(i) A size-effect [see experimental Cosserat figure] is predicted in the torsion of circular cylinders and of square section bars of Cosserat elastic materials. Slender cylinders appear more stiff then expected classically. A similar size effect is also predicted in the bending of plates and of beams. In experimental Cosserat elasticity, one uses these size effects to determine the characteristic lengths. No size effect is predicted in tension.
(ii) The stress concentration factor for a circular hole, is smaller than the classical value. Small holes exhibit less stress concentration than larger ones. This gives rise to enhanced toughness.
(iii) The wave speed of plane dilatational waves in an unbounded Cosserat elastic medium is independent of frequency as in the classical case. The speed of shear waves depends on frequency in a Cosserat solid. A new kind of wave associated with the micro- rotation is predicted to occur in Cosserat solids.
(iv) The range in Poisson's ratio based on stability considerations for an object with free surfaces is from -1 to +0.5, the same as in the classical case.
Articles , mostly experimental
Osteon Article - Strong Cosserat effects seen in the torsion of bone.
Lakes, R. S., "The role of gradient effects in the piezoelectricity of bone", IEEE Trans. Biomed. Eng., BME-27 (5), 282-283, (1980). Stress gradient effects in piezoelectricity are obtained from general nonlocality considerations. A nonlocal elastic and nonlocal piezoelectric continuum representation of bone is appropriate in view of bone's structure. More recently, gradient effects in piezoelectricity have been called "flexoelectricity" of "flexoelectric" materials. Get pdf
J. F. C. Yang and Roderic Lakes, "Transient study of couple stress in compact bone: torsion", Journal of Biomechanical Engineering, 103, 275-279, (1981).
Couple stress theory, which admits an internal moment per unit area as well as the usual force per unit area, is a generalization of classical elasticity. Experimentally we have demonstrated the existence of couple stress by measuring the effect of size on apparent stiffness of compact bone in quasi-static torsion. From these measurements we obtain the characteristic length for bone in couple stress theory. The characteristic length is comparable to the diameter of osteons.
Lakes, R. S., "Dynamical study of couple stress effects in human compact bone",
Journal of Biomechanical Engineering, 104, 6-11, (1982).
Torsional resonance experiments performed on wet human compact bone disclose effects due to couple stress. The characteristic length, which is an additional material coefficient which appears in couple stress theory, is of the order of the size of osteons and appears to be smaller at high frequencies than at low frequencies. The presence of couple stress effects implies a reduction in the stress concentration factor around holes, particularly small holes.
Yang, J. F. C., and Lakes, R. S., "Experimental study of micropolar and couple stress elasticity in bone in bending", Journal of Biomechanics, 15, 91-98, (1982).
Generalized continuum theories such as couple stress theory and micropolar theories have degrees of freedom in addition to those of classical elasticity. Such theories are thought to be applicable to materials with a fibrous or granular structure. In this study we observe size effects in quasistatic bending of compact bone. The effects are consistent with micropolar elasticity. From them we evaluate two non-classical elastic constants.
Rod Lakes and Robert Benedict, "
Noncentrosymmetry in micropolar elasticity",
International Journal of Engineering Science, 29 (10), 1161-1167, (1982).
A solid which is isotropic with respect to coordinate rotations but not with respect to inversions is called noncentrosymmetric, acentric, hemitropic, or chiral. Chirality has no effect upon the classical elastic modulus tensor. In Cosserat elasticity, chirality (hemitropy) has an effect. A chiral Cosserat solid has three new elastic constants in addition to the six considered in the fully isotropic micropolar solid. The chiral micropolar solid is predicted to undergo torsional deformation when subjected to tensile load. Thus chiral solids have different mechanical behavior from solids with a center of symmetry, as allowed by the more general Cosserat elastic theory.
Lakes, R. S., "Size effects and micromechanics of a porous solid",
J. Materials Science, 18 2572-2581, (1983).
The rigidity of rods of a polymeric foam in bending and torsion is measured as a function of diameter. The dependence of rigidity upon specimen size is found to be inconsistent with a classically viscoelastic continuum model. The Cosserat continuum, which admits additional degrees of freedom associated with rotation of the microstructure, describes the foam more accurately than the classical continuum. Evidence is presented that additional degrees of freedom, associated with deformation of the microstructure, must be incorporated in a complete continuum model of foamed materials. Specifically a cut off frequency is observed, consistent with the theory of elasticity with microstructure after Mindlin, called micromorphic by Eringen. The material exhibits a stop band of frequency. Get pdf
Nakamura, S., Benedict, R. L. and Lakes, R. S., "Finite element method for orthotropic micropolar elasticity", International Journal of Engineering Science, 22 319-330 (1984).
