Preface

Professor Beatty has contributed a wide variety of research papers and book articles on topics in finite elasticity, continuum mechanics and classical mechanics, including some fundamental experimental work. His works are clear and informative and expose a didactic quality. In the following, we briefly touch upon some of the highlights of his research involvement throughout the years.

1 Synopsis of the Works of Millard Beatty

Beatty’s earliest work concerns elastic stability theory in which he studied the uniqueness of solutions to problems of small static and dynamic motions superimposed on a possibly finitely deformed equilibrium configuration of an arbitrary hyperelastic body under dead load forces and subject to a zero moment condition for boundary conditions of place and tractions. He showed that the conditions of dead load stability of the underlying equilibrium state yielded a uniqueness theorem for the superimposed problem, and guaranteed that dynamically superimposed stress waves are admissible. He then showed that similar results follow within couple stress theory, based upon the introduction of a generalized Betti reciprocal theorem for small, superimposed deformations. Within this context, he then developed an extended form of Castigliano’s classical theorems which relate the body surface point displacement and rotation to the corresponding surface force and couple acting thereon. This led him to a generalization of the theory to study certain hyperelastic Cosserat continua with constrained material directors and to explore hyperelastic materials of grade 2. Applications of these and various other aspects of finite elasticity and elastic stability theory were reported in several following articles, including the estimation of safe loads in special cases, specific results for incompressible materials, and problems of instability of a normally loaded half-space. Specific examples included the study of a fiber-reinforced thick slab and a thick-walled cylindrical tube under axial loading.

Several experimental studies in collaboration with Beatty’s students explored the compression buckling of rectangular and circular rubber rods and cylinders, small, superimposed transverse vibrations of a finitely stretched rubber cord, axial buckling of thick-walled, helically wound composite paper tubes, stress-softening of rubber materials, and the inflation of balloons. James Bell’s influence in the art of experimental mechanics is evident throughout these works as is exhibited in Beatty’s 1977 review article “Elastic Stability of Rubber Bodies in Compression”, in the ASME volume “Finite Elasticity”. In addition to a thorough critique of the empirical shape factor theory prevalent in engineering design treatments, Beatty presented new data that confirmed that the critical axial load for rubber columns that fail by bending with large deflections is the classical Southwell critical load.

Beatty continued with his formulation of the theory of stress-softening in both compressible and incompressible isotropic materials, and he illustrated this theory with examples. He developed a general constitutive equation to account for the stress-softening Mullins effect in uniaxial experiments and studied its effects on the transverse vibration of a rubber string and membrane, and on the inflation of balloons. The familiar effect of preconditioning by stretching on the primary inflation of a balloon was characterized in his study of the Mullins effect in equibiaxial deformation. Applications in pure shear, simple shear, triaxial stretch of rubberlike and biological materials, and the torsion, extension and inflation of cylindrical tubes were described. He then introduced a phenomenological model containing an exponential front factor softening parameter in the constitutive equation for isotropic, incompressible rubberlike materials and showed that the results for the Arruda-Boyce and James-Guth parent material models compared favorably with simple extension data by Johnson and by Mullins and Tobin. The appreciable error that may be introduced into test data due to the reinforcement effect of a strain gage applied to a soft material was studied numerically and a method to reduce it was described.

Beatty then turned his attention to the interesting subject of universal relations for isotropic materials, including those with certain inextensible fibers, and universal quasi-static motions for certain viscoelastic materials of differential type. The physical properties of rubberlike, cellulosic, polymeric, and biological materials were studied, which led him to introduce the “Poisson function” to better characterize the finite simple extension of isotropic, elastic solids in terms of the data that was reported for several rubber-like materials. Extending the scope of this work, Beatty went on to develop necessary and sufficient conditions on the form of the strain energy function for a general compressible, isotropic hyperelastic material in order that antiplane, axisymmetric and helical shear deformations may be produced by application of surface tractions alone, a problem related to one that his thesis adviser, Jerry Ericksen, introduced years earlier; results are illustrated in several examples. He discovered that while these controllable axisymmetric, antiplane shear deformations are possible in a particular Hadamard material, they cannot be sustained in a certain Blatz-Ko compressible, foamed polyurethane rubber material. This is the same polyurethane material for which Beatty later showed that unstable equilibrium states exist in the small amplitude superimposed horizontal motion of a load supported symmetrically by rubber springs.

