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On Universal Deformations of Compressible Cauchy Elastic Solids Reinforced by Inextensible Fibers

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Pith's one-line read For compressible isotropic solids reinforced by parallel inextensible fibers, the paper proves that when deformed fibers stay straight, only homogeneous Z-isometric mappings and one bending-and-stretching family are universal, and that…

desk verdict Useful explicit results, but the completeness proof rests on a mean-curvature formula that requires an isothermal parametrization the paper never establishes. read the letter →

arxiv 2506.11203 v3 pith:HETC5ZQ6 submitted 2025-06-12 math-ph math.MP

Arash Yavari This is my paper · ORCID
classification math-phmath.MP MSC 74B2074E1074E3053A4553Z05
keywords Universaldeformationsfiber-reinforcedsolidsinextensiblefibersinextensibilityconstraintCauchyelasticityhyperelasticitynonlinear
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks which deformations of a fiber-reinforced elastic solid can be produced by boundary tractions alone, no matter what compressible isotropic material fills the matrix. For fibers that are initially straight and parallel, the author shows that if the deformed fibers remain straight lines, exactly two families of universal deformations exist: all homogeneous deformations that keep the fiber direction unstretched, and one genuinely inhomogeneous family that bends horizontal planes into concentric cylinders while stretching the block along the cylinder axis. The proof derives the universality constraints from the requirement that a tension field exist along the fibers for every admissible response function. The paper also proves that Cauchy elastic solids (no energy function assumed) and hyperelastic solids with the same fiber reinforcement have identical universal deformations, and it leaves the case of curved deformed fibers as an open problem.

What carries the argument

The argument runs through the Lagrange-multiplier tension field T associated with the inextensibility constraint. Equilibrium without body forces reduces to the overdetermined system (T n),Z = f, so a universal deformation must make f lie in the span of the deformed fiber direction n (straight case) or in span{n, n,Z} (curved case) for every admissible response function; setting the response-function coefficients to zero independently yields the universality constraints div b = beta n, div c = gamma n, and gradient conditions on the invariants. Once n is shown to be an eigenvector of the Finger tensor with eigenvalue 1, the constraints force the surfaces orthogonal to the fibers — parametrized by the base surface a(X,Y) of the straight-line deformation x = a(X,Y) + n(X,Y) Z — to have constant Gaussian and mean curvature, which limits them to planes, cylinders, or spheres.

What would settle it

Test the completeness claim by checking the mean-curvature step: verify whether the base surface a(X,Y) inherits an isothermal parametrization from the material coordinates; if it does not, recompute the mean curvature of the orthogonal surfaces with the conformal scale factor and test whether a cone with constant Gaussian curvature K = c6/c4 satisfies the universality constraints (3.66)-(3.73) and div b = beta n, div c = gamma n. A cone-type solution would be a universal deformation outside Families 0Z and Z1, contradicting Proposition 3.7.

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Extended reading notes

Core claim

The central discovery is that the structure of universal deformations is governed entirely by the geometry of the deformed fibers and of the surfaces orthogonal to them. When the deformed fibers are straight lines and at least one principal invariant is nonconstant, the fiber direction must be an eigenvector of the Finger tensor with eigenvalue 1, the invariants may depend only on the fiber arclength coordinate Z, and the surfaces normal to the fibers must have constant mean and Gaussian curvature, forcing them to be planes, circular cylinders, or spheres. Planes lead to homogeneous universal deformations (Family 0Z); cylinders lead to the single inhomogeneous universal family Z1, r = Z+Z0, theta = alpha0 X + beta0 Y + theta0, z = k1 Y + z0, which is non-isochoric; spheres produce no admissible solutions. When all principal invariants are constant and fibers remain straight, only homogeneous deformations are universal. The paper further establishes that with the same fiber reinforcement, compressible isotropic Cauchy elastic solids and hyperelastic solids share the same universality constraints, hence the same universal deformations.

Load-bearing premise

The completeness of the two-family list depends on the base surface a(X,Y) allowing an isothermal parametrization so that its mean curvature is H = -1/2(a,X·n,X + a,Y·n,Y); if the material coordinates are not isothermal, the formula gains a scale factor and cones are not excluded, so the classification could miss universal deformations.

