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This paper introduces a novel deep learning approach for generating fine resolution structures that preserve all the information from the topology optimization (TO). The proposed approach utilizes neural networks (NNs) that map the desired engineering properties to seed for determining optimized structure. The main novelty of this framework is the utilization of parameters such as density and nodal deflections to predict optimized topologies for a wide range of design space. In this research, a three-stage NN framework is employed, wherein the first stage is a feedforward deep neural network. The second stage uses a convolutional neural network (CNN), and the final stage is the post-processing of results obtained from CNN. The first stage maps input design variables vector to output density distribution image. The image output of the first stage and design variables vector inputted to the first network serve as inputs to the second stage, which outputs optimized structure. Finally, post-processing of this optimized structure is done to ensure volume fraction constraint and optimal solution. These structures can be tessellated per the nodal displacement of an FEA solved larger structure. Because of this framework, the proposed approach makes TO computations faster and captures essential physics while using NNs to determine the final structure. The application of the proposed model to three-dimensional problems is also demonstrated.

Summary In order to investigate size-dependent creeping plane microfluidic flow, a boundary element method is implemented that involves the calculation of interior quantities and multi-domain problems. The governing equations are formulated using the skew-symmetric character of the couple-stress tensor and its energy conjugate mean-curvature tensor, as established in consistent couple stress theory. This theoretical formulation includes one characteristic material length scale parameter l that represents the size-dependency of the problem. Here, we present the boundary integral representation and numerical implementation for two-dimensional size-dependent steady state creeping incompressible flow, in which velocities, angular velocities, force- and couple-tractions are the primary variables. Beyond the previously known singular fundamental solution kernels for point force and point couple in the velocity and angular velocity integral equations, here we present the integral equations for calculating internal mean curvatures and couple-stresses, along with their corresponding kernels. The formulation is then applied to solve some computational problems both to investigate the size effects on the response resulting from the theory, and to validate the strength of the numerical method. For this purpose, we study the important effect of different boundary conditions on the flow pattern and consider a multi-fluid domain Couette flow problem.

The focus of this paper is on the use of polynomial chaos (PC) for developing surrogate models for differential algebraic equations (DAEs) with time-invariant uncertainties. Intrusive and nonintrusive approaches to synthesize PC surrogate models are presented including the use of Lagrange interpolation polynomials as basis functions. Unlike ordinary differential equations (ODEs), if the algebraic constraints are a function of the stochastic variable, some initial conditions of the DAEs are also random. A benchmark RLC circuit which is used as a benchmark for linear models is used to illustrate the development of a PC-based surrogate model. A nonlinear example of a simple pendulum also serves as a benchmark to illustrate the potential of the proposed approach. Statistics of the results of the PC models are validated using Monte Carlo (MC) simulations in addition to estimating the evolving probably density functions (PDFs) of the states of the pendulum.

We address the numerical simulation of periodic solids (phononic crystals) within the framework of couple stress elasticity. The additional terms in the elastic potential energy lead to dispersive behavior in shear waves, even in the absence of material periodicity. To study the bulk waves in these materials, we establish an action principle in the frequency domain and present a finite element formulation for the wave propagation problem related to couple stress theory subject to an extended set of Bloch-periodic boundary conditions. A major difference from the traditional finite element formulation for phononic crystals is the appearance of higher-order derivatives. We solve this problem with the use of a Lagrange-multiplier approach. After presenting the variational principle and general finite element treatment, we particularize it to the problem of finding dispersion relations in elastic bodies with periodic material properties. The resulting implementation is used to determine the dispersion curves for homogeneous and porous couple stress solids, in which the latter is found to exhibit an interesting bandgap structure., Comment: 29 pages, 8 figures

A new variational formulation is proposed for time-domain analysis of initial/boundary value problems of dynamic thermoelasticity. By using the concept of mixed convolved action, the difficulties with dissipative phenomena and proper representations of the temporal end point conditions can be overcome to create a true stationary variational principle. After an elementary illustrative development for a lumped parameter thermoelastic model, the convolved action functional for linear thermoelastic continuum is written directly in terms of mixed variables, which include displacements and the impulses of stress, temperature and heat flux. Unlike previous variational approaches, based for example upon a generalization of Hamilton's principle , the present mixed convolved action formulation allows direct application of finite element methodology in both space and time. Here, simple linear shape functions are employed for the temporal representations. Meanwhile, standard three-noded triangular elements are used in the present two-dimensional numerical implementation. Several computational examples are considered to test this original approach and to investigate interesting aspects of coupled dynamic thermoelastic response.