Further investigations of Beatty included: his work “On the Foundation Principles of General Classical Mechanics” in which the fundamental principles are deduced from Noll’s principle of frame indifference of mechanical power; his development of the Arruda-Boyce constitutive equation for elastomers from a full network molecular based model; his derivation of the constitutive equation for the phenomenological Gent model of rubber elasticity as a Padé approximant within a general constitutive framework for limited elastic molecular based materials; and his demonstration of universal relations that follow from the constitutive equation for isotropic, nonlinear elastic materials. Along the way, Beatty also developed a number of special mathematical results which dealt with: determinants having complex conjugate elements; general integrals arising in plane finite elasticity; and an integral identity applicable to continuum mechanics. His 1987 review article on “Topics in Finite Elasticity: Hyperelasticity of Rubbers, Elastomers and Biological Tissues, with Examples,” updated in 1996 as a book chapter titled “Introduction to Nonlinear Elasticity,” continues to be widely referenced as a basic source for readers in nonlinear elasticity. In addition, Beatty’s “Seven Lectures on Finite Elasticity” in the CISM volume “Topics in Finite Elasticity” presented at Udine in 2000 is an informative supplement to this article with its inclusion of additional applications to cellular solids, controlled deformations of compressible hyperelastic materials, vibrations of a load on viscohyperelastic shear mounts, hyperelastic Bell materials, and stress-softening in rubberlike materials.

Beatty showed broad interest in, and contributed to a basic understanding of the nonlinear dynamical behavior of finite amplitude free and driven oscillations of a load supported by various rubberlike supports, including members exhibiting stress-softening and limited extensibility. In related work, he reported on problems of small superimposed oscillation together with extensive experiments on the small transverse vibration of stretched rubber cords. His analysis regarding the transverse impact of a hyperelastic stretched string showed remarkably good agreement with the high velocity impact tests of Haddow and Wagner. In 1983, Beatty published the interesting paper “Finite Amplitude Oscillations of a Simple Rubber Support System” the exact solution to the longitudinal free vibration problem for a neo-Hookean oscillator and therein the general oscillatory motion and the period in terms of a complementary Heuman lambda function. The result delivered simple upper and lower bounds on the period expressed in terms of this special function. This seminal work led to exact and approximate solutions to other physically based dynamical nonlinear elasticity problems, including the finite amplitude oscillations of a load supported by simple shear mounts, the coupled oscillations of a load under shearing and combined torsion and extension, the radial oscillations of limited elastic thick-walled tubes, the finite amplitude oscillations of a load supported by a highly elastic tubular shear spring, and the small radial oscillations superimposed on the finite inflation of spherical and cylindrical shells. In following related studies Beatty included viscohyperelastic, stress-softening, limited elastic, and driven vibrational effects.

In a series of fundamental papers, the first of which, Part 1, appeared in 1992 in this journal, Beatty, together with Michael Hayes, developed the constitutive theory of an isotropic Bell constrained material. They were motivated by the empirical internal kinematic material constraint that was grounded in the great body of large deformation experiments by James Bell on a variety of annealed polycrystalline metals within the context of finite strain plasticity. Part 1 studied the homogeneous deformation of isotropic, hyperelastic Bell constrained materials. In this constrained theory, the kinematics alone showed that isochoric deformations are not possible - the material volume must decrease in every deformation. Still, they found that every such material behaves in small deformations like an incompressible material whose Poisson function in every simple extension, however large, has the constant value 1/2. The theory delivered some remarkable results that seem to validate some empirical results of Bell for plasticity. Overall, where comparisons were possible, the results of Beatty and Hayes concurred with Bell’s empirical conclusions. The book chapter “Hyperelastic Bell Materials: retrospection, experiment, theory” published in 2001 provides an overview of Bell’s experimental results and the aforementioned theoretical work.

Part 2 of the joint work on Bell constrained materials studied nonhomogeneous deformations. This included the bending and stretching of a rectangular block; the bending, extension and azimuthal shearing of an annular wedge; and the radial deformation of a spherical shell and membrane. A surprising result confirmed that eversion of a Bell constrained thick spherical shell is possible only when the wall thickness is less than its inside radius. Development of the theory of small superimposed deformations and waves and the propagation of small amplitude torsional waves followed in Part 3. For an infinitesimal twist superimposed on a pure homogeneous uniaxial extension or compression of a uniform prism, they found that the torsional couple may vanish in compression but not in extension if the prism is neither a solid circular cylinder nor a circular cylindrical tube. While this three-part work concluded their joint research on Bell materials, later, in 2005, Beatty and Hayes got together again and arranged for the production, and served as co-editors of, the special volume: “Mechanics and Mathematics of Crystals – Selected Papers of J. L. Ericksen”.

