Numerical Method
This chapter is concerned with a numerical method for calculating the strain rate intensity factor in plane strain deformation of rigid perfectly plastic material.
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Numerical Method
This chapter is concerned with a numerical method for calculating the strain rate intensity factor in plane strain deformation of rigid perfectly plastic material. It is evident that standard commercial packages are not capable for supplying solutions that satisfy the exact asymptotic expansion (2.20). For example, using ABAQUS a ring upsetting process has been analyzed in [5]. All the finite element analyses presented in this paper failed to converge in the case of the maximum friction law. The extended finite element method [9] might provide solutions in which Eq. (2.20) is satisfied. However, to the best of author’s knowledge, no attempt has been made to develop this numerical technique for calculating the strain rate intensity factor. Moreover, it is believed that the method of characteristics is the best choice for models described by hyperbolic systems of equations. In particular, general numerical schemes based on the method of characteristics are available for plane strain deformation of rigid perfectly plastic material [3, 4, 7, 8, 10, 11]. In order to calculate the strain rate intensity factor, it is just necessary to supplement these schemes with a corresponding procedure. To the best of authors knowledge, the only available procedure has been proposed in [2]. Most of the present chapter is based on this work.
5.1 The Strain Rate Intensity Factor in Characteristic Coordinates The system of equations consisting of Eqs. (1.1), (1.2), (1.15), and (1.16) is hyperbolic [10]. The characteristics for the stresses and the velocities coincide and, therefore, there are only two distinct characteristic directions at a point. The characteristic directions are orthogonal. Let the two families of characteristics be labeled by the parameters α and β. In general, one or two families of characteristics can be straight. © The Author(s) 2018 S. Alexandrov, Singular Solutions in Plasticity, SpringerBriefs in Continuum Mechanics, DOI 10.1007/978-981-10-5227-9_5
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5 Numerical Method
Fig. 5.1 Cartesian (x, y) and characteristic (α, β) coordinates
These special cases are excluded from consideration. The α- and β- lines are regarded as a pair of right-handed curvilinear orthogonal coordinates. By the convention, the orientation of these lines is chosen such that the algebraically greatest principal stress σ1 falls in the first and third quadrants. Let φ be the anti-clockwise angular rotation of the α- line from the x- axis of a Cartesian coordinate system (x, y) (Fig. 5.1). Then [10], φ = α + β, ∂φ 1 = , R ∂sα
∂φ 1 =− S ∂sβ
(5.1) (5.2)
where ∂/∂sα and ∂/∂sβ are space derivatives taken along the α- and β- lines, respectively. Note that ∂φ/∂sα = ∂α/∂sα since β is constant along the α- lines and ∂φ/∂sβ = ∂β/∂sβ since α is constant along the β- lines. Also, R is the radius of curvature of the α- lines and S is the radius of curvature of the β- lines. Those are algebraic quantities whose sign depends on the sense of the derivatives ∂/∂sα and ∂/∂sβ . Let θ be the anti-clockwise rotation from the α- dire
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