Consistent skew-symmetric couple stress theory, which introduces an intrinsic characteristic length scale parameter l into the formulation, is applied here to model the Rayleigh–Benard convection problem. The focus is on the potential effect of size-dependency on the stability of Rayleigh–Benard convection. We consider a two-dimensional flow and study the effects of couple-stresses by developing and then applying a stream function-vorticity-temperature computational fluid dynamics formulation. Details are provided on the governing equations for size-dependent flow, based on the Boussinesq approximation. Afterwards, the formulation is applied to the Rayleigh–Benard convection problem in a square cavity to examine numerically the stability of the flow as a function of the length scale parameter l. The investigation covers a range of Rayleigh numbers, and includes an evaluation of the critical value beyond which convective instabilities start to appear, based upon an overall kinetic energy norm. The additional boundary conditions associated with consistent couple stress theory are found to play an important role in determining this critical Rayleigh number and also affect the flow patterns. Furthermore, the developments here may encourage the study of size-dependent multiphysics flow phenomena in general, which in turn may reveal new features and mechanisms of energy dissipation and storage at the smallest continuum scales.

The bonded interface between dissimilar materials is often the site of premature failures, especially when there is a stress singularity at the intersection of the interface and the free edge or surface. However, the singularities, which depend upon the elastic constants and the interior angles of the two materials, may vanish for certain ranges of these angles. Thus, proper selection of the interior angles offers the possibility of enhancing joint strength. In this work, the focus is on the shear strength of bonded aluminum-epoxy cylinders of uniform diameter under torsional loading and the effect of selecting angles for which the stress singularity at the edge disappears. A new boundary element framework is developed for pure shear loading to determine the power of the singularity for general interface angles and the value of the associated generalized stress intensity factor (GSIF), using a weighted traction axisymmetric torsional mode formulation. Physical experiments are conducted to examine whether changes in the interior angles produce differences in measured joint strength and failure patterns. Three different geometries are considered: a “concave” angle of 37° for the aluminum cylinder, a butt joint with 90° angles for both cylinders and a “convex” aluminum cylinder with an interior angle of 143°. Based upon elastic theory, the first two cases are associated with singularities, while the third is singularity-free. The experiments demonstrate that the joint shear strength can be greatly improved by avoiding stress singularities through the selection of a proper edge angle. From a physical perspective, this strength improvement is due to crack tortuosity, extensive adherence of epoxy to aluminum, and epoxy matrix spalling. Low strength specimens show rapid debonding and minimal adherence.

A boundary element method is developed to examine two-dimensional size-dependent thermoelastic response in isotropic solids. The formulation is founded on the recently established consistent couple stress theory, in which both the couple-stress tensor and its energy conjugate mean curvature tensor are skew-symmetric. For isotropic materials, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, size-dependency is quantified by one characteristic material length scale parameter l , while the thermal coupling is activated through the classical thermal expansion coefficient α . Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. Here, we develop the boundary integral representation and numerical implementation for this size-dependent thermoelastic boundary element method (BEM) for plane problems, which involves temperatures, displacements, rotations, normal heat fluxes, force-tractions and couple-tractions as primary variables within a boundary-only formulation. Then, we apply this new BEM formulation to several basic computational problems in an effort to validate the robustness of the numerical implementation and to examine size-dependent response.

