OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 6 4 4 Ta b l e 1 Formulas for calculating normal contact stresses px(hx) The fi rst elastic section with a length of х1 The second elastic section with a length of х4 1 1 1 1 2 1.15 1 x x S i i i h p E h − − − ⎧ ⎛ ⎞ ⎪ = ⋅ ⋅ − ⋅ + ⎨ ⎜ ⎟ δ δ + ⎪ ⎝ ⎠ ⎩ ( ) 1 1 1 1 1 1 1 1 1.15 i i i i x i i S h h E − δ − − − − − ⎫ ⎡ ⎤ ⎛ ⎞ δ − σ ⎪ + − ⎢ ⎥⎬ ⎜ ⎟ δ + ⋅ δ ⋅ ⎢ ⎥ ⎝ ⎠ ⎪ ⎣ ⎦⎭ , where 1 tg 2 i i − μ δ = ⎛ α ⎞ ⎜ ⎟ ⎝ ⎠ 1 2 1.15 1 x x S i i i h p E h ⎧ ⎛ ⎞ ⎪ = ⋅ ⋅ − ⋅ + ⎨ ⎜ ⎟ δ δ + ⎪ ⎝ ⎠ ⎩ 1 ( 1) 1.15 i i i i x i i S h h E δ ⎫ ⎛ ⎞ ⎡ ⎤ δ − σ ⎪ + − ⎬ ⎜ ⎟ ⎢ ⎥ δ + ⋅ δ ⋅ ⎪ ⎝ ⎠ ⎣ ⎦⎭ , where ( ) tg i i μ δ = β Plastic section ( ) ( ) 1 1 1 1 1 ( ) 0.5 0.5 1.15 1 ln( ) ln( ) ( ) 1.15 tg tg 2 2 x n x pl x n n pl h h h p p h h h h h h ⎡ ⎤ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ − ⎢ ⎥ ⎜ ⎟ = ⋅ σ ⋅ ⋅ + + ⋅ + − + ⎢ α α ⎥ − ⎜ − ⎟ ⋅ σ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ Fig. 1. Distribution pattern of normal contact stresses in the deformation zone during hot rolling: 1 – deformable strip; 2 – working roll; hi−1, hi – thickness of the strip before and after rolling, mm; h1, h4 – thickness of the strip at the boundaries of the fi rst and second elastic sections, mm; hn – thickness in neutral section, mm; x1, x4 – lengths of elastic sections, mm; x2, x3 – lengths of the plastic sections of lag and advance, mm; τi – contact tangential stresses, MPa; рi – normal contact stresses, MPa; α – angle of nip of the strip, β – angle of inclination of the deformation zone on the second elastic section, deg; σi-1, σi – back and front tensions, MPa
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