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### Guest Editorial

J. Appl. Mech. 2013;80(2):020301-020301-1. doi:10.1115/1.4007965.

Mécanique des Milieux Poreux (1991) is Professor Olivier Coussy's pioneering book, written in French on the mechanics of porous media, that propelled this topic in to our regular teaching environment. Here, I quote from the book's preface, written by the late Paul Germain, President of IUTAM and the perpetual secretary of the French Academy of Sciences, “Si l'on veut être rigoureux, si l'on veut atteindre la même cohérence et la même clarté, la même qualité d'exposé pour les matériaux poreux que pour les matériaux pris en compte dans la Mécanique des Milieux Continus classique, c'est tout l’édifice qu'il faut reconstruire complètement. C'est ce que fait notre auteur; sans doute est-il l'un des tout premiers à le faire; et il le fait avec une rigueur, une élégance et une maîtrise que je me plais ici à souligner.” That is, “If we have to be rigorous and we want to attain the exact coherence and the same quality of the work for porous material continuum as it is the case of the solid continuum material we have to totally reconstruct the whole edifice. That is exactly what our author has done. Without any doubt he is one of the very first to have done it; and he does it with rigor, elegance and a mastery that I am happy to herein outline.” It is these words, and in particular those in the last sentence of the quote, that have marked the life of our late colleague, Professor Olivier Coussy. It is with elegance, rigor, and mastery that he conducted his life as a scientist, colleague, and a friend. He always quoted to me the famous Bernard of Chartres (twelfth-century philosopher), “We stand on the shoulders of giants.” Coussy believed that we carry on what great scientists have already started. However, “giants” like Professor Coussy, who, in reformulating and reworking the “edifice” of the mechanics of porous media, brought it into the classical realm of teaching materials, has forever marked his legacy in science. In addition, Olivier not only reformulated Biot's theory of consolidation in his 1991 book, but in his publications that followed. Dr. Coussy established the physical and mathematical foundations of the mechanics of porous media now known as the poromechanics theory.

Commentary by Dr. Valentin Fuster

### Research Articles

J. Appl. Mech. 2013;80(2):020901-020901-8. doi:10.1115/1.4023481.

The parallel linear equations solver capable of effectively using 1000+ processors becomes the bottleneck of large-scale implicit engineering simulations. In this paper, we present a new hierarchical parallel master-slave-structural iterative algorithm for the solution of super large-scale sparse linear equations in a distributed memory computer cluster. Through alternatively performing global equilibrium computation and local relaxation, the specific accuracy requirement can be met in a few iterations. Moreover, each set/slave-processor majorly communicates with its nearest neighbors, and the transferring data between sets/slave-processors and the master-processor is always far below the communication between neighboring sets/slave-processors. The corresponding algorithm for implicit finite element analysis has been implemented based on the MPI library, and a super large 2-dimension square system of triangle-lattice truss structure under randomly distributed loadings is simulated with over 1 × 109 degrees of freedom (DOF) on up to 2001 processors of the “Exploration 100” cluster in Tsinghua University. The numerical experiments demonstrate that this algorithm has excellent parallel efficiency and high scalability, and it may have broad applications in other implicit simulations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020903-020903-14. doi:10.1115/1.4023011.

We propose a new two-scale model to compute the swelling pressure in colloidal systems with microstructure sensitive to pH changes from an outer bulk fluid in thermodynamic equilibrium with the electrolyte solution in the nanopores. The model is based on establishing the microscopic pore scale governing equations for a biphasic porous medium composed of surface charged macromolecules saturated by the aqueous electrolyte solution containing four monovalent ions $(Na+,Cl-,H+,OH-)$. Ion exchange reactions occur at the surface of the particles leading to a pH-dependent surface charge density, giving rise to a nonlinear Neumann condition for the Poisson–Boltzmann problem for the electric double layer potential. The homogenization procedure, based on formal matched asymptotic expansions, is applied to up-scale the pore-scale model to the macroscale. Modified forms of Terzaghi's effective stress principle and mass balance of the solid phase, including a disjoining stress tensor and electrochemical compressibility, are rigorously derived from the upscaling procedure. New constitutive laws are constructed for these quantities incorporating the pH-dependency. The two-scale model is discretized by the finite element method and applied to numerically simulate a free swelling experiment induced by chemical stimulation of the external bulk solution.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020904-020904-9. doi:10.1115/1.4007907.

This paper investigates the hydroionic transport processes at the near surface of cement-based porous materials under external drying-wetting (D-W) actions. A basic multiphase model is retained and reviewed critically for moisture transport under D-W actions. The multiphase model fails to account for the substantial difference between moisture diffusivities during drying and wetting. The multiphase model is adapted for moisture transport under D-W actions through the respective mechanisms of moisture transport during drying and wetting. Together with the associated ionic transport, a global hydroionic model is established and the corresponding numerical scheme is developed to solve the near surface transport problem. Then, systematic experiments are performed on two concretes with high and low porosities for transport properties and hydroionic transport under D-W actions with pure water and salt solution. Experimental data validate the global model, while some fundamental aspects of hydroionic modeling are discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020905-020905-12. doi:10.1115/1.4007922.

There are lots of ceramic geological and biological materials whose microscopic load carrying behavior is not dominated by bending of structural units, but by the three-dimensional interaction of disorderedly arranged single crystals. A particularly interesting solution to capture this so-called polycrystalline behavior has emerged in the form of self-consistent homogenization methods based on an infinite amount of nonspherical (needle or disk-shaped) solid crystal phases and one spherical pore phase. Based on eigenstressed matrix-inclusion problems, together with the concentration and influence tensor concept, we arrive at the following results: Young’s modulus and the poroelastic Biot modulus of the porous polycrystal scale linearly with the Young’s modulus of the single crystals, the former independently of the Poisson’s ratio of the single crystals. Biot coefficients are independent of the single crystals’ Young’s modulus. The uniaxial strength of a pore pressure-free porous polycrystal, as well as the blasting pore pressure of a macroscopic stress-free polycrystal, scale linearly with the tensile strength of the single crystals, independently of all other elastic and strength properties of the single crystals. This is confirmed by experiments on a wide range of bio- and geomaterials, and it is of great interest for numerical simulations of structures built up by such polycrystals.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020906-020906-5. doi:10.1115/1.4007923.

