An analytical solution is derived for the time-dependent flow and deformation coupling of a saturated isotropic homogeneous incompressible poroelastic media within a two-dimensional (2D) finite domain due to a point source at some arbitrary position. In this study, the pore pressure field is assumed to conform to the second type of boundary conditions. Boundary conditions of the displacement field are chosen with care to match the appropriate finite sine and cosine transforms and simplify the resulting solution. It is found that the analytical solution is always independent of the Poisson’s ratio. The detailed solutions are given for the case of a periodic point source with zero pressure derivatives on the boundaries and for an imposed pressure derivative on the lower edge in the absence of a source. The presented analytical solutions are highly applicable for calibrating numerical codes, and meanwhile they can be used to further investigate the transient behavior of flow and deformation coupling induced by fluid withdrawal within a 2D finite poroelastic media.

A helical spring that is constrained to no rotation has a compliance that is typically more than 95% of the compliance of springs constrained to free rotation when restricted to symmetric wires made from materials with Poisson’s ratio between 0 and 1/2. It is shown that the shape of the spring wire can be designed so the spring will not twist when it is extended nor extend when it is twisted. The constrained spring versus a freely rotating spring with the helix angle equal to π / 4 has the largest reduction in compliance in the limits of beam theory. Spring compliances for torsion and extension with quite complex helical spring geometries are found to be related by a dimensionless ratio of compliances in a very simple equation that only depends on Poisson’s ratio and the helical, spring angle, ψ . Springs made from materials with negative Poisson’s ratio, however, can have a very substantial reduction in compliance; the no rotation compliance is zero when Poisson’s ratio is −1. There are large changes in spring compliances for springs with geometric coils that are elongated rectangles or flattened ellipses.

In this work, Green’s functions for unbounded elastic domain in a functionally graded material with a quadratic variation of elastic moduli and constant Poisson’s ratio of 0.25 are derived for both two-dimensional (2D) and three-dimensional (3D) cases. The displacement fields caused by a point force are derived using the logarithmic potential and the Kelvin solution for 2D and 3D cases, respectively. For a circular (2D) or spherical (3D) bounded domain, analytical solutions are provided by superposing the above solutions and corresponding elastic general solutions. This closed form solution is valuable for elastic analysis with material stiffness variations caused by temperature, moisture, aging effect, or material composition, and it can be used to perform early stage verification of more complex models of functionally graded materials. Comparison of theoretical solution and finite element method results demonstrates the application and accuracy of this solution.

The mechanical system considered is a bilayer cantilever plate. The substrate and the film are linear elastic. The film is subjected to isotropic uniform prestresses due for instance to volume variation associated with cooling, heating, or drying. This loading yields deflection of the plate. We recall Stoney’s analytical formula linking the total mechanical stresses to this deflection. We also derive a relationship between the prestresses and the deflection. We relax Stoney’s assumption of very thin films. The analytical formulas are derived by assuming that the stress and curvature states are uniform and biaxial. To quantify the validity of these assumptions, finite element calculations of the three-dimensional elasticity problem are performed for a wide range of plate geometries, Young’s and Poisson’s moduli. One purpose is to help any user of the formulas to estimate their accuracy. In particular, we show that for very thin films, both formulas written either on the total mechanical stresses or on the prestresses, are equivalent and accurate. The error associated with the misfit between our theorical study and numerical results are also presented. For thicker films, the observed deflection is satisfactorily reproduced by the expression involving the prestresses and not the total mechanical stresses.

The path-dependence of the J -integral is investigated numerically via the finite-element method, for a range of loadings, Poisson’s ratios, and hardening exponents within the context of J 2 -flow plasticity. Small-scale yielding assumptions are employed using Dirichlet-to-Neumann map boundary conditions on a circular boundary that encloses the plastic zone. This construct allows for a dense finite-element mesh within the plastic zone and accurate far-field boundary conditions. Details of the crack tip field that have been computed previously by others, including the existence of an elastic sector in mode I loading, are confirmed. The somewhat unexpected result is that J for a contour approaching zero radius around the crack tip is approximately 18% lower than the far-field value for mode I loading for Poisson’s ratios characteristic of metals. In contrast, practically no path-dependence is found for mode II. The applications of T - or S -stress, whether applied proportionally with the K -field or prior to K , have only a modest effect on the path-dependence.