The total potential energy for a body composed of an anisotropic micropolar linear elastic material is developed and used to formulate a displacement type finite element method of analysis. As an example of this formulation triangular plane stress (and plane couple stress) elements are used to analyze several problems. The program is verified by computing the stress concentration factor around a hole in an isotropic micropolar material for which an exact analytical solution exists. Several anisotropic material cases are presented which demonstrate the dependence of the stress concentration factor on the micropolar material constants.
Lakes, R. S., "A pathological situation in micropolar elasticity, J. Applied Mechanics, 52 234-235 (1985).
The case of zero coupling number N in micropolar elasticity is considered. This situation has been examined by several investigators to simplify the analysis of micropolar materials. We show that the case N = 0 is pathological and present a physical example.
Lakes, R. S., Gorman, D., and Bonfield, W., "Holographic screening method for microelastic solids", J. Materials Science, 20 2882-2888 (1985).
An experimental method is presented for the rapid evaluation of structured solids with microelastic degrees of freedom associated with the microstructure. By contrast with earlier methods based on size effect studies, the present method makes use of holographic interferometry. Results are presented for polymethyl methacrylate (PMMA) and for a dense polyurethane foam, which in previous studies were demonstrated to behave, respectively, as classical elastic and Cosserat solids. The method is based on study of deformation of a square cross section bar, in torsion, with a small crack at the corner. Get pdf.
The image on the left shows holographic fringes for a PMMA bar in torsion. The notches at the cross section corner are indicated by yellow arrows. There is no displacement across either notch as indicated by the continuance of fringe order across the notch. This is as expected from classical elasticity.
The image on the left shows holographic fringes for a dense polyurethane foam bar in torsion. The notches at the cross section corner are indicated by yellow arrows. There is a displacement across each notch as indicated by the jump in fringe order across the notch. This is as expected from Cosserat elasticity.
Lakes, R. S., "Demonstration of consequences of the continuum hypothesis", Mechanics Monograph, M5 1-5 (1985).
The significance of several crucial assumptions within the theory of deformable bodies can be demonstrated via simple observations of motion of a crack in a bar of a structured material. The demonstration is based on a null experiment of a sort requiring no instrumentation. The demonstration is therefore ideally suited for the classroom environment.
Consequences of the continuum hypothesis and its failure may be illustrated by means of a bar of flexible material, of square or rectangular cross section. Such a bar, if sufficiently large, is also useful for demonstrations of principal and anticlastic curvatures in bending, and warp of the cross sections in torsion. For the purposes of this article, consider two such bars, identical in size and shape. Let one be made of rubber and the other of polymeric foam such as that used as packing material.
Consider a differential element at the corner of the bar. The surface traction upon the lateral surface is zero for torsional loading of the bar. By virtue of the symmetry of the stress tensor, the complementary shear stress must be zero. Since all components of the shear stress vanish at the corner, the shear strain must also vanish. By such arguments one can show that the cross sections of a rectangular bar in torsion must undergo warp: see, e.g. F.P. Beer and E.R. Johnston, Jr., Mechanics of Materials, McGraw Hill, New York, 1981. Consider, however, a similar micro-element in a material with a lattice type structure. The struts in the lattice can support a moment as well as a force, as shown in this diagram. The moment must be nonzero since the transverse struts undergo a twist. For the micro-element to be in equilibrium, a nonzero shear force must be transmitted through the longitudinal struts.
