Variational formulations of non-linear and non-smooth structural problems

A variational formulation of the boundary value problem on the motion of a viscous incompressible fluid with a free surface and changing dynamic boundary angles is proposed. The mathematical description of the process is based on the equations of motion, continuity and natural boundary conditions on the free surface. The traditional feature of the mathematical model on the three-phase contact lines (LTPC) is eliminated by the slip condition. The boundary angle on the LTPC is included in the variational formulation of the problem by replacing the curvature of the free boundary by the Laplace - Beltrami operator and using integration in parts. To describe the dynamic conditions on the LTPC, linking the speed of the LTPC and the dynamic edge angles on the solid walls of the cylinders, the empirical Jiang ratio is used. The numerical solution of the problem is based on the mixed finite element method with approximation of the main variables of the problem (velocity and pressure vector) satisfying the compatibility condition (LBB - condition). In addition, we use a singular finite element and a discontinuous approximation for the pressure to reduce the pressure oscillations in the vicinity of the LTPC. The numerical implementation of the kinematic condition of the free surface motion is performed according to the predictor-corrector scheme. The algorithm is tested on the problems with an analytical solution. Numerical studies of the flow kinematics and the free surface behavior when filling the coaxial gap in terms of the determining parameters of the Reynolds numbers (Re) in the range from 0 to 5, Stokes (W) in the range from 0 to 300 and capillary number (Ca) in the range from 0.0001 to 10 are carried out. The influence of the main parameters of the problem and dynamic conditions on LTPC on the evolution and maximum deflection of the free boundary is shown. For slow filling conditions, gravitational and capillary forces have the greatest influence on the kinematic characteristics of the filling. In this case, the dominant influence on the evolution of the free surface of the capillary forces begins with the values of the capillary numbers less than CA < 0,1. Increasing the flow rate (Re > 1) leads to a significant increase in the deflection of the free surface.

1R40. Microcontinuum Field Theories II: Fluent Media. - AC Eringen (15 Red Tail Dr, Littleton CO 80126-5001). Springer-Verlag, New York. 2001. 340 pp. ISBN 0-387-98969-2. $149.00.Reviewed by ME Ryan (Dept of Chem Eng, SUNY, 505 Furnas Hall, Buffalo NY 14260).The book, Microcontinuum Field Theories II. Fluent Media, is a mathematically-based exposition of the continuum mechanics of structured fluid media, such as anisotropic fluids, liquid crystals, suspensions, etc. Volume II builds upon and extends the theoretical foundation of the subject developed in the first volume, Microcontinuum Field Theories I. Foundations and Solids. The development in Volume II extends the classical field theories of elasticity, fluid dynamics, and electromagnetism to continua constituted of material points or internal structures having directors that may be deformable as well as orientable. The book is largely based on the author’s contribution to the subject over the past few decades. The book is a reference source that assumes the reader has sufficient background in the subject to follow the derivation and application of the field equations to a variety of different situations. The book is intended for research scientists and graduate students in the disciplines of mathematics, physics, engineering, and related fields. This volume comprises nine chapters (Chs 9-17) which can be divided into four parts: I) Theory of micropolar fluids (Chs 9-11), II) Liquid crystals (Chs 12-14), III) Microstretch fluids (Chs 15-16), and IV) Micromorphic fluids (Ch 17). Electromag-netic interactions are also incorporated into the first three parts. Chapter 9 introduces basic measures of strain and rotation, constitutive equations, and the basic field equations. Analyses of pipe flow, flow in an orthogonal rheometer, lubrication in a journal bearing, Stokes flow around a sphere, stagnation flow, thermal instability, and boundary layer flow are provided. Mixed convection in vertical flow and a discussion of turbulence are also provided within the context of micropolar fluid dynamics. Chapter 10 develops the constitutive and field equations of magneto-hydrodynamics. Analyses of channel flow, shear flow, and thermal stability are presented. Chapter 11 addresses anisotropic fluids and suspensions. The basic theory is developed and applied to channel and couette flows.Chapter 12 gives a detailed presentation of the kinematics, balance laws, constitutive equations, and field equations of liquid crystals. The electrodynamics of liquid crystals is presented in Chapter 13. A variety of problems involving liquid crystals are analyzed in Chapter 14 including shear flow and heat conduction in a channel. The kinematics, constitutive equations, and field equations relating to microstretch fluids are provided in Chapter 15. Applications to the theory of acoustic waves in bubbly fluids and blood flow in small arteries are presented. Chapter 16 presents a continuum theory of the electrodynamics of polymeric liquid crystals. The last chapter addresses the subject of micromorphic fluids with applications of the theory to channel flow, the rheological behavior of blood in arteries, and the behavior of deformable suspensions. A few problems are briefly stated at the end of each chapter. A list of references, a subject index, and a list of errata for Volume I are provided at the end of the book. The treatment provided in this monograph will serve as a useful reference for the academician or research scientist interested in the fundamental theoretical development of this subject.

