Churilova Maria Saint-Petersburg State Polytechnical University Department of Applied Mathematics

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1 Churilova Maria Saint-Petersburg State Polytechnical University Deartment of Alied Mathematics Technology of EHIS (staming) alied to roduction of automotive arts The roblem described in this reort originated from the joint roject of General Motors, USA and the Laboratory of Pulsed Power Energy, and the Laboratory of Alied Mathematics and Mechanics, Saint Petersburg State Polytechnical University, Russia. The roject is in rogress. Motivation An alication of the discharge in water with the aim of metal forming was firstly described in 4s of the th century and then examined exerimentally and theoretically since the end of 5s in the Soviet Union and the USA. Since that time this technology has been develoed mostly in our country for the urose to roduce small series of items. Nowadays, GM is interested in industrial alication of Electro Hydraulic Imulse Staming (EHIS) to ut out small automotive arts. It is lanned to construct a laboratory bench for staming of small fragments of the automotive arts. An exerimental research of the EHIS rocess is conducted on this bench. A numerical methodology for comlex simulation of such rocess is develoed and validated on the basis of exerimental data. The analysis of the revious ublications and the available exerience give evidence that the electric discharge in water can be effectively used for staming of the flat-attern arts. There is a great number of ublications with descrition of the secific technological rocesses. An additional investigation is needed including research of the hysics of EHIS rocess, measuring arameters that determine the deformation rocess (energy, ressure field, material deformation rate, etc.), develoment of a theoretical model that can interret the obtained exerimental results and relate them to the ower suly arameters. The roject is aimed at:. Develoment and validation of numerical methodology, which is caable to simulate the EHIS rocess. The develoed mathematical model of the rocess should describe the lastic deformation of the late stock under the action of exanding water. For simulation of the EHIS rocess it is lanned to imlement LS-DNA code.. Testing the EHIS rocess with model hydro-ulse laboratory benches. It is lanned to erform exerimental investigations of the rocess of ressure field forming at the electrical discharge. Moreover, it is necessary to find similarity conditions of EHIS rocesses for ossible scaling of the rocess. 3. Develoment of an EHIS technological model for a given object (art). Verification of the numerical model and similarity conditions at realization of full scale technological rocess. Descrition of the technological rocess

2 A conventional scheme of the sark discharge exeriment in the water is shown in Fig.. After closing switch S, the voltage from the charged caacitor C is alied to electrodes located in the discharge chamber. The chamber is filled with water. The discharge causes develoment of a gas bubble and water exansion. The time of discharge develoment is determined by the 5 characteristics of the discharge ga. Usually, it is from to 4 seconds. Fig.. Exerimental lant Photograhy of the chamber in which the examination of discharges in water with deformation of flat metal secimens was held is shown in Fig.. Fig.. Work chamber The chamber is rovided with two sight holes of diameter 3 mm. Two electrodes are inserted into the same volume through insulators. Electrodes form the discharge ga in which the energy necessary for deformation releases. A ressure from to5 atmosheres roduced by water exansion deforms the metal samle which is on the to of the chamber. Numerical simulation of electrohydroimulse staming rocess Due to difficulties of roviding similarity conditions for the considered rocess, its accurate numerical simulation for the small-scale exerimental lant becomes very imortant for further construction of a large-scale one. For this reason, it is necessary to rovide the following research:. Review of mathematical models used for fast rocesses analysis;. Develoment of a mathematical model and comutational method for thin sheet dynamic deformation under secified ressure on its surface; 3. Test comutations (based on exerimental data); 4. Comarative analysis of exerimental and comutational results. In articular, the tasks below have extreme imortance