The corner elements of an elastic material and a lattice structure, then, are predicted to behave quite differently. To illustrate this difference, make a small nick in the corner of the rubber bar and a similar nick in the corner of the foam bar. When the rubber bar is twisted, the crack is not expected to open in mode III (in which the faces of the crack shear parallel to each other and parallel to the crack front) since there is no stress in the region of the crack. While this is strictly true only for an infinitesimal crack, the crack opening is negligibly small if the crack is sufficiently short compared with the bar width, as shown in as shown in figure 3. The situation is different in the foam bar shown in figure 4. The corner crack opens noticeably in mode III when the bar is twisted. Here the corner of the bar has been made more visible by gluing black thread to it with rubber cement. A line can also be drawn with ink as was done with the rubber bar, but the line tends to be rather broad, due to the structure of the foam. The continuum view is shown in figure 5
Lakes, R. S., "Experimental microelasticity of two porous solids",
International Journal of Solids and Structures, 22 55-63 (1986).
Experiments are performed to determine the dependence of torsional and bending rigidity upon diameter for rod-shaped specimens of dense polyurethane foam and of syntactic foam. Results show an effect due to the microstructure. Size effects are observed in the linear regime of small strain. Results are describable by a Cosserat elastic model. All six Cosserat elastic constants are determined from experiment.
Get pdf of this article.
Park, H. C. and Lakes, R. S., "Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent", J. Biomechanics, 19 385-397 (1986). The diagram at the right shows the interpretation of reduction of warp in the context of Cosserat elasticity.
Experimental mechanics determination of the strain distribution in prismatic, square cross section bars of human compact bone in torsion disclosed nonclassical effects associated with the microstructure. Specifically, in wet bone at small strain, significant deviations from the classically predicted strain distribution were observed. The measured strain distribution in wet bone followed predictions based on Cosserat (micropolar) elasticity. Predictive power of Cosserat elasticity is illustrated by the fact the correct strain distribution is predicted assuming the characteristic lengths obtained from size effect studies. In dry bone, the strain distribution was very close to the prediction of classical elasticity. The interaction between Haversian osteons and the cement substance between them was hypothesized to be the principal mechanism for the phenomena. To evaluate this hypothesis, additional specimens were subjected to prolonged torsional load and the cement lines were observed by reflected light microscopy. Localized deformation at the cement lines was observed, but it was less than values reported earlier for bovine plexiform bone. Get pdf.
Park, H. C. and Lakes, R. S. "Torsion of a micropolar elastic prism of square cross section", Int. J. Solids, Structures, 23, 485-503 (1987).
An analytical solution is presented for the problem of torsion of a prismatical bar of square cross section of a micropolar (Cosserat) elastic solid. Warp of the cross sections is found to differ from the warp in a classically elastic solid. Contrary to the classical case, a nonzero shear strain is predicted to occur at the edge of the bar. Novel experimental modalities are suggested on the basis of this analytical solution. The strain distribution across the lateral surfaces depends on the Cosserat characteristic length as shown in this diagram Download pdf.
Lakes, R. S., "Negative Poisson's ratio materials", Science, 238 551 (1987).
In the Cosserat model for structured solids, many phenomena are predicted and observed that depend on the material size scale in relation to the length scale associated with the deformation. A simple tension deformation (in an isotropic material) however is uniform and has no associated length scale. Consequently both the Cosserat model and classical elasticity predict the Poisson effect to be independent of scale. As for causes, structural aspects may be responsible both for Cosserat effects and negative Poisson's ratio. Get pdf or gif from here.
Nakamura, S. and Lakes, R. S. "Finite element analysis of stress concentration around a blunt crack in a Cosserat elastic solid", Computer Methods in Applied Mechanics and Engineering, 66, 257-266 (1988).
The problem of a blunt edge notch of elliptic contour in a strip of Cosserat (micropolar) elastic material under tension has been analyzed via two-dimensional (plane stress) finite element analysis. Both 3-node constant strain triangular elements and 4-node isoparametric elements were used. Three cases were explored: Cosserat characteristic length much less than the crack root radius, equal to the crack root radius; and comparable to the crack length. Stress concentration factors were found to be reduced by as much as a factor of 2.6 in comparison with a classically elastic material. For the special case of a classically elastic material, the finite element results agreed well with the classical analytic solution. Get pdf.
Chen, C. P. and Lakes, R. S., "Dynamic wave dispersion and loss properties of conventional and negative Poisson's ratio polymeric cellular materials", Cellular Polymers, 8(5), 343-359 (1989).