The problem is considered for the indentation of a stratified elastic strip by a rigid punch of finite dimension with a surface microrelief. Boundary variational formulations of the problem are given using the Poincaré-Steklov operator that maps normal stresses to normal displacements. To approximate this operator the discrete Fourier transform is used. The fast Fourier transform algorithms are applied for numerical realization. A variational formulation of a boundary value problem for transforms of displacements is used to calculate a transfer function. A quadratic programming problem with equality and inequality restrictions is obtained by approximating the original contact problem. To solve this problem numerically an algorithm based on the conjugate gradient method is used. Some regularities of contact interaction have been established.

Fuzzy reliability analysis can be implemented using two discrete optimization maps in the processes of reliability and fuzzy analysis. Actually, the efficiency and robustness of the iterative reliability methods are two main factors in the fuzzy-based reliability analysis due to the huge computational burdens and unstable results. In the structural fuzzy reliability analysis, the first-order reliability method (FORM) using discrete nonlinear map can provide a C membership function. In this paper, a discrete nonlinear conjugate map is proposed using a relaxed-finite step size method for fuzzy structural reliability analysis, namely Fuzzy conjugate relaxed-finite step size method fuzzy CRS. A discrete conjugate map is stabilized using two adaptive factors to compute the relaxed factor and step size in FORM. The framework of the proposed fuzzy structural reliability method is established using two linked iterative discrete maps as an outer loop, which constructs the membership function of the response using alpha level set optimization based on genetic operator, and the inner loop, implemented for reliability analysis using proposed conjugate relaxed-finite step size method. The fuzzy CRS and fuzzy HL-RF methods are compared to evaluate the membership functions of five structural problems with highly nonlinear limit state functions. Results demonstrated that the fuzzy CRS method is computationally more efficient and is strongly more robust than the HL-RF for fuzzy-based reliability analysis of the nonlinear structural reliability problems.

In the field of solid mechanics it is often necessary to use constitutive (physical) relations in the rate form, for example, for formulation of the boundary value problem in the rate form with contact conditions when a contact area is a priori unknown and changes in a deformation process.The paper considers some questions of formulating geometrically nonlinear constitutive relations of elastic material in the rate form and the relationship of these equations and the constitutive relations in the finite form.In many existing elastic and elasto-plastic models Hooke's law (written in terms of the actual configuration) is used as the constitutive relation.As a rule, the rate measure of the strain is deformation rate tensor (the symmetrical part of velocity gradient) and the stress rate measure is some objective derivative (convective or corotational) of weighted Kirchhoff stress tensor.The use of the convective derivative leads to certain difficulties, for example, analysis of evolution stress state in a deformable basis is complicated; consequently, convective derivatives are excluded from consideration.Using a stress tensor corotational derivative instead of the material derivative (derivative by time) allows satisfying the principle of material indifference (the principle of independence of constitutive relation from the choice of reference frame), however a choice of the derivative type can be implemented by many ways.An arbitrary selection of stress corotational derivative leads to undesirable effects: stress oscillations for the simple shear monotonic loading (for example, for Jaumann derivative), "not closed" stress trajectories and nonzero stress work for a closed deformation trajectory.In papers of A. Meyers, H. Xiao, O. Bruhns corotational derivative (with logarithmic spin or logspin) was proposed.The derivative of the right Hencky's strain tensor is exactly equal to deformation rate tensor.When using this derivative in the constitutive relation the described effects are missing, on the basis of this remarked logspin authors state the logspin exclusivity and recommend it only for using in rate form constitutive relations.The article shows full compliance between Hooke's law in the rate and finite form under conditions: existence of the material basis in which properties of the body (in this case -elastic) remain unchanged and the use of the same type corotational derivative for stress and strain measures.Computational experiments for illustration of the proved assertion about the equivalence of different Hooke's law forms were conducted: various

We provide a general abstract theory for the solvability and Galerkin approximation of nonlinear twofold saddle point problems. In particular, a Strang error estimate containing the consistency terms arising from the approximation of the continuous operators involved is deduced. Then we apply these results to analyse a fully discrete Galerkin scheme for a twofold saddle point formulation of a nonlinear elliptic boundary value problem in divergence form. Some numerical results are also presented.