3 . Determination of constitutive relationshis for the materials to be used;. Adatation of a comutational model in accordance to the secific material characteristics; 3. Estimation of influence of ossible inaccuracy in model arameters on the difference between comutational and exerimental results. Research method The electrohydroimulse staming of metals is a comlicated rocess, which is characterized by develoment of several hysical henomena and their interaction in time. An additional comlexity in simulation of such rocess is related to the fact that the deformation of a blank may lead to significant changes in the model geometry. Elaboration of any staming technology requires some estimation for staming rocess arameters and develoment of corresonding mathematical models, which are able to reresent real-life hysical henomena adequately. Caabilities of modern engineering software (ANSS, LS-DNA, etc.) for roviding accurate and adequate simulations to the rocess of elastolastic metal deformation are wide. It is well known, that LS-DNA is able to rovide accurate results for simulation of a fast metal staming rocess, but there are only a few articles where such an aroach is described in alication to electrohydroimulse forming of metals. In our oinion, usage of LS-DNA for all stages of simulation (such as selection of a material model, loading, solution, and ost rocessing of the results) based on comarison to exerimental data is a right way of constructing methods for mathematical modeling of EHIS. Physical henomena of metal deformation, which are necessary to take into account For successful staming, it is necessary to achieve a high level of lastic deformations, therefore a model of lasticity becomes very imortant. This model should be suitable for simulation of lastic behavior of a metal as well as for simulation of a hardening. Governing system of equations for descrition of a blank motion is the following: ρu&& = divσ + ρ f, where u is the dislacement vector, ρ is the current density (a function of satial coordinates only), σ is the Cauchy stress tensor, and f is the body force density. For any model, this system should be comleted by boundary conditions for tractions and dislacements, and by a constitutive relationshi of the form σ = σε ( ), where ε is the total strain tensor, which is the sum of elastic and lastic arts, i.e. e ε = ε + ε. For examle, for a linear elastic material the constitutive relationshi has the form e σ = Dε, ε = ε, where D is the tensor of elastic moduli. Plasticity A well-known tye of tests to analyze the behavior of a material is the tension of a rod (see Fig. ). The result of such test is reresented by lotting the ratio of tensile stress to the initial crosssectional area, against some measure of the total strain. If the length of a tensile secimen is increased from l to l the amount of deformation is measured as ε = ln( l / l ) (called logarithmic or natural strain) or as ε = ( l l ) / l (engineering or

4 conventional strain); they are aroximately identical when the change in length is small, but as the strain increases natural strain becomes rogressively less than engineering strain. Fig. 3. Tension of a rod Most common engineering materials exhibit a linear stress-strain relationshi u to a stress level known as the roortional limit (Point A). Beyond this limit, the material behavior becomes nonlinear, but not necessarily inelastic. Plastic behavior begins when stresses exceed the material's yield oint (i.e., the stress level corresonding to Point B). Fig. 4. Tyical stress-strain relationshis for metals (a) soft metal (coer), (b) hard metal (steel) A roortional limit B yield oint C oint where hardening starts DE unloading, EF reeated loading Because there is usually a small difference between the yield oint and the roortional limit, it is tyically assumed that these two oints are coincident in lasticity analyses. From Point B u to Point C the stress remains stable and only deformation increases (see diagram (a)). For some metals there is no curve interval BC and after Point B the hardening of a material starts immediately (see diagram (b)). It is called hardening because after unloading and reeated loading yielding limit increases. Processes of unloading and reeated loading are also shown in both diagrams (curves DE and EF, resectively). For simlified mathematical models of lasticity it is usually assumed that these curves are arallel to the starting linear curve OA. ield criterion The yield criterion determines the stress level at which yielding begins. For multi-comonent stresses, it is reresented through the definition of the yield function F (σ ) of the stress tensor σ.

5 For many cases, this function is interreted as a function of the equivalent (von Mises) 3 D D stressσe = σ. Here σ = σ + PΙ is the deviator of the tensor σ, where Ι is the identity trσ D D D tensor, P = is the ressure and σ = σij σij. When the equivalent stress is equal to a 3 material yield arameterσ, the material will develo lastic strains. If σ E is less thanσ, the material is elastic, and stresses develo according to the elastic stress-strain relationshi. Fig. 5. Stress-strain relations for the simlest mathematical models of lasticity Case (a) corresonds to the erfect lasticity (no hardening), and Case (b) reresents behavior with linear hardening where there are two linear arts of the stress-strain curve. ield surface In case of a comlicated stress state for descrition of a yielding start conditions it is necessary to inut yield function F. Stress is considered to be accetable if F(σ ). That surfaces, for which F ( σ ) =, are called yield surfaces. Any stress state corresonding to the oint inside yield surface can be reached urely with elastic deformations. Plastic deformations occur only if F( σ ) = (on the yield surface). In case of a erfect lasticity yield surface remains constant. Area where F ( σ ) < is called the elastic area, and area where F ( σ ) = is called the lastic one. A material element is said to be in an elastic state if F ( σ ) <, and in a lastic state when F ( σ ) =. For lastic yielding, the element needs to be in a lastic state ( F = ), and to remain in a lastic state ( F & = ); otherwise the lastic strain rate & ε = & ε & ij εij vanishes (in common lasticity theory time is not taken into account, but it is used as a arameter which controls the loading rocess, then dot means the time derivative). Hence & ε = for F < or ( F = and F & < ); otherwise there is a yielding. The first condition corresonds to the case when element is in an elastic state, second to the case when element asses from a lastic state to an elastic state (unloading). For metal under not very high ressure yield function deend not on stress tensor but on its deviator. In this case, yield surface can be defined with equation (von Mises criterion) D σ F( σ) = σ =. 3 In rincial stress sace σ, σ, σ 3 yield surface will look like a cylinder with axis σ = σ = σ 3 which is erendicular to the deviatoric lane (where trσ = ). In rojection on a deviatoric lane, yield surface looks like a circle with center in the origin.