Beatty continued with his investigation of the instability of a Bell constrained thick- and thin-walled cylindrical tube subject to averaged compressive dead loads over its ends, a thick plate under compression and a half-space under biaxial loading. His estimate for the critical buckling load of thin plates derived from the thick plate analysis for hyperelastic Bell constrained materials were found to coincide with the corresponding classical relation. Based upon the Hadamard criterion ensuring that propagating incremental wave speeds are real, Beatty studied the stability of a Bell constrained material subject to a pure homogeneous deformation. Among other things, he showed that the natural state is the only materially stable equilibrium state of a Bell constrained material under pressure loading.

Finally, Professor Beatty’s passion for teaching is exhibited in a number of his didactic related articles on classical mechanics and is emphasized further in two volumes on the principles of engineering mechanics. The book “Dynamics – The Analysis of Motion” presents many illustrative examples that emphasize the predictive value of the principles of mechanics. Notable examples include the discovery of the planet Neptune, the U.S. Navy torpedo failures in World War II, the Foucault’s pendulum experiments, the erratic ballistics in the 1914 battle of the Falkland Islands, and the general law of mutual internal action as derived by Noll using the principle of material frame indifference from continuum mechanics. This book contains a foundation for studies in advanced dynamics including Euler’s formulation of the general principles of mechanics for all material bodies and Lagrange’s invariant formulation of the equations of classical mechanics. Each chapter concludes with a list of annotated references for expanded study.

On every appropriate happening, Millard Beatty has expressed profound gratitude to his thesis advisor and principal teacher Jerry Ericksen in nonlinear elasticity, stability theory, and tensor analysis, to Clifford Truesdell his thesis reader and teacher in the non-linear field theories of mechanics, and to James Bell his senior research advisor and motivational teacher in vibrations, nonlinear mechanics, solid mechanics and experimental solid mechanics. Other influential scholars whose classes or lectures he attended when he was a Graduate Student at the Johns Hopkins University are Bernard Coleman, Oscar Dillon, Walter Noll, Owen Phillips, Ronald Rivlin, James Serrin, and Richard Toupin. In all, a most remarkable experience.

Beatty has noted on occasion that much of his research and didactic activity would not have been possible without the collaboration of some remarkable colleagues, talented post-doctoral fellows, and exceptional former students both undergraduate and graduate, especially their contribution to his various experimental programs. His list of publications exemplifies these mutual and cooperative accomplishments. But beyond that, Millard served as a mentor to many faculty and young scientists, and was a much beloved and respected chairman who always stood ready to protect the academic excellence in the Engineering Mechanics department at the University of Nebraska, Lincoln.

2 Principal Journal and Conference Publications of Millard Beatty

• 1957: Undergraduate student publication: Transducer Applications. The Vector 12, No. 1 (November 1957), 18-19, 64-70. The Johns Hopkins University, Baltimore, Maryland.

• 1963: Graduate student publication: Vector Representation of Rigid Body Rotation. Am. J Phys. 31, 134-135.

• 1965: Some Static and Dynamic Implications of the General Theory of Elastic Stability. Arch. Rational Mech. Anal. 19, 167-183.

• 1966: Kinematics of Finite Rigid Body Displacements. Am. J. Physics 34, 949-954.

• 1967: A Theory of Elastic Stability for Incompressible, Hyperelastic Bodies. Int. J. Solids Struc. 3, 23-37.

  On the Foundation Principles of General Classical Mechanics. Arch. Rational Mech. Anal. 24, 264-273.

  A Reciprocal Theorem in the Linearized Theory of Couple-Stresses. Acta Mechanica $\boldsymbol{3}$, 154-166.

• 1968: Some Experiments on the Stability of Rubber-like, Circular Bars Under End Thrust. (with D.E. Hook) Int. J. Solids Struc. 4, 623-635.

  Stability of the Undistorted States of an Isotropic Elastic Body. Int. J. Nonlinear Mech. 3, 337-349.

• 1970: Stability of Hyperelastic Bodies Subject to Hydrostatic Loading. Int. J. Nonlinear Mech. 5, 367-383.

  Surface Centroid and Some Theorems on Equipollent Force Systems. (with D.C. Leigh) Bull. Mech. Engng. $Ed$. 9, 57-60.

  A Mathematical Theory for the Traction on a Body in Motion in a Continuum: Part I The Foundation Principles. (with D.C. Leigh) Arch. Rational Mech. Anal. 38, 81-106.

  A Theory of Elastic Stability for Constrained, Hyperelastic Cosserat Continua. Arch. Mech. Stos. 22, 585-606.

• 1971: A Theory of Elastic Stability for Perfectly Elastic Materials with Couple-Stresses Proc. IUTAM Symposium, Herrenalb 1969. Instability of Continuous Systems, Edited by H. Leipholz, New York: Springer-Verlag, 85-89.