An electrodynamic model of the electric field dynamics behavior in the atmospheric surface layer due to the action of local factors is presented. The model was obtained by reducing the differential equations system of the electrode effect to the so-called total current equation, which is a second-order equation of parabolic type, considered in a two-dimensional space-time region. The total current equation allows us to connect the set of main factors influencing on the atmospheric surface layer electric field state: conduction current, turbulent current and current arising as a result of convective processes in the atmosphere with the so-called total current in the nearground layer, reflecting changes in the ionospheric potential. The described method provides significant advantages in research, since within the framework of one model it allows the formulation of various problems of electrodynamics of the surface layer and a comparative analysis of the influence on the behavior of the electric field in the surface layer of both individual factors and their combinations. Based on meteorological data at the Peak Cheget mountain station in the Elbrus region, a daily variation in the values of the turbulent diffusion coefficient was constructed. The resulting analytical dependence was used to solve the equation for the total current in the surface layer, provided that it is constant at the upper boundary of the turbulent electrode layer. Solutions obtained by the Fourier method describe the daily variation of the electric field strength depending on the degree of turbulent mixing in the atmosphere. The appearance of a time shift in daily extrema, a change in their amplitude, and the appearance of additional extrema depending on the elec-tric field values have been established. All of these effects are comparable to the global unitary variation and increase with increasing electric field.

Background The theory of linear elasticity is insufficient at small length scales, e.g., when dealing with micro-devices. In particular, it cannot predict the “size effect” observed at the micro- and nanometer scales. In order to design at such small scales an improvement of the theory of elasticity is necessary, which is referred to as strain gradient elasticity. Methods There are various approaches in literature, especially for small deformations. In order to include geometric nonlinearities we start by discussing the necessary balance equations. Then we present a generic approach for obtaining adequate constitutive equations. By combining balance equations and constitutive relations nonlinear field equations result. We apply a variational formulation to the nonlinear field equations in order to find a weak form, which can be solved numerically by using open-source codes. Results By using balances of linear and angular momentum we obtain the so-called stress and couple stress as tensors of rank two and three, respectively. Since dealing with tensors an adequate representation theorem can be applied. We propose for an isotropic material a stress with two and a couple stress with three material parameters. For understanding their impact during deformation the numerical solution procedure is performed. By successfully simulating the size effect known from experiments, we verify the proposed theory and its numerical implementation. Conclusion Based on representation theorems a self consistent strain gradient theory is presented, discussed, and implemented into a computational reality.