Length scale based continuum mechanics can account for size dependence - a feature absent in classical theory that is becoming increasingly important in analyzing behavior of matter in micro- and nano-technology. Starting from a foundational mechanics and thermodynamics perspective, recent work has provided a self-consistent formulation to resolve certain issues associated with previous couple stress theory. In the current paper, a penalty-based finite element framework is developed to enable solution of general problems for linear elastic isotropic materials under plane strain conditions within this consistent couple stress theory (C-CST). The penalty parameter is incorporated in the formulation to satisfy continuity requirements, while allowing convergence of the method with only C0 elements. The efficacy of the finite element analysis is verified by studying three example problems for which closed form solutions are known. Furthermore, a fourth example problem for which a closed form solution is unknown is solved numerically to bring out nontrivial features of couple stress theory. The main objective of this work is to demonstrate that a simple finite element analysis-based framework can be effective in exploring the interesting features of C-CST. Additionally, detailed finite element analysis results provided in this work can be used to benchmark future computational development for size-dependent mechanics theory. Underlying couple stress theory resolves indeterminacies in previous formulations.Penalty formulation reduces continuity requirements from C1 to C0.Numerical examples demonstrate robustness of formulation for range of length scales.Bending example illustrates Saint-Venant principle for couple stress elasticity.Stress concentration factors may reduce dramatically due to couple stresses.

One of the major, long-standing challenges in analytical mechanics involves the inability to address systems with dissipation in a rigorous manner. In this paper, we overcome that difficulty by formulating a novel temperature-based stationary variational principle for transient heat diffusion based upon a temporal convolution operator and fractional derivatives. The associated Euler-Lagrange equations provide the governing heat equation, along with the initial conditions on temperature and specified heat flux boundary conditions. A further integration-by-parts then leads to a formulation that is somewhat less symmetric but can be written without introducing fractional calculus. Finally, the resulting principle is used to solve two basic one-dimensional problems, as an illustration of a Ritz-type approach.

Based upon the principle of minimum total potential energy for consistent couple stress theory, a Ritz variational approach is developed using tensor product B-splines for both two- and three-dimensional elastostatic analysis. The underlying theory is size-dependent due to incorporation of the energy conjugate skew-symmetric couple-stress and mean curvature tensors, in addition to the force-stress and strain conjugate pair of classical theory. The use of B-splines as the basis functions can assure the required C 1 continuity of the displacement field, while also permitting higher order representations. Both displacements and rotations are essential boundary conditions in this theory. Displacements can be enforced in the usual way, but rotations require special treatment to maintain symmetry and positive definiteness of the stiffness matrix for well-posed elastostatic problems . Several computational examples are considered to validate the formulation, illustrate convergence characteristics, and investigate mechanical behavior under consistent couple stress theory. All previous numerical analysis of consistent couple stress theory has been limited to plane strain problems . Thus, the extension here to three-dimensions is of great importance, especially because the behavior in several cases shows significant deviation from the plane strain solutions. Consequently, the variational formulations and computational methodology presented in this paper can play a critical role in assessing the predictive capability of consistent couple stress theory and in understanding size-dependent elastic response .

A true variational formulation is developed for dissipative processes based upon the concept of mixed convolved action. Here the focus is on continuum problems associated with heat diffusion, as well as related second sound phenomena. The convolved action can overcome the shortcomings of typical action principles, such as Hamilton’s principle, to address dissipative processes without the need for separate dissipation functionals and ad hoc variational operations. In addition, the mixed convolved action is compatible with the initial and boundary conditions of a well-posed heat problem. In fact, the stationarity of the mixed convolved action is shown to provide the governing partial differential equations, the initial conditions and the boundary conditions as its Euler–Lagrange equations. Thus, the mixed convolved action encapsulates the entire description of the initial/boundary value heat problem. In addition to the theoretical significance, this new formulation can establish the basis for effective numerical methods, for example, involving finite element representations over both space and time. One particular two-dimensional formulation is developed here and then applied to two example problems to illustrate the viability of the proposed approach.