Editorial extensions

If this is right

  • Family Z1, the new inhomogeneous family r = Z+Z0, theta = alpha0 X + beta0 Y + theta0, z = k1 Y + z0, gives the first non-homogeneous universal deformation for compressible fiber-reinforced solids and can serve as a benchmark for mixed finite-element codes that handle inextensible fibers.
  • When the principal invariants are constant and deformed fibers stay straight, no inhomogeneous universal deformation exists; only homogeneous deformations qualify.
  • The classical Family 5 universal deformations of incompressible elasticity, when restricted by inextensibility, cease to be universal in fiber-reinforced solids.
  • Compressible isotropic Cauchy elastic solids and hyperelastic solids with the same fiber reinforcement share exactly the same universal deformations.
  • For curved deformed fibers, the three principal invariants must be functionally dependent and the fiber binormal must be an eigenvector of the Finger tensor; whether any universal deformation exists with curved fibers remains open.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The short list gives a practical completeness test: any straight-fiber universal deformation found in a numerical experiment must have a right Cauchy-Green tensor whose ZZ component is 1 and whose orthogonal surfaces are planar or cylindrical, otherwise it falls outside the classified families.
  • The Cauchy-hyperelastic coincidence is plausibly generic: because the argument uses only the algebraic form of the isotropic stress representation and the tension integrability conditions, the same equivalence should hold for solids reinforced by multiple fiber families or by inextensible surfaces.
  • In the limit of very stiff (nearly inextensible) fibers, the two universal families of this paper should survive as the leading-order solution set; the extent to which nearby approximate universal deformations exist is a perturbation question the paper does not address.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies universal deformations of compressible isotropic Cauchy elastic solids reinforced by a single family of inextensible straight fibers. It derives the universality constraints for the cases of straight and curved deformed fibers, claims a complete classification when the deformed fibers are straight, and introduces Family 0Z (homogeneous Z-isometric deformations) and Family Z1 (an inhomogeneous bending-stretching family). It also proves that the universal deformations of the corresponding hyperelastic solids coincide with those of Cauchy elasticity (Proposition 4.1) and shows that Family 5 of incompressible elasticity is not universal under the inextensibility constraint.

Significance. If the completeness result were established, this would be a valuable contribution: it would be the first systematic classification of universal deformations for compressible isotropic fiber-reinforced Cauchy elasticity, would add a new inhomogeneous family, and would extend the known Cauchy/hyperelastic universality equivalence to fiber-reinforced solids. The derivation of the universality constraints and the explicit candidate deformation families are credible, and Proposition 4.1 appears sound. However, the central completeness claim (Proposition 3.7) rests on an invalid mean-curvature identity, and the classification is therefore not proven as written.

major comments (2)
  1. [§3.2.3, Eq. (3.79)] The formula H = −1/2(a,X·n,X + a,Y·n,Y) is not the mean curvature for a general parametrization. The correct expression is H = (eG − 2fF + gE)/(2(EG − F²)), and the displayed quantity equals (e+g)/2, which coincides with H only when E = G = 1 and F = 0. The constraints (3.68), (3.70), and (3.71) do not force isothermal coordinates, so Eq. (3.69) shows only that e+g is constant, not that H is constant. Consequently, the exclusion of cones and non-circular cylinders among K = 0 surfaces is unsupported, and the completeness of Families 0Z and Z1 in Proposition 3.7 is not established. This is a load-bearing gap in the central classification claim.
  2. [§3.2.3, Eqs. (3.77)–(3.80) and the cylinder subsection] The surface-classification step is also not valid as written. The list of complete surfaces with K = 0 as 'planes, cylinders, cones' is inaccurate: complete flat surfaces include non-circular cylinders, and a cone with its apex removed is not complete. More importantly, the exclusion of cones and non-circular cylinders relies on the assertion that H is constant, which is not established (see previous comment). In the subsequent cylinder analysis, the paper assumes without proof that the cylinders are circular, coaxial with the z-axis, and that one may take n3 = 0 and a3 = a3(Y) (footnote 11). Even if the mean-curvature issue were repaired, the analysis would cover only a subfamily of the possible cylinder-type surfaces. Thus the reduction to Family Z1 does not follow from the presented arguments.
minor comments (4)
  1. [Proposition 3.7] The proposition states that 'the only universal deformations are those belonging to the Family Z1', which contradicts the earlier statement that Family 0Z homogeneous deformations are universal. The statement should be clarified as 'the only inhomogeneous universal deformations' or should list Family 0Z as well.
  2. [§3.2.2, case (iii)] The deduction that n must be constant uses the condition curl n = 0, which is not shown to hold in this case. The text should either prove this condition or state it as an additional assumption.
  3. [Footnote 11] The 'without loss of generality' reduction to n3 = 0 and a3 = a3(Y) for cylindrical surfaces is not rigorously justified and should be replaced by a precise argument.
  4. [General presentation] There are minor typographical and notational issues, including the unclear phrase 'the referential coordinate Z is the arc length parametrization' in §3.2.3 and some missing parentheses in the displayed constraints (3.92)–(3.96).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the universality derivation is self-contained, and the flagged mean-curvature issue is a mathematical-gap concern rather than a circular step.

