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Existence of solutions for the anti-plane stress for a new class of “strain-limiting” elastic bodies

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Abstract

The main purpose of this study is to establish the existence of a weak solution to the anti-plane stress problem on V-notch domains for a class of recently proposed new models that could describe elastic materials in which the stress can increase unboundedly while the strain yet remains small. We shall also investigate the qualitative properties of the solution that is established. Although the equations governing the deformation that are being considered share certain similarities with the minimal surface problem, the boundary conditions and the presence of an additional model parameter that appears in the equation and its specific range makes the problem, as well as the result, different from those associated with the minimal surface problem.

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Notes

  1. Galileo refers to Aristotle’s interest in the problem.

  2. There have been additional approaches to modeling fracture by introducing additional forces and additional balance laws. Unfortunately, many of the quantities introduced in such theories have never been measured and the need to introduce such additional quantities and an additional balance law are far from compelling. As no specific boundary value problem has been solved within the context of such recondite theories one can not even evaluate the usefulness of such theories even from the point of view of merely carrying out a parametric study.

  3. Beatty and Hayes have studied an elastic material that obeys the Bell constraint in great detail (see [24]).

  4. The various interpretations of what was considered as an elastic material has been discussed in a review article titled ”Conspectus of concepts of elasticity” by Rajagopal [38].

  5. Naturally, \(\varvec{g}\) has to fulfil the necessary condition guaranteeing that the body can take an equilibrium state. If \(B\) is bounded, then (2.1) and (2.2) imply, by applying Gauss’ theorem to (2.1) that \(\int _{\partial B} \varvec{g} dS = \varvec{0}\).

  6. Note that by Gauss’ theorem we observe that \(g\) has to meet the condition \(\int _{\partial \Omega } g \, dS = 0\).

  7. See also the papers by Nečas [31, 32] and Stará [49] that can serve as the source for the complete proofs on which our proof is based.

  8. Here, we use the abbreviation \(D_{ij}:=\frac{\partial ^2}{\partial x_j \partial x_i}\).

  9. Note that in case of minimal surface equation, one does not need to assume the uniform convexity, due to the ability of finding proper barriers. However, in the setting studied in this paper we do not know yet, in particular for \(a<1\), whether such barriers exist.

References

  1. Antipov, Y.A., Schiavone, P.: Integro-differential equation for a finite crack in a strip with surface effects. Quart. J. Mech. Appl. Math. 64, 87–106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beatty, M.F., Hayes, M.A.: Deformations of an elastic internally constrained materials. 1. homogeneous deformation. J. Elast. 29, 1–84 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beatty, M.F., Hayes, M.A.: Deformations of an elastic internally constrained materials. 2. non-homogeneous deformation. Q. J. Appl. Math. 45, 663–709 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beatty, M.F., Hayes, M.A.: Deformations of an elastic internally constrained materials. 3. small superimposed deformations and waves. Zeitschrift fur Angewandte Mathematick und Physik 46, 72–106 (1995)

  5. Bell, J.F.: Contemporary perspectives in finite strain plasticity. Int. J. Plast. 1, 3–27 (1985)

    Article  MATH  Google Scholar 

  6. Bell, J.F.: Experiments in the kinematics of large plastic strain in ordered materials. Int. J. Solids Struct. 25, 268–278 (1989)

    Article  Google Scholar 

  7. Bildhauer, M.: A priori gradient estimates for bounded generalized solutions of a class of variational problems with linear growth. J. Convex Anal. 9, 117–137 (2002)

    MathSciNet  MATH  Google Scholar 

  8. Bildhauer, M., Fuchs, M.: Relaxation of convex variational problems with linear growth defined on classes of vector-valued functions. Algebra i Analiz 14, 26–45 (2002)

    MathSciNet  Google Scholar 

  9. Broberg, K.B.: Cracks and fracture. Academic Press, San Diego (1999)

    Google Scholar 

  10. Bustamante, R., Rajagopal, K.R.: Solutions of some simple boundary value problems within the context of a new class of elastic materials. Int. J. Non-Linear Mech. 46, 376–386 (2011)