The relationship between the macro- and microvelocity fields in a poroelastic representative volume element (RVE) has not being fully investigated. This relationship is considered to be a function of the tortuosity: a quantitative measure of the effect of the deviation of the pore fluid streamlines from straight (not tortuous) paths in fluid-saturated porous media. There are different expressions for tortuosity based on the deviation from straight pores, harmonic wave excitation, or from a kinetic energy loss analysis. The objective of the work presented is to determine the best expression for tortuosity of a multiply interconnected open pore architecture in an anisotropic porous media. The procedures for averaging the pore microvelocity over the RVE of poroelastic media by Coussy and by Biot were reviewed as part of this study, and the significant connection between these two procedures was established. Success was achieved in identifying the Coussy kinetic energy loss in the pore fluid approach as the most attractive expression for the tortuosity of porous media based on pore fluid viscosity, porosity, and the pore architecture. The fabric tensor, a 3D measure of the architecture of pore structure, was introduced in the expression of the tortuosity tensor for anisotropic porous media. Practical considerations for the measurement of the key parameters in the models of Coussy and Biot are discussed. In this study, we used cancellous bone as an example of interconnected pores and as a motivator for this study, but the results achieved are much more general and have a far broader application than just to cancellous bone.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020907-020907-12. doi:10.1115/1.4007904.

Shales, clays, hydrogels, and tissues swell and shrink under changing osmotic conditions, which may lead to failure. The relationship between the presence of cracks and fluid flow has had little attention. The relationship between failure and osmotic conditions has had even less attention. The aim of this research is to study the effect of osmotic conditions on propagating discontinuities under different types of loads for saturated ionized porous media using the finite element method (FEM). Discontinuous functions are introduced in the shape functions of the FEM by partition of unity method, independently of the underlying mesh. Damage ahead of the crack-tip is introduced by a cohesive zone model. Tensile loading of a crack in an osmoelastic medium results in opening of the crack and high pressure gradients between the crack and the formation. The fluid flow in the crack is approximated by Couette flow. Results show that failure behavior depends highly on the load, permeability, (osmotic) prestress and the stiffness of the material. In some cases it is seen that when the crack propagation initiates, fluid is attracted to the crack-tip from the crack rather than from the surrounding medium causing the crack to close. The results show reasonable mesh-independent crack propagation for materials with a high stiffness. Stepwise crack propagation through the medium is seen due to consolidation, i.e., crack propagation alternates with pauses in which the fluid redistributes. This physical phenomenon challenges the numerical scheme. Furthermore, propagation is shown to depend on the osmotic prestressing of the medium. This mechanism may explain the tears observed in intervertebral disks as degeneration progresses.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020908-020908-6. doi:10.1115/1.4007906.

Everyone can observe the peculiar effect of water on a sponge: upon drying, a sponge shrinks and stiffens; it swells and softens upon wetting. In this work, we aim to explain and model this behavior by using the Biot–Coussy poromechanical framework. We measure the volume and the bulk modulus of sponges at different water contents. Upon drying, the volume of the sponge decreases by more than half and its bulk modulus increases by almost two orders of magnitude. We develop a partially saturated microporomechanical model of the sponge undergoing finite transformations. The model compares well with the experimental data. We show that about half of the stiffening of the sponge upon drying is due to geometrical nonlinearities induced by a closing of the pores under the action of capillary pressure. The other half of the stiffening can be explained by the nonlinear elastic properties of the cellulose material itself.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020909-020909-9. doi:10.1115/1.4007921.

A nonlinear poroelastic constitutive model for unsaturated porous materials is formulated based on a higher order formulation of free energy including mechanical and moisture contributions and the coupling between moisture and mechanics. This orthotropic model leads to the explicit formulation of the dependence of the compliance, moisture capacity, and coupling coefficient on stress and liquid pressure. The nonlinear poroelastic material properties can be easily determined from mechanical testing at different moisture content and free swelling/sorption tests. An academic example illustrates the capacity of the proposed model to describe nonlinear moisture dependent elasticity, stress dependent sorption, and swelling, also called mechano-sorption and moisture expel during mechanical loading. Two materials are analyzed in detail: wood and Berea sandstone. The poroelastic model shows a good agreement with measurements. Different moisture dependence of the elastic properties is found, with wood showing a more complex moisture/mechanical interaction. Berea sandstone is found to show an important nonlinear elastic behavior dependent on stress, similar in dry and wet conditions.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020910-020910-8. doi:10.1115/1.4007908.

We revisit the poromechanics set up by Olivier Coussy for better understanding of the mechanical effect of partial freezing in cohesive porous materials. This approach proves to be able to quantitatively predict swelling even if the in-pore liquid does not expand when solidifying. In this case, dilation results from the so-called cryosuction process that dominates thermal shrinkage under cooling, as shown in our analysis conducted on the historical experiment run by Beaudoin and MacInnis (1974, “The Mechanism of Frost Damage in Hardened Cement Paste,” Cem. Concr. Res., 4, pp. 139–147) on benzene saturated 24-h old cement paste. Both mechanisms are also at work when freezing water saturated early age cement paste with air voids. In this case, the cryosuction process results in shrinkage and should be added to the thermal shrinkage, their respective amplitudes being temperature dependent but, a priori, of the same order of magnitude.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020911-020911-7. doi:10.1115/1.4007924.