full rationale

The paper's central derivation starts from the constitutive representation sigma = T n⊗n + αg + βb + γc, computes the divergence of the stress, and extracts universality constraints by requiring the coefficients of arbitrary response functions to vanish. The straight-fiber classification then solves the resulting overdetermined PDE system (3.66)-(3.73) together with the divergence constraints; Family Z1 appears as an explicit solution family, not as an assumed input. Proposition 4.1 compares the hyperelastic and Cauchy universality constraints term-by-term, so the claimed coincidence is established by direct calculation rather than by importing a conclusion from a prior self-citation. The paper's self-citations, including Yavari (2024a), provide background and analogous results but are not load-bearing: the equivalence proof in Section 4 is carried out within the paper. The reviewer-flagged issue with Eq. (3.79) is a potential correctness or completeness gap, because the mean-curvature formula used assumes an isothermal parametrization that is not established; however, that is not a circularity in the sense of a fitted parameter being renamed as a prediction or a result being equivalent to its input by construction. No circular step can be exhibited, so the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim depends on the standard constitutive representation of constrained Cauchy elasticity, the arbitrary-response-function argument, and the geometric classification of surfaces. There are no fitted parameters; the constants in Family Z1 are solution parameters.

assumptions (3)
  • domain assumption The surfaces normal to the fibers in the deformed configuration are complete.
    Stated in footnote 10 as 'reasonable' and used to apply the classification of complete surfaces of constant Gaussian curvature, thereby excluding cones. This is not derived from the balance laws or constitutive assumptions.
  • domain assumption The response functions alpha, beta, gamma and their derivatives are arbitrary independent functions of the invariants.
    This is the standard universality requirement: constraints must hold for all materials in the class. It is invoked in Section 3.1 when passing from f x n = 0 to the separate constraints (3.11)-(3.15).
  • standard math All deformation fields and response functions are smooth.
    Smoothness is assumed throughout for the geometric arguments and is standard in nonlinear elasticity.

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Pith. "Pith review of On Universal Deformations of Compressible Cauchy Elastic Solids Reinforced by Inextensible Fibers." pith.science (2026). https://pith.science/paper/HETC5ZQ6

@misc{pith2026250611203,
  author       = {Pith},
  title        = {Pith review of: On Universal Deformations of Compressible Cauchy Elastic Solids Reinforced by Inextensible Fibers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HETC5ZQ6}},
  note         = {Machine review of arXiv:2506.11203}
}
read the original abstract