Computational mechanics codes include contact or over-closure models to account for contact stiffness, which can be related to surface roughness and nominal pressure from Greenwood and Williamson. We present results of wear simulation studies with adjustable contact parameters for a ring-on-ring sliding configuration. A boundary element code is used to solve the non-linear contact problem with an axisymmetric thermoelastic representation of the rings, along with a localized Archard wear model. Parametric studies indicate the extent to which the wear tracks can be affected by the roughness and wear history. The effects range from minor to moderately significant, with areas of maximum wear shifting radially. Contact parameters influence pressure, temperature and wear distributions, with stiffer contacts resulting in greater localization.

Based upon the extended framework of Hamilton's principle, a variational formulation in heat diffusion, as well as second sound phenomena is developed. This formulation is compatible with the initial and boundary conditions of a well-posed heat problem, and it correctly accounts for the governing partial differential equations, as its Euler-Lagrange equations. Thus, this new formulation provides a sound base to develop various unified space-time finite element methods. In order to validate the finite element representation over both space and time in the context of this formulation, two-dimensional lower-order space-time finite element methods are also developed with numerical investigations on representative examples.

In this paper, a mixed Lagrangian multiplier formulation is developed for consistent size-dependent couple stress elastodynamic response. This extends previous work on quasistatic couple stress response that also requires the rotation to be one half of the curl of displacement. A Lagrange multiplier, which turns out to be related to the skew-symmetric part of the force stress, is used to constrain the relation between rotation and displacement, so that rotation becomes an independent variable and $$\hbox {C}^{1}$$ continuity is avoided in the weak form. The finite element method is applied first to obtain the matrix form of the Lagrangian formulation in space, and then, discrete variational time integration is introduced to produce the discrete action. Afterward, the variation of discrete action is applied to derive the governing equations in algebraic form, which are used, along with the initial condition relations, to obtain the solutions at finite time steps. Finally, the problems of uniaxial deformation of a square plate, transverse deformation of a cantilever, and uniform traction on a square plate with a hole are investigated under this formulation, and the results are compared to existing methods where possible.

The classical theory of elasticity is an idealized model of a continuum, which works well for many engineering applications. However, with careful experiments one finds that it may fail in describing behavior in fatigue, at small scales and in structures having high stress concentration factors. Many size-dependent theories have been developed to capture these effects, one of which is the consistent couple stress theory. In this theory, couple stress μ i j is present in addition to force stress σ i j and its tensor form is shown to have skew symmetry. The mean curvature κ i j , which is defined as the skew-symmetric part of the gradient of rotations, is the correct energy conjugate of the couple stress. This mean curvature κ i j and strain e i j together contribute to the elastic energy. The scope of this paper is to extend the work to study anisotropic materials and present a corresponding finite element method. A fully displacement based finite element method for couple stress elasticity requires C 1 continuity. To avoid this, a mixed formulation is presented with primary variables of displacements u i and couple stress μ i vectors, both of which require only C 0 continuity. Centrosymmetric classes of materials are considered here for which force stress and strain are decoupled from couple stress and mean curvature in the constitutive relations. Details regarding the numerical implementation are discussed and the effect of couple stress elasticity on anisotropic materials is examined through several computational examples.

The vibrations of ultrathin silicon cantilever microbeams are studied using consistent couple-stress theory to investigate size-dependent effects. The corresponding Euler-Bernoulli beam the...

