Lecture #10: 151-0735: Dynamic behavior of materials and structures Anisotropic plasticity Crashworthiness Basics of shell elements by Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing 2015 1 1 1
ANISOTROPIC PLASTICITY 2 2 2
Hill 48 yield function Hill (1948) proposed an anisotropic quadratic yield function. For plane stress conditions, the Hill 48 function reads 3 3 3
Experimental Characterization of flow anisotropy 4 4 4
Experimental Characterization of flow anisotropy In addition to the axial strain, the width strain is measured in uniaxial tension experiments using digital image correlation. 5 5 5
Experimental Characterization of flow anisotropy 6 6 6
Experimental Characterization of flow anisotropy 7 7 7
Experimental Characterization of flow anisotropy 8 8 8
Experimental Characterization of flow anisotropy 9 9 9
Hill 48 yield function For steel sheets, it is recommended to combine the isotropic von Mises yield function with an anisotropic Hill 48 flow potential. The Hill parameters can then be conveniently determined from the Lankford ratio measurements. 10 10 10
Crashworthiness 11 11 11
Axial crushing of square tubes P Results from Kohar et al. (Thin-walled structures, 2015) 12 12 12
SEA of a structure The Specific Energy Absorption (S.E.A.) is often used to evaluate the crashworthiness of thin-walled structures. It is defined as the ratio of the Work W performed ( absorbed energy ) during the crushing/crashing of a structure and the total mass m of the structure. The SEA of structures is typically given in J/g. W Pdu SEA W m P 13 13 13
Folding Mode Consider a hollow square box structure of with B and wall-thickness t subject to an axial force P. In our simplified analysis, it is assumed that the structure deforms in a so-called symmetric axial folding mode. 14 14 14
Kinematics Each of the four initially flat constituent plates, needs to form three plastic hinge lines in order to create a fold. To ensure kinematic compatibility (i.e. no formation of opening cracks along the plate intersections), each plate needs to be stretched along the circumferential direction. Plastic hinge lines 15 15 15
Plastic hinge lines For an ideal plastic material, the plastic work performed during the folding of a plate of width B/2 reads E b M 4 0 B 2 with M 0 denoting the fully plastic bending moment. For completing an entire fold, we have =p/2 and hence E b M pb 0 B / 2 16 16 16
Fully Plastic Bending Moment The fully plastic bending moment (per unit width) for a plate of thickness t is M 0 1 0t 4 2 assuming an ideal plastic material behavior and a uniaxial stress state in the sheet material (strong simplifications). 0 0 t 0 0 elastic Partially-plastic fully-plastic 17 17 17
Corner stretching The amount of circumferential stretching required depends on the plastic folding wavelength 2H. B / 2 18 18 18
Corner stretching H B / 2 b H H b 3D-view B / 2 top view (after completing fold) For a plastic folding wave length of 2H, we have a maximum hypothetical corner opening of b 2H 19 19 19
Corner stretching B / 2 B / 2 H / 2 H H b H top view (after completing fold) The plastic work required to stretch a plate of initial dimension of 2H x B/2 to the unfolded geometry shown above (right) would be E m H / 2 1 4 ( t 0 ) d t 0H 2 0 Unfolded side view The above is only an estimate of the plastic work knowing that the real kinematics are more complicated. 20 20 20 2
Work balance A square column is composed of eight plates of width B/2. The balance of external work performed by the force P and the internal work then reads: 2H 1 2 p Pd 8 0tH 0t 2 0 4 while the mean crushing force is P m P m 1 2H 2H 0 0t2H Pd 2 tb p H B 21 21 21
Folding wave length The folding wave length is an unknown in our problem. It is common practice to assume that the folding wave length adjusts itself such as to minimize the mean crushing force (Alexander s postulate), H arg min P m [ H ] The minimization problems reads which yields H P m H 0t2 p 2 ptb 2 tb H 22 22 22 0 According to this simple model, the folding wave length of homogenous columns is independent of the material properties. It is monotonically related to the wall thickness and profile width.