This article describes experimental investigations of the dynamical behaviour of conventional and negative Poisson's ratio foamed materials in torsional vibration. Dispersion of standing waves and cut-off frequencies were observed. Consequently, foamed materials do not obey the classical theory of elasticity or viscoelasticity. The dynamical effects were attributed to micro-vibrations of the cell ribs in a structural view and were associated with Mindlin microstructure elasticity or micromorphic elasticity in a continuum view. Cut-off frequencies were lower in re-entrant foams with negative Poisson's ratios than in the conventional foams from which they were derived. An analytical structural model was developed in which the ribs of the conventional foams were modeled as free-free vibrating beams. The predicted cut-off frequencies were comparable to those observed experimentally. Such stop bands can block waves. Get pdf
Lakes, R. S., Nakamura, S., Behiri, J. C. and Bonfield, W., "Fracture mechanics of bone with short cracks", Journal of Biomechanics, 23, 967-975 (1990).
Tensile fracture experiments were performed upon specimens of wet mature bovine Haversian bone, with short, controlled notches. Stress concentration factors were found to be significantly less than values predicted using a maximum stress criterion in the theory of elasticity. Results were also modeled with the aid of linear elastic fracture mechanics. Agreement of experiment with theory was better in this case, however deviations were seen for short notches. Two mechanisms were evaluated for the behavior: plasticity near the crack tip, and effects of the Haversian microstructure, modelled by Cosserat elasticity, a generalized continuum theory. Plastic zone effects were found to be insignificant. Cosserat elasticity, by contrast, predicted stress concentration factors which better approximated observed values. To explore strain redistribution processes, further experiments were conducted upon notched specimens in torsion at small strain. They disclosed a strain redistribution effect consistent with Cosserat elasticity. These microelastic effects interpreted within a micropolar continuum model, are attributed to the Haversian architecture of bone. Get pdf.
Lakes, R. S., "Experimental micro mechanics methods for conventional and negative Poisson's ratio cellular solids as Cosserat continua", J. Engineering Materials and Technology, 113, 148-155 (1991).
Continuum representations of micromechanical phenomena in structured materials are described, with emphasis on cellular solids. These phenomena are interpreted in light of Cosserat elasticity, a generalized continuum theory which admits degrees of freedom not present in classical elasticity. These are the rotation of points in the material, and a couple per unit area or couple stress. Experimental work in this area is reviewed, and other interpretation schemes are discussed. The applicability of Cosserat or micropolar elasticity to cellular solids and fibrous composite materials is considered as is the application of related generalized continuum theories. New experimental results are presented for foam materials with negative Poisson's ratios. Get pdf
A heavily twisted bar of negative Poisson's ratio foam exhibits minimal warp, in contrast to expectation of classical elasticity as shown in the image.
Chen, C. P. and Lakes, R. S., "Holographic study of conventional and negative Poisson's ratio metallic foams: elasticity, yield, and micro-deformation", J. Materials Science, 26, 5397-5402 (1991)
This article presents an experimental study by holographic interferometry of the following material properties of conventional and negative Poisson's ratio copper foams: Young's moduli, Poisson's ratios, yield strengths, and characteristic lengths associated with inhomogeneous deformation. The Young's modulus and yield strength of the conventional copper foam were comparable to those predicted by microstructural modelling on the basis of cellular rib bending. The re-entrant copper foam exhibited a negative Poisson's ratio as indicated by the elliptic contour fringes on the specimen surface in the bending tests. Inhomogeneous, non-affine deformation was observed holographically in both foam materials. Get pdf
Lakes, R. S., "Strongly Cosserat elastic lattice and foam materials for
enhanced toughness", Cellular Polymers, 12, 17-30 (1993).
Some foams exhibit size effects and other phenomena not describable by classical elasticity. These foams are describable by Cosserat elasticity, which is a continuum theory with more freedom than classical elasticity. Cosserat solids have a characteristic length which is greater than zero. Strongly Cosserat elastic materials are considered to be those materials for which the Cosserat characteristic length is substantially greater than the structure size and for which the coupling number is large. Such materials are predicted to exhibit superior toughness. A mechanically isotropic lattice model is presented for the study of foams. Ordinary open cell foams are shown to be weakly Cosserat elastic. If cell rib properties are modified, strongly Cosserat elastic effects can occur in the foam. Anisotropic laminate and fibrous materials can also be made to exhibit strongly Cosserat elastic effects.
