New Mathematical Stress Analysis in the Compressive Stage of the High-Pressure Torsion Process
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New Mathematical Stress Analysis in the Compressive Stage of the High‑Pressure Torsion Process S. Hadi1 · F. Rahimzadeh lotfabad1 · M. H. Paydar1 · R. Ebrahimi1 Received: 7 June 2020 / Accepted: 7 September 2020 © The Korean Institute of Metals and Materials 2020
Abstract In this study, a mathematical analysis based on the slab method is used to drive a formula to calculate the average hydrostatic pressure in the compressive stage of the HPT process. The analytical method used here leads to a relationship for describing the average hydrostatic pressure as a function of the material shear strength, sample volume, cavity volume and force. Based on the results obtained and by employing the Sturm’s theorem for polynomials, a general expression for the minimum force required to start the deformation of the material under compressive force and creation of flash is developed. The analytical results were confirmed by carrying out some experiments by using commercial pure aluminum and copper as test materials. The experimental results showed good congruence with the analytical calculations. The most important outcome of this analysis is that with an appropriate selection of the sample volume, considering the capacity of the HPT press facility and the proper choice of the die cavity volume, favorable average hydrostatic pressure, necessary for the purpose of application can be achieved. Keywords Average hydrostatic pressure · Severe plastic deformation · High pressure torsion · Mathematical analysis · Stress analysis · Slab method analysis List of Symbols F or Ftotal Magnitude of the compressive force applied in the HPT process Ainitial Initial contact area of the material and the die Afinal Final contact area of the material and the die 𝜎̄ Equivalent stress 𝜀̄ Equivalent strain k Material’s shear strength τrz, τrɵ and τzɵ Shear stress components in cylindrical coordinate σr, σɵ and σz Normal stress components in cylindrical coordinate m Constant friction factor Ri Die cavity radius Rf Flash material radius t Thickness of the flash material σH Hydrostatic pressure * R. Ebrahimi [email protected] 1
Department of Materials Science and Engineering, School of Engineering, Shiraz University, Shiraz 7134851154, Iran
𝜎̃ H Average hydrostatic pressure in cavity zone Vs Sample volume Vc Cavity volume Vcritical Critical sample volume Fcavityy and Fflash Applied force to the material in the cavity zone and the flash material respectively
1 Introduction Severe plastic deformation (SPD) processing is defined as any method of metal forming under an extensive hydrostatic pressure that may be used to impart a very high strain to a bulk solid without any significant change in the overall dimensions of the sample and having the advantage of producing exceptional grain refinement. It is now established that materials with ultra fine grained (UFG) microstructures may be fabricated using two different approaches which are generally termed the ‘‘bottom-up” and ‘‘top-down” procedures. In the ‘‘bottom-up” procedure,
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