We investigate the problem of an N -phase elliptical inhomogeneity in plane elasticity. The elliptical inhomogeneity is bonded to the unbounded matrix through the intermediate ( N − 2 ) interphases, and the matrix is subjected to remote uniform stresses. We observe that the stress field inside the elliptical inhomogeneity is still uniform when the following two conditions are satisfied: (i) The formed interfaces are ( N − 1 ) confocal ellipses, and (ii) the interphases and the matrix possess the same shear modulus but different Poisson’s ratios. In Appendixes A xB , we also discuss an arbitrary number of interacting arbitrary shaped inhomogeneities embedded in an infinite matrix, and an N -phase inhomogeneity with ( N − 1 ) interfaces of arbitrary shape. Here all the phases comprising the composite possess the same shear modulus but different Poisson’s ratios. The results in the main body and in Appendixes A xB are further extended in Appendix C to finite plane strain deformations of compressible hyperelastic harmonic materials.

The two small parameters that appear in the final equations developed in Part I ( Simmonds and Hosseinbor, 2010, “The Free and Forced Vibrations of a Closed Elastic Spherical Shell Fixed to an Equatorial Beam—Part I: The Governing Equations and Special Solutions,” ASME J. Appl. Mech., 77, p. 021017 ), namely, h / R , the ratio of the constant shell thickness to the radius of curvature of the shell’s reference surface and H / R , where H is the depth (or width) of the equatorial beam, are exploited using perturbation techniques (including the WKB method). The natural frequencies depend not only on these parameters, but also on the ratio of the mass densities of the shell and beam, the ratio of the Young’s moduli of the shell and beam, Poisson’s ratio, and the circumferential wave number m . Short tables for typical parameter values are given for those cases where the frequency equation is not explicit.

This brief note presents a rigorous proof of the general solutions obtained by Aderogba [1977, “On Eigenstresses in Dissimilar Media,” Philos. Mag., 35, pp. 281–292 ] for a defect in elastic bimaterial solids. The derivation is based on the fact that some derivative and integral forms of the biharmonic functions are also biharmonic. The associated solutions can be expressed as the linear combinations of these biharmonic functions and their unknown coefficients can be determined by the interface conditions. It is found that the coefficients only depend on the contrasts of material constants across the interface.

An examination of the effect of Poisson’s ratio on stress distribution is important to interpret the results of a stress-strain analysis by using experimental methods because the material of the model frequently has a different Poisson’s ratio from that of the prototype. In linear elasticity, the effect of Poisson’s ratio on three-dimensional stress distribution is theoretically explained for simply connected bodies by using static methods in this study. It is proven that the stress components are independent from Poisson’s ratio in sections of the body where the stress components arising are in equilibrium only with surface tractions. This result is useful in interpreting three-dimensional photoelasticity and other experiments and even in guiding the design.

In 1970, F. Y. M. Wan derived a single, complex-valued ordinary differential equation for an elastically isotropic right circular conical shell ( “On the Equations of the Linear Theory of Elastic Conical Shells,” Studies Appl. Math., 49, pp. 69–83 ). The unknown was the n th Fourier component of a complex combination of the midsurface normal displacement and its static-geometric dual, a stress function. However, an attempt to formally replace the Fourier index n by a partial derivative in the circumferential angle θ results in a partial differential equation, which is eighth order in θ . The present paper takes as unknowns the traces of the bending strain and stress resultant tensors, respectively, and derives static-geometric dual partial differential equations of fourth order in both the axial and circumferential variables. Because of the explicit appearance of Poisson ratios of bending and stretching, these two equations cannot be combined into a single complex-valued equation. Reduced equations for beamlike (axisymmetric and lateral) deformations are also derived.

Knowledge of the flexural vibration frequencies of thin rectangular cantilever plates forms the basis for numerous applications in sensing and instrumentation. Despite the seemingly simple nature of the problem, an accurate formula for the fundamental resonant frequency that is valid for all aspect ratios and Poisson’s ratios is notably lacking in the literature. In this article, we present such a result using a variational and singular perturbation formulation. This yields a simple analytical formula that exhibits a maximum error of 2%.

Current methodologies used for the inference of thin film stress through curvature measurements are strictly restricted to stress and curvature states that are assumed to remain uniform over the entire film/substrate system. By considering a circular thin film/substrate system subject to nonuniform and nonaxisymmetric temperature distributions, we derive relations between the film stresses and temperature, and between the plate system’s curvatures and the temperature. These relations featured a “local” part that involves a direct dependence of the stress or curvature components on the temperature at the same point, and a “nonlocal” part that reflects the effect of temperature of other points on the location of scrutiny. Most notably, we also derive relations between the polar components of the film stress and those of system curvatures which allow for the experimental inference of such stresses from full-field curvature measurements in the presence of arbitrary nonuniformities. These relations also feature a “nonlocal” dependence on curvatures making full-field measurements of curvature a necessity for the correct inference of stress. Finally, it is shown that the interfacial shear tractions between the film and the substrate are related to the gradients of the first curvature invariant and can also be inferred experimentally.