Introduction Generalized Transport Phenomena Approach to Problem Analysis General Content Conservation Laws and Field Equations Concentrations, Velocities, and Fluxes Thermodynamics of Purely Viscous Fluid Mixtures Conservation of Mass for a One-Component System Conservation of Mass for a Mixture Modification of Field Equations for Mass Transfer Conservation of Linear Momentum for One-Component Systems Conservation of Linear Momentum for a Mixture Conservation of Moment of Momentum for One-Component Systems Conservation of Moment of Momentum for a Mixture Strategies for the Solution of Mass Transfer Problems Boundary Conditions Definitions Jump Balances for Mass Conservation Jump Balances for Linear Momentum Conservation Postulated Boundary Conditions at Phase Interfaces Boundary Conditions in the Absence of Mass Transfer Utilization of Jump Balances Additional Comments on Boundary Conditions Boundary Conditions and Uniqueness of Solutions Constitutive Equations Constitutive Principles First-Order Theory for Binary Systems Combined Field and Constitutive Equations for First-Order Binary Theory First-Order Theory for Ternary Systems Special Second-Order Theory for Binary Systems Viscoelastic Effects in Flow and Diffusion Validity of Constitutive Equations Parameters in Constitutive Equations General Approach in Parameter Determination Diffusion in Polymer-Solvent Mixtures Diffusion in Infinitely Dilute Polymer Solutions Diffusion in Dilute Polymer Solutions Diffusion in Concentrated Polymer Solutions - Free-Volume Theory for Self-Diffusion Diffusion in Concentrated Polymer Solutions - Mutual Diffusion Process Diffusion in Crosslinked Polymers Additional Properties of Diffusion Coefficients Special Behaviors of Polymer-Penetrant Systems Volumetric Behavior of Polymer-Penetrant Systems Sorption Behavior of Polymer-Penetrant Systems Antiplasticization Nonequilibrium at Polymer-Penetrant Interfaces Mathematical Apparatus Basic Definitions Classification of Second-Order Partial Differential Equations Specification of Boundary Conditions Sturm-Liouville Theory Series and Integral Representations of Functions Solution Methods for Partial Differential Equations Separation of Variables Method Separation of Variables Solutions Integral Transforms Similarity Transformations Green's Functions for Ordinary Differential Equations Green's Functions for Elliptic Equations Green's Functions for Parabolic Equations Perturbation Solutions Weighted Residual Method Solution Strategy for Mass Transfer Problems Proposed Solution Methods Induced Convection Solutions of a General Set of Mass Transfer Problems Mixing of Two Ideal Gases Steady Evaporation of a Liquid in a Tube Unsteady-State Evaporation Analysis of Free Diffusion Experiments Dissolution of a Rubbery Polymer Bubble Growth from Zero Initial Size Stability Behavior and Negative Concentrations in Ternary Systems Analysis of Impurity Migration in Plastic Containers Efficiency of Green's Function Solution Method Mass Transfer in Tube Flow Time-Dependent Interfacial Resistance Laminar Liquid Jet Diffusion Analysis Analysis of the Diaphragm Cell Dissolved Organic Carbon Removal from Marine Aquariums Unsteady Diffusion in a Block Copolymer Drying of Solvent-Coated Polymer Films Flow and Diffusion Past a Flat Plate with Solid Dissolution Gas Absorption in Vertical Laminar Liquid Jet Utilization of Polymers in Drug Delivery Gas Absorption and Diffusion into a Falling Liquid Film Perturbation Solutions of Mass Transfer Moving Boundary Problems Dissolution of a Plane Surface of a Pure Gas Phase Bubble Dissolution Singular Perturbations in Moving Boundary Problems Dropping Mercury Electrode Sorption in Thin Films Numerical Analysis of Mass Transfer Moving Boundary Problems Diffusion and Reaction Design of a Tubular Polymerization Reactor Transport Effects in Low-Pressure CVD Reactors Solution of Reaction Problems with First-Order Reactions Plug Flow Reactors with Variable Mass Density Bubble Dissolution and Chemical Reaction Danckwerts Boundary Conditions for Chemical Reactors Transport in Nonporous Membranes Assumptions Used in the Theory for Membrane Transport Steady Mass Transport in Binary Membranes Steady Mass Transport in Ternary Membranes Unsteady Mass Transport in Binary Membranes Phase Inversion Process for Forming Asymmetric Membranes Pressure Effects in Membranes Analysis of Sorption and Desorption Derivation of a Short-Time Solution Form for Sorption in Thin Films Sorption to a Film from a Pure Fluid of Finite Volume A General Analysis of Sorption in Thin Films Analysis of Step-Change Sorption Experiments Integral Sorption in Glassy Polymers Integral Sorption in Rubbery Polymers Oscillatory Diffusion and Diffusion Waves Dispersion and Chromatography Formulation of Taylor Dispersion Problem Dispersion in Laminar Tube Flow for Low Peclet Numbers Dispersion in Laminar Tube Flow for Long Times Dispersion in Laminar Tube Flow for Short Times Analysis of an Inverse Gas Chromatography Experiment Effects of Pressure Gradients on Diffusion: Wave Behavior and Sedimentation Wave Propagation in Binary Fluid Mixtures Hyperbolic Waves Dispersive Waves Time Effects for Parabolic and Hyperbolic Equations Sedimentation Equilibrium Viscoelastic Diffusion Experimental Results for Sorption Experiments Viscoelastic Effects in Step-Change Sorption Experiments Slow Bubble Dissolution in a Viscoelastic Fluid Transport with Moving Reference Frames Relationships Between Fixed and Moving Reference Frames Field Equations in Moving Reference Frames Steady Diffusion in an Ultracentrifuge Material Time Derivative Operators Frame Indifference of Material Time Derivatives Frame Indifference of Velocity Gradient Tensor Rheological Implications Appendix: Vector and Tensor Notation General Notation Conventions Vectors Tensors Results for Curvilinear Coordinates Material and Spatial Representations Reynolds' Transport Theorem

In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation.

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields various distinct solutions to the field equations. Our proof, which fills various gaps in the existing literature, is based on the solution of a second Riemann-Hilbert problem and highlights the deep role of the spectral curve, the normalization condition in the factorization and the choice of the contour. This approach allows us to construct explicit solutions, including new ones, to the non-linear gravitational field equations, using simple complex analytic results.