6 Hardening Fig. 6. ield surface for von Mises criterion For metals under lastic deformations, yield limit increases with deformation growth. It means that the value of the arameter σ increases. Metal gains additional elastic roerties, and loses ability to deform lastically. This henomenon is called hardening. The hardening rule describes how the yield surface changes with rogressive yielding, so that the conditions (i.e. stress states) for subsequent yielding can be established. In work hardening, the yield surface remains centered about its initial centerline and exand in size as the lastic strains develo (see Fig. 7 where the rojection of the yield surface to the deviatoric lane is shown). This otion is referable for our analysis due to the necessity to include large strains into a model. Fig. 7. ield surface exansion in the model with isotroic hardening ANSS LS-DNA lasticity models Bilinear Isotroic Model (BISO) This classical strain rate indeendent bilinear isotroic (work) hardening model uses two sloes (elastic and lastic) to reresent the stress-strain behavior of a material (see Fig. 5(b)). Stressstrain behavior can be secified at only one temerature (a temerature deendent bilinear isotroic model is also available). Inut elastic arameters are elastic modulus E (the tangent sloe to the first art of the stress-strain curve in Fig. 5(b)), Poisson s ratio, and density of a material. The rogram calculates the bulk modulus using the elastic modulus and Poisson s ratio values. Inut arameters for the simulation of a lastic behavior are the initial yield limit σ and the tangent sloe E tan to the second (lastic) art of the stress-strain curve in Fig. 5(b). A material anisotroy is not suorted for this model. The constitutive relationshis for this model are given for rates (or small increments) of dislacements, stresses and strains. The von Mises criterion of yielding is used. The system of corresonding relations have the form

7 e T dε = dε + dε = ( du+ du ), dσ = D( dε dε ), F dε = λ, σ D where λ is a lastic multilier which vanishes for elastic state and unloading. For any iteration of the solution rocess, using the so-called lastic tangent modulus EEtan EP =, E Etan ANSS LS-DNA calculates the current value of the hardening arameter as σ ( ε ) = σ + E ε, where ε eff t dε eff eff eff = is the current value of the effective lastic strain (which is also called the equivalent lastic strain), and the effective lastic strain increment is given by the formula dε eff = dεij dεij. 3 From two formulas above follows, that during the calculation rocess it is necessary to save into memory the time history only for the effective lastic strain. The value of λ for every small subste of loading comes from the lastic yielding condition F D F df = dσ dσ D + ij =, σ ìj σ where F D = σ, dσ D = Edεe ff. σ The last relation is also used for calculation of increment for hardening arameter. Taking into account the above-mentioned exlicit form of the function F and the relation D D D D D σ ij dσij = σ : dσ = σ : dσ, we arrive at the exression D D σ : dσ σedεeff = σ : D( dε dε ) σedεeff = 3 3 or D D D σ : Ddε λ D = Dσ : σ + σe σ 3 3. Therefore D σ : Ddε λ =. D D D Dσ : σ + σe σ 3 3 Johnson-Cook model This model, also called the viscolastic model, is a strain-rate and adiabatic (heat conduction is neglected) temerature-deendent lasticity model. This model is suitable for roblems where strain rates vary over a large range.