In this study, we show that the key to understand why the same salt can cause damage in some conditions and not in others is the kinetics of crystallization. We present experiments assessing the impact of the recrystallization dynamics of sodium sulfate on damage observed in sandstone after repeated cycles of rewetting/drying and humidification/drying. Macroscopic and microscopic scale experiments using magnetic resonance imaging and phase contrast microscopy demonstrate that sodium sulfate that has both hydrated and anhydrous phases can lead to severe damage in sandstone during rewetting/drying cycles, but not during humidity cycling. During rewetting (a rapid process) in regions (pores) that are highly concentrated in salt, anhydrous microcrystals dissolve only partially, giving rise to a heterogeneous salt solution that is supersaturated with respect to the hydrated phase. The remaining anhydrous crystals then act as seeds for the formation of large amounts of hydrated crystals, creating grape-like structures that expand rapidly. These clusters can generate stresses larger than the tensile strength of the stone, leading to damage. On the other hand, with humidification (a slow process) and after complete deliquescence of salt crystals, the homogeneous sodium sulfate solution can reach high concentrations during evaporation without any nucleation, favoring the formation of isolated anhydrous crystals (thenardite). The crystallization of the anhydrous salt generates only very small stresses compared to the hydrated clusters and therefore causes hardly any damage to the stone.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):020912-020912-14. doi:10.1115/1.4007925.

The porochemoelectroelastic analytical models have been used to describe the response of chemically active and electrically charged saturated porous media such as clay soils, shales, and biological tissues. However, existing studies have ignored the anisotropic nature commonly observed on these porous media. In this work, the anisotropic porochemoelectroelastic theory is presented. Then, the solution for an inclined wellbore drilled in transversely isotropic shale formations subjected to anisotropic far-field stresses with time-dependent down-hole fluid pressure and fluid activity is derived. Numerical examples illustrating the combined effects of porochemoelectroelastic behavior and anisotropy on wellbore responses are also included. The analysis shows that ignoring either the porochemoelectroelastic effects or the formation anisotropy leads to inaccurate prediction of the near-wellbore pore pressure and effective stress distributions. Finally, wellbore responses during a leak-off test conducted soon after drilling are analyzed to demonstrate the versatility of the solution in simulating complex down-hole conditions.

Topics: Pressure , Fluids , Drilling , Stress
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021001-021001-8. doi:10.1115/1.4007424.

Based on the finite element method, the numerical solution of the shallow-water equation for one-dimensional (1D) unsteady flows was established. To respect the stability criteria, the time step of the method was dependent on the space step and flow velocity. This method was used to avoid the restriction due to the wave celerity variation in the computational analysis when using the method of characteristics. Furthermore, boundary conditions are deduced directly from the scheme without using characteristics equations. For the numerical solution, a general-purpose computer program, based on the finite element method (FEM), is coded in fortran to analyze the dynamic response of the open channel flow. This program is able to handle rectangular, triangular, or trapezoidal sections. Some examples solved with the finite element method are reported herein. The first involves routing a discharge hydrograph down a rectangular channel. The second example consists of routing a sudden shutoff of all flow at the downstream end of a rectangular channel. The third one deals with routing a discharge hydrograph down a trapezoidal channel. These examples are taken from the quoted literature text book. Numerical results agree well with those obtained by these authors and show that the proposed method is consistent, accurate, and highly stable in capturing discontinuities propagation in free surface flows.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021002-021002-9. doi:10.1115/1.4007681.

The buckling of thin films with natural nonlinearity can provide a useful tool in many applications. In the present paper, the mechanical properties of controllable buckling of thin films are investigated by accounting for both geometric nonlinearity and surface effects at nanoscale. The effects of surface elasticity and residual surface tension on both static and dynamic behaviors of buckled thin films are discussed based on the surface-layer-based model. The dynamic design strategy for buckled thin films as interconnects in flexible electronics is proposed to avoid resonance in a given noise environment based on the above analysis. Further discussion shows that the thermal and piezoelectric effects on mechanical behavior of buckled thin film are equivalent to that of residual surface tension.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021003-021003-7. doi:10.1115/1.4007721.

Piezoelectric bimorph benders are a particular class of piezoelectric devices, which are characterized by the ability to produce flexural deformation greatly larger than the length or thickness deformation of a single piezoelectric layer. Due to extensive dimensional reduction of devices and to the high accuracy and repeatability requested, the effect of erroneous parameter estimation and the fluctuation of parameters due to external reasons, sometimes, cannot be omitted. As such, we consider mechanical, electrical and piezoelectric parameters as uniformly distributed around a nominal value and we calculate the distribution of natural frequencies of the device. We consider an analytical model for the piezoelectric bimorph proposed in literature. The results show how the parameters errors are reflected on the natural frequencies and how an increment of the error is able to change the shape of the frequencies distribution.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021004-021004-6. doi:10.1115/1.4007226.

The objective of this paper is to investigate the effects of geometrical parameters such as the edge distance-to-hole diameter ratio {e/d}, plate width-to-hole diameter ratio {w/d}, and the distance between two holes-to-hole diameter ratio {l/d} on stress distribution in a unidirectional composite laminate with two serial pin-loaded holes, analytically and numerically. It is assumed that all short and long fibers lie in one direction while loaded by a force po at infinity. To derive differential equations based on a shear lag model, a hexagonal fiber-array model is considered. The resulting pin loads on composite plate are modeled through a series of spring elements accounting for pin elasticity. The analytical solutions are, moreover, compared with the detailed 3D finite element values. A close match is observed between the two methods. The presence of the pins on shear stress distribution in the laminate is also examined for various pin diameters.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021005-021005-8. doi:10.1115/1.4007431.

