On Shearing, Stretching and Spin

An analysis is presented of stretching, shearing and spin of material line elements in a continuous medium. It is shown how to determine all pairs of material line elements at a point x, at time t, which instantaneously are not subject to shearing. For a given pair not subject to shearing, a formula is presented for the determination of a third material line element such that all three form a triad not subject to shearing, instantaneously. It is seen that there is an infinity of such triads not subject to shearing. A new decomposition of the velocity gradient L is introduced. In place of the classical decomposition of Cauchy and Stokes, L=d+w, where d is the stretching tensor and w is the spin tensor, the new decomposition is L=?+ℳ, where ?, called the ldquo;modified” stretching tensor, is not symmetric, and ℳ, called the “modified” spin tensor, is skew-symmetric – the tensor ? being chosen so that it has three linearly independent real right (and left) eigenvectors. The physical interpretation of this decomposition is that the material line elements along the three linearly independent right eigenvectors of ? instantaneously form a triad not subject to shearing. They spin as a rigid body with angular velocity μ (say) associated with ℳ. Also, for each decomposition L=?+ℳ, there is a decomposition L=? ^T +, where is also skew-symmetric. The triad of material line elements along the right eigenvectors of ? ^T (the set reciprocal to the right eigenvectors of ?) is also instantaneously not subject to shearing and rotates with angular velocity (say) associated with . It is seen that the vorticity vector ω is the mean of the two angular velocities μ and , ω =(μ+)/2. For irrotational motion, ω =0, so that μ=-; any triad of material line elements suffering no shearing rotates with angular velocity equal and opposite to that of the reciprocal triad of material line elements. It is proved that provided d is not spherical, there is an infinity of choices for ? and ℳ in the decomposition L=?+ℳ. Two special types of decompositions are introduced. The first type is called “CCS-decomposition” (where CCS is an abbreviation for Central Circular Section). It is associated with the infinite family of triads (not subject to shearing) with a common edge along the normal to one plane of central circular section of an ellipsoid ? associated with the stretching tensor, and the two other edges arbitrary in the other plane of central circular section of ?. There are two such CCS-decompositions. The second type is called “triangular decomposition”, because, in a rectangular cartesian coordinate system, ? has three off-diagonal zero elements. There are six such decompositions. Received 14 November 2000 and accepted 2 August 2001     

THE HAMILTONIAN SYSTEM AND COMPLETENESS OF SYMPLECTIC ORTHOGONAL SYSTEM

I.IntroductionThemethodofseparationofvariablesisimportanttosolvethesoluti0n0fprobIem0fmathematicalphysics,butmanyproblen1sofmathematicalphysicscannotseparatet'ariab1es,thereforeitrestrictstheranget0appIicatemethodofseparationofvariable.Inthepaperlll,Zhong…     

Inhomogeneous electromagnetic plane waves in crystals

Summary The propagation of inhomogeneous, time harmonic, elliptically polarised, electromagnetic plane waves in non-absorbing, magnetically isotropic, but electrically anisotropic, crystals is considered. The electric displacement and the magnetic induction are assumed to have the forms D exp l(S · x–t) and B exp l(S · x–t), respectively, at the place x and time t, where D, S, B are Gibbs bivectors (complex vectors) and is real. The implications of Maxwell's equations for the various field quantities are interpreted simply and directly through the use of bivectors and their associated ellipses.The propagation of circularly polarised waves is considered in detail. For such waves the electric displacement bivector is isotropic: D · D = 0. In order that such waves may propagate it is found that either (i) D is parallel to the slowness bivector S, so that both D and S are isotropic and coplanar, or (ii) D is parallel to the magnetic induction bivector B, so that both D and B are isotropic and coplanar. It is shown that for type (ii) the secular equation must have a double root for the slowness and conversely if the secular equation has a double root then there exists an isotropic electric displacement right eigenbivector of the optical tensor.Both types of waves are possible in a biaxial crystal. They complement each other in the following way. For type (i) all but two great circles on the unit sphere are possible circles of polarisation for circularly polarised waves with D and S parallel. Each of the exceptional great circles is such that an optic axis is normal to the plane of the circle. These two exceptional circles are the only possible circles of polarisation for circularly polarised waves of type (ii) when D and B are parallel.The situation for uniaxial crystals is similar—the only essential difference being that for uniaxial crystals there is only one exceptional circle since there is only one optic axis.For isotropic crystals the situation is quite different. Circularly polarised waves of type (i) are not possible. All great circles on the unit sphere are possible circles of polarisation for circularly polarised waves of type (ii) with D and B parallel.     

