Solution of impact tasks, assessment of result reliability

1. Solution of impact tasks,assessment of result reliability M. Okrouhlík Institute of Thermomechanics

2. Scope of the lecture • What is a ‘good agreement’ in dynamical transient analysis in solid continuum mechanics • Vehicle for comparison • Continuum model and its limits • Experimental and FE analysis • Assessment of agreement quality • Synergy of FE and experimental analyses • Conclusions

3. A good agreement of experimental and FE results Experimental and FE treatment are limited by cut-off frequencies, which are generally different – in this case the filter applied on FE results has the same frequency limit as raw experimental data.

4. Impact loading of a tube with four spiral slots

5. Dimensions and points of interest

6. 3D elements and node numbering for mesh1 Location Nodes in unfold location Strain gauges Strain gauges Nodes in one layer

7. Geometry, material and elements G1R90L_NM ... four slots with 15 deg. indentation, axial length = 17.28 mm, slots and mesh rotate 90 deg Geometry and elements first part of the tube = 800 mm 800 layers, 144 elem in one layer second part, ie. axial length of slots = 17.28 mm 18 layers, 120 elem in one layer third part of the tube = 800 mm 800 layers, 144 elem in one layer total lenght of tube = 1617.28 mm 1618 total number of layers outer to inner diameters D/d = 22/16 mm number of spiral slots = 4 Element properties trilinear bricks elements, 8 nodes, Gauss order quadrature NG = 3 consistent mass matrix for NM (Newmark) diagonal mass matrix for CD (central differences) number of elements 232 560 number of nodes 310 576 number of degrees of freedom 931 728 max. front width 606 Material properties Young modulus = 2.05e11 Pa Poisson's ratio = 0.24 density = 7800 kg/m^3

8. Loading and FE technology Loading Vertical arrangement of the loading is assumed. In experiment the upper face of the tube is loaded by a vertically falling striker, which has been released from a certain height. Lower face of the tube is fixed. In FE model only axial displacements of the lower face are constrained. The striker is made of the same material as the tube, has the same outer and inner diameters. For computational purposes the upper side of the tube is loaded by uniform pressure, whose time dependence is given by a rectangular pulse. The loading pressure corresponds to the height from which the striker was released. The time of the pulse corresponds to the length of the striker. In this case the loading pressure is 88.5198 Mpa, which corresponds to the height of h = 1m. Velocity of the striker just before the impact is sqrt(2*g*h) = 4.42944 m/s. Material particle velocity immediately after the impact is v = 0.5*sqrt(2*g*h) = 2.2147 m/s. The time length of the pulse is 15.6 microsec, which corresponds to a medium length striker, Ls = 40 mm. Pressure is evaluated from p = E*v/c0, with c0 = sqrt(E/ro). Input energy from the striker is mgh = 0.548086 J. m = 05587 kg. FE technology Newmark with no algorithmic damping was used with consistent mass matrix Central differences with diagonal mass matrix timestep = 0.1 microsec = 1e-7 s total number of time steps = 3500 (Newmark); 5000 (central differences) this corresponds to total time = 350 microsec (Newmark); 500 microsec (central differences)

9. Wavefront timetable

10. FE strain distribution in location 1

11. Raw comparison – no tricks Are FE or experimental results closer to reality? Where is the truth?

12. Where is the truth? Are experimental or FE results closer to reality? • When trying to reveal the ‘true’ behavior of a mechanical system we are using the experiment • When trying to predict the ‘true’ behavior of a mechanical system we are accepting a certain model of it and then solve it analytically and/or numerically • Physical laws (models) as • Newton’s • Energy conservation • Theory of relativity cannot be proved (in mathematical sense) • Often we say that it is the experiment which ultimately confirms the model • But experiments, as well as the numerical treatment of models describing the nature, have observational thresholds. Sometimes, the computational threshold of computational analysis are narrower than those of experiment • Let’s ponder about limits of applicability of continuum model as well as about limits of modern computational approaches when applied to approximate solution of continuum mechanics

13. Continuum mechanics • Deals with response of solid and/or fluid medium to external influence • By response we mean description of motion, displacement, force, strain and stress expressed as functions of time and space • By external influence we mean loading, constraints, etc. Expressed as functions of time and space

14. Solid continuum mechanics • Macroscopic model disregarding the corpuscular structure of matter • Continuous distribution of matter is assumed – continuity hypothesis • All considered material properties within the observed infinitesimal element are identical with those of a specimen of finite size • Quantities describing the continuum behaviour are expressed as piecewice continuous functions of time and space

15. Governing equations • Cauchy equations of motion • Kinematic relations • Constitutive relations For transient problems, we are interested in, the inertia forces are to be taken into account

16. In linear continuum mechanicsthe equations are simplified to • Inertia forces are considered • loading is • localized in space • of short duration with a short rise time • The high frequency components are of utmost importance

17. Wave equation 2D - plane stress, Lamé equations – expressed in displacements only Nondispersive and uncoupled solutions in unbound region Transversal shear rotational distortion equivolumetrical Longitudinal dilatational irrotational extension P … primary S … shear

18. Velocities for 2D plane strain and 3D P … primary Within the scope of linear theory of elasticity, P and S waves are uncoupled.