BackgroundComputer modeling of endovascular coiling intervention for intracranial aneurysm could enable a priori patient-specific treatment evaluation. To that end, we previously developed a finite element method (FEM) coiling technique, which incorporated simplified assumptions. To improve accuracy in capturing real-life coiling, we aimed to enhance the modeling strategies and experimentally test whether improvements lead to more accurate coiling simulations.MethodsWe previously modeled coils using a pre-shape based on mathematical curves and mechanical properties based on those of platinum wires. In the improved version, to better represent the physical properties of coils, we model coil pre-shapes based on how they are manufactured, and their mechanical properties based on their spring-like geometric structures. To enhance the deployment mechanics, we include coil advancement to the aneurysm in FEM simulations. To test if these new strategies produce more accurate coil deployments, we fabricated silicone phantoms of 2 patient-specific aneurysms in duplicate, deployed coils in each, and quantified coil distributions from intra-aneurysmal cross-sections using coil density (CD) and lacunarity (L). These deployments were simulated 9 times each using the original and improved techniques, and CD and L were calculated for cross-sections matching those in the experiments. To compare the 2 simulation techniques, Euclidean distances (dMin, dMax, and dAvg) between experimental and simulation points in standardized CD-L space were evaluated. Univariate tests were performed to determine if these distances were significantly different between the 2 simulations.ResultsCoil deployments using the improved technique agreed better with experiments than the original technique. All dMin, dMax, and dAvg values were smaller for the improved technique, and the average values across all simulations for the improved technique were significantly smaller than those from the original technique (dMin: p = 0.014, dMax: p = 0.013, dAvg: p = 0.045).ConclusionIncorporating coil-specific physical properties and mechanics improves accuracy of FEM simulations of endovascular intracranial aneurysm coiling.

In this paper, we examine the mathematical and physical consistencies of the three primary couple stress theories: original Mindlin-Tiersten-Koiter couple stress theory (MTK-CST), modified couple stress theory (M-CST) and consistent couple stress theory (C-CST). As has been known for many years, MTK-CST suffers from some fundamental inconsistencies, such as the indeterminacy of the couple-stress tensor. Therefore, despite the fact that MTK-CST has a fundamental position in the evolution of size-dependent continuum mechanics, it is not a reliable theory within continuum mechanics, for example, in developing new size-dependent multi-physics formulations. We also observe that M-CST not only inherits all inconsistences from the original MTK-CST, but also suffers from new additional inconsistencies, such as the introduction of a new non-physical governing equation. These inconsistencies refute the claim of those who state that the couple-stress tensor may be chosen symmetric. Therefore, the apparent success of MTK-CST and M-CST in describing a size-effect for some problems, such as two-dimensional plate and beam bending, is not enough to justify these theories as suitable for general cases. In fact, the symmetric couple-stresses in M-CST create torsional or anticlastic deformation, not bending. On the other hand, C-CST, with a skew-symmetric couple-stress tensor, is the consistent continuum mechanics suitable for solving different size-dependent solid, fluid and multi-physics problems.&nbsp

Stress waves scatter upon entering a new medium. This occurs due to the reflection and transmission of the waves, which depends on the impedance mismatch between the two materials and the angle of incidence. For a bi-layered structure with finite dimensions and constant impedance ratio, the scattering and intensity of the stress waves may be varied by changing the interface profile between the two layers. In this paper, a methodology is proposed for optimizing the interface profile between the layers of a finite bi-layered plate for the objective of planar stress wave attenuation. The bi-layered plates are subjected at one end to highly impulsive loadings with various durations, and the geometry of the internal interface is optimized for the purpose of minimizing the amplitude of the maximum reaction force at the opposite fixed end. The optimization methodology is based on a genetic algorithm, which is coupled with a finite element method for analyzing the wave propagation behavior of the plates. It is observed that the interface profile and the amount of stress wave attenuation depend on the duration of the applied impulsive loading, with higher amounts of attenuation obtained when the wavelength associated with the impulsive load is small compared to the dimensions of the bi-layered plates.