On Anisotropic Elastic Materials for which Young''s Modulus E ( n ) is Independent of n or the Shear Modulus G ( n , m ) is Independent of n and m

It is shown that, among anisotropic elastic materials, only certain orthotropic and hexagonal materials can have Young modulus E(n) independent of the direction n or the shear modulus G(n,m) independent of n and m. Thus the direction surface for E(n) can be a sphere for certain orthotropic and hexagonal materials. The structure of the elastic compliance for these materials is presented, and condition for identifying if the material is orthotropic or hexagonal is given. We also study the case in which n of E(n) and n, m of G(n,m) are restricted to a plane. When E(n) is a constant on a plane so are G(n,m) and Poisson's ratio ν(n,m). The converse, however, does not necessarily hold. A plane on which E(n) is a constant can exist for all anisotropic elastic materials. In particular, existence of such a plane is assured for trigonal, hexagonal and cubic materials. In fact there are four such planes for a cubic material. For these materials, not only E(n) is a constant, two other Young's moduli, the three shear moduli and the six Poisson's ratio on the plane are also constant.     

On Finite Shear

If a pair of material line elements, passing through a typical particle P in a body, subtend an angle Θ before deformation, and Θ+γ after deformation, the pair of material elements is said to be sheared by the amount γ. Here all pairs of material elements at P are considered for arbitrary deformations. Two main problems are addressed and solved. The first is the determination of all pairs of material line elements at P which are unsheared. The second is the determination of that pair of material line elements at P which suffers the maximum shear. All unsheared pairs of material elements in a given plane π(S) with normal S passing through P are considered. Provided π(S) is not a plane of central circular section of the C-ellipsoid at P (where C is the right Cauchy-Green strain tensor), it is seen that corresponding to any material element in π(S) there is, in general, one companion material element in π(S) such that the element and its companion are unsheared. There are, however, two elements in π(S) which have no companions. We call their corresponding directions \textit{limiting directions.} Equally inclined to the direction of least stretch in the plane π(S), the limiting directions play a central role. It is seen that, in a given plane π(S), the pair of material line elements which suffer the maximum shear lie along the limiting directions in π(S). If Θ _L is the acute angle subtended by the limitig directions in π(S) before deformation, then this angle is sheared into its supplement π−Θ _L so that the maximum shear γ^*;(S) is γ^*=π− 2 Θ _L . If S is given and C is known, then Θ _L may be determined immediately. Its calculation does not involve knowing the eigenvectors or eigenvalues of C. When all possible planes through P are considered, it is seen that the global maximum shear γ^* _G occurs for material elements lying along the limiting directions in the plane spanned by the eigenvectors of C corresponding to the greatest principal stretch λ₃ and the least λ₁. The limiting directions in this principal plane of C subtend the angle and . Generally the maximum shear does not occur for a pair of material elements which are originally orthogonal. For a given material element along the unit vector N, there is, in general, in each plane π(S passing through N at P, a companion vector M such that material elements along N and M are unsheared. A formula, originally due to Joly (1905), is presented for M in terms of N and S. Given an unsheared pair π(S), the limiting directions in π(S) are seen to be easily determined, either analytically or geometrically. Planar shear, the change in the angle between the normals of a pair of material planar elements at X, is also considered. The theory of planar shear runs parallel to the theory of shear of material line elements. Corresponding results are presented. Finally, another concept of shear used in the geology literature, and apparently due to Jaeger, is considered. The connection is shown between Cauchy shear, the change in the angle of a pair of material elements, and the Jaeger shear, the change in the angle between the normal N to a planar element and a material element along the normal N. Although Jaeger's shear is described in terms of one direction N, it is seen to implicitly include a second material line element orthogonal to N. Accepted: May 25, 1999     

Convexity conditions for strain-dependent energy functions for membranes

The forms that the convexity, polyconvexity, and rank-one convexity inequalities take when the strain energy is required to be a function of the strain G are studied. It is shown in particular that W(G) must be an increasing function of G, in the sense that W(G)W(G) if G – G is non-negative definite. Relatively simple sufficient conditions in terms of G alone are given. Necessary and sufficient conditions in terms of G alone are found to be rather complex.     