8 Johnson and Cook exress the yielding limit as σ y where * m ( ( ) ) & ε n eff = ( A + B( ε eff ) ) + c ln + T ε & A, B, c, n, m, T melt are material constants, & ε eff, T * T T = T T melt room room is the dimensionless lastic strain rate, ε& & ε the initial strain rate (usually is treated as. s - ). Inut arameters for the simulation are the elastic modulus, Poisson s ratio, density of a material and all above-mentioned material constants. There is also a art of the model for fracture, but in our case, no fracture was detected. Temerature deendence is neglected due to exerimental data that show no heating of a metal after deformation (it is also often neglected because of insufficient exerimental material data). Equation of state (EOS) In the case of the Johnson-Cook model, an equation of state describes relationshi between ρ the ressure P, the secific internal energy E and the relative volume V =, where ρ is the initial ρ density. If V and E are indeendent thermodynamic functions it is sufficient to write this equation of state using only one function P = P( V, E). There are two suitable tyes of equations in ANSS LS-DNA. They are called linear olynomial and Gruneisen. The linear olynomial equation of state has the form P = Cμ + Cμ + ( C3 + C4μ) E, ρ where μ = is a dimensionless arameter. For exansion rocess ( μ < ), term C μ is set to ρ zero and equation transforms into P = C μ + C + C ) E. ( 3 4μ Gruneisen equation of state is used for high ressures. For comressed materials it has the following form: γ a ρc μ + ( ) μ μ P= + ( γ + aμ) E, 3 μ μ ( S ) μ S S3 μ+ ( μ+ ) where γ is the Gruneisen gamma, and S, S, S3, c, a material constants. For exanded materials the equation becomes P = ρ C μ + γ + a ) E. ( μ If the arameter μ is small, linear olynomial and Gruneisen equations of state are almost identical to each other. It is known from the literature that to achieve the value of μ =. it is necessary to rovide a ressure about 3 GPa, which is much higher than the range of ressures in our exeriments. Therefore, the difference between these two models is insignificant (that is aroved by comutations).,

9 Below we resent some results of numerical simulation for coer staming at three regimes of ressure distributions in time. We also comare numerical results with exerimental results obtained on the small-scale lant. Required data for simulation. Geometrical model;. Material model; 3. Material roerties for a selected material model (density, Poisson s ratio, elastic modulus, lastic roerties and other material constants); 4. Finite Element model based on a selected tye of element; 5. Boundary conditions; 6. Loading as a function of time and satial coordinates; 7. The termination time for simulation; Geometrical and finite element model The model consists of tree arts: the deformable metal blank (cyan), rigid holder (blue) and rigid die (red), /4 th symmetry was used to reduce the amount of calculations. ANSS. APR 8 9 5:48:57 COMPONENTS Set of _PART (Elems) _PART (Elems) _PART3 (Elems) Blank size: thickness.5 mm radius mm The SOLID64 element is used Mesh Near the die edge mesh was refined for accurate comutations in area where lastic deformations mostly occur ANSS. APR 8 9 5:49:59 COMPONENTS Set of _PART (Elems) _PART (Elems) _PART3 (Elems) Mesh: nodes 5 58 elements Mesh refinement Pressure on the deformable metal blank was measured exerimentally and then aroximated via shar or smooth curves. Results of such an aroximation are deicted below. Loading () Maximum ressure:.4 MPa

10 (x**3) Plot Exlicit Dynamics Curve 4 ANSS. APR 8 9 5:5:4 mics PAR V = *DIST=.75 *F =.5 *F =.5 *F =.5 -BUFFER PAR (x**-3) PAR loading curve Fig. 8. Exerimental ressure curve (left), ANSS LS-DNA ressure curve (right) The ressure via time curve has tooth-saw eeks. Loading begins not from the zero time but from time about.7 milliseconds, but we begin loading from zero. Loading () Maximum ressure:.5 MPa (x**3) Plot Exlicit Dynamics Curve 4 ANSS. APR 8 9 5:5: mics PAR V = DIST=.75 F =.5 F =.5 F =.5 -BUFFER 5 PAR (x**-3) PAR loading curve Fig. 9. Exerimental ressure curve (left), ANSS LS-DNA ressure curve (right) Curve was aroximated smoothly because of its comlex structure. Loading (3) Maximum ressure: 3.3 MPa (x**3) Plot Exlicit Dynamics Curve 4 ANSS. APR 8 9 5:5:34 mics PAR V = DIST=.75 F =.5 F =.5 F =.5 -BUFFER 4 PAR (x**-3) PAR loading curve Fig.. Exerimental ressure curve (left), ANSS LS-DNA ressure curve (right)