  Estimation of Ultimate Safe Loads in Elastic Stability Theory. J. Elasticity 1, 95-120.

• 1972: Lagrange’s Theorem on the Center of Mass of a System of Particles. Am. J. Phys. 40, 205-207.

• 1974: On the Surface Instability of a Highly Elastic Half-Space. (with S.A. Usmani). J. Elasticity 4, 249-263.

• 1975: On the Mathematical Theory of the Mechanical Behavior of Some Non-Simple Materials. (with K.J. Cheverton). Arch. Rational Mech. Anal. 60, 1-16.

  On the Indentation of a Highly Elastic Half-Space. (with S.A. Usmani). Quar. J. Mech. Appl. Math. 28, 47-62.

• 1976: Some Experiments on the Elastic Stability of Some Highly Elastic Bodies. (with P. Dadras). Int. J. Engng. Sci. 14, 233-238. Ibid. Errata 15 (1977), 219.

  An Integral Identity with Application in Continuum Mechanics. (with K.J. Cheverton). J. Elasticity 6, 81-82.

  The Basic Equations for Materials of Grade 2 Viewed as an Oriented Continuum. (with K.J. Cheverton). Arch. Mech. Stos. 28, 205-213.

  Some Theorems in the General Theory of Small Deformations Superimposed on a Finite Deformation of a Hyperelastic Material of Grade 2. (with K.J. Cheverton). Int. J. Solids Struc. 12, 339-351.

• 1977: Kinematics of Finite Rigid Body Rotations: Revisited. Am. J. Phys. 45, 1006.

  Vector Analysis of Finite Rigid Rotations. J. Appl. Mech. 44, 501-502.

  Elastic Stability of Rubber Bodies in Compression. ASME Symposium on Finite Elasticity, Appl. Mech. Div. - Vol. 27, 125-150.

• 1978: General Solutions in the Equilibrium Theory of Inextensible Elastic Materials. Acta Mech. 29, 119-126.

  Reinforcement Effect of a Strain Gage Bonded to a Soft Material. Recent Advances in Engineering Science. Prod. 15th Meet. Soc. Engng. Sci., U. Of Florida, Gainesville, December 1978, 163-165.

• 1979: Numerical Analysis of the Reinforcement Effect of a Strain Gage Applied to a Soft Material. (with S.W. Chewning) Int. J. Engng. Sci. 17, 907-915.

  Axial Buckling of a Short Helically Structured Tube. Proc. 3rd ASCE Engng. Mech. Div. Spec. Conf., U. of Texas, Austin, September 1979, 85-88.

• 1980: Axial Compression of a Helically Wound Laminated Paper Tube. (with W.C. Cheng). J. Composite Matls. 14, Suppl., 42-56.

• 1981: Extension, Torsion and Expansion of an Incompressible, Hemitropic Cosserat Circular Cylinder. (with K.J. Cheverton). J. Elasticity 11, 207-227.

• 1982: An Experimental Investigation of the Transverse Vibrational Frequency Ratio for Identical Loaded and Unloaded Rubber Strings. (with J.H. Lienhard, V). Am. J. Phys. 50, 113-119.

• 1983: On the Transverse Vibration of a Rubber String. (with A.C. Chow). J. Elasticity 13, 317-344.

  Finite Amplitude Oscillations of a Simple Rubber Support System. Arch. Rational Mech. Anal. 83, 195-219.

• 1984: Free Vibrations of a Loaded Rubber String. (with A.C. Chow) Int. J. Nonlinear Mech. 19, 69-81.

  Finite Amplitude Vibrations of a Mass Supported by Simple Shear Springs. J. Appl. Mech. 51, 361-366.

• 1985: Transverse Impact of a Hyperelastic Stretched String. (with J.B. Haddow) J. Appl. Mech. 52, 137-143.

• 1986: Finite Amplitude Vibrations of a Neo-Hookean Oscillator. Quar. Applied Math. 44, 19-34.

  The Poisson Function of Finite Elasticity. (with D.O. Stalnaker) J. Appl. Mech. 53, 807-813.

  Finite Amplitude Oscillations of a Simple Rubber Support System. The Breadth and Depth of Continuum Mechanics, A collection of (previously published) papers dedicated to J.L. Ericksen, Editors: C.M. Dafermos, D.D. Joseph, and F.M. Leslie, Springer-Verlag, New York, 43-67.

• 1987: Some Dynamical Problems in Continuum Physics. IMA Volumes in Mathematics and its Applications, Editors: J.L. Bona, C. Dafermos, J.L. Ericksen, and D. Kinderlehrer, Springer-Verlag, New York, Dynamical Problems in Continuum Physics, Vol. 4, 43-78.