Universal deformations are those that can be maintained in the absence of body forces and with boundary tractions alone, for all materials within a given constitutive class. We study the universal deformations of compressible isotropic Cauchy elastic solids reinforced by a single family of inextensible fibers. We consider straight fibers parallel to the Cartesian Z-axis in the reference configuration and derive the associated universality constraints, which depend explicitly on the geometry of the deformed fibers. We study universal deformations in two cases: (i) deformed fibers are straight lines, and (ii) deformed fibers have non-vanishing curvature. For case (i), we provide a complete classification. The universality constraints reduce to geometric restrictions on the orthogonal surfaces, which must be planes, circular cylinders, or spheres. This gives one inhomogeneous universal deformation family: the non-isochoric Family Z1 of combined bending and stretching deformations. In addition, Family 0Z consists of homogeneous deformations that respect the inextensibility constraint. We further show that if all principal invariants are constant and deformed fibers remain straight, then only homogeneous universal deformations are possible. For case (ii), when deformed fibers have non-vanishing curvature, the universality constraints become significantly more complex. The existence of universal deformations in this case remains an open problem. In particular, we demonstrate that Family 5 universal deformations of incompressible elasticity, when restricted to satisfy the inextensibility constraint, are no longer universal in fiber-reinforced solids. Finally, we prove that the universal deformations of Cauchy and hyperelastic solids with the same fiber reinforcement coincide.

Figures

Figures reproduced from arXiv: 2506.11203 by the authors.

Figure 1
Figure 1. A universal deformation from Family Z1 combining bending about the z-axis and uniaxial stretch along the z-axis. (a) Undeformed configuration: a rectangular block with inextensible fibers parallel to the Z-axis. (b) Deformed configuration (α0 = 1.5, β0 = 0, θ0 = 1.5, z0 = 1, k1 = 1.5, A = B = C = 1): horizontal planes Z = const. are bent into concentric circular cylinders centered on the z-axis, and material lines a… view at source ↗

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Works this paper leans on

70 extracted references · 59 canonical work pages

  1. [1]

    J. E. Adkins. Finite plane deformation of thin elastic sheets reinforced with inextensible cords. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 249 0 (961): 0 125--150, 1956

    work page 1956

  2. [2]

    J. E. Adkins and R. S. Rivlin. Large elastic deformations of isotropic materials X . R einforcement by inextensible cords. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 248 0 (944): 0 201--223, 1955

    work page 1955

  3. [3]

    V. I. Arnold. Mathematical Methods of Classical Mechanics, volume 60 of Graduate Texts in Mathematics. Springer, New York, 2nd edition, 1989. ISBN 978-0-387-96890-2. doi:10.1007/978-1-4757-2063-1

    doi:10.1007/978-1-4757-2063-1 1989

  4. [4]

    M. F. Beatty. General solutions in the equilibrium theory of inextensible elastic materials. Acta Mechanica, 29 0 (1): 0 119--126, 1978

    work page 1978

  5. [5]

    M. F. Beatty. A class of universal relations for constrained, isotropic elastic materials. Acta Mechanica, 80 0 (3): 0 299--312, 1989

    work page 1989

  6. [6]

    M. Berger. A Panoramic View of Riemannian Geometry. Springer, 2003. ISBN 978-3-540-65317-2. doi:10.1007/978-3-642-18245-7

    doi:10.1007/978-3-642-18245-7 2003

  7. [7]

    D. E. Beskos. Universal solutions for fiber-reinforced compressible isotropic elastic materials. Journal of Elasticity, 2 0 (3): 0 153--168, 1972

    work page 1972

  8. [8]

    D. E. Beskos. Universal solutions for fiber-reinforced incompressible isotropic elastic materials. International Journal of Solids and Structures, 9 0 (4): 0 553--567, 1973

    work page 1973

Show all 70 references
  1. [9]

    J. A. Blume. Compatibility conditions for a left C auchy- G reen strain field. Journal of Elasticity, 21 0 (3): 0 271--308, 1989

    1989

  2. [10]

    J.-P. Boehler. On irreducible representations for isotropic scalar functions. Zeitschrift f \"u r Angewandte Mathematik und Mechanik , 57 0 (6): 0 323--327, 1977

    1977

  3. [11]

    B \"o hm, J

    C. B \"o hm, J. Korelc, B. Hudobivnik, A. Kraus, and P. Wriggers. Mixed virtual element formulations for incompressible and inextensible problems. Computational Mechanics, 72 0 (6): 0 1141--1174, 2023

    2023

  4. [12]