    Article  Google Scholar 

  11. Cherepanov, G.P.: Fracture. Krieger Publishing Company, Malabar (1998)

    MATH  Google Scholar 

  12. De Giorgi, E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3(3), 25–43 (1957)

  13. Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.: Minimal surfaces. I, vol. 295 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Boundary value problems. Springer, Berlin (1992)

  14. Dierkes, U., Hildebrandt, S., Küster, A., Wohlrab, O.: Minimal surfaces. II, vol. 296 of Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Boundary regularity. Springer, Berlin (1992)

  15. Finn, R.: Remarks relevant to minimal surfaces, and to surfaces of prescribed mean curvature. J. Anal. Math. 14, 139–160 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  16. Galilei, G.: Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla mecanica ed ei movimenti locali, Dover, New York, 1954. original published in 1638. Engl. transl. “Dialogue concerning two new sciences”

  17. Giusti, E.: Minimal surfaces and functions of bounded variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)

  18. Giusti, E.: Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge (2003)

    Book  MATH  Google Scholar 

  19. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kaninnen, M.F., Popelar, C.H.: Advanced fracture mechanics. Oxford University Press, New York (1985)

    Google Scholar 

  21. Kim, C.I., Schiavone, P., Ru, C.-Q.: Analysis of a mode-III crack in the presence of surface elasticity and a prescribed non-uniform surface traction. Zeitschrift fur Angewandte Mathematik und Physik 61, 555–564 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kim, C.I., Schiavone, P., Ru, C.-Q.: The effects of surface elasticity on an elastic solid with mode-III crack: complete solution. J. Appl. Mech. 77 (2010)

  23. Kim, C.I., Schiavone, P., Ru, C.-Q.: Analysis of plane-strain crack problems (mode-i and mode-ii) in the presence of surface elasticity. J. Elast. 104, 397–420 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kim, C.I., Schiavone, P., Ru, C.-Q.: Effect of surface elasticity on an interface crack in plane deformations. Proc. R. Soc. A 467, 3530–3549 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Knowles, J.K.: The finite anti-plane shear field near the tip of the crack for a class of incompressible elastic solids. Int. J. Fract. 13, 611–639 (1997)

    Article  MathSciNet  Google Scholar 

  26. Kulvait, V., Málek, J., Rajagopal, K.R.: Anti-plane stress state of a plate with a V-notch for a new class of elastic solids. Int. J. Fract. 179, 59–73 (2013)

    Article  Google Scholar 

  27. Li, T., Morris Jr., J.W., Nagasako, N., Kuramoto, S., Chrzan, D.C.: Ideal engineering alloys. Phys. Rev. Lett. 98 (2007)

  28. Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Nadai, A.: Theory of flow and fracture of solids. McGraw-Hill, New York (1950)

    Google Scholar 

  30. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nečas, J.: Sur la régularité des solutions variationelles des équations elliptiques non-linéaires d’ordre \(2k\) en deux dimensions. Ann. Scuola Norm. Sup. Pisa 21(3), 427–457 (1967)

  32. Nečas, J.: Sur la régularité des solutions faibles des équations elliptiques non linéaires. Comment. Math. Univ. Carolinae 9, 365–413 (1968)

    MathSciNet  MATH  Google Scholar 

  33. Nitsche, J.C.C.: On the non-solvability of Dirichlet’s problem for the minimal surface equation. J. Math. Mech. 14, 779–788 (1965)

    MathSciNet  MATH  Google Scholar 

  34. Oh, E.S., Walton, J.R., Slattery, J.C.: A theory of fracture based upon an extension of continuum mechanics to the nanoscale. J. Appl. Mech. 73, 792–798 (2006)

    Article  MATH  Google Scholar 

  35. Rajagopal, K.R.: On the nonlinear elastic response of bodies in the small strain range. Acta Mechanica 225, 1545–1553 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rajagopal, K.R.: On implicit constitutive theories. Appl. Math. 48, 279–319 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Rajagopal, K.R.: Elasticity of elasticity. Zeitschrift fur Angewandte Math Phys 58, 309–417 (2007)

