Meinders, T.

Snippe, Corijn H.C.

English

Meinders, Timo

University of Twente Research Information

IX International Conference on Computational Plasticity COMPLAS IX E. Oñate and D. R. J. Owen (Eds) © CIMNE, Barcelona, 2007  FE-BASED RADIATION LENGTH ANALYSIS OF SUPERPLASTICALLY FORMED PRODUCTS Corijn H.C. Snippe*, T. Meinders† * National Institute for Nuclear and High-Energy Physics (Nikhef) Kruislaan 409, 1098 SJ Amsterdam, The Netherlands e-mail: csnippe@nikhef.nl, web page: http://www.nikhef.nl  † University of Twente Mechanics of Forming Processes, Department of Mechanical Engineering Drienerlolaan 5, 7522 NB Enschede, The Netherlands e-mail: V.T.Meinders@ctw.utwente.nl, web page: http://www.tm.ctw.utwente.nl Key words: Superplasticity, strain rate sensitivity, cavity formation, radiation length.  1 INTRODUCTION In subatomic particle detection, it is important that structural members have a minimal effect on the trajectory of the particles. Since every piece of material acts as a barrier in which particles are decelerated (defined in a material constant called radiation length), the amount of material that can be traversed has to be minimised. Since all particles originate from a single point source, the problem can be written in terms of an averaged traversed path.  In shell-like structures, the thickness distribution determines the average traversed path, which must be minimised. Therefore, the process of superplastic forming seems an appropriate way of manufacturing some of these parts. In order to use superplasticity in an FE program, an accurate constitutive model is necessary. The FE thickness results serve as a basis to determine per element the average traversed path. Combined with the element view factor, a percentual radiation length can be calculated for the whole part. 2 SUPERPLASTICITY Superplastic materials exhibit very high plastic strains before failure, generally in the order of more than 100%. Failure of superplastic materials is preceded by cavity initiation, growth and coalescence at high plastic strains. Experiments are necessary to obtain the stress-strain behaviour in the superplastic regime of the material. A simple relationship is one in which the flow stress is dependent on the strain rate only: mKεσ &=y  (1)The value of the strain rate sensitivity m is roughly 0.7 at the optimal strain rate for superplasticity. To perform accurate FE simulations on superplastic forming, a more detailed material model is necessary, in this case it is chosen to build the model up of three parts: initial flow stress determination, hardening evolution and softening. Corijn H.C. Snippe, T. Meinders 2.1 Initial flow stress Equation (1) does not account for strain hardening. This relation can be used in a limited range of strain rates around the optimum, m is not constant. In fact, m depends on the strain rate and can be visualised in a plot showing log(σy) versus log( ε& ), which is close to a sigmoidal curve. This curve has an inflection point, at which the slope m is at a maximum. Experiments on an 5083 aluminium alloy type resulted in the curve shown in Figure 1, four parameters a, b, c and d describe the initial flow stress:  −3.5 −3.0 −2.5 −2.0 −1.5 −1.00.40.60.81.0experimenttheory:a = 0.728b = 3.79c = -2.49d = 0.382log strain ratelog flow stress−3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.50.00.20.40.6experimenttheorylog strain ratem(a) (b)( ) dcba++=)logexp(1log y εσ& Figure 1: (a) Experimental stress-strain values fitted by a sigmoidal curve, (b) strain rate sensitivity m 2.2 Strain hardening Equations (1) and (2) do not account for strain hardening. In a simplified situation it can be said that hardening will not take place if the grains do not grow. In this phenomenological material model, grain growth1 will be responsible for strain hardening, which is dependent on the strain, and the strain rate. The strain hardening can be expressed in terms of the initial flow stress multiplied by a factor which, in combination with a parameter k, can be multiplied with the value of the initial flow stress, to refine the model: y0refd 1 σεεσkrqddp⎟⎟⎠⎞⎜⎜⎝⎛+−=& (2)2.3 Softening With increasing plastic strain, internal cavities will initiate and grow, Figure 2 shows three pictures with the material containing cavities for three increasing values of the strain. The overall cavity volume fraction is determined by the sum of two effects, initiation and growth2. An increase in the void volume fraction ξ adds a softening behaviour into the material’s constitutive behaviour that can phenomenologically be described as ( ) 321dcav 1 nnn ξσσ −=  (3) 2Corijn H.C. Snippe, T. Meinders  Figure 2: Cavity formation at increasing plastic strains Cavity coalescence can be interpreted as the onset of fracture in the material, leading to failure. In the literature, several viewpoints are mentioned: a coalescence condition can be defined by a critical value3 σcr or ξcr  for which a value of 0.15 is suggested4. 