    E. Cartan. Leçons sur la géométrie des espaces de Riemann. Gauthier-Villars, Paris, 1928. Reprinted by Éditions Jacques Gabay, 1988

    1928

  5. [13]

    A.-L. Cauchy. Sur les \'e quations qui expriment les conditions d' \'e quilibre ou les lois du mouvement int \'e rieur d'un corps solide, \'e lastique ou non \'e lastique. Exercises de Math \'e matiques , 3: 0 160--187, 1828

  6. [14]

    H. Chi, C. Talischi, O. Lopez-Pamies, and G. H. Paulino. Polygonal finite elements for finite elasticity. International Journal for Numerical Methods in Engineering, 101 0 (4): 0 305--328, 2015

    2015

  7. [15]

    M. P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs, NJ, 1976. ISBN 9780132125895

    1976

  8. [16]

    T. C. Doyle and J. L. Ericksen. Nonlinear elasticity. Advances in Applied Mechanics, 4: 0 53--115, 1956

    1956

  9. [17]

    E. Dragoni. The radial compaction of a hyperelastic tube as a benchmark in compressible finite elasticity. International Journal of Non-Linear Mechanics, 31 0 (4): 0 483--493, 1996

    1996

  10. [18]

    Erdemir and M

    E. Erdemir and M. M. Carroll. Finite deformations and motions of radially inextensible hollow spheres. Journal of Elasticity, 88 0 (3): 0 193--205, 2007

    2007

  11. [19]

    J. L. Ericksen. Deformations possible in every isotropic, incompressible, perfectly elastic body. Zeitschrift f \"u r Angewandte Mathematik und Physik (ZAMP) , 5 0 (6): 0 466--489, 1954

    1954

  12. [20]

    J. L. Ericksen. Deformations possible in every compressible, isotropic, perfectly elastic material. Journal of Mathematics and Physics, 34 0 (1-4): 0 126--128, 1955

    1955

  13. [21]

    R. L. Fosdick. Remarks on C ompatibility. Modern Developments in the Mechanics of Continua, pages 109--127, 1966

    1966

  14. [22]

    Goodbrake, A

    C. Goodbrake, A. Yavari, and A. Goriely. The anelastic E ricksen problem: U niversal deformations and universal eigenstrains in incompressible nonlinear anelasticity. Journal of Elasticity, 142 0 (2): 0 291--381, 2020

    2020

  15. [23]

    A. Goriely. The Mathematics and Mechanics of Biological Growth. Springer, 2017

    2017

  16. [24]

    M. E. Gurtin. The linear theory of elasticity. In Handbuch der Physik, Band VIa/2. Springer-Verlag, Berlin, 1972

    1972

  17. [25]

    B. C. Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, volume 222 of Graduate Texts in Mathematics. Springer, New York, 2nd edition, 2015. ISBN 978-3-319-13466-7. doi:10.1007/978-3-319-13467-4

    2015 doi

  18. [26]

    F. W. Hehl and Y. N. Obukhov. Foundations of Classical Electrodynamics: Charge, Flux, and Metric, volume 33 of Progress in Mathematical Physics. Birkh \"a user, 2003. ISBN 978-0-8176-4222-8. doi:10.1007/978-1-4612-0051-2. URL https://doi.org/10.1007/978-1-4612-0051-2

    2003 doi

  19. [27]

    W. W. Klingbeil and R. T. Shield. On a class of solutions in plane finite elasticity. Zeitschrift f \"u r angewandte Mathematik und Physik ZAMP , 17 0 (4): 0 489--511, 1966

    1966

  20. [28]

    J. E. Marsden and T. J. R. Hughes. Mathematical Foundations of Elasticity. Dover Publications, New York, 1994

    1994

  21. [29]

    A. J. A. Morgan. Some properties of media defined by constitutive equations in implicit form. International Journal of Engineering Science, 4 0 (2): 0 155--178, 1966

    1966

  22. [30]

    Motaghian

    S. Motaghian. A note on the universal deformations of incompressible isotropic bodies. Journal of Elasticity, 157 0 (51), 2025. doi:10.1007/s10659-025-10141-9. URL https://link.springer.com/article/10.1007/s10659-025-10141-9

    2025 doi

  23. [31]