A model for nonfrictional power loss in derailleur-type, bicycle chain drives is developed to identify factors that influence transmission efficiency. Existing treatments of chain drive efficiency consider frictional losses but these do not explain the measured tension dependence of power losses and efficiencies for derailleur-type systems. Based on a nonlinear, spring-mass, mechanical transmission line, the model developed in this work shows that losses can be related to harmonic generation and dispersion in the chain. The nonlinear response leading to harmonic generation results from elastic contact at pin-bushing interfaces while dispersion is related to the periodic nature of the chain construction. Using this approach, the tension-dependence of power loss and efficiency are modeled and the influences of various chain-related characteristics on efficiency are assessed. If Hertzian contact descriptions are used, then the dependence of loss and efficiency on pin-bushing clearance, contact length and modulus can be estimated. Modeled results agree with experiment and show that power loss decreases with increasing chain tension and that efficiency varies nearly linearly with the reciprocal of the chain tension under operational conditions that are typical for bicycle chain drives. Significant increases to the power transmission efficiency of bicycle chain drives in derailleur-based systems could be achieved by altering the geometries and materials of current chain components.

Topics: Chain , Bicycles , Bushings , Tension
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021006-021006-12. doi:10.1115/1.4007432.

In this paper, the vibrational behavior of double-walled carbon nanotubes (DWCNTs) is studied by a nonlocal elastic shell model. The nonlocal continuum model accounting for the small scale effects encompasses its classical continuum counterpart as a particular case. Based upon the constitutive equations of nonlocal elasticity, the displacement field equations coupled by van der Waals forces are derived. The set of governing equations of motion are then numerically solved by a novel method emerged from incorporating the radial point interpolation approximation within the framework of the generalized differential quadrature method. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions. The influences of small scale factor, layerwise boundary conditions and geometrical parameters on the mechanical behavior of DWCNTs are fully investigated. Explicit expressions for the nonlocal frequencies of DWCNTs with all edges simply supported are also analytically obtained by a nonlocal elastic beam model. Some new intertube resonant frequencies and the corresponding noncoaxial vibrational modes are identified due to incorporating circumferential modes into the shell model. A shift in noncoaxial mode numbers, not predictable by the beam model, is also observed when the radius of DWCNTs is varied. The results generated also provide valuable information concerning the applicability of the beam model and new noncoaxial modes affecting the physical properties of nested nanotubes.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021007-021007-5. doi:10.1115/1.4007476.

A simple and easy-to-use analytical (“mathematical”) predictive model has been developed for the assessment of the size of an inelastic zone, if any, in a ball-grid-array (BGA) assembly. The BGA material is considered linearly elastic at the strain level below the yield point and ideally plastic above the yield strain. The analysis is carried out under the major assumptions that, as far as the estimated size of an inelastic zone is concerned, (1) the inhomogeneous (“discrete”) BGA structure can be substituted by a homogeneous (continuous) bonding layer of the same thickness (height) and (2) only the longitudinal cross-section of the package-substrate assembly can be considered. The numerical example carried out for a 30 mm long surface-mount package and a $200 μm$ thick lead-free solder indicated that, in the case of a high expansion PCB substrate, about 7.5% of the interface's size experienced inelastic strains, while no such strains could possibly occur in the case of a low expansion ceramic substrate. The suggested model can be used to check if the zone of inelastic strains exists in the design of interest and, if inelastic strains cannot be avoided, how large this zone is, before applying a Coffin-Manson-type of an equation (such as, say, Anand's model in the ANSYS software) with an objective to evaluate the BGA material lifetime.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021008-021008-10. doi:10.1115/1.4007477.

Low-temperature thermally induced stresses in a trimaterial assembly subjected to the change in temperature are predicted based on an approximate structural analysis (strength-of-materials) analytical (“mathematical”) model. The addressed stresses include normal stresses acting in the cross-sections of the assembly components and determining their short- and long-term reliability, as well as the interfacial shearing and peeling stresses responsible for the adhesive and cohesive strength of the assembly. The model is applied to a preframed crystalline silicon photovoltaic module (PVM) assembly. It is concluded that the interfacial thermal stresses, and especially the peeling stresses, can be rather high, so that the structural integrity of the module could be compromised, unless appropriate design for reliability measures are taken. The developed model can be helpful in the stress analysis and physical (structural) design of the PVM and other trimaterial assemblies.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021009-021009-10. doi:10.1115/1.4007254.

A controllable experimental method using a two-hemispherical-bead setup and a split Hopkinson pressure bar (SHPB) apparatus is implemented to study the dynamic elasto-plastic contact laws between ductile beads in contact. Beads made of four different metals, either rate sensitive (stainless steel 302 and 440C) or rate insensitive (Al alloy 2017 and brass alloy 260), are used. The experimental elasto-plastic contact force-displacement curves are obtained under different loading rates. The effects of material rate sensitivity and bead pair size on the contact laws are studied, and the way that the rate sensitivity of the materials translates to rate sensitivity contact force-displacement relations is explored. The transmitted energy ratio, which is related to the macroscale concept of a coefficient of restitution, is also calculated and, for all materials, shows a decrease with increasing impact speed. In addition, the experimental contact force–displacement data, residual compressive displacement, and diameter of yield area are compared with predictions from several widely-used theoretical models to generalize these experimental results to arbitrary contact situations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021010-021010-13. doi:10.1115/1.4007722.