  A Class of Universal Relations in Isotropic Elasticity Theory. J. Elasticity 17, 113-121.

  Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues, with Examples. Appl. Mech. Revs. 40, 1699-1734.

• 1988: Finite Amplitude, Periodic Motion of a Body Supported by Arbitrary Isotropic Elastic Shear Mountings. J. Elasticity 20, 203-230.

  Finite Amplitude Vibrations of a Mooney-Rivlin Oscillator. (with A.C. Chow) Arch. Rational Mech. Anal. 102, 141-166.

• 1989: Stability of a Body Supported by a Simple Vehicular Shear Suspension System. Int. J. Non-Linear Mech. 24, 65-77.

  Gent-Thomas and Blatz-Ko models for foamed elastomers. In: Mechanics of Cellulosic and Polymeric Materials. Ed. R.W. Perkins, ASME AMD-Vol. 99, 75-78.

  A Class of universal relations for constrained, isotropic elastic materials. Acta Mech. 80, 299-312.

  Stability of the Free Vibrational Motion of a Vehicular Body Supported by Rubber Shear Mountings with Quadratic Response. (with R. Bhattacharyya) Int. J. Non-Linear Mech. 24, 401-414.

• 1990: Instability of a Fiber Reinforced Thick Plate Under Axial Loading. Int. J. Non-Linear Mech. 25, 343-362.

  Poynting oscillations of a rigid disk supported by a neo-Hookean rubber shaft. (with R. Bhattacharyya) J. Elasticity 24, 135-186.

• 1991: Finite amplitude, free vibrations of a body supported by incompressible, nonlinear viscoelastic shear mountings. (with Z. Zhou) Int. J. Solids Struc. 27, 355-370.

  Universal motions for a class of viscoelastic materials of differential type. (with Z. Zhou) Cont. Mech. Thermo., 3, 169-191.

  Torsion of a Bell material. (with M.A. Hayes) in Recent Developments in Elasticity, Eds. R.C. Batra and G.P. MacSithigh, ASME-AMD Vol. 124, 55-60.

• 1992: Deformations of an internally constrained elastic material. Part 1: Homogeneous deformations. (with M.A. Hayes) J. Elasticity 29, 1-84.

  Deformations of an internally constrained elastic material. Part 2: Honhomogeneous deformations. (with M.A. Hayes) Quar. J. Mech. Appl. Math. 45, 663-709.

• 1993: The Mullins effect in uniaxial extension and its influence on the transverse vibration of a rubber string. (with M.A. Johnson) Cont. Mech. Thermo. 5, 83-115.

  A constitutive equation for the Mullins effect in stress controlled uniaxial extension experiments. (with M.A. Johnson) Cont. Mech. Thermo 5, 301-318.

  On Bell’s constraint in finite elasticity. (with M.A. Hayes) in Advances in Modern Continuum Dynamics. International Conference in Memory of Antonio Signorini, Elba Island, July 6-11, 1991. Ed. G. Ferrarese, Pitagora Editrice Bologna (1993), 183-190.

• 1994: Simple shearing of an incompressible viscoelastic quadratic material. (with Z. Zhou) Int. J. Solids Struc. 31, 3201-3215.

  Deformations of an internally constrained elastic material. Part 2: Nonhomogeneous deformations. (with M.A. Hayes) in Nonlinear Elasticity and Theoretical Mechanics. A collection of (previously published) papers dedicated to A.E. Green. Editors: P.M. Naghdi, A.J.M. Spencer, and A.H. England, Oxford University Press, New York, 147-193.

• 1995: Finite amplitude, free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring. (with R.A. Khan) J. Elasticity 37, 179-242.

  The Mullins effect in equibiaxial extension and its influence on the inflation of a balloon. (with M.A. Johnson) Int. J. Engng. Sci. 33, 223-245.

  Small amplitude torsional waves propagating in a Bell material. (with M.A. Hayes) Proceedings of the IUTAM Symposium Nonlinear Waves in Solids, Eds. J.L. Wegner and F.R. Norwood, University of Victoria, Victoria, B.C., August 15-20, 1993, Appl. Mech. Revs. Book No. AMR 137, 67-72.

  Solución de la Ecuación de Duffing Aplicando el Método de Balanceo Elíptico. (Solution of the Duffing Equation by Use of the Elliptic Balance Method) (Prepared by A. E. Zúñiga), Memorais de la XXV Reunión de Investigación y Desarrollo Technológico del Sistema ITESM (Proceedings of the 25th Meeting on the Investigation and Technological Development of the ITESM System), Instituto Tecnológico y de Estudios Superiores de Monterrey, Mexico, January 13, 1995, 108-116.