The paper begins with an overview of several of the classical integral formulations of elastodynamics, which highlights the natural appearance of temporal convolutions in the reciprocal theorem for such problems. This leads first to the formulation of a principle of virtual convolved action, as an extension of the principle of virtual work to dynamical problems. Then, to overcome the key shortcomings of Hamilton’s principle, the concept of mixed convolved action is developed for linear dynamical problems within the context of continuum solid mechanics. This new approach is broadly applicable to both reversible and irreversible phenomena without the need for special treatments, such as the artificial definition of Rayleigh dissipation functionals. The focus here is on linear elastic and viscoelastic media, which in the latter case is represented by classical Kelvin–Voigt and Maxwell models. Remarkably, for each problem type, the stationarity of the mixed convolved action provides not only the governing partial differential equations, but also the specified boundary and initial conditions, as its Euler–Lagrange equations. Thus, the entire initial/boundary value problem definition is encapsulated in a scalar mixed convolved action functional written in terms of displacements and stress impulses. The resulting formulations possess an elegant structure that provides a versatile framework for the development of novel computational methods, involving finite element representations in both space and time. We present perhaps the simplest approach by employing linear three-node triangular elements for two-dimensional analysis, along with linear shape functions over the temporal domain. Numerical examples are included to verify the formulation and to explore concepts of stress wave attenuation.

Dynamic Ratcheting (DR) is a phenomenon in which plastic deformation in a hysteretic damping system increases asymmetrically in successive cycles. In the present paper, DR in single degree-of-freedom (SDOF) systems under earthquake excitation is investigated based on strong motion data collected in California between 2002 and 2012. The characteristics of earthquakes in the time–frequency domain are studied on spectrograms; furthermore, linkages between the spectrogram, displacement time history, and the Fourier Transform are identified to locate intervals during which DR is expected to occur. It is observed that DR occurs for 10 out of 160 studied earthquakes according to the proposed DR investigation procedure. The likelihood of DR seems to mainly depend on local site conditions; however, the likelihood significantly increases as the magnitude of earthquakes increases.

Some physical experiments exhibit size-dependency for fluid flows at small scales. This in turn necessitates the introduction of couple-stresses in the corresponding continuum theory. The resulting size-dependent couple stress fluid mechanics can be used to explore a range of such non-Newtonian flow behavior at micro- and nano-scales, and also to bridge between atomistic and classical continuum theories. Here we concentrate on two-dimensional flow and examine the effects of couple-stresses by developing and then applying a stream function-vorticity computational fluid dynamics formulation. Details are provided both on the governing equations for size-dependent flow and on the corresponding numerical implementation. Afterwards, the formulation is applied to the lid-driven cavity problem to examine the behavior of the flow as a function of the length scale parameter l . The investigation covers a range of Reynolds numbers, and includes an evaluation of the critical value beyond which a stationary response is no longer possible. The additional boundary conditions associated with consistent couple stress theory are found to play an important role in determining the flow pattern and critical Reynolds number.

In the bonding of dissimilar materials, there is usually a stress singularity at the edge surface whose effects cause failure at the surface, resulting in a premature failure of the joint. In this work, the corresponding generalized stress intensity factor (GSIF) for an aluminum–epoxy joint is studied to determine the angles for which the stress singularity at the edge disappears. Experimental results are gathered for three different edge angles: a “concave” angle of 37° and a butt joint with 90°, both associated with interface stress singularities, and a “convex” angle of 143° that from theory is singularity free. It is demonstrated that the joint strength can be greatly improved by avoiding stress singularities through the selection of the proper edge angle, still preserving the external surfaces of the cylindrical samples. Testing with a larger diameter rod joint for the convex angle geometry shows the same nominal fracture strength, indicating that the measured strength is a material property independent of size when the stress singularity is absent.

The classical theory of piezoelectricity defines linear size-independent electromechanical response in non-centrosymmetric dielectrics that involves coupling between the electric field and the mechanical strains. However, with the continuing push to develop novel micro- and nano-scale materials, structures and devices, there is a need to refine and explore size-dependent electro-mechanical coupling phenomena, which have been observed in experiments on centrosymmetric dielectrics. Here a finite element variational formulation is developed based upon a recent consistent size-dependent theory that incorporates the interactions between the electric field and the mechanical mean curvatures in dielectrics, including those with centrosymmetric structure. The underlying formulation is theoretically consistent in several important aspects. In particular, the electric field equations are consistent with Maxwell's equations, while the mechanical field equations are based upon the recent consistent couple stress theory, involving skew-symmetric mean curvature and couple stress tensors. This, in turn, permits the development of a fully-consistent finite element method for the solution of size-dependent piezoelectric boundary value problems. In this paper, an overview of size-dependent piezoelectricity is first provided, followed by the development of the variational formulation and finite element representation specialized for the planar response of centrosymmetric cubic and isotropic materials. The new formulation is then applied to several illustrative examples to bring out important characteristics predicted by this consistent size-dependent piezoelectric theory.

