А. Yu. Muizemnek, I. V. Denisov, О. L. Pervukhina, А. Е. Rosen, I. S. Los and Yu. A. Gordopolov DEFORMATION OF LONG- LENGTH EXPLOCLAD SHEETS: MATHEMATICAL MODELING

Object: Experimentally-theoretical research of longitudinal deformations of making layers of a multilayer material at explosion welding Research Technique 1. Computer simulation of deformation process of large-size sheets at explosion welding by means of LS-DYNA. 2. Experimental research of large-size sheets deformation by means of fixed points method. 3. Analysis of computer simulation and experimental results. 2

Explosion welding scheme 1 – clad plate ; 2 – base plate ; 3 – air technological gap ; 4 - sand background The geometrical sizes Thickness, mm Length, mm Width, mm clad plate base plate gap8 explosive sand background PHYSICOMECHANICAL PROPERTIES OF MATERIALS Steel plate: – Density ρ = 7800 kg/m 3 ; – Young modulus E = 192 GPa; – Yield strength σ т = 350 MPa; – ultimate strength σ в = 500 MPa; – unit elongation δ = 21%; – coefficient of thermal expansion α = 11,4 ºС-1 (100ºС). Explosive - apparent density ρ ВВ = 740 kg/m 3 - velocity of detonation D = 2100 m/s. Sand - apparent density ρпес = 2800 kg/m 3 - compression strength σсж = 140 MPa Initial data 3

Mathematical simulation by means of LS-DYNA software for the following situations : 1.Porous background, dissimilar metals (steel+stainless steel) under the assumption that both plates material behaves as a solid body, technological gap between clad and base plates. 2.Porous background, dissimilar metals, under the assumption that a clad material behaves as a liquid, a base material behaves as a solid body, technological gap between clad and base plates. 4

Finite-element mesh The description of used finite-element mesh – Quantity of elements ~ ; – Quantity of units ~ ; – The maximal size of an element – 0.5 mm. Used models of materials and state equations. Explosive : – material model - #9 (Wilkins-Geyrouch); – state equation - # 2 (JWL). Metal plate : – material model - #15 (Johnson – Cook); – state equation - # 4 (Mi – Gruneisen); Sand background : – zero-material - #9; – state equation of porous material - # 8. Technological gap : – vacuum model - #140. 5

Distribution of material density in calculation area at t = 3 ms: a – the beginning clad process; b – the termination clad process The first variant of calculation Porous background, dissimilar metals under the assumption that both plates material behaves as a solid body, technological gap between clad and base plates. b a It is established that the left butt of clad and base plates is extended by 16,1 mm and 29 mm. The right butt of clad and base plates is extended by 71 mm and 61,3 mm accordingly. 6

b a The second variant of calculation Porous background, dissimilar metals, under the assumption that a clad material behaves as a liquid (base material behaves as a solid body ), technological gap between clad and base plates It is established that clad plate isnt extended and base plate is extended by 35 mm. 7 Distribution of material density in calculation area at t = 3 ms a – the beginning clad process; b – the termination clad process

clad platebase plate 12 t = 1000 μs t = 1500 μs t = 2000 μs 8 Change of pressure longitudinal along sheets

t = 2500 μs t = 2750 μs t = 3000 μs clad plate 1 base plate 2 9 Change of longitudinal stress along the sheets

The scheme of sheet deformation revealing after explosion welding 1010 Before explosion welding After explosion welding clad plate base plate The beginning of initiation Matching clad and base plates Places of matching clad and base plates Labels

Results of experiments1 Matching clad and base plates Before explosion welding After explosion welding The top view

The generalized results of explosion welding simulation Plate Moving from the initiation point Variant of calculation Experimental data at V 0 =2100 km/s 12 base to the right, mm16,100 clad to the right, mm2900 base to the left, mm61,33525 – 28 clad to the left, mm7100 base the beginning of process of lengthening, mm

Conclusions: 1. On the deformation behavior and change of geometric sizes of clad and base sheet influence the next parameters: – the initial geometric size of plates; – characteristics of physical-mechanical properties of welded plates materials and explosive. 2. The residual elongation of plates occurs nonuniformly from 80% of sheet length. The maximal residual deformation is near the opposite butt from the initiation point. 3. Calculation and experimental results showed that the clad sheet behaves as a viscous liquid and the base sheet behaves as a metal in solid state. 4. Tensile deformation of base sheet due to the impact of clad sheet goes ahead of the contact point along the full thickness to the joint formation. Consequently explosion welding at the end areas goes along the moving surface of base sheet.

Анимация 2. Движение материала в расчётной области (Начало процесса сварки)

Анимация 3. Движение материала в расчётной области (окончание процесса сварки)

Для описания поведения материалов листов была использована модель Джонсона-Кука со следующими значениями параметров модели: $ *MAT_JOHNSON_COOK $ $ mid ro g e pr dtf vp $ $ A B n c m tm tr epso E E e-5 $ $ cp pc SPALL IT D1 D2 D3 D E $ $ D5 0.0 $ *EOS_LINEAR_POLYNOMIAL $ $ eosid c0 c1 c2 c3 c4 c5 c $ $ e0 v $ MID – идентификатор материала в виде уникального номера; RO – массовая плотность; G – модуль сдвига; SIGY – предел текучести; PC – предельное давление при растяжении; SPALL – тип разрушения; EPS – эффективная пластическая деформация; ES – эффективное напряжение; EOSID – метка уравнения состояния; Е0 – начальная внутренняя энергия; V0 – начальный относительный объем.

Для описания поведения ВВ была использована модель MAT_HIGH_EXPLOSIVE_BURN и уравнение состояния JWL со следующими значениями параметров модели: $ *MAT_HIGH_EXPLOSIVE_BURN $ $ mid ro D PCJ BETA K G SIGY $ *EOS_JWL $ $ eosid a b r1 r2 omeg e0 v $ MID – идентификатор материала в виде уникального числа; RO – массовая плотность; D – скорость детонации; PCJ – давление Чэпмена-Жуге; EOSID – метка уровня состояния; V0 – начальный относительный объем.

Для описания поведения песка была использована модель MAT_NULL для пористого материала со следующими значениями параметров: $ *MAT_NULL $ $ mid ro pc mu terod cerod ym pr e+3 1.0e $ *EOS_TABULATED_COMPACTION $ $ eosid gama e0 v $ $ ev1 ev2 ev3 ev4 ev $ ev6 ev7 ev8 ev9 ev $ c1 c2 c3 c4 c5 0.8e e-4 2.4e-4 5.6e e-4 $ c6 c7 c8 c9 c e e e e e-4 $ t1 t2 t3 t4 t5 0.0e+6 0.0e+6 0.0e+6 0.0e+6 0.0e+6 $ t6 t7 t8 t9 t10 0.0e+6 0.0e+6 0.0e+6 0.0e+6 0.0e+6 $ k1 k2 k3 k4 k5 40.0e e e e e-4 $ k6 k7 k8 k9 k e e e e e-4 $ MID – идентификатор материала в виде уникального номера; RO – массовая плотность; PC – предельное давление при растяжении; MU – коэффициент вязкости; TEROD – относительный объем для разрушения при растяжении; GEROD – относительный объем для разрушения при сжатии; YM – модуль Юнга (используется только для нулевых балочных и оболочечных элементов); PR – коэффициент Пуассона (используется только для нулевых балочных и оболочечных элементов).

Выражение Джонсона (Johnson) и Кука (Cook) для напряжения текучести Уравнение состояния JWL задает давление в виде