Effect of density parameter in a disk made of orthotropic material and rubber
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ORIGINAL PAPER
Effect of density parameter in a disk made of orthotropic material and rubber Pankaj Thakur1 · Neeru Gupta1 · Monika Sethi1 · Kanav Gupta2 Received: 15 January 2020 / Accepted: 19 August 2020 © The Malaysian Rubber Board 2020
Abstract The objective of this paper is to present the study of density parameter in a disk made of orthotropic material and rubber by using Seth’s transition theory. This theory includes classical macroscopic solving problems in plasticity, creep relaxation, and semi-empirical yield conditions. It has been observed that disk made of rubber material requires higher angular speed to yield at the internal surface in comparison to disk made Barytes material at the transition state and for the full-plastic state having density parameter m > 0 . The value of circumferential stress is maximum at the internal surface of the rotating disk made of rubber and Barytes materials having density m > 0, whereas the reverse results obtained for m ≤ 0. Keywords Density · Stresses · Disk · Orthotropic material · Rubber · Angular speed · Yielding List of symbols v Poisson ratio A0 Constant of integration cijkl Elastic constants a, b Inner and outer radii of the disk c Compressibility factor 𝜌 Variable density of material u, v, w Displacement components r, 𝜃, z Radial, circumferential, and axial directions 𝜔 Angular velocity of rotation 𝜌0 Constant density Y, Y ∗ Yield stress 𝜆, 𝜇 Lame’s constant 𝜏ij , 𝜀ij Stress and strain components m Density parameter Ω21 , Ω22 Angular speed for initial and fully plastic stage (orthotropic material) Ω23 , Ω24 Angular speed for initial and fully plastic stage (isotropic material)
* Pankaj Thakur [email protected] 1
Department of Mathematics, ICFAI University Himachal Pradesh, Baddi 174105, India
Department of Physics, ICFAI University Himachal Pradesh, Baddi, India
2
Non dimensional components R = r∕b, R0 = a∕b Radii ratio 𝜎r Radial stress component ( 𝜏rr ∕Y ∗) 𝜎𝜃 Circumferential stress component ( 𝜏𝜃𝜃 ∕Y ∗) / ∗ 2 2 2 Ω = 𝜌0 𝜔 b Y Speed factors
Introduction Theoretical investigation of elasto-plastic deformations in a disk induced by centrifugal forces is an important topic due to its various applications in engineering components. The proposed models in this paper are used in mechanical and electronic devices and have extensive practical engineering applications such as turbo generators, steam turbines, gas turbines, flywheel of internal combustion engines, disk brake, reciprocating engines, centrifugal compressors and disk drive, etc. In this context, numerical investigations have been extensively used to predict the deformation, failure, and stress and strain fields in a uniform rotating disk under different loading conditions. The analytical solution in elastic–plastic deformation in rotating disks for isotropic materials can be found in many textbooks [1–4], and the stress analysis in the curvilinearly orthotropic disk and cylinders under pressure can also be found in the literature [5]. Arya et al. [6] solved t
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