A fuzzy fiber reinforced composite (FFRC) reinforced with wavy zig-zag single-walled carbon nanotubes (CNTs) and carbon fibers is analyzed in this study. The distinct constructional feature of this composite is that the wavy CNTs are radially grown on the surface of carbon fibers. To study the effect of the waviness of CNTs on the elastic properties of the FFRC, analytical models based on the mechanics of materials (MOM) approach is derived. Effective elastic properties of the FFRC incorporating the wavy CNTs estimated by the MOM approach have been compared with those predicted by the Mori–Tanaka (MT) method. The values of the effective elastic properties of this composite are estimated in the presence of an interphase between the CNT and the polymer matrix which models the nonbonded van dar Waals interaction between the CNT and the polymer matrix. The effect of waviness of CNTs on the effective properties of the FFRC is investigated when the wavy CNTs are coplanar with two mutually orthogonal planes. The results demonstrate that the axial effective elastic properties of the FFRC containing wavy CNTs can be improved over those of the FFRC with straight CNTs.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021011-021011-8. doi:10.1115/1.4007434.

There is currently a trend toward increased usage of polymeric materials as functional materials because they are likely to experience force in the nanometer range. Thus, we describe here nanoindentation experiments in a polymer nanocomposite system with different nanoclay content and compare with the pristine counterpart. A Berkovich nanoindenter was used to conduct nanoscale deformation experiments using a load of 1–5 mN. The nanoindentation contact properties of relevance to functional applications notably hardness, modulus, and adhesion forces were studied. The addition of 8 wt% of nanoclay to high density polyethylene led to an increase in the indentation hardness by ∼30% and modulus by 25%. Furthermore, using load-displacement plots, the adhesion force between the indenter tip and the material's surface was measured. The adhesion force that is related to the stickiness of the surface was observed to decrease on the introduction of nanoclay in the polymer because of an increase in hardness and modulus of the nanocomposite, leading to a decrease in the area of interaction between the indenter tip and the probed surface. The resistance to nanoindentation of the nanocomposite is explained in terms of a shift in von Mises stress from the surface to the subsurface in the nanocomposite.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021012-021012-11. doi:10.1115/1.4007524.

A physically meaningful analytical (mathematical) model is developed for the prediction of the interfacial shearing thermal stress in an assembly comprised of two identical components, which are subjected to different temperatures. The bonding system is comprised of a plurality of identical columnlike supports located at equal distances (spaces) from each other. The model is developed in application to a thermoelectric module (TEM) design where bonding is provided by multiple thermoelectric material supports (legs). We show that thinner (dimension in the horizontal direction) and longer (dimension in the vertical direction) TEM legs could result in a significant stress relief, and that such a relief could be achieved even if shorter legs are employed, as long as they are thin and the spacing between them is significant. It is imperative, of course, that if thin legs are employed for lower stresses, there is still enough interfacial “real estate,” so that the adhesive strength of the assembly is not compromised. On the other hand, owing to a lower stress level in an assembly with thin legs and large spacing, assurance of its interfacial strength is less of a challenge than for a conventional assembly with stiff, thick, and closely positioned legs. We show also that the thermal stresses not only in conventional TEM designs (using $Be2Te3$ as the thermoelectric material, and Sn-Sb solder), but also in the future high-power (and high operating temperatures) TEM design (using Si or SiGe as the thermoelectric material and Gold100 as the appropriate solder), might be low enough, so that the short- and long-term reliability of the TEM structure could still be assured. We have found, however, that thin-and-long legs should be considered for lower stresses, but not to an extent that appreciable bending deformations of the legs become possible. Future work will include, but might not be limited to, the finite-element computations and to experimental evaluations (e.g., shear-off testing) of the stress-at-failure for the TEMs of interest.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021013-021013-11. doi:10.1115/1.4007221.

In a conductor carrying electric current, the Lorentz force gives rise to mechanical stresses. Here, we study a long elastic cylindrical conductor that moves axially with constant velocity through two electrode plates. The aims are to explore how the stresses in the conductor depend on the velocity in the stationary case of constant current and to assess the validity of the analytic method used. The diffusion equation for the magnetic flux density is solved by use of Fourier transform, and the current density is determined. The stresses, due to the Lorentz force, are found by use of an analytic method combining the solutions of a quasi-static radial problem of plane deformation and a dynamic axial problem of uniaxial stress. They are also determined through FE analysis. Radial field profiles between the plates indicate a velocity skin effect signifying that the current and the magnetic field are concentrated near the cylindrical surface up-stream and are more uniformly distributed downstream. The radial and hoop stresses are compressive, while the axial stress is tensile. The von Mises effective stress increases towards the symmetry axis, in the downstream direction, and with velocity. There are circumstances under which a large current can produce an effective stress in a copper conductor of the order of the yield stress without causing a significant temperature rise. The stresses obtained with the two methods agree well, even relatively near the electrode plates. The analytical method should be useful in similar cases as well as for the provision of test cases for more general simulation tools.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021014-021014-9. doi:10.1115/1.4007228.

We study the short time transient stress and pore pressure fields near the tip of a stationary crack when a sudden load is applied to a poroelastic solid. These fields are determined using a small scale “yielding” (SSY) analysis where the stress relaxation due to fluid flow is confined to a small region near the crack tip. They are found to exhibit the usual inverse square root singularity characteristic of cracks in linear elastic solids. Analysis shows that these fields are self-similar; the region of stress relaxation that propagates outward from the crack tip is proportional to $Dct$, where $Dc$ is the cooperative diffusion coefficient and t is time. The pore pressure at the crack tip vanishes immediately after loading. The stress intensity factor at the crack tip is found to be reduced by a factor of $1/[2(1-v)]$, where $v$ is the Poisson's ratio of the drained solid. Closed form approximations are found for the pore pressure and the trace of the effective stress. These approximate analytical solutions compare well with finite element results.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021015-021015-13. doi:10.1115/1.4007440.