A variational temperature-dependent formulation is proposed and developed for dynamical problems of thermoplasticity. The variational method is based on a Hamiltonian approach using a weak ...

There can be no doubt as to the importance of vortical motion in fluid mechanics. Yet, very little attention is given typically to the balance law of angular momentum and to its role in defining the fundamental character of stress, which as a result is usually assumed as a symmetric tensor. Here, we allow for the possibility of couple-stresses, along with general non-symmetric force–stresses, and develop a self-consistent size-dependent theory within the context of classical continuum mechanics. This development relies upon the identification of the following key components for the dynamic response of three-dimensional fluid continua: (i) fundamental, uniquely defined kinematical measures of flow, (ii) an independent set of energy conjugate variables, (iii) the corresponding permissible natural and essential boundary conditions, and (iv) a non-redundant set of body-force and inertial contributions. Based upon this formulation, one can recognize that the previous couple-stress theory for fluids suffers from some inconsistencies, which may have restricted its applicability in the study of viscous flows. After presenting the general formulation of the new consistent theory, we specialize for incompressible viscous flow and consider the problem of generalized Poiseuille flow within this size-dependent fluid mechanics. Finally, we conclude that the theory presented here may provide a basis for a broad range of fluid mechanics applications and for fundamental studies of flows at the finest scales for which a continuum representation is valid.

We develop a variational principle based on recent advances in couple-stress theory and the introduction of an engineering mean curvature vector as energy conjugate to the couple stresses. This new variational formulation provides a base for developing a couple-stress finite element approach. By considering the total potential energy functional to be not only a function of displacement, but of an independent rotation as well, we avoid the necessity to maintain C1 continuity in the finite element method that we develop here. The result is a mixed formulation, which uses Lagrange multipliers to constrain the rotation field to be compatible with the displacement field. Interestingly, this formulation has the noteworthy advantage that the Lagrange multipliers can be shown to be equal to the skew-symmetric part of the force-stress, which otherwise would be cumbersome to calculate. Creating a new consistent couple-stress finite element formulation from this variational principle is then a matter of discretizing the variational statement and using appropriate mixed isoparametric elements to represent the domain of interest. Finally, problems of a hole in a plate with finite dimensions, the planar deformation of a ring, and the transverse deflection of a cantilever are explored using this finite element formulation to show some of the interesting effects of couple stress. Where possible, results are compared to existing solutions to validate the formulation developed here.

The objective of this study is to apply a constitutive damage model for cyclic response of E-shaped dampers as well as metal and metallic plate dampers in order to be used in more broad areas such as material and component levels. The seismic isolation system used as sliding bearings and metallic dampers undergoes almost elastoplastic behavior due to seismic excitation. Accordingly, a two surface damage model is applied to understand inelastic behavior and to evaluate the potential damaged state of E-shaped dampers in the seismic isolation system. This damage model is originally formulated through a thermodynamic approach based on the entropy production equation. The cyclic damage model proposed in this study can be implemented into a user subroutine in the finite element (FE) software ABAQUS. Finally, the numerical results for E-shaped energy dissipation devices in the seismic isolation system are compared with the experimental test results so as to check the validity of the proposed model.

The objective of this study is to develop a new constitutive model for cyclic response of metals with much broader applicability. Accordingly, a two-surface damage thermoplasticity model is proposed to understand inelastic behavior and to evaluate a potential damaged state of the metals. This model, which derived from small strain theory, is formulated through a thermodynamic approach to damage mechanics based on entropy production. A simple shear problem was utilized to examine several effects of this model, such as fatigue by cyclic loading and temperature, and to allow for the thermal effects on metals. Following this, the proposed cyclic damage model is implemented as a user subroutine in the finite element software ABAQUS. Finally, numerical results of energy dissipation devices are compared with experimental data for validity of this model.