This paper deals with the derivation of exact solutions for the static analysis of functionally graded (FG) plates integrated with a layer of piezoelectric fiber reinforced composite (PFRC) material. The layer of the PFRC material acts as the distributed actuator of the FG plates. The Young’s modulus of the FG plate is assumed to vary exponentially along the thickness of the plate while the Poisson’s ratio is assumed to be constant over the domain of the plate. The numerical values of the exact solutions are presented for both thick and thin smart FG plates and indicate that the activated PFRC layer potentially counteracts the deformations of the FG plates due to mechanical load. The through-thickness behavior of the plates revealed that the coupling of bending and extension takes place in the FG plates even if the PFRC layer is not subjected to the applied voltage. The solutions also revealed that the activated PFRC layer is more effective in controlling the deformations of the FG plates when the layer is attached to the surface of the FG plate with minimum stiffness than when it is attached to the surface of the same with maximum stiffness. The solutions of this benchmark problem may be useful for verifying the other approximate and numerical models of the smart functionally graded plates for which exact solutions cannot be derived.

The virtual internal bond (VIB) method was developed for the numerical simulation of fracture processes. In contrast to the traditional approach of fracture mechanics where stress analysis is separated from a description of the actual process of material failure, the VIB method naturally allows for crack nucleation, branching, kinking, and arrest. The idea of the method is to use atomic-like bond potentials in combination with the Cauchy-Born rule for establishing continuum constitutive equations which allow for the material separation–strain localization. While the conventional VIB formulation stimulated successful computational studies with applications to structural and biological materials, it suffers from the following theoretical inconsistency. When the constitutive relations of the VIB model are linearized for an isotropic homogeneous material, the Poisson ratio is found equal to 1 ∕ 4 so that there is only one independent elastic constant—Young’s modulus. Such restriction is not suitable for many materials. In this paper, we propose a modified VIB (MVIB) formulation, which allows for two independent linear elastic constants. It is also argued that the discrepancy of the conventional formulation is a result of using only two-body interaction potentials in the microstructural setting of the VIB method. When many-body interactions in “bond bending” are accounted for, as in the MVIB approach, the resulting formulation becomes consistent with the classical theory of isotropic linear elasticity.

This paper presents the development of a theoretical damage mechanics model applicable to random short glass fiber reinforced composites. This model is based on a macroscopic approach using internal variables together with a thermodynamic potential expressed in the stress space. Induced anisotropic damage, nonsymmetric tensile/compressive behavior (unilateral effect) and residual effects (permanent strain) are taken into account. The anisotropic damage is represented with second-order tensorial internal variables D . The unilateral effect due to microcrack closure in compression is introduced by generalizing the hypothesis of the complementary elastic energy equivalence. In the case of the permanent strain, a new term related to frozen energy, which is a function of the damage variable, the stress tensor, and some materials constants to be identified, is added to the basic thermodynamic potential. Using laboratory test results, parameter identification has been performed to illustrate the applicability of the proposed model.

Poisson’s ratio for an anisotropic linear elastic material depends on two orthogonal directions n and m . Materials with negative Poisson’s ratios for all ( n , m ) pairs are called completely auxetic while those with positive Poisson’s ratios for all ( n , m ) pairs are called nonauxetic. Simple necessary and sufficient conditions on elastic compliances are derived to identify if any given material of cubic or hexagonal symmetry is completely auxetic or nonauxetic. When these conditions are not satisfied, the medium is auxetic for some ( n , m ) pairs. Several simple necessary conditions for completely auxetic or nonauxetic media are derived for a general anisotropic elastic material.

An acoustic model is developed for transient wave propagation in a weak layer excited by prescribed pressure or prescribed acceleration at the boundary. The validity of the acoustic model is investigated for the two excitations. A comparison of transient response from the acoustic model and a 3D axisymmetric elastic model reveals that for prescribed acceleration the acoustic model fails to capture important features of the elastic model even as Poisson ratio ν approaches 1 ∕ 2 . However for prescribed pressure, the two models agree since shear stress is reduced. For prescribed acceleration adopting the modal approach, the mixed boundary-value problem on the excited boundary is converted to a pure traction problem utilizing the influence method. To validate the elaborate modal approach a finite difference model is also developed.

The effective compliance moduli of a plate with a doubly periodic set of traction-free holes are considered. Attention is drawn to the perturbation form in which they are expressed by applying the complex variable methods in two-dimensional elasticity. This permits one to derive specific dimensionless combinations of the effective moduli, which are independent of the solid Poisson ratio. Using them saves computations of the structure moduli by FEM-like methods and helps one to evaluate their practical accuracy. Thus far, the only result of this kind has been observed numerically by Day, Snyder, Garboczi, and Thorpe (J. Mech. Phys. Solids. 40, pp. 1031–1051, 1992) and later proved by Cherkaev, Lurie, and Milton (Proc. R. Soc. London, Ser. A 458, pp. 519–529, 1992) .