Extended Polar Decompositions for Plane Strain

In a finite deformation at a particle of a continuous body, a triad of infinitesimal material line elements is said to be “unsheared” when the angles between the three pairs of line elements of the triad suffer no change. In a previous paper, it has been shown that there is an infinity of unsheared oblique triads. With each oblique unsheared triad may be associated an “extended polar decomposition” F = QG = HQ of the deformation gradient F, in which Q is a rotation tensor, and G, H are not symmetric. Both G and H have the same real eigenvalues which are the stretches of the elements of the triad. In this paper, a detailed analysis of extended polar decompositions is presented in the case when the finite deformation is that of plane strain. Then, we may deal with a 2 × 2 deformation gradient F′ = Q′G′ = H′Q′ instead of the full 3 × 3 tensor F. In this case, the extended polar decompositions are associated with “unsheared pairs,” i.e., pairs of infinitesimal material line elements in the plane of strain which suffer no change in angle in the deformation. If one arm of an unsheared pair is chosen in the plane of strain, then, in general, its companion in the plane is determined. It follows that all possible extended polar decompositions may then be described in terms of a single parameter, the angle that the chosen arm makes with a coordinate axis in the plane. Explicit expressions for G′ and H′ are obtained, and various special cases are discussed. In particular, we note that the expressions for G′ and H′ remain valid even when the chosen arm is along a “limiting direction,” that is the direction of a line element which has no companion element in the plane forming an unsheared pair with it. The results are illustrated by considering the cases of simple shear and of pure shear.Dedicated to Professor Piero Villaggio as a symbol of our friendship and esteem.     

An Eigenvalue Criterion for Stability of a Steady Navier–Stokes Flow in {\mathbb{R}}^3

We study the resolvent equation associated with a linear operator L{\mathcal{L}} arising from the linearized equation for perturbations of a steady Navier–Stokes flow U^*{\mathbf{U^*}}. We derive estimates which, together with a stability criterion from [33], show that the stability of U^*{\mathbf{U^*}} (in the L²-norm) depends only on the position of the eigenvalues of L{\mathcal{L}}, regardless the presence of the essential spectrum.     

Certain reducible helical hydromagnetic flows

Summary Intrinsic formulations are obtained of the equations governing three-dimensional steady flows of an inviscid magnetic fluid with infinite electrical conductivity both when the streamlines of the magnetic field H and the fluid velocity v everywhere have the same unit tangent, and when H lies in the normal plane to the fluid streamlines at every point, so that v×H=0 and v·H=0 respectively. Using these intrinsic equations certain results concerning hydromagnetic helical flows are derived, and particular classes of such flows are reduced (in the sense of Power and Walker¹)) to associated non-magnetic two-dimensional flows. Finally, reciprocal relations are used to link classes of hydromagnetic helical flows to four-parameter classes of ordinary fluid flows.     

Displacement Uncertainty Quantifications in T3-Stereocorrelation

Background Uncertainty quantifications are required for any measurement result to be meaningful.

Objective The present work aims at deriving and comparing a priori estimates of displacement uncertainties in T3-stereocorrelation for a setup to perform high temperature tests.

Methods Images acquired prior to the actual experiment (i.e.,at room temperature) were registered using 3-noded triangular elements (T3-stereocorrelation) to determine displacement uncertainties for different positions of the experimental setup.

Results The displacement uncertainties were then compared to their a priori estimates.

Conclusions For the analyzed experiment, it is shown that noise floor estimates only differed by a factor 2 when compared to a posteriori measurements of standard displacement uncertainties.

     

Relaxation and Recovery in Hydrogel Friction on Smooth Surfaces

Background

Hydrogels are crosslinked polymer networks that can absorb and retain a large fraction of liquid. Near a critical sliding velocity, hydrogels pressed against smooth surfaces exhibit time-dependent frictional behavior occurring over multiple timescales. The origin of these dynamics is unresolved

Objective

Here, we characterize this time-dependent regime and show that it is consistent with two distinct molecular processes: sliding-induced relaxation and quiescent recovery.

Methods

Our experiments use a custom pin-on-disk tribometer to examine poly(acrylic acid) hydrogels on smooth poly(methyl methacrylate) surfaces over a variety of sliding conditions, from minutes to hours.

Results

We show that at a fixed sliding velocity, the friction coefficient decays exponentially and reaches a steady-state value. The time constant associated with this decay varies exponentially with the sliding velocity, and is sensitive to any precedent frictional shearing of the interface. This process is reversible; upon cessation of sliding, the friction coefficient recovers to its original state. We also show that the initial direction of shear can be imprinted as an observable “memory”, and is visible after 24 hrs of repeated frictional shearing.