Surface modifications are known as efficient technologies for advanced carbon fibers to achieve significant improvement of interface adhesion in composites, one of which is to increase the surface roughness in the fiber's longitudinal direction in practice. As a result, many microridges and grooves are produced on carbon fiber's surfaces. How does the surface roughness influence the carbon fiber's pull-out behavior? Are there any restrictions on the relation between the aspect ratio and surface roughness of fibers in order to obtain an optimal interface? Considering the real morphology on carbon fiber's surface, i.e., longitudinal roughness, an improved shear-lag theoretical model is developed in this paper in order to investigate the interface characteristics and fiber pull-out for carbon fiber-reinforced thermosetting epoxy resin (brittle) composites. Closed-form solutions to the carbon fiber stress are obtained as well as the analytical load-displacement relation during pullout, and the apparent interfacial shear strength (IFSS). It is found that the interfacial adhesion and the apparent IFSS are effectively strengthened and improved due to the surface roughness of carbon fibers. Under a given tensile load, an increasing roughness will result in a decreasing fiber stress in the debonded zone and a decreasing debonded length. Furthermore, it is interesting to find that, for a determined surface roughness, an optimal aspect ratio, about 30∼45, of carbon fibers exists, at which the apparent IFSS could achieve the maximum. Comparison to the existing experiments shows that the theoretical model is feasible and reasonable to predict the experimental results, and the theoretical results should have an instructive significance for practical designs of carbon/epoxy composites.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021016-021016-10. doi:10.1115/1.4007475.

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021017-021017-8. doi:10.1115/1.4007586.

In this paper, analytical solutions are derived for the case when an elastic water-backed plate (WBP) is subject to an exponential shock loading near a fixed solid boundary. Two cases, a rigid plate and an elastic plate represented by two mass elements connected by a spring and a dashpot, are studied. The analytical solution is extended from Taylor's (1963, “The Pressure and Impulse of Submarine Explosion Waves on Plates,” Scientific Papers of Sir Geoffrey Ingram Taylor, Vol. 3, G. K. Batchelor, ed., Cambridge University Press, Cambridge, UK, pp. 287–303) floating air-backed plate (ABP) model and the water-backed plate model of Liu and Young (2008, “Transient Response of Submerged Plates Subject to Underwater Shock Loading: An Analytical Perspective,” J. Appl. Mech., 75(4), 044504; 2010, “Shock-Structure Interaction Considering Pressure Precursor,” Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, CA). The influences of five parameters are studied: (a) the distance of the fixed boundary from the back plate $d$, (b) the fluid structure interaction ($FSI$) parameter $φ$ of the plate, (c) the stiffness of the plate as represented by the natural frequency of the system $T$, (d) the material damping coefficient $CD$ of the plate, and (e) the pressure precursor (rise) time $θr$. The results show that the pressure responses at the front and back surfaces of the plate are greatly affected by the proximity to the fixed boundary, the fluid-structure interaction parameter, the ratio of the shock decay time to the natural period of the structure, and the rise time of incident pressure. The effect of structural damping (assuming a typical material damping coefficient of 5%) is found to be practically negligible compared to the other four parameters.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021018-021018-7. doi:10.1115/1.4007212.

The problem of stress concentrations in the vicinity of pin-loaded holes is of particular importance in the design of multilayered composite structures made of triangular or circular glass fibers. It is assumed that all of the fibers in the laminate lie in one direction while loaded by a force p0 at infinity, parallel to the direction of the fibers. According to the shear lag model, equilibrium equations are derived for both types of fibers. A rectangular arrangement is postulated in either case. Upon the proper use of boundary and bondness conditions, stress fields are derived within the laminate, along with the surrounding pinhole. The analytical results are compared to those of the finite element values. A very good agreement is observed between the two methods. According to the results, composite structures made of triangular glass fibers result in lower values of stress concentrations around the pin, as opposed to those of circular glass fibers.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021019-021019-13. doi:10.1115/1.4007577.

A mechanical system is often modeled as a set of particles and rigid bodies, some of which are constrained in one way or another. A concise method is proposed for identifying a set of constraint forces needed to ensure the restrictions are met. Identification consists of determining the direction of each constraint force and the point at which it must be applied, as well as the direction of the torque of each constraint force couple, together with the body on which the couple acts. This important information can be determined simply by inspecting constraint equations written in vector form. For the kinds of constraints commonly encountered, the constraint equations are expressed in terms of dot products involving velocities of the affected points or particles and angular velocities of the bodies concerned. The technique of expressing constraint equations in vector form and identifying constraint forces by inspection is useful when one is deriving explicit, analytical equations of motion by hand or with the aid of symbolic algebra software, as demonstrated with several examples.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021020-021020-11. doi:10.1115/1.4007779.

The Girsanov linearization method (GLM), proposed earlier in Saha, N., and Roy, D., 2007, “The Girsanov Linearisation Method for Stochastically Driven Nonlinear Oscillators,” J. Appl. Mech.,74, pp. 885–897, is reformulated to arrive at a nearly exact, semianalytical, weak and explicit scheme for nonlinear mechanical oscillators under additive stochastic excitations. At the heart of the reformulated linearization is a temporally localized rejection sampling strategy that, combined with a resampling scheme, enables selecting from and appropriately modifying an ensemble of locally linearized trajectories while weakly applying the Girsanov correction (the Radon–Nikodym derivative) for the linearization errors. The semianalyticity is due to an explicit linearization of the nonlinear drift terms and it plays a crucial role in keeping the Radon–Nikodym derivative “nearly bounded” above by the inverse of the linearization time step (which means that only a subset of linearized trajectories with low, yet finite, probability exceeds this bound). Drift linearization is conveniently accomplished via the first few (lower order) terms in the associated stochastic (Ito) Taylor expansion to exclude (multiple) stochastic integrals from the numerical treatment. Similarly, the Radon–Nikodym derivative, which is a strictly positive, exponential (super-) martingale, is converted to a canonical form and evaluated over each time step without directly computing the stochastic integrals appearing in its argument. Through their numeric implementations for a few low-dimensional nonlinear oscillators, the proposed variants of the scheme, presently referred to as the Girsanov corrected linearization method (GCLM), are shown to exhibit remarkably higher numerical accuracy over a much larger range of the time step size than is possible with the local drift-linearization schemes on their own.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021021-021021-7. doi:10.1115/1.4007255.

