In this paper, we study the stress distribution around a circular hole in transversely isotropic elastic bodies, whose material moduli depend on density under small deformations. Two special constitutive relations are considered within the framework of implicit elasticity to describe the nonlinear stress–strain response of such bodies. We studied the stress concentration problem in a thin plate under uniaxial tension. The constitutive relations presented herein exhibit stiffening or softening mechanical response under tension and compression depending on the values of the material parameters. While the stress distribution at the circumference of the hole resembles that of a classical linearized elastic body, the peak value of the normalized hoop stress at the hole is not constant and varies as a function of the applied stress. Furthermore, we perform a parametric study to understand the influence of different material parameters on the stress distribution around the hole.
RajagopalKR.An implicit constitutive relation for describing the small strain response of porous elastic solids whose material moduli are dependent on the density. Math Mech Solids2021; 26(8): 1138–1146.
2.
RajagopalKBustamanteR.Constitutive relations for anisotropic porous solids undergoing small strains whose material moduli depend on the density and the pressure. Int J Eng Sci2024; 195: 104005.
3.
BustamanteRRajagopalKRWinemanAS.Residual stresses for a new class of transversely isotropic nonlinear elastic solid. Proc R Soc A2021; 477(2252): 20210330.
4.
BellJF.Mechanics of solids. Vol. 1: The experimental foundations of solid mechanics. Cham: Springer, 1973.
5.
RivlinRS.Torsion of a rubber cylinder. J Appl Phys1947; 18(5): 444–449.
6.
RivlinR. Large elastic deformations of isotropic materials. I. Fundamental concepts. Philos Trans R Soc London1948; 240(822): 459–490.
7.
RivlinRSSaundersD.Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Philos Trans R Soc London1951; 243(865): 251–288.
8.
RivlinRSaundersD.The free energy of deformation for vulcanized rubber. Trans Faraday Soc1952; 48: 200–206.
9.
TruesdellCToupinR. The classical field theories. In: FlüggeS (ed.) Principles of classical mechanics and field theory. Cham: Springer, 1960, pp. 226–858.
10.
SpencerAJM. Constitutive theory for strongly anisotropic solids. In: SpencerA (ed.) Continuum theory of the mechanics of fibre-reinforced composites. Cham: Springer, 1984, pp. 1–32.
11.
SpencerASoldatosK.Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. Int J Non Linear Mech2007; 42(2): 355–368.
12.
SoldatosKP.A complete set of independent and physically meaningful invariants in the mechanics of solids reinforced by two families of fibres. Mech Res Comm2018; 94: 49–52.
13.
RajagopalK.Conspectus of concepts of elasticity. Math Mech Solids2011; 16(5): 536–562.
14.
RajagopalK.The elasticity of elasticity. Z Angew Math Mech2007; 58: 309–317.
15.
KirschC.Die theorie der elastizitat und die bedurfnisse der festigkeitslehre. Z Ver Dtsch Ing1898; 42: 797–807.
16.
InglisCE.Stresses in a plate due to the presence of cracks and sharp corners. Trans Inst Naval Archit1913; 55: 219–241.
17.
LekhnitskiiSG.Anisotropic plates. New York: Gordon and Breach, Science Publishers, Inc., 1968.
18.
SavinGN.Stress distribution around holes. Washington, DC: NASA, 1968.
19.
TingTCT.Anisotropic elasticity. theory and applications. Oxford: Oxford University Press, 1996.
20.
ColemanBDNollW.Material symmetry and thermostatic inequalities in finite elastic deformations. Arch Rational Mech Anal1964; 5: 87–111.
21.
HogerA.On the residual stress possible in an elastic body with material symmetry. Arch Ration Mech Anal1985; 88: 271–289.
22.
RajagopalKWinemanA.Residual stress and material symmetry. Int J Eng Sci2024; 197: 104013.
23.
NguyenLBeaucourALOrtolaS, et al. Influence of the volume fraction and the nature of fine lightweight aggregates on the thermal and mechanical properties of structural concrete. Constr Build Mater2014; 51: 121–132.
24.
CristescuN.Rock rheology. Singapore: Springer Science & Business Media, 2012.
25.
ManoylovABorodichFMEvansHP.Modelling of elastic properties of sintered porous materials. Proc R Soc A2013; 469(2154): 20120689.
26.
BordiaRKZuoRGuillonO, et al. Anisotropic constitutive laws for sintering bodies. Acta Mater2006; 54(1): 111–118.
27.
MorganEFUnnikrisnanGUHusseinAI.Bone mechanical properties in healthy and diseased states. Annu Rev Biomed Eng2018; 20: 119–143.
28.
MurruPTTorrenceCGrasleyZ, et al. Density-driven damage mechanics (D3-M) model for concrete I: mechanical damage. Int J Pavement Eng2022; 23(4): 1161–1174.
29.
ArumugamJAlagappanPBirdJ, et al. A new constitutive relation to describe the response of bones. Int J Non Linear Mech2024; 161: 104664.
30.
MurruPTRajagopalKR.Stress concentration due to the presence of a hole within the context of elastic bodies. Mater Des Process Commun2021; 3(5): e219.
31.
MurruPTRajagopalK.Stress concentration due to the bi-axial deformation of a plate of a porous elastic body with a hole. Z Angew Math Mech2021; 101(11): e202100103.
32.
VajipeyajulaBMurruPRajagopalKR.Stress concentration due to an elliptic hole in a porous elastic plate. Math Mech Solids2023; 28(3): 854–869.
33.
OftadehRPerez-ViloriaMVilla-CamachoJC, et al. Biomechanics and mechanobiology of trabecular bone: a review. J Biomech Eng2015; 137(1): 010802.
34.
MorganEFBayraktarHHKeavenyTM.Trabecular bone modulus–density relationships depend on anatomic site. J Biomech2003; 36(7): 897–904.
35.
KeavenyTMMorganEFNieburGL, et al. Biomechanics of trabecular bone. Ann Rev Biomed Eng2001; 3(1): 307–333.
36.
MorganEFYehOCChangWC, et al. Nonlinear behavior of trabecular bone at small strains. J Biomech Eng2001; 123(1): 1–9.
37.
MyneniMBenjaminCCRajagopalKR.Stress concentration factors around a circular hole in two fiber reinforced materials under large deformations. Mech Mater2021; 163: 104089.
38.
MyneniMRajagopalKR.Stress concentration around a circular rigid inclusion in incompressible nonlinearly elastic transversely isotropic and orthotropic thin sheets under uniaxial extension. Math Mech Solids2023; 28(1): 394–410.