3 RADIATION LENGTH In order to calculate the percentual radiation length of the formed sheet, the output must contain element thickness and nodal displacement data. In case of linear shell elements with reduced integration, the thickness result is constant within an element. When the plastic strains are high, the cavity volume fraction can influence the percentual radiation length.  3.1 Thickness distribution If the simulation uses full-integration linear elements, then the thickness distribution along the element is linear. Generally this leads to more accurate results in most simulations, but since the elements will distort heavily in superplastic forming simulations, shear locking can negatively infuence the results. Reduced-integration results in elements with constant thickness, but they are more accurate than the results from full integrated elements. 3.2 Averaged traversed path The averaged traversed path through an element can be explained best in a one-dimensional situation. Figure 3 shows this situation in case of an element with length l e which changes in thickness from t 0 to t l. This change in thickness can be explained as a thickness change resulting from full-integrated elements, or from averaged nodal results from reduced-integration elements. A ray emerging from point IP through the element at angle θ  results in a traversing path of t(θ). The average traversed path through the element can then be defined as ( )∫==αθθθα01 dtp)  (4) If the coordinates of the interaction point IP and the nodal coordinates are known from simulation, then an analytical solution can be found. Numerical integration is necessary, a calculation showed that (in two dimensions) the element may be rotated such that its normal points towards the interaction point, without considerably affecting the averaged traversed  3Corijn H.C. Snippe, T. Meinders path; also the numerical integration has to be carried out with four Gauss integration points in case of quadrilateral elements. This procedure gives accurate results if the distance from IP to the element center is more than twice the element dimensions. αθIPt lt 0tθ2134 Figure 3: Visualisation of the traversed path t(θ) 3.3 Determination of the radiation length To determine the percentual radiation length, the average traversed path has to be multiplied with the rotated element view factor5 and divided by the material radiation length. The calculation of the radiation length by means of the average traversed path as described in the previous section is carried out by a FORTRAN program which reads the thickness results  and the nodal coordinates of the output data file. A summation over all elements results in the percentual radiation length of the analysed part. This number is very important in structures where measurements are carried out involving subatomic particles. 4 FUTURE WORK The establishment of the material constitutive behaviour is necessary in order to optimise superplastically formed sheets with respect to radiation length. An important constraint is the gas tightness of the sheet, which means that internal cavities are allowed to occur, but may not coalesce, this can lead to serious gas leaks through the shee thickness. REFERENCES [1] D.H. Bae and A.K. Ghosh, “Grain size and temperature dependence of superplastic deformation in an Al-Mg alloy under isostructural condition”, Acta Materialia, 48, 1207-1224 (2000). [2] M.A. Khaleel, H.M. Zbib and E.A. Nyberg, “Constitutive modeling of deformation and damage in superplastic materials”, Inernationa’l Journal of Plasticity, 17, 277-296 (2001). [3] D. Lassance, F. Scheyvaerts and T. Pardoen, “Growth and coalescence of penny-shaped voids in metallic alloys”, Engineering Fracture Mechanics, 73, 1009-1034 (2006). [4] R.H.J. Peerlings,”Plasticity of Porous Materials”, EM Micromechanics Course lecture notes (2004). [5] J.J. MacFarlane, “VISRAD⎯A 3-D view factor code and design tool for high-energy density physics experiments”, Journal of quantitative Spectroscopy & Radiative Transfer, 81, 287-300 (2003).  4

FE-based radiation length analysis of superplastically formed products

Meinders, Vincent T.

NARCIS 

Constitutive modeling of deformation and damage in superplastic materials”,

Grain size and temperature dependence of superplastic deformation in an AlMg alloy under isostructural condition”,

Growth and coalescence of penny-shaped voids in metallic alloys”,

VISRAD⎯A 3-D view factor code and design tool for high-energy density physics experiments”,

https://ris.utwente.nl/ws/files/6163373/Snippe_COMPLAS07_paper.pdf

FE-based radiation length analysis of superplastically formed products

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