On the use of convected coordinate systems in the mechanics of continuous media

1951 ◽  
Vol 47 (3) ◽  
pp. 575-584 ◽  
Author(s):  
A. S. Lodge
Keyword(s):  
Coordinate System ◽  
Strain Energy ◽  
Equations Of State ◽  
Equations Of Motion ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Moving Medium ◽  
Coordinate Systems ◽  
Invariance Properties ◽  
The Right

The use of a coordinate system convected with the moving medium for describing its mechanics, first proposed by Hencky (5), has since been extended by several authors, and has several advantages over the more conventional use of a coordinate system fixed in space; Brillouin(1) has shown that the relation between the strain-energy function for an ideally elastic solid and the stress tensor takes a very simple form when the latter is referred to a convected coordinate system; Oldroyd(8) has given a very general discussion of the formulation of rheological equations of state and has shown that the right invariance properties are most readily recognized when the equations are referred to a convected coordinate system; Green and Zerna (4) have similarly expressed the equations of motion and boundary conditions; and Gleyzal (2), and Green and Shield (3) have applied the formalism to certain problems in elasticity theory.

Finite strains in an anisotropic elastic continuum

1950 ◽  
Vol 202 (1070) ◽  
pp. 345-358 ◽  
Keyword(s):  
Equations Of State ◽  
Elastic Solid ◽  
Finite Strains ◽  
Free Energy Function ◽  
Elastic Solids ◽  
Elastic Continuum ◽  
Invariance Properties ◽  
Anisotropic Elastic ◽  
Necessary And Sufficient ◽  
Thermodynamic Restrictions

The discussion in a previous paper (Oldroyd 1950), on the invariance properties required of the equations of state of a homogeneous continuum, is extended by taking into account thermodynamic restrictions on the form of the equations, in the case of an elastic solid deformed from an unstressed equilibrium configuration. The general form of the finite strainstress-temperature relations, expressed in terms of a free-energy function, is deduced without assuming that the material is isotropic. The results of other authors based on the assumption of isotropy are shown to follow as particular cases. The equations of state are derived by considering quasi-static changes in an elastic solid continuum; the results then apply to non-ideally elastic solids in equilibrium, or subjected to quasi-static changes only, and to ideally elastic solids in general motion. A necessary and sufficient compatibility condition for the finite strains at different points of a continuum is also derived. As a simple illustration of the derivation and use of equations of state involving anisotropic physical constants, the torsion of an anisotropic cylinder is discussed briefly.

Small Indentations of Rubber Blocks: Effect of Size and Shape of Block and of Lateral Compression

2006 ◽  
Vol 79 (4) ◽  
pp. 674-693 ◽  
Author(s):  
A. N. Gent ◽  
O. H. Yeoh
Keyword(s):  
Cross Section ◽  
Strain Energy ◽  
Rectangular Cross Section ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Biaxial Compression ◽  
Rigid Surface ◽  
Lateral Compression ◽  
Block Width ◽  
Rubber Block

Abstract Many gaskets and seals consist of a long rubber strip or thin-walled ring, placed on a flat rigid surface and indented by a flat-ended rigid indenter. We have examined their resistance to small indentations by FEA. They are treated as infinitely-long elastic blocks of rectangular cross-section, resting on a rigid frictionless base. The indentation stiffness is calculated for various ratios of indenter tip width to block width and to block thickness, using two restraint conditions on the outer surfaces: frictionless walls (zero outwards displacement), as for a gasket placed in a recess; or stress-free, as for a gasket with no lateral restraint. For an infinitely-wide and infinitely-thick block, the theoretical resistance to indentation is zero. For comparison, the indentation stiffness is calculated for cylindrical rubber blocks of varied radius and thickness, indented by a flat-ended cylindrical indenter. In this case the result for an infinitely-large block is finite. A second study treats indentation of a rubber block, pre-compressed in the surface plane. Biot showed that the indentation stiffness of a half-space becomes zero at a critical compression, about 33% for equi-biaxial compression and 44 % for plane strain compression, for both a neo-Hookean and a Mooney-Rivlin elastic solid. FEA calculations were made of the indentation stiffness of neo-Hookean blocks of various sizes, pre-compressed to various degrees. The results are compared with Biot's result. In an Appendix, the critical degree of compression is calculated for a more realistic strain energy function than either the neo-Hookean or the Mooney-Rivlin approximation.