  Deformations of an internally constrained elastic material. Part 3: Small superimposed deformations and waves. (with M.A. Hayes), Z. Angew. Math Phys. (ZAMP) 46, S72-S106.

  Compressible materials capable of sustaining axisymmetric shear deformations. Part 1: Anti-plane shear of isotropic hyperelastic materials. (with Q. Jiang) J. Elasticity 39, 75-95.

• 1996: Compressible materials capable of sustaining axisymmetric, anti-plane shear deformations. (with Q. Jiang) in the J.L. Ericksen Anniversary Volume: Contemporary Research in the Mechanics and Mathematics of Materials, Eds. R.C. Batra and M.F. Beatty, International Center for Numerical methods in Engineering (CIMNE) Barcelona, Spain, 1996, 133-144.

  Método de Balanceo Elíptico Aplicando a la Solución de la Ecuación Amortiguada de Duffing con Término Forzante de Tipo Elíptico. (Application of the Elliptic Balance Method to the Solution of the Damped Duffing Equation) (Prepared by A. E. Zúñiga), Avances en Ingeniería Mecánica Memoria. (Proc. 1$^{st}$ Internat. Cong. Engng. Electromech. Sys.), Instituto Politecnico Nacional, Mexico, November 11-15, 1996, 228-234.

• 1997: On the motion of lineal bodies subject to linear dam**, J. Appl. Mech. 64, 227-229.

  On Compressible materials capable of sustaining axisymmetric shear deformations. Part 2: Rotational shear of isotropic hyperelastic materials. (with Q. Jiang), Quar. J. Mech. Appl. Math. 50, 211-237.

  On determinants having complex conjugate columns and rows. (with F. Pan) J. Elasticity, 47, 69-72.

  Remarks on the instability of an incompressible and isotropic, hyperelastic thick-walled cylindrical tube. (with F. Pan) J. Elasticity, 48, 217-239.

  Instability of a Bell constrained cylindrical tube under end thrust - Part 1: Theoretical Development. (with F. Pan) Math. Mech. Solids 2, 243-273.

• 1998: Stability of an internally constrained, hyperelastic slab. (with F. Pan) Int. J. Non-Linear Mech., 33, 867-906.

• 1999: The Mullins effect in incompressible elastomers. Applied Mechanics in the Americas, Volume 7, Proceedings of the 6^th Pan-American Congress of Applied Mechanics, Eds. P.B. Concalver, I. Jasiuh, D. Pamplona, C. Steele, H.I. Weber, and L. Bevilacqua, Rio De Janeiro, Brazil, January 4-9, 1999, 1095-1098.

  On compressible materials capable of sustaining axisymmetric shear deformations. Part 3. Helical shear of isotropic hyperelastic materials. (with Q. Jiang) Quart. Appl. Math. 57, 681-697.

  Instability of an internally constrained elastic material. (with F. Pan) Int. J. Non-Linear Mech. 34, 169-177.

  Instability of a Bell constrained cylindrical tube under end thrust - Part 2: Examples, Thin tube analysis. (with F. Pan), Math. Mech. Solids 4, 227-250.

  Stress-softening of elastomers. (with S. Krishnaswamy) $Pro. 1^{st}$ Canadian Conference on Nonlinear Solid Mechanics, Ed. E.M. Croitoro, University of Victoria, British Columbia, Canada, June 16-20, 1999, University of Victoria Press, 139-151.

• 2000: Stress-softening in combined triaxial stretch and simple shear of a block. In: Contributions to Continuum Mechanics, Anniversary Volume for Krzysztof Wilmanski, Weierstrass Institute for Applied Analysis and Stochastics, Report 18, Ed. B. Albers, Berlin, 25-36.

  A theory of stress-softening in incompressible isotropic materials. (with S. Krishnaswamy) J. Mech. Phys. Solids 48, 1931-1965.

  The Mullins effect in compressible solids. (with S. Krishnaswamy) Int. J. Engng. Sci. 38, 1397-1414.

  The Mullins effect in equibiaxial deformation. (with S. Krishnaswamy) J. Appl. Math. Physics (ZAMP) 51, 1-33.

  A note on some general integrals arising in plane finite elasticity. (with J.M. Hill) Quar. J. Mech. Appl. Math. 53, 421-428.

• 2001: The Mullins effect in a pure shear. J. Elasticity. 59, 369-392.

  Damage induced stress-softening in the combined torsion, extension, and inflation of a cylindrical tube. (with S. Krishnaswamy) Quar. J. Mech. Appl. Math. 54, 295-327.