In 1967 Roger Penrose introduced what is now called twistor geometry (Penrose 1967). This geometry is an amplification of the geometry of lines in projective space studied extensively by Felix Klein and others in the late 19th century. Penrose made the fundamental observation that this classical geometry could be used in conjunction with a Radon transform style integral geometry to provide rather complete sets of solutions of numerous partial differential equations, both linear and nonlinear, which arise naturally in theoretical physics. The Radon transform, which involves integration over straight lines in R 3, was used by F. John to generate solutions of hyperbolic partial differential equations in 1942. Penrose integrates over complex projective lines, real 2-spheres, and the data of the integration is complex-analytic functions of 3 complex variables. This is analogous to (and indeed related to) Weierstrass' representation of solutions of the minimal surface equation in terms of triples of holomorphic functions of a single complex variable which satisfy an algebraic relation. In this paper we discuss the origins of the differential equations which are amiable to twistor-geometric study from the realms of theoretical physics. Twistor geometry is most suitable for physical theories which are relativistically invariant, including Maxwell's description of electromagnetic phenomena from the 19th century (Maxwell's equations), Einstein's theory of gravitation (the equations of general relativity), and the contemporary descriptions of elementary particles (quantum field theory). Twistor geometry interacts in a nontrivial manner with the mathematical models involved in all of the above physical theories. In particular, it has led to new solutions of some of the nonlinear partial differential equations involved (Einstein's equations, Yang-Mills equations, equations of a magnetic monopole). We will describe some recent progress on twistorgeometric representations of solutions of classical field equations. There are two aspects to this. First, there is the problem of translating a given system of differential equations into twistor-geometric language. This has been done successfully in many different cases, and will be described in Section 3. Once a given problem has been transferred to a twistor-geometric context, it becomes, in general, a specific problem involving algebraic geometry, several complex variables, and algebraic topology involving holomorphic data on complex manifolds. The only partial differential equations remaining in the twistor context are the CauchyRiemann equations characterizing holomorphic functions. In effect, the partial differential equation has been transformed into a geometric problem. This is analogous to the Fourier transform transforming a partial differential equation with constant coefficients into an algebraic problem, where differentiation is replaced by the (algebraic) symbol of the differential operator in question. The second aspect of the recent progress on the twistor-geometric representation of the solutions of the classical field equations concerns itself with solving the geometric problem in question and transforming back to space-time (or Euclidean space), generating (sometimes new) solutions of the classical field equations.

AλΦ4 solitary-wave propagator, a propagator constructed from exact particular solutions (solitary-wave solutions) to a nonlinear field equation is used in the usual derivation of a two-nucleon potential. The field equation is a Klein-Gordon equation modified by the addition of a cubic self-interaction term (λΦ3) and describes massive spinless persistently self-interacting systems. The solitary-wave propagator has poles at integral multiple [(2n + 1)m] mass states. Contributions from these poles to the two-nucleon potential in co-ordinate space appear as a superposition of Yukawa and exponential terms with residues that depend on the self-interaction coupling constant (λ). In the limit of vanishing λ, the solitary-wave exchange potential (SWEP) reduces to the standard OPEP. For a two-nucleon in ans-state, the SWEP is very strongly attractive at intermediate ranges (0.5 ≲mr ≲ 1.5) and smoothly gives rise to a repulsive core at very short ranges (mr ≲ 0.5). At very large ranges (mr ≳ 1.5) the SWEP approaches the OPEP tail. Overall, the SWEP exhibits characteristic features of OBEP and phenomenological potential models, but requires only a small number of parameters even in contrast to OBEP.

Galileons are related to other implicitly integrable equations such as the first order nonlinear wave equation and the universal field equation, which is a Lagrangian for the Galileon.

Chameleon gravity is an example of a model that gives rise to interesting phenomenology on cosmological scales while simultaneously possessing a screening mechanism, allowing it to avoid solar system constraints. Such models result in non-linear field equations, which can be solved analytically only in simple highly symmetric systems. In this work we study the equation of motion of a scalar-tensor theory with chameleon screening using the finite element method. More specifically, we solve the field equation for spherical and triaxial NFW cluster-sized halos. This allows a detailed investigation of the relationship between the NFW concentration and the virial mass parameters and the magnitude of the chameleon acceleration, as measured at the virial radius. In addition, we investigate the effects on the chameleon acceleration due to halo triaxiality. We focus on the parameter space regions that are still allowed by the observational constraints. We find that given our dataset, the largest allowed value for the chameleon-to-NFW acceleration ratio at the virial radius is ∼ 10-7. This result strongly indicates that the chameleon models that are still allowed by the observational constraints would not lead to any measurable effects on galaxy cluster scales. Nonetheless, we also find that there is a direct relationship between the NFW potential and the chameleon-to-NFW acceleration ratio at the virial radius. Similarly, there is a direct (yet a much more complicated) relationship between the NFW concentration, the virial mass and the acceleration ratios at the virial radius. Finally, we find that triaxiality introduces extra directional effects on the acceleration measurements. These effects in combination could potentially be used in future observational searches for fifth forces.