Specific Energy Absorption The corresponding mean crushing force then reads P m 2 0 t 2ptB The mass of the column section of height 2H is m 4tB(2H ) 8tBH And hence we have a specific energy absorption of SEA 0 2 2p t B This result suggest that the structural efficiency for absorbing energy is a monotonic function of the t/b ratio. In other words, thick-walled box columns are the best energy absorbers, provided that they deform in a symmetric axial folding mode. 23 23 23
SEA of a material The specific energy absorption can also be evaluated at the material level. The material SEA can be defined as the integral of the stress-strain curve (work density) as normalized by the mass density. The units are therefore the same as those of the structural SEA, i.e. J/g. W V m V d p SEA W m 1 d p p 24 24 24
SEA of a ideal plastic material 0 SEA 1 d 0 p 2 0.5 p 25 25 25
Structural efficiency The folding mode is not very efficient as far as the mass specific absorption of energy is concerned. This can be seen when comparing the structural SEA with the material SEA: SEA 0 2 2p t B Material SEA Example: t=1mm B=60mm SEA.32 2 0 0 26 26 26
Metallic Honeycombs Honeycombs man-made low-density materials that feature a 2D hexagonal microstructure. The porosity of metallic honeycombs made from aluminum foil is typically greater than 95%. 27 27 27
Axial crushing of honeycombs Under uniaxial compression, honeycombs specimens exhibit two distinct phases: (1) crushed/folded material, and (2) undeformed material. These two phases are separated by a crushing front which travels through the specimen. Crushed material Crushed/folded Crushing front undeformed 28 28 28
Axial crushing of honeycombs The stress-strain response exhibits (1) an initial peak, followed by (2) a plateau regime, followed by (3) a densification regime. Constant width (no plastic Poisson s effect) 29 29 29
Detailed FE modeling The axial folding of honeycombs can be described through a detailed FE model of the characteristic unit cell of their periodic microstructure. 30 30 30
Detailed FE modeling A plot of the plastic work density shows that most of the energy is absorbed near the intersection lines of neighboring cell walls. However, large proportions of the folded microstructure contribute only little (blue areas) to the overall energy absorption. a b c d e f g h i j k 31 31 31
Detailed FE modeling A longitudinal cut elucidates the progressive folding mechanism. 32 32 32
Comment on the SEA of honeycombs As shown for thin-walled tubes, structures form folds to decrease the amount of work required to accommodate an applied axial deformation. Many portions of the structures do not contribute to the energy absorption. Consequently, the material SEA of thin-walled honeycombs is significantly lower than that of nonporous homogeneous materials! Honeycombs are nonetheless advertised as excellent materials for energy absorption purposes. This statement can be justified by the fact that many the deceleration may not exceed a given threshold value. F ma crit Due to the constant stress (plateau regime) and zero-poisson effect, it is easy to design protective structures with honeycombs that deform at a constant force (and hence constant deceleration). 33 33 33
FE Analysis with Shell Elements 34 34 34
Basic Shell Elements The main feature of shell elements is that all quantities are expressed with respect to the shell mid-surface. Solid element Shell element It is typically assumed that cross-sections remain straight (Bernoulli hypothesis). In addition to displacement Degrees-Of-Freedom (DOF), the rotation of the cross-section is included as an additional DOF. Dynamore (2013) 35 35 35
Basic Shell Elements There exist numerous shell element formulation and their review is far beyond the scope of this class. Here, we will just comment on a few features of standard shell elements which are based on the Mindlin-Reissner kinematic assumptions (e.g. element S4R in Abaqus or element type 2 in LS-DYNA). The main features are: Dynamore (2013) In the global coordinate system, the shell element features 6 nodal DOF The stresses are computed using the plane stress formulation of the constitutive model Thickness is updated using the Poisson behavior 36 36 36
Basic Shell Elements The virtual work (see FEA class) is computed using numerical integration at the element level. The constitutive law (and hence the stresses) are only evaluated at the location of the integration points. The number of thickness integration points usually needs to be specified by the user. Dynamore (2013) 37 37 37
Reading Materials for Lecture #10 Kocks, Argon and Ashby (1975), Thermodynamics and kinetics of slips M.A. Meyers, Dynamic behavior of Materials C. Roth and (2015), Ductile fracture experiments with locally proportional loading histories, Int. J. Plasticity, http://www.sciencedirect.com/science/article/pii/s0749641915001412 38 38 38