A micromechanical analysis for the linear elastic behavior of a low-density foam with open cells is presented. The foam structure is based on the geometry of a Kelvin soap froth with flat faces: 14-sided polyhedral cells contain six squares and eight hexagons. Four struts meet at every joint in the perfectly ordered, spatially periodic, open-cell structure. All of the struts and joints have identical shape. Strut-level force-displacement relations are expressed by compliances for stretching, bending, and twisting. We consider arbitrary homogeneous deformations of the foam and present analytic results for the force, moment, and displacement at each strut midpoint and the rotation at each joint. The effective stress-strain relations for the foam, which has cubic symmetry, are represented by three elastic constants, a bulk modulus, and two shear moduli, that depend on the strut compliances. When these compliances are evaluated for specific strut geometries, the shear moduli are nearly equal and therefore the elastic response is nearly isotropic. The variational results of Hashin and Shtrikman are used to calculate the effective isotropic shear modulus of a polycrystal that contain grains of Kelvin foam.

Three-dimensional effects of symmetrically compressing an O-ring with rigid plates are investigated. The O-ring is assumed to be an incompressible elastic material and the mixed boundary value problem is formulated within the framework of classical the linear elasticity theory. Transversely compressing the O-ring between flat rigid plates increases the toroidal radius and induces a reduction in the O-ring cross-sectional radius. This reduction depends strongly on the conditions of contact between plate and O-ring, and the reduced contact pressure may threaten the integrity of the seal. For O-rings compressed radially inward compressed radially inward or outward, most of the radial displacement is accommodated by a change in the toroidal radius with very little actual compression of the ring material. If this effect is not accounted for in the design of the seal, the contact pressure may not be sufficient to secure the seal.

A constitutive model for the nonlinear elastic behavior of isotropic low-density opencell foams with three-dimensional structure is formulated in terms of a strain energy function. The theory is based on micromechanical analysis of an idealized tetrahedral unit cell of arbitrary orientation that contains four half-struts joining at equal angles. The force-displacement relations for each strut are expressed by compliances for bending and stretching that do not depend on the magnitude of applied force. Contributions to the strain energy from large deformation effects are assumed to depend on strut reorientation and stretching, and are determined by analyzing a pin-jointed structure. The analysis is considered to be valid for finite strains below the onset of yielding associated with strut buckling.

The plane elastic deformation of a notched half-plane which is compressed by a flat rigid surface is investigated. The stress and displacement fields of this notched halfplane are felt to be representative of the conditions occurring in the neighborhood of a flaw on the surface of a compressed circular O-ring. O-rings are often installed as environmental seals in electronic components, and the results of this analysis indicate that minor surface flaws do not threaten the integrity of O-ring seals under normal installation conditions. Compressing the O-ring reduces the notch surface opening and pushes the notch surface toward the rigid compressing plate. Both of these effects reduce the notch void volume and the possibility of seal leakage.

This work provides a theoretical analysis of the elastic behavior of an O-ring compressed between two rigid plates with irregular surfaces. Relations between deflection, contact force and contact pressure are obtained. The contact pressure, which is of fundamental importance in establishing criteria for effective sealing, is dependent upon both the amplitude and wavelength of the surface irregularity. This analysis suggests that surfaces in contact with O-ring seals should be characterized by the root mean square slope Δq in addition to the usual R a which depends on amplitude only.

A theoretical model for the linear elastic properties of three-dimensional open-cell foams is developed. We consider a tetrahedral unit cell, which contains four identical half-struts that join at equal angles, to represent the essential microstructural features of a foam. The effective continuum stress is obtained for an individual tetrahedral element arbitrarily oriented with respect to the principal directions of strain. The effective elastic constants for a foam are determined under the assumption that all possible orientations of the unit cell are equally probable in a representative volume element. The elastic constants are expressed as functions of compliances for bending and stretching of a strut, whose cross section is permitted to vary with distance from the joint, so the effect of strut morphology on effective elastic properties can be determined. Strut bending is the primary distortional mechanism for low-density foams with tetrahedral microstructure. For uniform strut cross section, the effective Young’s modulus is proportional to the volume fraction of solid material squared, and the coefficient of proportionality depends upon the specific strut shape. A similar analysis for cellular materials with cubic microstructure indicates that strut extension is the dominant distortional mechanism and that the effective Young’s modulus is linear in volume fraction. Our results emphasize the essential role of microstructure in determining the linear elastic properties of cellular materials and provide a theoretical framework for investigating nonlinear behavior.

Several problems in analysis can arise in estimating in-situ stresses from standard hydraulic fracturing operations if the borehole is not aligned with one of the principal stress directions. In these nonaligned situations, the possibility of fracturing a spherical cavity for estimating the in-situ stresses is investigated. The theory utilizes all the advantages of direct stress measurements associated with hydraulic fracturing and eliminates the geometrical problems associated with the analysis of hydraulic fractures in cylindrical boreholes.

Well bore stresses induced by inflatable packers during hydraulic fracturing operations are investigated. The geologic formation is modeled as an unbounded homogeneous isotropic linear elastic solid containing an infinitely long circular cavity, while the packer is modeled as a semi-infinite thin-walled circular cylindrical shell. For given packer properties, these induced stresses are shown to depend on the difference between packer pressure and fracturing pressure and can become significant. Typical numerical results are obtained and presented graphically. Analytical approximations for the maximum values of these stresses are also presented. While these effects are of no importance in the usual application of hydraulic fracturing to enhance oil and gas recovery, they are crucial in attempts to estimate in-situ stresses from hydraulic fracturing pressure data.

A linear two-temperature theory of thermoelasticity is used to investigate the stresses arising from the aerodynamic heating of a solid isotropic sphere. The major effect of this theory as contrasted with the classical one-temperature theory is to mitigate the maximum compressive stress occurring on the surface of the sphere and to retard its occurrence.

Results of the thermoelastic analysis of an infinite elastic medium containing two arbitrary-sized spherical cavities with arbitrary separation distance are presented graphically. Of particular interest is the interference effect of one cavity on the other. The cavity surfaces are assumed to be held at uniform temperature while zero temperature prevails at infinity. Examples treated include those for which the cavity surfaces are stress-free and those for which individually self-equilibrated rigid liners are bonded to the cavity surfaces.

Analytical methods are developed for treating steady-state axisymmetric thermoelastic problems defined in bispherical coordinates. Possible geometrical configurations include the infinite space with two spherical cavities of arbitrary radii and separation distance, the half-space with a spherical cavity, and the thick-walled shell having eccentric spherical boundaries. Thermal conditions must be prescribed at the surface of the body such that the temperature distribution is uniquely determined. The surfaces of the body are traction free. Numerical results for a half-space containing a spherical cavity heated to constant temperature with zero temperature on the plane and at infinity are presented in graphical form for representative geometrical variations.