also geometric hypotheses. According to the latter the elongational deformations of the shell along lines parallel to the generator of its middle surface and the shear deformations in the middle surface, as ma+gitudes having . little effect on the state of the fundamental internal forces of the shell, are taken equal to zero. The deformations of the shell in our computational model are such that in the first place the lines of this surface perpendicular to the generator are inextensible at each point end in the second place the angles between the lines of principal curvature (the coordinate lines) which are straight before the deformation remain straight after the deformation.

Vlasov, V. S.↗