Finite Elastic Deformation of an Annular Rotating Disk

1976 ◽  
Vol 98 (4) ◽  
pp. 375-379 ◽  
Author(s):  
J. B. Haddow ◽  
M. G. Faulkner
Keyword(s):  
Elastic Deformation ◽  
Strain Energy ◽  
Equations Of Motion ◽  
Strain Energy Function ◽  
Rotating Disk ◽  
Force Balance ◽  
Annular Disk ◽  
Incompressible Materials ◽  
Plane Stress Problem ◽  
Governing Differential Equation

An accurate approximate method for the solution of the generalized plane stress problem of finite elastic deformation of a rotating annular disk is given. The method is applicable to both compressible and incompressible materials. Use of the nonlinear governing differential equation is avoided by considering the disk to be an aggregate of discrete coaxial rings, equations of motion, force balance and compatibility relations being formulated for the rings. Results are given for neo-Hookean disks and disks of a compressible material which has a strain energy function proposed by Blatz and Ko [7].

scholarly journals Residually Stressed Fiber Reinforced Solids: A Spectral Approach

Materials ◽  
2020 ◽  
Vol 13 (18) ◽  
pp. 4076
Author(s):  
Mohd Halim Bin Mohd Shariff ◽  
Jose Merodio
Keyword(s):  
Constitutive Equation ◽  
Energy Function ◽  
Strain Energy ◽  
Finite Strain ◽  
Strain Energy Function ◽  
Spectral Approach ◽  
Fiber Reinforced ◽  
Elastic Solids ◽  
Preferred Direction ◽  
The Right

We use a spectral approach to model residually stressed elastic solids that can be applied to carbon fiber reinforced solids with a preferred direction; since the spectral formulation is more general than the classical-invariant formulation, it facilitates the search for an adequate constitutive equation for these solids. The constitutive equation is governed by spectral invariants, where each of them has a direct meaning, and are functions of the preferred direction, the residual stress tensor and the right stretch tensor. Invariants that have a transparent interpretation are useful in assisting the construction of a stringent experiment to seek a specific form of strain energy function. A separable nonlinear (finite strain) strain energy function containing single-variable functions is postulated and the associated infinitesimal strain energy function is straightforwardly obtained from its finite strain counterpart. We prove that only 11 invariants are independent. Some illustrative boundary value calculations are given. The proposed strain energy function can be simply transformed to admit the mechanical influence of compressed fibers to be partially or fully excluded.

The finite compression of elastic solid cylinders in the presence of gravity

1971 ◽  
Vol 321 (1546) ◽  
pp. 381-396 ◽  
Keyword(s):  
Axial Compression ◽  
Energy Function ◽  
Strain Energy ◽  
Body Force ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Infinitesimal Deformation ◽  
Material Constants ◽  
Gravity Effects ◽  
Theoretical Predictions

The deformation induced by gravity in a solid cylinder is considered. The cylinder is placed on a horizontal frictionless surface with the axis vertical and is treated as an isotropic incompressible elastic material. Further distortion is produced by finite axial compression. Thus the overall deformation is predominantly uniform with a small perturbation superimposed to account for the gravity effects. A particular case is when there is no finite axial compression. The solution then describes the shortening and general infinitesimal deformation associated with the gravitational body force. The results may be used for determining the material constants of soft elastic materials. In particular, the equivalent of Young’s modulus for gelatin has been found by measuring the changes in height of cylindrical specimens when removed from rigid containers and using the appropriate formula derived in the analysis. The results obtained were extremely consistent. In addition, the behaviour of gelatin under finite compression has been examined. By comparing the theoretical predictions with the experimental measurements it is shown that gelatin behaves more as a Mooney material than as a material which has a quadratic form for the associated strain energy function. Values of the two material constants occurring in the Mooney form of the strain energy function are obtained.

Nature of the Requirements for Reference Coordinate Systems

1975 ◽  
Vol 26 ◽  
pp. 49-62
Author(s):  
C. A. Lundquist
Keyword(s):  
Rigid Body ◽  
Coordinate System ◽  
Equations Of Motion ◽  
Inertial Reference Frame ◽  
Earth System ◽  
Coordinate Systems ◽  
Dynamical Equations ◽  
The Earth ◽  
Measurement Uncertainties ◽  
Rotational Motions

AbstractThe current need for more precisely defined reference coordinate systems arises for geodynamics because the Earth can certainly not be treated as a rigid body when measurement uncertainties reach the few centimeter scale or its angular equivalent. At least two coordinate systems seem to be required. The first is a system defined in space relative to appropriate astronomical objects. This system should approximate an inertial reference frame, or be accurately related to such a reference, because only such a coordinate system is suitable for ultimately expressing the dynamical equations of motion for the Earth. The second required coordinate system must be associated with the nonrigid Earth in some well defined way so that the rotational motions of the whole Earth are meaningfully represented by the transformation parameters relating the Earth system to the space-inertial system. The Earth system should be defined so that the dynamical equations for relative motions of the various internal mechanical components of the Earth and accurate measurements of these motions are conveniently expressed in this system.

Universal deformations in inhomogeneous isotropic nonlinear elastic solids

2021 ◽  
Vol 477 (2253) ◽  
Author(s):  
Arash Yavari
Keyword(s):  
Strain Energy ◽  
Elastic Solid ◽  
Necessary Condition ◽  
Density Functions ◽  
Elastic Solids ◽  
Universal Deformations ◽  
Green Strain ◽  
The Right ◽  
Nonlinear Elastic ◽  
Isotropic Solids

Universal (controllable) deformations of an elastic solid are those deformations that can be maintained for all possible strain-energy density functions and suitable boundary tractions. Universal deformations have played a central role in nonlinear elasticity and anelasticity. However, their classification has been mostly established for homogeneous isotropic solids following the seminal works of Ericksen. In this article, we extend Ericksen’s analysis of universal deformations to inhomogeneous compressible and incompressible isotropic solids. We show that a necessary condition for the known universal deformations of homogeneous isotropic solids to be universal for inhomogeneous solids is that inhomogeneities respect the symmetries of the deformations. Symmetries of a deformation are encoded in the symmetries of its pulled-back metric (the right Cauchy–Green strain). We show that this necessary condition is sufficient as well for all the known families of universal deformations except for Family 5.