The effectiveness of a non-homogeneous soil profile for horizontal ground source heat pumps (GSHPs), defined as natural backfill with an intermediate layer of material having different thermal characteristics, is investigated. Steps toward development of a comprehensive model to consider the effects of the non-homogeneous layer are described. The developed model is utilized successfully in conjunction with a genetic algorithm (GA) search method to obtain the optimized operational parameters for a GSHP in three different climate conditions. A properly sized and engineered non-homogeneous soil profile demonstrated the potential to increase the energy extraction/dissipation rates from/to the ground to a significant level. The potential benefit of a recycled product, Tire Derived Aggregate (TDA), as an insulating non-homogeneous layer is assessed. TDA is demonstrated to be more effective in the cold climate (Buffalo) by increasing the energy extraction rates from the ground approximately 15% annually. TDA's effectiveness is less pronounced in a relatively moderate climate (Dallas) by increasing the energy extraction rates from the ground about 4% annually. For the cooling only scenario (Miami), a high conductive intermediate layer of saturated sand exhibited greater potential to increase the energy dissipation to the ground.

Within the framework of continuum fluid dynamics, couple-stresses appear as an inevitable consequence of non-central forces and the discrete character of matter at the finest scales. As a result, the force-stress tensor becomes non-symmetric and classical theory may not accurately predict the behavior. Recent theoretical work has shown that the couple-stress tensor must be skew-symmetric and that mean curvature rate is the energy-conjugate kinematical measure. The resulting fully consistent couple stress theory incorporates a characteristic material length scale for the fluid that becomes increasingly important, as the characteristic geometric dimension of the problem approaches that level. This size-dependent non-Newtonian theory is essential to understand a range of behavior at micro-scales with potential applications to blood flow and lubrication, among others. Here we concentrate on steady creeping flow within this newly developed fully determinate linear couple stress theory and formulate a boundary integral representation for two-dimensional problems. These boundary integral equations are written in terms of velocities, angular velocities, force-tractions and moment-tractions as primary variables. Details are provided on the derivation of fundamental solutions and on the corresponding boundary element implementation. Afterwards, the new boundary element method is applied for the solution of two basic problems to explore some consequences of this size-dependent couple stress fluid mechanics.

Fundamental solutions for two- and three-dimensional linear isotropic size-dependent couple stress elasticity are derived, based upon the decomposition of displacement fields into dilatational and solenoidal components. While several fundamental solutions have appeared previously in the literature, the present version is for the newly developed fully determinate couple stress theory. Within this theory, the couple stress tensor is skewsymmetrical and thus possesses vectorial character. The present derivation provides solutions for infinite domains of elastic materials under the influence of unit concentrated forces and couples. Unlike all previous work, unique solutions for displacements, rotations, force-stresses and couple-stresses are established, along with the corresponding force-tractions and couple-tractions. These fundamental solutions are central in analysis methods based on Green's functions for infinite domains and are required as kernels in the corresponding boundary integral formulations for size-dependent couple stress elastic materials., 35 pages

A variational approach for fully coupled dynamic irreversible thermoelasticity is developed for continua, which considers both the conservative and dissipative character in terms of mixed variables. By introducing a consistent variational scheme for the spatial and temporal discretization of the governing equations, a mixed continuum element is established under the Hamiltonian-Lagrangian formalism. The proposed method leads to the development of minimum principles in discrete form with the proper selection of state variables and temporal action sum operators. Consequently, this novel mixed variational formulation can provide the basis for a class of optimization-based methods for irreversible thermomechanics. Several applications are considered to demonstrate the robustness of the proposed variational approach, including transient dynamic response of thermoelastic media due to surface heating caused by ramp- and step-type heat fluxes, and a sequence of laser pulses.