19. Typical values for steel in m/s For E = 2.1e11 Pa, r = 7800 kgm^-3, m = 0.3

20. Analytical solutions for bounded regions are difficult and exist for bodies with simple initial and boundary conditions only One way to solve the problem is to apply the Fourier transform in space and the Laplace transform in time on equations of motion. The subsequent inverse procedure leads to infinite series of improper integrals. To indicate amount of effort to be exerted, the formulas derived by F. Valeš are shown.

21. A typical expression for a stress component in a transiently loaded thin elastic strip is shown

22. Relations between integrand quantities and integral variable Roots of these so called dispersive relations have to be evaluated numerically before the integration itself.

23. Roots of dispersive relations

24. After all, the analytical solution ends up with a numerical evaluation • Only finite number of terms canbe summed up • Integration itself has to be provided numerically • This led to development of approximate numerical solutions

27. Unbound frequency response of continuum For fast transient problems as shock and impact the high frequency components of solutions are of utmost importance. In continuum, there is no upper limit of the frequency range of the response. In this respect continuum is able to deal with infinitely high frequencies. This is a sort of singularity deeply embedded in the continuum model. As soon as we apply any of discrete methods for the approximate treatment of transient tasks in continuum mechanics, the value of upper cut-off frequency is to be known in order to ‘safely’ describe the frequencies of interest.

28. Dispersive propertiesof 1D and 2D constant strain elements

29. When looking for the upper frequency limit of a discrete approach to continuum problems, we could proceed as follows • Characteristic element size • Wavelength to be registered • How many elements into the wavelength • Wavelength to period relation • Wave velocity in steel • Frequency to period relation • The limit frequency • For 1 mm element we get

30. Limits of continuum, FE analysis and experiment

31. Validity limits of a model • Model is a purposefully simplified concept of a studied phenomenon invented with the intention to predict – what would happen if .. • Accepted assumptions (simplifications) specify the validity limits of the model • Model is neither true nor false • Regardless of being simple or complicated, it is good, if it is approved by an experiment

32. Using a model outside its limits is a blunder • Using a model outside of its validity limits leads to erroneous results and conclusions • This is not, however, the fault of the model, but pure consequence of a poor judgment of the model’s user • Model gives no warning. Lot of checks might be satisfied and still …

33. Blunders are easy to commit • Point force is • prohibited in continuum, • frequently used in FE analysis • Employing smaller and smaller elements, leads to singularity, since we are coming closer and closer to ‘continuum’ revealing thus unacceptable behavior of the point force in continuum • An example follow

34. Example of a transient problem Elements L, Q, full int. Consistent mass axisymmetric Mesh Coarse 20x20 Medium 40x40 Fine 80x80 Newmark Loading a point force equiv. pressure L or Q

35. Primary wave

36. Lamb, presure loading, rectangular pulse, velocity distribution, FEA R waves S waves P waves Again, where is the first nonzero P-wave appearance?

38. Pollution-free energy production

39. How to avoid blunders By knowing and understanding • the assumptions • instrumental limitations By providing • validity checks • within the model itself • comparing with other methods

40. FE self-check considerations

41. Numerický experiment může mez rozlišení simulovat a napomoci tak experimentu. Dá se ukázat, jak zjištěná rychlost šíření rozruchu závisí na hodnotě meze pozorování V pozorovaném místě známe - vzdálenost od aplikace budící síly, - čas, kdy dorazí „nenulový“ signál můžeme tedy vypočítat rychlost šíření rozruchu. Experimental threshold

42. Numerical analysis should be robust It should inform us about its limits Should be independent of • mass matrix formulation • the method of integration • meshsize • element type It is not always so, very often our results are method dependent. See the next example.

43. Validity self-assessments, NM vs. CD

45. Instrumental limitation Floating point representation of real numbers threshold Memory size limitation Meshsize and time step limitations

46. machine epsilon

47. FFT frequency analysis Now, let’s concentrate on the frequency analysis frequency analysis of the loading pulse and of axial and radial displacements obtained in the outer corner node of location C by means of NM and CD operators for the mesh1. The normalized power spectra are plotted in the range from 0 to Nyquist frequency together with the power spectrum of the loading pulse. timestep [s] sampling rate [MHz] Nyquist frequency [MHz] mesh1 1e-7 10 5 mesh2 1e-7/2 20 10 mesh3 1e-7/4 40 20 mesh4 1e-7/8 80 40

48. Assessment by frequency analysis

49. Validity self-assessments Mesh- and timestep refinement

50. Synergy of experiment and FE analysis FE analysis needs input data for computational models from experiment Experimental analysis could benefit from FE in ‘proper’ settings of observational thresholds