    Article  MATH  Google Scholar 

  38. Rajagopal, K.R.: Conspectus of concepts of elasticity. Math. Mech. Solids 16, 536–562 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rajagopal, K.R.: Non-linear elastic bodies exhibiting limiting small strain. Math. Mech. Solids 16, 122–139 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rajagopal, K.R.: On a new class of models in elasticity. J. Math. Comp. Appl. 15, 506–528 (2011)

    Google Scholar 

  41. Rajagopal, K.R., Srinivasa, A.R.: On the response of non-dissipative solids. Proc. R. Soc. Lond. A 463, 357–367 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Rajagopal, K.R., Srinivasa, A.R.: On a class of non-dissipative materials that are not hyperelastic. Proc. R. Soc. Lond. A 465, 493–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rajagopal, K.R., Tao, L.: On the response of non-dissipative solids. Commun. Nonlinear Sci. Numer. Simul. 13, 1089–1100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rajagopal, K.R., Walton, J.: Modeling fracture in the context of strain-limiting theory of elasticity. Int. J. Fract. 169, 39–48 (2011)

    Article  MATH  Google Scholar 

  45. Saito, T., Furuta, T., Hwang, J.-H., Kuramoto, S., Nishino, K., Suzuki, N., Chen, R., Yamada, A., Ito, K., Seno, Y., Nonaka, T., Ikehata, H., Nagasako, N., Iwamoto, C., Ikuhara, Y., Sakuma, T.: Multifunctional alloys obtained via a dislocation-free plastic deformation mechanism. Science 300, 464–467 (2003)

    Article  Google Scholar 

  46. Sendova, T., Walton, J.R.: A new approach to the modeling and analysis of fracture through extension of continuum mechanics to the nanoscale. Math. Mech. Solids 15, 368–413 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  47. Slattery, J.C., Sagis, L., Oh, E.-S.: Interfacial transport phenomena. Springer, Berlin (2007)

    MATH  Google Scholar 

  48. Souček, V.: The nonexistence of a weak solution of Dirichlet’s problem for the functional of minimal surface on nonconvex domains. Comment. Math. Univ. Carolinae 12, 723–736 (1971)

    MathSciNet  MATH  Google Scholar 

  49. Stará, J.: Regularity results for non-linear elliptic systems in two dimensions. Ann. Scuola Norm. Sup. Pisa 25(3), 163–190 (1971)

  50. Sternberg, E., Knowles, J.K.: Finite-deformation analysis of elastostatic field near tip of a crack-reconsideration and higher order results. J. Elast. 4, 201–233 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  51. Sternberg, E., Knowles, J.K.: Failure of ellipticity and emergence of discontinuous deformations gradients in plane finite elastostatics. J. Elast. 8, 329–379 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  52. Sternberg, E., Knowles, J.K.: Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear-example. J. Elast. 10, 81–110 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sternberg, E., Knowles, J.K.: Anti-plane shear fields with discontinueous deformation gradients near the tip of a crack in finite elastostatics. J. Elast. 11, 129–164 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  54. Talling, R.J., Dashwood, R.J., Jackson, M., Kuramoto, S., Dye, D.: Determination of (c11–c12) in ti-36nb-2ta-3zr-0.3o (wt.%) (gum metal). Scripta Materialia 59, 669–672 (2008)

    Article  Google Scholar 

  55. Tarantino, A.M.: Nonlinear fracture mechanics for an elastic Bell material. Quart. J. Mech. Appl. Math. 50, 436–456 (1997)

    Article  MathSciNet  Google Scholar 

  56. Walton, J.: A note on fracture models incorporating surface elasticity. J. Elast. 1–8 (2011)

  57. Withey, E., Jin, M., Minor, A., Kuramoto, S., Chrzan, D.C., Morris Jr, J.W.: The deformation of “gum metal” in nanoindentation. Mater. Sci. Eng. A 493, 26–32 (2008)

    Article  Google Scholar 

  58. Zhang, S.Q., Li, S.J., Jia, M.T., Hao, Y.L., Yang, R.: Fatigue properties of a multifunctional titanium alloy exhibiting nonlinear elastic deformation behavior. Scripta Materialia 60, 733–736 (2009)

    Article  Google Scholar 

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Correspondence to Josef Málek.