Chen, C. P. and Lakes, R. S., "Holographic study of non-affine deformation in copper foam with a negative Poisson's ratio -0.8", Scripta Metall et Mater., 29, 395-399, (1993).
Micro-deformation studies of conventional and negative Poisson's ratio copper foams were conducted holographically. Inhomogeneous, non-affine deformation was observed holographically in both foam materials. The negative Poisson's ratio material with a permanent volumetric compression ratio 2.2 exhibited a substantially greater non-affine deformation than the conventional material, in contrast to foam with compression ratio 3.0 examined earlier. Get pdf
Anderson, W. B., Chen, C. P., and Lakes, R. S., "Experimental study of size effects and surface damage of polymethacrylimide closed-cell foam", Cellular Polymers, 13, 1-15 (1994).
Anderson, W. B. and Lakes, R. S., "Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam", Journal of Materials Science, 29, 6413-6419, (1994).
This article describes the experimental investigation of Cosserat (or micropolar) elasticity and surface damage effects in closed cell polymethacrylimide foams of different densities. The method of size effects was used to find the degree of Cosserat behavior for both cylindrical and square cross section specimens in bending and torsion. The foams were found to behave as Cosserat materials (slender specimens appear less stiff than thick ones), provided sufficient care is taken when machining the specimens. Surface damage caused by the machining process may cause the apparent stiffness to decrease with decreasing specimen size, giving an opposite softening size effect. Get pdf
Anderson, W. B., Lakes, R. S., and Smith, M. C., "Holographic evaluation of warp in the torsion of a bar of cellular solid", Cellular Polymers, 14, 1-13, (1995).
Holographic interferometry methods are utilized to examine deviations from classical elasticity in a cellular solid, polymethacrylamide closed cell foam. A square cross section bar is subjected to static torsional deformation. The warp deformation is observed to be less in a foam bar than in a homogeneous polymeric bar used as a control. The homogeneous bar obeys the predictions of classical elasticity. Behavior of the foam bar is consistent with Cosserat elasticity. In a Cosserat solid, points in the continuum to rotate as well as translate, and the material supports couple per unit area as well as force per unit area. Cosserat effects can lead to enhanced toughness even in all brittle composites since stress concentrations are reduced. The image on the right shows holographic fringes associated with warp. This image shows computer generated fringe patterns associated with warp. The degree of warp depends on the Cosserat elastic constants. Get pdf
Cellular solids are two phase composite materials in which one phase is solid and the other is a fluid, most often air. If the size scale becomes large enough, the material may no longer be assumed to be continuous. Some researchers have found that classical elasticity theory does not always adequately describe the behavior of cellular materials. In composite materials with stress concentrations due to holes or cracks, the observed fracture behavior is not correctly predicted by the classical theory of anisotropic elasticity. The experimental stress concentrations are consistently less than the theoretical ones . The non-classical fracture behavior has been dealt with using point stress and average stress criteria, however that approach cannot account for non-classical strain distributions in objects under small load. Strain distributions have been observed in fibrous composites and cellular solids which differ from the predictions of classical elasticity, particularly near small holes and small cracks. Observed concentrations of strain are less than predicted values. Strain fields around large holes, by contrast, follow classical predictions. A more general continuum theory such as Cosserat elasticity (micropolar continuum theory) or nonlocal elasticity, may be of use in predicting non-classical strain distributions. In this study, size effects in the mechanical rigidity of foams are examined experimentally. Analysis of the results is via generalized continuum mechanics and a model of surface damage.
Lakes, R. S., "On the torsional properties of single osteons", J. Biomechanics, 28, 1409-1410, (1995).
Lakes, R. S., "Experimental methods for study of Cosserat elastic solids
and other generalized continua", in Continuum models for materials with micro-structure, ed. H. Mühlhaus, J. Wiley, N. Y. Ch. 1, p. 1-22, (1995).