On the formulation of rheological equations of state

1950 ◽  
Vol 200 (1063) ◽  
pp. 523-541 ◽  
Keyword(s):  
Equations Of State ◽  
Anisotropic Materials ◽  
Structural Theory ◽  
Frame Of Reference ◽  
Simple Experiment ◽  
Invariance Properties ◽  
Fixed Frame ◽  
The Right ◽  
Universal Application

The invariant forms of rheological equations of state for a homogeneous continuum, suitable for application to all conditions of motion and stress, are discussed. The right invariance properties can most readily be recognized if the frame of reference is a co-ordinate system convected with the material, but it is necessary to transform to a fixed frame of reference in order to solve the equations of state simultaneously with the equations of continuity and of motion. An illustration is given of the process of formulating equations of state suitable for universal application, based on non-invariant equations obtained from a simple experiment or structural theory. Anisotropic materials, and materials whose properties depend on previous rheological history, are included within the scope of the paper.

Collagen and elastin fibers in human pulmonary alveolar mouths and ducts

1987 ◽  
Vol 63 (3) ◽  
pp. 1185-1194 ◽  
Author(s):  
M. Matsuda ◽  
Y. C. Fung ◽  
S. S. Sobin
Keyword(s):  
Strain Energy ◽  
Unit Length ◽  
Strain Energy Function ◽  
Lung Parenchyma ◽  
Fiber Bundles ◽  
Elastin Fiber ◽  
Analytic Expressions ◽  
Quantitative Basis ◽  
The Mean ◽  
The Right

To provide a quantitative basis for the understanding of the mechanical properties of the lung tissue, the morphology of the pulmonary alveolar ducts was studied and the dimensions of the collagen and elastin fiber bundles in the alveolar mouths were measured. Statistical data are presented in this report. It was found that the probability frequency functions of the widths of both the collagen and elastin fibers are skewed to the right, but the fourth roots of the widths are normally distributed. Hence, knowing the mean and standard deviation of (width)1/4, the probability of finding fiber bundles of width D is known. On the other hand, we already know the analytic expressions of the strain energy per unit length of fibers of given width. With the probability distribution of width and an estimation of the length of alveolar mouths, the strain energy of the alveolar mouths can be computed. Then the contribution of the alveolar mouths to the stress in the lung parenchyma due to any strain can be obtained by a differentiation of the strain energy function with respect to that strain.

Learned Dynamics of Reaching Movements Generalize From Dominant to Nondominant Arm

2003 ◽  
Vol 89 (1) ◽  
pp. 168-176 ◽  
Author(s):  
Sarah E. Criscimagna-Hemminger ◽  
Opher Donchin ◽  
Michael S. Gazzaniga ◽  
Reza Shadmehr
Keyword(s):  
Coordinate System ◽  
Mirror Symmetry ◽  
Reaching Movements ◽  
Control Group ◽  
Design Matrix ◽  
Coordinate Systems ◽  
Accurate Performance ◽  
The Right ◽  
Hemispheric Communication ◽  
Transfer Direction

Accurate performance of reaching movements depends on adaptable neural circuitry that learns to predict forces and compensate for limb dynamics. In earlier experiments, we quantified generalization from training at one arm position to another position. The generalization patterns suggested that neural elements learning to predict forces coded a limb's state in an intrinsic, muscle-like coordinate system. Here, we test the sensitivity of these elements to the other arm by quantifying inter-arm generalization. We considered two possible coordinate systems: an intrinsic (joint) representation should generalize with mirror symmetry reflecting the joint's symmetry and an extrinsic representation should preserve the task's structure in extrinsic coordinates. Both coordinate systems of generalization were compared with a naı̈ve control group. We tested transfer in right-handed subjects both from dominant to nondominant arm (D→ND) and vice versa (ND→D). This led to a 2 × 3 experimental design matrix: transfer direction (D→ND/ND→D) by coordinate system (extrinsic, intrinsic, control). Generalization occurred only from dominant to nondominant arm and only in extrinsic coordinates. To assess the dependence of generalization on callosal inter-hemispheric communication, we tested commissurotomy patient JW. JWshowed generalization from dominant to nondominant arm in extrinsic coordinates. The results suggest that when the dominant right arm is used in learning dynamics, the information could be represented in the left hemisphere with neural elements tuned to both the right arm and the left arm. In contrast, learning with the nondominant arm seems to rely on the elements in the nondominant hemisphere tuned only to movements of that arm.

Sign in / Sign up

Export Citation Format

Share Document