Additional information

Communicated by Y. Giga.

M. Bulíček and J. Málek acknowledge the support of the project LL1202 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic. M. Bulíček is a researcher in the Charles University Centre for Mathematical Modelling, Applied Analysis and Computational Mathematics (Math MAC). K. R. Rajagopal thanks the National Science Foundation and the Office of Naval Research for their support of his work. He and J. R. Walton acknowledge support by Award No. KUS-C1-016-04 from King Abdullah University of Science and Technology.

We are thankful to Emilio Acerbi for several valuable comments and suggestions.

Appendix

Appendix

1.1 Derivation of the uniform estimate (4.3)

We shall prove that there is a \(C>0\) independent of \(\delta \) such that

$$\begin{aligned} \Vert \nabla U^{\delta }\Vert _{1+\delta } \le C\Vert \nabla \tilde{U}_0 \Vert _{1+\delta } + C\,. \end{aligned}$$
(8.1)

We provide two different proofs.

First Proof of (8.1) In what follows, we write \(U\) instead of \(U^{\delta }\). We also use frequently the elementary inequality

$$\begin{aligned} (1+ x^a)^{\frac{1}{a}} \le c(a) (1+x), \quad \text { where } c(a) = {\left\{ \begin{array}{ll} 1 &{}\text { if } a\ge 1 \\ 2^{\frac{1}{a} - 1} &{}\text { if } a\in (0,1) \end{array}\right. }\,. \end{aligned}$$
(8.2)

Multiplying (4.1) by \(U - \tilde{U}_0\) and integrating the result over \(\Omega \), we arrive at

$$\begin{aligned} 0= \left( \frac{\nabla U}{(1+|\nabla U|^a)^{\frac{1-\delta }{a}}}, \nabla (U - \tilde{U}_0) \right) , \end{aligned}$$

which leads to

$$\begin{aligned} I:= & {} \int _{\Omega } \frac{|\nabla U|^2}{(1+|\nabla U|^a)^{\frac{1-\delta }{a}}} = \int _{\Omega } \frac{\nabla U \cdot \nabla \tilde{U}_{0}}{(1+|\nabla U|^a)^{\frac{1-\delta }{a}}} \nonumber \\\le & {} \int _{\Omega } (1+|\nabla U|^a)^{\frac{\delta }{a}} |\nabla \tilde{U}_{0}| \le \Vert \nabla \tilde{U}_0\Vert _{1+\delta } \left( \int _{\Omega } (1 + |\nabla U|^a)^{\frac{1+\delta }{a}} \right) ^{\frac{\delta }{1+\delta }} \nonumber \\\le & {} [c(a)]^{\delta } \Vert \nabla \tilde{U}_0\Vert _{1+\delta } \Vert 1+|\nabla U| \, \Vert _{1+\delta }^{\delta }. \end{aligned}$$
(8.3)

Next,

$$\begin{aligned} \Vert \nabla U \Vert _{1+\delta }^{1+\delta }= & {} \int _{\Omega } |\nabla U|^{1+\delta } = \int _{\Omega } \left( \frac{|\nabla U|^2}{(1+ |\nabla U|^a)^{\frac{1-\delta }{a}}} \right) ^{\frac{1+\delta }{2}} (1+ |\nabla U|^a)^{\frac{1-\delta }{a}\frac{1+\delta }{2}} \nonumber \\\le & {} I^{\frac{1+\delta }{2}} \left( \int _{\Omega } (1+ |\nabla U|^a)^{\frac{1+\delta }{a}} \right) ^{\frac{1-\delta }{2}}. \end{aligned}$$
(8.4)