11 Material constants for coer Elastic modulus.e9 N/m Poisson s ratio.343 Density 893 kg/m 3 Constants for BISO model ielding limit 7.e6 N/m Tangent sloe.e9 N/m Constants for Johnson-Cook model Room temerature 7 C Melt temerature 83 C Initial strain rate. Secific heat 385 Joule / kg C A 89.63e6 N/m B 9.64e6 N/m n.3 c.5 Constants for EOS C 4e9 N/m C.8e9 N/m C 3.96 C 4.47 Results Maximum deflection for coer blank: max ressure, MPa Exeriment, mm BISO, mm Accuracy, % Johnson-Cook, mm Time of a tyical comutation about 5 hours (AMD Athlon(tm) 64x Dual Core Processor GHz, 3 Gb RAM) Average accuracy for BISO model 5.% Johnson-Cook model rovides unsatisfactory results. Let consider the last examles in more details. M ANSS. APR 9 9 :: STEP= SUB =5 TIME=.66E-4 USUM (AVG) RSS= EFACET= DM =.3E-3 SM =.3E-3.47E-4.93E-4.44E-4.586E-4.733E-4.879E-4.3E-3.7E-3.3E-3 M ANSS. APR 9 9 ::39 STEP= SUB = TIME=.48E-3 USUM (AVG) RSS= EFACET= DM =.688 SM = E-3.375E-3.563E-3.75E-3.938E USUM for the material and set secified Ste 5 Time.66e-4 sec USUM for the material and set secified Ste Time.48e-3 sec

12 M ANSS. APR 9 9 ::4 STEP= SUB =3 TIME=.98E-3 USUM (AVG) RSS= EFACET= DM =.997 SM = E-3.666E-3.999E M ANSS. APR 9 9 :: STEP= SUB =6 TIME=.47E-3 USUM (AVG) RSS= EFACET= DM =.3476 SM = E-3.77E USUM for the material and set secified USUM for the material and set secified Ste 3 Time.98e-3 sec Ste 6 Time.47e-3 sec M ANSS. APR 9 9 :3:9 STEP= SUB = TIME=.33E-3 USUM (AVG) RSS= EFACET= DM =.343 SM = E-3.76E M ANSS. APR 9 9 :3:35 STEP= SUB =5 TIME=.88E-3 USUM (AVG) RSS= EFACET= DM =.344 SM = E-3.765E USUM for the material and set secified Ste Time.33e-3 sec USUM for the material and set secified Ste 5 Time.88e-3 sec Fig.. Total dislacement via time On the Fig. we can see how the total dislacement changes in time. From these ictures, it follows that the deformation mostly grows u in the eriod of 6 time stes, but the termination time for calculations corresonds to the set number. Analyzing the results, we conclude, that from the time ste (.33e-3 sec) only elastic damed oscillations are resent. But the amlitude of such oscillations is small enough and we leave them out of account. The accuracy of calculation of the final total dislacement is about 3% that is good for engineering uroses. M ANSS. APR 9 9 :8: STEP= SUB =5 TIME=.66E-4 EPPLEQV (AVG) EFACET= DM =.3E-3 S =.6E-4 SM =.347.6E-4.375E-3.734E M ANSS. APR 9 9 :8:37 STEP= SUB = TIME=.48E-3 EPPLEQV (AVG) EFACET= DM =.688 S =.563E-3 SM = E Von Mises lastic strain Ste 5 Time.66e-4 sec Von Mises lastic strain Ste Time.48e-3 sec

13 M ANSS. APR 9 9 :9: STEP= SUB =3 TIME=.98E-3 EPPLEQV (AVG) EFACET= DM =.997 S =.69 SM = M ANSS. APR 9 9 :9: STEP= SUB =6 TIME=.47E-3 EPPLEQV (AVG) EFACET= DM =.3476 S =.3887 SM = Von Mises lastic strain Von Mises lastic strain Ste 3 Time.98e-3 sec Ste 6 Time.47e-3 sec M ANSS. APR 9 9 :3:35 STEP= SUB = TIME=.33E-3 EPPLEQV (AVG) EFACET= DM =.343 S =.34 SM = M ANSS. APR 9 9 :3:48 STEP= SUB =5 TIME=.88E-3 EPPLEQV (AVG) EFACET= DM =.344 S =.34 SM = Von Mises lastic strain Ste Time.33e-3 sec Von Mises lastic strain Ste 5 Time.88e-3 sec Fig.. Von Mises lastic strain via time The same situation occurs for lastic deformations. They are stable beginning from the time ste (.33e-3 sec). U to this moment, the total level of lastic deformations is achieved. Conclusions. BISO model is more adequate for descrition of slow rocesses with a moderate ressure.. Mean error of the comutations with BISO model is about 5%. Plastic deformations occur where they were exected theoretically. 3. Deformation of a blank in small-scale exerimental lant is well simulated by BISO model. 4. Johnson-Cook model is a more comlicated and, in our case, it rovides worse correlation with exerimental results. Acknowledgments Finally, I wish to thank the head of the Laboratory of Pulsed Power Energy Prof. German Shneerson and his colleagues, and also Assoc. Prof. Sergey Luuleac and Assoc. Prof. Maxim Frolov, who rovided continuous guidance of the resent research work.