    R. W. Ogden. Non-Linear Elastic Deformations. Dover, 1984

    1984

  24. [32]

    A. C. Pipkin. Generalized plane deformations of ideal fiber-reinforced materials. Quarterly of Applied Mathematics, 32 0 (3): 0 253--263, 1974

    1974

  25. [33]

    A. C. Pipkin. Stress analysis for fiber-reinforced materials. In Advances in applied mechanics, volume 19, pages 1--51. Elsevier, 1979

    1979

  26. [34]

    A. C. Pipkin. Some developments in the theory of inextensible networks. Quarterly of Applied Mathematics, 38 0 (3): 0 343--355, 1980

    1980

  27. [35]

    A. C. Pipkin and T. G. Rogers. Plane deformations of incompressible fiber-reinforced materials. 1971

    1971

  28. [36]

    K. R. Rajagopal. On implicit constitutive theories. Applications of Mathematics, 48: 0 279--319, 2003

    2003

  29. [37]

    K. R. Rajagopal. The elasticity of elasticity. Zeitschrift f \"u r angewandte Mathematik und Physik , 58: 0 309--317, 2007

    2007

  30. [38]

    R. S. Rivlin. Large elastic deformations of isotropic materials IV . F urther developments of the general theory. Philosophical Transactions of the Royal Society of London A, 241 0 (835): 0 379--397, 1948

    1948

  31. [39]

    R. S. Rivlin. Large elastic deformations of isotropic materials. V . the problem of flexure. Proceedings of the Royal Society of London A, 195 0 (1043): 0 463--473, 1949 a

    1949

  32. [40]

    R. S. Rivlin. A note on the torsion of an incompressible highly elastic cylinder. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 45, pages 485--487. Cambridge University Press, 1949 b

    1949

  33. [41]

    R. S. Rivlin. Plane strain of a net formed by inextensible cords. Journal of Rational Mechanics and Analysis, 4: 0 951--974, 1955

    1955

  34. [42]

    R. S. Rivlin and J. L. Ericksen. Stress-deformation relations for isotropic materials. Journal of Rational Mechanics and Analysis, 4: 0 323--425, 1955

    1955

  35. [43]

    R. S. Rivlin and D. W. Saunders. Large elastic deformations of isotropic materials VII . E xperiments on the deformation of rubber. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 243 0 (865): 0 251--288, 1951

    1951

  36. [44]

    Saccomandi

    G. Saccomandi. Universal solutions and relations in finite elasticity. In Topics in finite elasticity, pages 95--130. Springer, 2001

    2001

  37. [45]

    Saccomandi and M

    G. Saccomandi and M. F. Beatty. Universal relations for fiber-reinforced elastic materials. Mathematics and Mechanics of Solids, 7 0 (1): 0 95--110, 2002

    2002

  38. [46]

    R. T. Shield. The rotation associated with large strains. SIAM Journal on Applied Mathematics, 25 0 (3): 0 483--491, 1973

    1973

  39. [47]

    M. F. Shojaei and A. Yavari. Compatible-strain mixed finite element methods for incompressible nonlinear elasticity. Journal of Computational Physics, 361: 0 247--279, 2018

    2018

  40. [48]

    J. C. Simo and J. E. Marsden. On the rotated stress tensor and the material version of the D oyle- E ricksen formula. Archive for Rational Mechanics and Analysis, 86 0 (3): 0 213--231, 1984

    1984

  41. [49]

    Singh and A

    M. Singh and A. C. Pipkin. Note on E ricksen's problem. Zeitschrift f \"u r angewandte Mathematik und Physik ZAMP , 16 0 (5): 0 706--709, 1965

    1965

  42. [50]

    Sternberg

    S. Sternberg. Semi-Riemannian Geometry and General Relativity. Harvard University, Department of Mathematics, 2003. Available online: https://people.math.harvard.edu/ shlomo/docs/semi_riemannian_geometry.pdf

    2003

  43. [51]

    E. B. Tadmor, R. E. Miller, and R. S. Elliott. Continuum Mechanics and Thermodynamics: From Fundamental Concepts to Governing Equations. Cambridge University Press, 2012