  Compressible materials capable of sustaining axisymmetric shear deformations. Part 4: Helical shear of anisotropic hyperelastic materials. (with Q. Jiang) J. Elasticity 62, 47-83.

  Forced vibrations of a body supported by hyperelastic shear mountings (with A. E. Zúñiga) Mech. Res. Comm. 28, 429-446.

  Forced vibrations of a body supported by viscohyperelastic shear mountings (with A.E. Zúñiga) J. Engng. Math. 40, 333-353.

• 2002: Universal relations for fiber reinforced materials. (with G. Saccomandi) Math. Mech. Solids 7, 95-110.

  A new phenomenological model for stress-softening in elastomers (with A.E. Zúñiga) J. Appl. Math. Phys. (ZAMP) 53, 794-814.

  The Mullins effect in the transverse vibration of a rubber cord. In Volume 2: Proc. 2$^{nd}$ Can. Conf. Nonlinear Solid Mech. Ed. E. M. Croitoto, Simon Fraser University, Vancouver, British Columbia, June 19-23, 2002, 341-351.

  On the motion of thin plates and shells subject to Stokes dam**. Mech. Res. Comm. 29, 321-326.

  Constitutive equations for amended non-Gaussian network models of rubber elasticity. (with A.E. Zúñiga) Int. J. Engng. Sci. 40, 2265-2294.

• 2003: An average-stretch full-network model for rubber elasticity. J. Elasticity 70, 65-86.

  The Mullins effect in the vibration of a stretched rubber membrane. Math. Mech. Solids 8, 435-445.

  Stress-softening effects in the vibration of a non-Gaussian rubber membrane. (with A.E. Zúñiga). Math. Mech. Solids 8, 481-495.

  Dead loading of a unit cube of compressible isotropic elastic material. (with R.S. Rivlin). J. Appl. Math. Phys. (Z. Angew. Math. Phys.) (ZAMP) 54, 954-963.

  Stress-softening effects in the transverse vibration of a non-Gaussian rubber string. (with A.E. Zúñiga). Mecannica 38, 419-433.

• 2005: Constitutive equations for the back stress in amorphous glassy polymers. Math. Mech. Solids 10, 167-181.

• 2007: On the radial oscillations of incompressible, isotropic, elastic and limited elastic thick-walled tubes. Int. J. Non-Linear Mech. 42, 283-297.

  Elliptic balance solution to two degree of freedom, undamped, homogeneous systems having cubic nonlinearities. (with A.E. Zúñiga), J. Sound Vibra. 304, 175-185.

  Elliptic balance solution of two degree-of-freedom, undamped, forced systems with cubic nonlinearity. (with A.E. Zúñiga), Nonlinear Dynam. 49, 151-161.

• 2008: On constitutive models for limited elastic, molecular based materials. Math. Mech. Solids 13, 375-387; https://doi.org/10.1117/1081286507076405.

  Effect of stress-softening on the dynamics of a load supported by a rubber string. (with S. Sarangi and R. Bhattacharyya) J. Elasticity: The Physical and Mathematical Science of Solids 92, 115-149; https://doi.org/10.1007/s10659-007-9154-9.

  Oscillations of a load supported by incompressible, isotropic limited elastic shear mounts. Quart. J. Mech. Appl. Math. 61, 373-394. https://doi.org/10.1093/qjmam/hbn007.

• 2009: Small amplitude, free longitudinal vibrations of a load on a finitely deformed stress-softening spring with limiting extensibility. (with R. Bhattacharyya and S. Sarangi) ZAMP 60, 971-1006; https://doi.org/10.1007/s00033-008-8127-6.

  Small longitudinal oscillations of a load on an incompressible, isotropic limited elastic spring. Int. J. Engng. Sci. 47, 1110-1118; https://doi.org/10.1016/j.ijengsci.2008.06.009.

• 2011: Small amplitude radial oscillations of an incompressible, isotropic elastic spherical shell. Math. Mech. Solids 16, 492-512; https://doi.org/10.1177/1081286510387407.

  Infinitesimal stability of the equilibrium states of an incompressible, isotropic elastic tube under pressure. J. Elast. 104, 71-90; https://doi.org/10.1007/s10659-011-9321-x.

  Finite amplitude, horizontal motion of a load symmetrically supported between isotropic hyperelastic springs. (with T.R. Young) Int. J. Non-Linear Mech. 47, 166-172; https://doi.org/10.1016/j.ijnonlinmec.2011.04.004.

• 2020: Small-amplitude superimposed horizontal motion of a load supported symmetrically by rubber springs. (with T.R. Young) Math. Mech. Solids 25, 597-618; https://doi.org/10.1177/1081286519885179.