Using (8.3) and (8.4) we then get

$$\begin{aligned} \Vert \nabla U\Vert _{1+\delta }\le & {} I^{1/2} \left( \int _{\Omega }(1+ |\nabla U|^a)^{\frac{1+\delta }{a}} \right) ^{\frac{1-\delta }{2(1+\delta )}} \le I^{1/2} (c(a))^{\frac{1-\delta }{2}} \Vert 1+|\nabla U| \, \Vert _{1+\delta }^{\frac{1-\delta }{2}} \nonumber \\\le & {} (c(a))^{\frac{1}{2}} \Vert \nabla \tilde{U}_0\Vert _{1+\delta }^{\frac{1}{2}} \left( \int _{\Omega } (1+ |\nabla U|)^{1+\delta } \right) ^{\frac{1}{2(1+\delta )}} \nonumber \\\le & {} 2^{\frac{\delta }{2(1+\delta )}} (c(a))^{\frac{1}{2}} \Vert \nabla \tilde{U}_0\Vert _{1+\delta }^{\frac{1}{2}} (|\Omega | + \Vert \nabla U\Vert _{1+\delta }^{1+\delta } )^{\frac{1}{2(1+\delta )}} \nonumber \\\le & {} 2 (c(a))^{\frac{1}{2}} \max (1,|\Omega |)^{\frac{1}{2(1+\delta )}} \Vert \nabla \tilde{U}_0\Vert _{1+\delta }^{\frac{1}{2}} (1+ \Vert \nabla U\Vert _{1+\delta }^{1+\delta })^{\frac{1}{2(1+\delta )}} \nonumber \\\le & {} 2 (c(a))^{\frac{1}{2}} \Vert \nabla \tilde{U}_0\Vert _{1+\delta }^{\frac{1}{2}} \max (1,|\Omega |)^{\frac{1}{2(1+\delta )}} (1+ \Vert \nabla U\Vert _{1+\delta })^{1/2} \nonumber \\=: & {} \sqrt{C} \Vert \nabla \tilde{U}_0\Vert _{1+\delta }^{\frac{1}{2}} (1+ \Vert \nabla U\Vert _{1+\delta })^{1/2} . \end{aligned}$$
(8.5)

Use of the inequality \(AB \le \tfrac{1}{2} A^2 + \tfrac{1}{2} B^2\) completes the proof of (8.1).

Second Proof of (8.1) Defining

$$\begin{aligned} G(s):=\frac{1}{2} \int _0^s \frac{1}{(1+t^{\frac{a}{2}})^{\frac{1-\delta }{a}}} \; dt, \quad F(\nabla U):=G(|\nabla U|^2) \end{aligned}$$

and introducing the functional

$$\begin{aligned} J(u):=\int _{\Omega } F(\nabla U) \; dx, \end{aligned}$$

we consider the variational problem: to find \(U\in W^{1,1+\delta }(\Omega )\) such that \(U=\tilde{U}_0\) on \(\partial \Omega \) for which \(J\) attains minimum. Since \(F\) is strictly convex and coercive there is a unique minimum which is the only solution of (3.1)–(3.2). Consequently, we have

$$\begin{aligned} \int _{\Omega }F(\nabla U)\; dx \le \int _{\Omega } F(\nabla \tilde{U}_0)\; dx. \end{aligned}$$

Since

$$\begin{aligned} \frac{2^{\frac{\delta -1+a}{a}}}{1+\delta }(s^{\frac{1+\delta }{2}}-1)&\le \int _0^s \frac{1}{(2t^{\frac{a}{2}})^{\frac{1-\delta }{a}}}\chi _{\{t>1\}}+ \frac{1}{2^{\frac{1-\delta }{a}}}\chi _{\{t\le 1\}} \; dt\le 2G(s) \\&=\int _0^s \frac{1}{(1+t^{\frac{a}{2}})^{\frac{1-\delta }{a}}} \; dt \le \frac{2}{1+\delta }s^{\frac{1+\delta }{2}}, \end{aligned}$$

we observe that the minimizer \(U (= U^{\delta })\) fulfills

$$\begin{aligned} \int _{\Omega }\frac{2^{\frac{\delta -1+a}{a}}}{1+\delta }(|\nabla U|^{1+\delta }-1) \le \int _{\Omega }2F(\nabla U)\; dx \le \int _{\Omega } 2F(\nabla \tilde{U}_0)\; dx\le \int _{\Omega }\frac{2}{1+\delta }|\nabla \tilde{U}_0|^{1+\delta }, \end{aligned}$$

which gives the precise estimate

$$\begin{aligned} \int _{\Omega }|\nabla U|^{1+\delta } \le 2^{\frac{1-\delta }{a}} \int _{\Omega } |\nabla \tilde{U}_0|^{1+\delta }+ |\Omega |. \end{aligned}$$