The classical continuum theory cannot be directly used to describe the behavior of nanostructures because of their size-dependent attribute. Surface stress effect is one of the most important size dependencies of structures at this submicron size, which is due to the high surface to volume ratio of nanoscale domain. In the present study, the nonclassical governing differential equation together with corresponding boundary conditions are derived using Hamilton's principle, into which the surface energies are incorporated through the Gurtin-Murdoch elasticity theory. The model developed herein contains intrinsic length scales to take the size effect into account and is used to analyze the free vibration response of circular nanoplates including surface stress effect. The generalized differential quadrature (GDQ) method is employed to discretize the governing size-dependent differential equation along with simply supported and clamped boundary conditions. The classical and nonclassical frequencies of circular nanoplates with various edge supports and thicknesses are calculated and are compared to each other. It is found that the influence of surface stress can be different for various circumferential mode numbers, boundary conditions, plate thicknesses, and surface elastic constants.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021022-021022-9. doi:10.1115/1.4007543.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021023-021023-8. doi:10.1115/1.4007478.

A neo-Hookean half-space, in equilibrium under uniform Cauchy stress, undergoes contact by a sliding rigid ellipsoid or a rolling rigid sphere. Sliding is resisted by friction, and sliding or rolling speed is subcritical. It is assumed that a dynamic steady state is achieved and that deformation induced by contact is infinitesimal. Transform methods, modified by introduction of quasi-polar coordinates, are used to obtain classical singular integral equations for this deformation. Assumptions of specific contact zone shape are not required. Signorini conditions and the requirement that resultant compressive load is stationary with respect to contact zone stress give an equation for any contact zone span in terms of a reference value and an algebraic formula for the latter. Calculations show that prestress can significantly alter the ratio of spans parallel and normal to the direction of die travel, an effect enhanced by increasing die speed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021024-021024-12. doi:10.1115/1.4007682.

A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters δ and α, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of δ and α. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-θ, and generalized-α methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-θ, and generalized-α methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021025-021025-17. doi:10.1115/1.4007787.

Transverse shaft cracks are one of the most dangerous malfunctions of the rotating machines, including turbo- and hydrogenerators, high-speed machine tool spindles, etc. The undetected crack may grow slowly and not disturb normal machine operation. However, if it extends to a critical depth, the immediate shaft fracture may completely damage the machine, resulting in a catastrophic accident. Therefore, in-depth knowledge of the crack propagation process is essential to ensure reliable and safe operation of rotating machinery. The article introduces a new model of the propagating shaft crack. The approach is based on the rigid finite element (RFE) method, which has previously proven its effectiveness in the dynamical analysis of numerous complicated machines and structures. The crack is modeled using several dozen spring-damping elements (SDEs), connecting the faces of the cracked section of the shaft. By controlling the exact behavior of individual SDEs, not only the breathing mechanism, but also the crack propagation process can be simply introduced. In order to accomplish this, the stress intensity factors (SIFs) along the crack edge are calculated using the novel approach based on the modified virtual crack closure technique (VCCT). Based on the SIF values, the crack propagation rate is calculated from the Paris law. If the number of load cycles is greater than the constantly updated threshold number, then the crack edge is shifted by a small increment. This way, starting from the first initially cracked SDE, the crack is extended little by little, continuously changing its shape. The approach is illustrated with numerical results, demonstrating the changes in the rotor vibration response and in the crack shape and also explaining some issues about the breathing mechanism due to the propagating shaft crack. The increasing amplitude of the 2X harmonic component is recognized as an evident propagating crack signature. The numerical results correspond well with the data reported in the literature. The RFE model of the rotor is validated by comparing the vibration responses obtained experimentally and numerically. A good agreement between these data confirms the correctness and accuracy of the proposed model. The suggested approach may be utilized for a more reliable dynamic analysis of the rotating shafts, having the potential to experience propagating transverse cracks.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021026-021026-11. doi:10.1115/1.4007435.

This paper is concerned with a solid shell finite element formulation to simulate the behavior of thin dielectric elastomer structures. Dielectric elastomers belong to the group of electroactive polymers. Due to efficient electromechanical coupling and the huge actuation strain, they are very interesting for actuator applications. The coupling effect in the material is mainly caused by polarization. In the present work, a simple constitutive relation, which is based on an elastic model involving one additional material constant to describe the polarization state, is incorporated in a solid shell formulation. It is based on a mixed variational principle of Hu-Washizu type. Thus, for quasi-stationary fields, the balance of linear momentum and Gauss' law are fulfilled in a weak sense. As independent fields, the displacements, electric potential, strains, electric field, mechanical stresses, and dielectric displacements are employed. The element has eight nodes with four nodal degrees of freedom, three mechanical displacements, and the electric potential. The surface oriented shell element models the bottom and the top surfaces of a thin structure. This allows for a simple modeling of layered structures by stacking the elements through the thickness. Some examples are presented to demonstrate the ability of the proposed formulation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021027-021027-11. doi:10.1115/1.4007723.