Conclusions

We attribute this behavior to nanoscale extension and relaxation dynamics of the near-surface polymer network, leading to a model of frictional relaxation and recovery with two parallel timescales.

     

Identities in Finite Strain

First of all the deformation is considered of two infinitesimal material line elements lying along vectors M,N emanating from a particle at X in a body. For all M,N lying in a given plane, an identity is derived relating the stretches along M,N and the angles of the pair of infinitesimal material line elements before and after deformation. Then, the deformation is considered of three non-coplanar infinitesimal material line elements lying along vectors M,N,P emanating from a particle at X in a body. An identity is derived relating the stretches along M,N,P and the angles between the three pairs of infinitesimal material line elements before and after deformation. The identity is factored leading to easy interpretation. The special case of infinitesimal strain is considered.      

The construction of dispersion approximations to the solution of the vector convective diffusion equation

This paper is the sequel to an earlier paper in which we discover how to construct dispersion approximations to the solution of the vector convective diffusion equation. In this paper we make the construction concrete by providing useful formulas for the important elements of our theory. For definiteness we investigate cylinders of circular cross-section.Notation re() real part of - im() imaginary part of - int[] greatest integer less than or equal to - Z* adjoint of Z - adj Z adjugate of Z - col Z column vector of Z - det Z determinant of Z - Z generalized inverse of Z - Im Z image of Z - [,] generalized inner product - Ker Z kernel of Z - Ker Z* subspace perpendicular to Ker Z* - rank Z rank of Z - (Z) spectral radius of Z - tr Z trace of Z     

An analysis tool for piezoelectric gages

The Piezo-electric Gage Analysis System U.S. (pegasus) couples a two-dimensional dynamic structural finite element code to a two-dimensional electrostatics code for analysis of piezoelectric gages. The method has a sound theoretical basis and is built around two powerful finite element anlysis codes. The analysis codes provide the solution of the time dependent stress state in the gage and the solution of the electrostatic equation for each time step. pegasus provides the link between the two codes and the steps required to carry the analysis through to prediction of gage currents. Post-processing of the results allows visual interpretation of the the electric fields within the gage. Here we briefly describe the code and show that it can be a valuable tool for understanding the nature of piezoelectric gages. Received 6 May 1996 / Accepted 31 October 1996     

On Aerodynamic Forces for Viscous Compressible Flow

The paper is aimed at reviewing and adding some new results to our recent work on a force theory for viscous compressible flows around a finite body. It has been proposed to analyze aerodynamic forces directly in terms of fluid elements of nonzero vorticity and density gradient. Let ρ denote the density, u the velocity, and ω the vorticity. It is demonstrated that for largely separated flows about bluff bodies, there are two major source elements: R _e(x) =−?u ²∇ρ·∇ϕ and V _e(x) =−u×ω·∇ϕ, where ϕ is an acyclic potential, generated by the solid body moving with unit velocity in the negative direction of the force considered. In particular, under mild conditions, the (unique) choice of ϕ enforces that the elements R _e(x) and V _e(x) decay rapidly away from the body. Four kinds of finite body are considered: a circular cylinder, a sphere, a hemi sphere-cylinder, and a delta wing of elliptic section—all in transonic-to-supersonic regimes. From an extensive numerical study carried out for these bodies, it is found that these two elements contribute to 95% or more of the total drag or lift for all the cases under consideration. Moreover, R _e(x) due to density gradient becomes progressively important relative to V _e(x) due to vorticity as the Mach number increases. The present method of force analysis enables effective analysis and assessment of relative importance of aerodynamics forces, contributed from individual flow structures. The analysis could therefore be very much useful in view of the rapid growth in numerical fluid dynamics; detailed (either local or global) flow information is often available. The paper is dedicated to Sir James Lighthill in honor of his great contributions to aeronautics on the occasion of the publication of his collected works. Received 3 January 1997 and accepted 11 April 1997     

Bending of Soft Micropatterns in Elastohydrodynamic Lubrication Tribology

Background

Soft tribology is increasingly important in the design and engineering of materials used in robotics, haptics, and biomechanics studies. When patterned surfaces are part of a lubricated tribopair that undergoes sliding and compressive deformation, the patterns experience a bending strain that affects the lubrication film thickness and elastohydrodynamic friction. The contribution of bending patterns to soft tribology is not well understood because earlier studies focused on hard tribopairs with effectively flat surfaces.

Objective

We investigate and model the differences in lubricated friction for poly(dimethyl siloxane) (PDMS) elastomer and PEGDA/alginate double network hydrogel patterns in order to determine the effect of height-to-width aspect ratio and bending angle on the elastohydrodynamic friction.