The behavior of solids can be represented by a variety of continuum theories. For example, Cosserat elasticity allows the points in the continuum to rotate as well as translate, and the continuum supports couple per unit area as well as force per unit area. We examine experimental methods for determining the six Cosserat elastic constants of an isotropic elastic solid, or the six Cosserat relaxation functions of a Cosserat viscoelastic solid. We also consider other generalized continuum theories (including micromorphic elasticity in which points rotate and deform, Cowin's void theory (theory of voids) in which the points dilate, and nonlocal elasticity). Ways of experimentally discriminating among various generalized continuum representations are presented. The applicability of Cosserat elasticity to cellular solids and fibrous composite materials is considered as is the application of related generalized continuum theories.
Get pdf of this review article.
Nakamura, S. and Lakes, R. S., "Finite element analysis of Saint Venant
end effects in micropolar elastic solids", Engineering Computations, 12, 571-587, (1995).
Distributions of stress and strain in composite and cellular materials can differ significantly from the predictions of classical elasticity. For example, concentration of stress and strain around holes and cracks is consistently less than classical predictions. Generalized continuum theories such as micropolar (Cosserat) elasticity offer improved predictive power. In this article Saint-Venant end effects for self equilibrated external forces in micropolar solids are investigated in two dimensions. A two dimensional finite element analysis is used which takes into account the extra degrees of freedom, to treat the problem of localized end loads acting upon a strip. The rate of decay of strain energy becomes slower in a two dimensional strip as the micropolar characteristic length l is increased (for l sufficiently less than the strip width). For the strip geometry a Cosserat solid exhibits slower stress decay than a classical solid. Get pdf
Lakes, R. S., "Elastic freedom in cellular solids and composite materials", in Mathematics of Multiscale Materials, p. 129-153, ed. K. Golden, G. Grimmert, R. James, G. Milton, P. Sen, IMA vol. 99, Springer, NY, Berlin, (1998).
Lakes, R. S., "Elastic and viscoelastic behaviour of chiral materials", Int. J. of Mechanical Sciences, 43, 1579-1589, June (2001).
Chiral materials are not invariant to inversions: there is a distinction between right and left handed material. Material properties such as piezoelectricity and pyroelectricity, represented by tensors of odd rank, can only occur in chiral materials. Chiral effects in elasticity cannot be expressed within classical elasticity since the modulus tensor, which is fourth rank, is unchanged under an inversion. We consider effects of chirality in elastic materials described by a generalized continuum representation, specifically Cosserat elasticity. Analysis of several configurations discloses a chiral material to generate reaction moments when compressed as a slab. A chiral plate bent to hyperbolic shape is predicted to exhibit size effects from the Cosserat characteristic length, and a shear force from the chirality. This analysis can be used for the interpretation of experiments on compliant chiral materials, in particular the evaluation of the elastic constants. Viscoelastic chiral solids are examined in the context of the correspondence principle. Download pdf .
Buechner, P. M., and Lakes, R. S., "Size effects in the elasticity and viscoelasticity of bone", Biomechanics and Modeling in Mechanobiology, 1 (4), 295-301 (2003). Size effects of large magnitude are observed in the torsional shear modulus and damping of bovine plexiform bone. Damping increases and stiffness decreases with specimen size over all sizes studied. Measurements were conducted in torsion using a laser based micromechanics apparatus capable of viscoelastic studies over a range of frequencies up to 100 kHz, upon samples of various size, with no parasitic friction or other errors that could mimic any size effect. Torsional tan delta at 1 Hz varies by about a factor of five over the size range 2.8 to 6.2 mm thick, and is more dependent on specimen thickness at 1 Hz than it is at higher frequency. The size effects are attributed to compliance and viscoelasticity of the interfaces between laminae. These laminae must be substantially stiffer than whole bone. Observed size effects are likely to play a role in understanding scaling laws of bones in living organisms. (Get pdf)
For recent interpretation of generalized continuum properties, see P. Neff site.
As for other experiments, waves have been recently used by Merkel, Tournat, Gusev "Experimental evidence of rotational elastic waves in a granular photonic crystal". The system here is an assembly of elastic spheres in contact. This is an ideal system, designed to exhibit only one Cosserat constant. Wave methods for Cosserat and other generalized continua are suitable for idealized elastic systems without dissipation as discussed in more detail in the linked page. For dissipative materials or systems, wave speed varies with frequency from both viscoelastic effects and dispersion from the microstructure. This complicates interpretation.