1.2 Derivation of the inequality (5.12)

Let \(f\ge 0\) be locally Lipschitz, non-decreasing. Setting \(F_1(s) = \int _0^s \frac{f(t) t^{a-1}}{1+t^a}\) and \(F_2(s) = \int _0^s \frac{(f(t) + (1-\delta ) F_1(t)) t}{(1+t^a)^{\frac{1-\delta }{a}}}\), there is a \(C>0\) such that for all \(w\ge 0\)

$$\begin{aligned} F_2(w) \le C(1+ w^{1+\delta } F_1(w)). \end{aligned}$$

Proof If \(w\in [0,1]\) then \(F_2(w) \le C:= \int _0^1 \frac{(f(t) + (1-\delta ) F_1(t)) t}{(1+t^a)^{\frac{1-\delta }{a}}}\). If \(w\ge 1\) we split \(F_2(w)\) in the following way

$$\begin{aligned} F_2(w)&= \int _0^1 \frac{(f(t) + (1-\delta ) F_1(t)) t}{(1+t^a)^{\frac{1-\delta }{a}}} + \int _1^w \frac{(f(t) + (1-\delta ) F_1(t)) t}{(1+t^a)^{\frac{1-\delta }{a}}} \\&= C + \int _1^w \frac{(f(t) + (1-\delta ) F_1(t)) t}{(1+t^a)^{\frac{1-\delta }{a}}}\,. \end{aligned}$$

Further, splitting the last integral into the sum of \(J_1(w)\) and \(J_2(w)\) so that \(F_2(w) = C + J_1(w) + J_2(w)\), whereas

$$\begin{aligned} J_1(w) = \int _1^w \frac{f(t) t}{ (1+ t^a)^{\frac{1-\delta }{a}}} \quad \text { and } J_2(w) = (1-\delta ) \int _1^w \frac{F_1 (t) t}{ (1+ t^a)^{\frac{1-\delta }{a}}}, \end{aligned}$$

we proceed by observing that

$$\begin{aligned} J_1&(w) = \int _1^w \frac{f(t) t}{ 1+ t^a} (1+ t^a)^{\frac{a+ \delta - 1}{a}} \\&{\left\{ \begin{array}{ll} \,\,\, \overset{a\ge 1}{\le } &{} 2^{\frac{a+\delta -1}{a}} \int _1^w \frac{f(t) t^{a-1}}{ 1+ t^a} t^{1+\delta } \le 2^{\frac{a+\delta -1}{a}} \omega ^{1+\delta } F_1(\omega )\,\\ \overset{a\in (0,1)}{\le } &{}\int _1^w \frac{f(t) t^{a-1}}{ 1+ t^a} t (1+ t^a)^{\frac{a+ \delta - 1}{a}} t^{1-a} \le \int _1^w \frac{f(t) t^{a-1}}{1+ t^a} t (1+ t^a)^{\frac{\delta }{a}} \le 2^{\frac{\delta }{a}} \omega ^{1+\delta } F_1(\omega ) \end{array}\right. } . \end{aligned}$$

Also, since \(F_1\) is increasing

$$\begin{aligned} J_2(w)&\le (1-\delta ) F_1(w) \int _1^w \frac{(1+ t^a)^{\frac{1}{a}}}{(1+ t^a)^{\frac{1-\delta }{a}}} \le \frac{2^{\frac{\delta }{a}}}{1+\delta } w^{1+\delta } F_1(w). \end{aligned}$$

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Bulíček, M., Málek, J., Rajagopal, K.R. et al. Existence of solutions for the anti-plane stress for a new class of “strain-limiting” elastic bodies. Calc. Var. 54, 2115–2147 (2015). https://doi.org/10.1007/s00526-015-0859-5

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