Methods

Photoresists of two different viscosities are spin coated onto silicon substrates to fabricate molds with pattern heights ranging from 20 μm to 50 μm.

Results

Tribological characterization of the tribopairs in the elastohydrodynamic lubrication regime shows that the patterns generate a friction peak that is independent of aspect ratio for short patterns but displays a “power-law fit” decrease with increasing aspect ratio for taller patterns. Two independent models are used to estimate the theoretical bending and deflection angles for the tribopairs.

Conclusions

The decrease in lubricated friction is attributed to taller patterns having large bending angles and a reduced effective surface for fluid load bearing. Results suggest that the bending of micropatterns could be harnessed to engineer lubricated friction in a variety of applications.

     

Stretching Graphene to 3.3% Strain Using Formvar-Reinforced Flexible Substrate

Background

As a one-atom-thick material, the mechanical loading of graphene in large scale remains a challenge, and the maximum tensile strain that can be realized is through a flexible substrate, but only with a value of 1.8% due to the weak interfacial stress transfer.

Objective

Aims to illustrate the interface reinforcement brought by formvar resins as a buffering layer between graphene and substrates.

Methods

Single crystal graphene transferred to different substrates, applied with uniaxial stretching to compare the interface strength, and finite element analysis was performed to simulate tensile process for studying the influence of Poisson’s ratio of the buffering layer for interface reinforcement.

Results

In this work we use formvar resins as a buffering layer to achieve a maximum uniaxial tensile strain of 3.3% in graphene, close to the theoretical limit (3.7%) that graphene can achieve by flexible substrate stretching. The interface reinforcement by formvar is significantly higher than that by other polymers, which is attributed to the liquid–solid phase transition of formvar for more conformal interfacial contact and its suitable Poisson’s ratio with graphene to avoid its buckling along the transverse direction.

Conclusions

We believe that these results can provide guidance for the design of substrates and interfaces for graphene loading, as well as the support for mechanics analysis of graphene-based flexible electronic devices.

     

Modeling Damage Induced Initiation of Explosives

Sandia National Laboratories developed the DaMaGe Initiated Reaction (DMGIR) model to numerically predict weakly supported shock waves that generate an initiation at energy levels below the shock-to-detonation transition (SDT) of an explosive. The DMGIR model couples the strain energy fluence contribution from damage to the initiation process. It does not attempt to include all possible specific initiation mechanisms at the mesoscale, but instead looks at cumulative damage incurred in the explosive through experimental calibration of a small number of material constants. Both shock pressure (hydrostatic) and shear stress (deviatoric) are accounted for in the summation. The model was designed to be robust and relatively simple to calibrate. The DMGIR model is implemented into the Sandia National Laboratories’ CTH shock wave and large deformation code. The model runs concurrently with the existing History Variable Reactive Burn (HVRB) model currently used for shock initiation.     

In-situ Strain Field Measurement and Mechano-electro-chemical Analysis of Graphite Electrodes Via Fluorescence Digital Image Correlation

Background

Mechano-electro-chemical coupling during the ion diffusion process is a core factor to determine the electrochemical performance of electrodes. However, relationship between the mechanics and the electrochemistry has not been clarified by experiments.

Objective

In this work, we conduct an in situ, visual, comprehensive characterization of strain field and Li concentration distribution to further explore the mechano-electro-chemical relationship.

Methods

The digital image correlation characterized by fluorescent speckle and active optical imaging is developed. Combined with electrochromic-based Li concentration detection, the spatiotemporal evolution of in-plane strain and Li concentration of a graphite electrode during the lithiation and delithiation processes are measured and displayed visually via a dual optical path acquisition system.

Results

The visual results show that in-plane strain and Li concentration possess a spatially non-uniform gradient distribution along the radial direction (i.e., diffusion path) with large values outside and small values inside, and that both present obvious temporal segmentation. And mechano-electro-chemical coupling analysis reveals that the in-plane strain is not always linearly related to the concentration and infers that a high strain limits the diffusion and lithiation. The strain–concentration evolution exhibits obvious asymmetric differences between lithiation and delithiation, wherein three equations are fitted to approximately represent the evolution process between in-plane strain and concentration during the lithiation and delithiation processes

Conclusions

This work overcomes the difficulties of fine strain measurements and collaborative concentration characterization during the electrochemical process, and provides an effective experimental method and data support for further exploration of mechano-electro-chemical coupling.