    2012

  44. [52]

    Truesdell

    C. Truesdell. The mechanical foundations of elasticity and fluid dynamics. Journal of Rational Mechanics and Analysis, 1 0 (1): 0 125--300, 1952

    1952

  45. [53]

    Truesdell

    C. Truesdell. The Elements of Continuum Mechanics. Springer-Verlag, 1966

    1966

  46. [54]

    Truesdell and W

    C. Truesdell and W. Noll. The Non-Linear Field Theories of Mechanics. Springer, 2004

    2004

  47. [55]

    C. C. Wang. On representations for isotropic functions: P art I . I sotropic functions of symmetric tensors and vectors. Archive for Rational Mechanics and Analysis, 33: 0 249--267, 1969

    1969

  48. [56]

    Wriggers, J

    P. Wriggers, J. Schr \"o der, and F. Auricchio. Finite element formulations for large strain anisotropic material with inextensible fibers. Advanced Modeling and Simulation in Engineering Sciences, 3: 0 1--18, 2016

    2016

  49. [57]

    A. Yavari. Compatibility equations of nonlinear elasticity for non-simply-connected bodies. Archive for Rational Mechanics and Analysis, 209 0 (1): 0 237--253, 2013

    2013

  50. [58]

    A. Yavari. Universal deformations in inhomogeneous isotropic nonlinear elastic solids. Proceedings of the Royal Society A, 477 0 (2253): 0 20210547, 2021

    2021

  51. [59]

    A. Yavari. Universal deformations and inhomogeneities in isotropic C auchy elasticity. Proceedings of the Royal Society A, 480 0 (2277): 0 20240229, 2024 a

    2024

  52. [60]

    A. Yavari. Universal displacements in inextensible fiber-reinforced linear elastic solids. Mathematics and Mechanics of Solids, 29 0 (6): 0 2224--2248, June 2024 b . doi:10.1177/10812865231179279

    2024 doi

  53. [61]

    Yavari and A

    A. Yavari and A. Goriely. Riemann-- C artan geometry of nonlinear dislocation mechanics. Archive for Rational Mechanics and Analysis, 205 0 (1): 0 59--118, 2012

    2012

  54. [62]

    Yavari and A

    A. Yavari and A. Goriely. The anelastic E ricksen problem: U niversal eigenstrains and deformations in compressible isotropic elastic solids. Proceedings of the Royal Society A, 472 0 (2196): 0 20160690, 2016

    2016

  55. [63]

    Yavari and A

    A. Yavari and A. Goriely. Universal deformations in anisotropic nonlinear elastic solids. Journal of the Mechanics and Physics of Solids, 156: 0 104598, 2021

    2021

  56. [64]

    Yavari and A

    A. Yavari and A. Goriely. Universality in anisotropic linear anelasticity. Journal of Elasticity, 150 0 (2): 0 241--259, 2022 a

    2022

  57. [65]

    Yavari and A

    A. Yavari and A. Goriely. The universal program of nonlinear hyperelasticity. Journal of Elasticity, pages 1--56, 2022 b

    2022

  58. [66]

    Yavari and A

    A. Yavari and A. Goriely. The universal program of linear elasticity. Mathematics and Mechanics of Solids, 2022 c

    2022

  59. [67]

    Yavari and A

    A. Yavari and A. Goriely. Controllable deformations in compressible isotropic implicit elasticity. Zeitschrift f \"u r angewandte Mathematik und Physik , 75 0 (5): 0 169, 2024 a

    2024

  60. [68]

    Yavari and A

    A. Yavari and A. Goriely. Nonlinear C auchy elasticity. arXiv preprint arXiv:2412.17090, 2024 b

    2024 arXiv

  61. [69]

    Yavari and D

    A. Yavari and D. Sfyris. Universal displacements in anisotropic linear C auchy elasticity. Journal of Elasticity, 157 0 (1): 0 1--15, 2025

    2025

  62. [70]

    Yavari, C

    A. Yavari, C. Goodbrake, and A. Goriely. Universal displacements in linear elasticity. Journal of the Mechanics and Physics of Solids, 135: 0 103782, 2020

    2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.