3 Book Articles

• 1971: A Theory of Elastic Stability for Perfectly Elastic Materials with Couple-Stresses. Proc. IUTAM Symposium, Herrenalb 1969. Instability of Continuous Systems, Edited by H. Leipholz, New York: Springer-Verlag, 85-89.

• 1977: Elastic Stability of Rubber Bodies in Compression. ASME Symposium on Finite Elasticity, Appl. Mech. Div. Vol. 27, 125-150.

• 1986: Finite Amplitude Oscillations of a Simple Rubber Support System. The Breadth and Depth of Continuum Mechanics, A Collection of (previously published) papers dedicated to J.L. Ericksen, Editors: C.M. Dafermos, D.D. Joseph, and F.M. Leslie, Springer-Verlag, New York, 43-67.

• 1987: Some Dynamical Problems in Continuum Physics. IMA Volumes in Mathematics and its Applications, Editors: J.L. Bona, C. Dafermos, J.L. Ericksen, and D. Kinderlehrer, Springer-Verlag, New York, Dynamical Problems in Continuum Physics, Vol. 4, 43-78.

• 1989: Gent-Thomas and Blatz-Ko models for foamed elastomers in Mechanics of Cellulosic and Polymeric Materials, Ed. R.W. Perkins, ASME AMD-Vol. 99, 75-78.

• 1991: Torsion of a Bell material. (with M.A. Hayes) in Recent Developments in Elasticity, Eds. R.C. Batra and G.P. MacSithigh, ASME-AMD Vol. 124, 55-60.

  On Bell’s constraint in finite elasticity. (with M.A. Hayes) in Advances in Modern Continuum Dynamics. International Conference in Memory of Antonio Signorini, Elba Island, July 6-11, 1991. Ed. G. Ferrarese, Pitagora Editrice Bologna (1993), 183-190.

• 1994: Deformations of an internally constrained elastic material. Part 2: Nonhomogeneous deformations. (with M.A. Hayes) in Nonlinear Elasticity and Theoretical Mechanics. A collection of (previously published) papers dedicated to A.E. Green, Editors: P.M. Naghdi, A.J.M. Spencer, and A.H. England, Oxford University Press, New York, 147-193.

• 1995: Deformations of an internally constrained elastic material. Part 3: Small super-imposed deformations and waves (with M.A. Hayes) in Theoretical, Experimental and Numerical Contributions to the Mechanics of Fluids and Solids. A collection of papers (also published in ZAMP) dedicated to Paul M. Naghdi, Eds. J. Casey and M. J. Crochet, Birkhäuser Verlag, Basel, Switzerland, S72-S106.

  Small amplitude torsional waves propagating in a Bell material. (with M.A. Hayes) Proceedings of the IUTAM Symposium Nonlinear Waves in Solids, Eds. J. L. Wegner and F. R. Norwood, University of Victoria, Victoria, B.C., August 15-20, 1993, Appl. Mech. Revs. Book No. AMR 137, 67-72.

• 1996: Introduction to Nonlinear Elasticity. In the Rivlin Anniversary Volume, Nonlinear Effects in Fluids and Solids, Eds. M.M. Carroll and M.A. Hayes, Plenum Press, New York, 13-112.

  Compressible materials capable of sustaining axisymmetric, antiplane shear deformations. (with Q. Jiang) In the J.L. Ericksen Anniversary Volume: Contemporary Research in the Mechanics and Mathematics of Materials, Eds. R.C. Batra and M.F. Beatty, International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain, 133-144.

• 2001: Hyperelastic Bell materials: Retrospection, experiment, theory. Chapter 3 in: Nonlinear Elasticity: Theory and Applications. Eds. Y. B. Fu & R. W. Ogden. Cambridge: Cambridge University Press, 58-96.

  Seven Lectures on Topics in Finite Elasticity. In: Courses and Lectures No. 424, International Centre for Mechanical Sciences, June 12-16, 2000, Udine, Italy. Eds. M. A. Hayes & G. Saccomandi, Springer-Verlag, 31-93.

4 Books

• 1986: Principles of Engineering Mechanics. Volume 1: Kinematics - The Geometry of Motion. Plenum Press, New York, London. (Volume 32: Mathematical Concepts and Methods in Science and Engineering, Series Ed. Angelo Miele.) Now owned by Springer.

• 2005: Principles of Engineering Mechanics. Volume 2: Dynamics - The Analysis of Motion. Springer, New York, London. (Volume 33: Mathematical Concepts and Methods in Science and Engineering, Series Ed. Angelo Miele.)