An integrable Eulerian rate formulation of finite deformation elasticity is developed, which relates the Jaumann or other objective corotational rate of the Kirchhoff stress with material spin to the same rate of the left Cauchy–Green deformation measure through a deformation dependent constitutive tensor. The proposed constitutive relationship can be written in terms of the rate of deformation tensor in the form of a hypoelastic material model. Integrability conditions, under which the proposed formulation yields (a) a Cauchy elastic and (b) a Green elastic material model are derived for the isotropic case. These determine the deformation dependent instantaneous elasticity tensor of the material. In particular, when the Cauchy integrability criterion is applied to the stress-strain relationship of a hyperelastic material model, an Eulerian rate formulation of hyperelasticity is obtained. This formulation proves crucial for the Eulerian finite strain elastoplastic model developed in part II of this work. The proposed model is formulated and integrated in the fixed background and extends the notion of an integrable hypoelastic model to arbitrary corotational objective rates and coordinates. Integrability was previously shown for the grade-zero hypoelastic model with use of the logarithmic (D) rate, the spin of which is formulated in principal coordinates. Uniform deformation examples of rectilinear shear, closed path four-step loading, and cyclic elliptical loading are presented. Contrary to classical grade-zero hypoelasticity, no shear oscillation, elastic dissipation, or ratcheting under cyclic load is observed when the simple Zaremba–Jaumann rate of stress is employed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):021028-021028-11. doi:10.1115/1.4007724.

An Eulerian rate formulation of finite strain elastoplasticity is developed based on a fully integrable rate form of hyperelasticity proposed in Part I of this work. A flow rule is proposed in the Eulerian framework, based on the principle of maximum plastic dissipation in six-dimensional stress space for the case of J2 isotropic plasticity. The proposed flow rule bypasses the need for additional evolution laws and/or simplifying assumptions for the skew-symmetric part of the plastic velocity gradient, known as the material plastic spin. Kinematic hardening is modeled with an evolution equation for the backstress tensor considering Prager’s yielding-stationarity criterion. Nonlinear evolution equations for the backstress and flow stress are proposed for an extension of the model to mixed nonlinear hardening. Furthermore, exact deviatoric/volumetric decoupled forms for kinematic and kinetic variables are obtained. The proposed model is implemented with the Zaremba–Jaumann rate and is used to solve the problem of rectilinear shear for a perfectly plastic and for a linear kinematic hardening material. Neither solution produces oscillatory stress or backstress components. The model is then used to predict the nonlinear hardening behavior of SUS 304 stainless steel under fixed-end finite torsion. Results obtained are in good agreement with reported experimental data. The Swift effect under finite torsion is well predicted by the proposed model.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):024503-024503-5. doi:10.1115/1.4007964.

The experimental behavior of natural Pisa clay under complex stress paths is simulated by an enhanced anisotropic elastoplastic bounding surface model. In its present application, the model has nine parameters and focuses on the basic features of clay behavior, such as yielding, critical state, overconsolidation and plastic anisotropy. The model is first calibrated against the test results obtained from tri-axial compression tests and subsequently used to predict the behavior of true tri-axial tests. The overall agreement between the model predictions and the experimental data is very good for proportional loading tests in both meridional and deviatoric stress spaces. The result of prediction is also compared with the original simulations that were conducted by an advanced clay model.

Commentary by Dr. Valentin Fuster

### Research Papers

J. Appl. Mech. 2013;80(2):020902-020902-14. doi:10.1115/1.4007905.

The strength homogenization of cohesive-frictional solids influenced by the presence of two pressurized pore spaces of different characteristic sizes is addressed in this study. A two-scale homogenization model is developed based on limit analysis and the second-order method (SOM) in linear comparison composite theory, which resolves the nonlinear strength behavior through the use of linear comparison composites with optimally chosen properties. For the scale of the classical configuration of a porous solid, the formulation employs a compressible thermoelastic comparison composite to deliver closed-form expressions of strength criteria. Comparisons with numerical results reveal that the proposed homogenization estimates for drained conditions are adequate except for high triaxialities in the mean compressive strength regime. At the macroscopic scale of the double-porosity material, the SOM results are in agreement with strength criteria predicted by alternative micromechanics solutions for materials with purely cohesive solid matrices and drained conditions. The model predictions for the cohesive-frictional case show that drained strength development in granularlike composites is affected by the partitioning of porosity between micro- and macropores. In contrast, the drained strength is virtually equivalent for single- and double-porosity materials with matrix-inclusion morphologies. Finally, the second-order linear comparison composite approach confirms the applicability of an effective stress concept, previously proposed in the literature of homogenization of cohesive-frictional porous solids, for double-porosity materials subjected to similar pressures in the two pore spaces. For dissimilar pore pressures, the model analytically resolves the complex interplays of microstructure, solid properties, and volume fractions of phases, which cannot be recapitulated by the effective stress concept.

Commentary by Dr. Valentin Fuster

### Technical Briefs

J. Appl. Mech. 2013;80(2):024501-024501-7. doi:10.1115/1.4007433.

The effects of wall properties on the peristaltic flow of an incompressible pseudoplastic fluid in a curved channel are investigated. The relevant equations are modeled. Long wavelength and low Reynolds number approximations are adopted. The stream function and axial velocity are derived. The variations of the embedding parameters into the problem are carefully discussed. It is noted that the velocity profiles are not symmetric about the central line of the curved channel.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2013;80(2):024502-024502-7. doi:10.1115/1.4007786.

A half-space containing transversely isotropic thermoelastic material with a depth-wise axis of material symmetry is considered to be under the effects of axisymmetric transient surface thermal and forced excitations. With the use of a new scalar potential function, the coupled equations of motion and energy equation are uncoupled, and the governing equation for the potential function, is solved with the use of Hankel and Laplace integral transforms. As a result, the displacements and temperature are represented in the form of improper double integrals. The solutions are also investigated in detail for surface traction and thermal pulses varying with time as Heaviside step function. It is also shown that the derived solutions degenerate to the results given in the literature for isotropic materials. Some numerical evaluations for displacement and temperature functions for two different transversely isotropic materials with different degree of anisotropy are presented to portray the dependency of response on the thermal properties as well as the degree of anisotropy of the medium.

Commentary by Dr. Valentin Fuster