PSI - Issue 28

Claudio Maruccio et al. / Procedia Structural Integrity 28 (2020) 2142–2147 Author name / Structural Integrity Procedia 00 (2020) 000–000

2144

3

the position vector of its normal (i.e. minimum distance) projection point onto the master surface, and ¯ n is the outer normal to the master surface at the projection point. The sign of the measured gap is used to discriminate between active and inactive contact conditions, a negative value of the gap leading to active contact. The electric field requires the definition of the contact electric potential jump: g φ = φ s − ¯ φ m where φ s and ¯ φ m are the electric potential values in the slave node and in its projection point on the master surface. If ρ mech and ρ el are the penalty parameters and g N and g φ are the mechanical and electrical gaps, the contact force term is: F N = ρ mech g N and the electric current term is: I N = ρ el g φ . The energy potential contributions due to mechanical contact is Π c M = 1 2 F N g N and due to electric contact: Π c E = 1 2 I N g φ . Therefore, the total potential energy contribution due to contact is: Π Contact = Π c M + Π c E . According to standard variational techniques, the global set of equations can be obtained by adding to the variation of the energy potential representing the continuum behavior the virtual work due to the electromechanical contact contribution pro vided by the active contact elements. The full set of equations is nonlinear due to the unilateral contact conditions. Consistent linearization of the contribution of the active contact elements is necessary to achieve an asymptotically quadratic convergence rate. Advanced symbolic computational tools available in the AceGen-AceFem finite element environment allow to automate the linearization process above introduced. In particular, if the global energy of the discretized system Π global is defined as: Π global = ∪ H ( u , φ ) + ∪ active Π Contact and if ˆ u is a set of degrees of freedom (DOFs) used to discretize the displacement field u and ˆ φ is a set of DOFs used to discretize the electric potential field φ and ˆ u ∪ ˆ φ is the vector of all nodal DOFs, the residual vector and the sti ff ness matrix terms resulting from the finite ele ment discretization are determined according to these formulas: R u i = δ Π global δ ˆ u i , R φ i = δ Π global δ ˆ φ i , K uu i , j = δ R u i δ ˆ u i , K φφ i , j = δ R φ i δ ˆ φ j , K u φ i , j = δ R u i δ ˆ φ j , K φ u i , j = δ R φ i δ ˆ u j . In this section, three di ff erent case studies are discussed. The systems are modelled with the finite element method in order to predict the voltage, displacement, strain and stress distributions in the solids under application of an external pressure or traction. The non-linear problems are consistently linearized with the automatic di ff erentiation technique and solved using incremental load or displacement control procedures with adaptive time-stepping. First of all, it is analyzed a simple 2D indentation problem with adhesion where a small elastic box is pull out from a large elastic box. The material parameters used in the analysis are given in [1] and a load control procedure is used. Figure 1 shows the mesh employed in the analysis for the upper (body 1, slave) and bottom (body 2, master) elements. The contour plots of the total displacement and the electrical voltage are also provided. Second, to benchmark the 3. Results

0.5

0.312e - 1 0.625e - 1 0.937e - 1 0.125 0.156 0.187 0.218

- 0.35e2 0.30e2 0.25e2 0.20e2 0.15e2 0.10e2 0.50e1

- 1.0

- 0.5

0.5

1.0

AceFEM 0.121e - 3 Min. 0.25 Max.

AceFEM 0.423e2 Min. 0 Max.

- 0.5

- 1.0

(a) Mesh and load conditions

(b) Displacement

(c) Voltage

Fig. 1. Countour levels of displacement and voltage distributions in the contact between two piezoelectric bodies due to traction (with adhesion) loads at the interface.

use of quadrilateral 3D contact elements with adhesion, we analyze a 3D indentation problem where again a small elastic box in adhesion with a substrate is subjected to a traction distributed load on his top. Again, the contour plots of the total displacement and the electrical voltage are given in Figure 2, showing the compressive case result for comparison. Third, a group of piezoelectric polymer fibers is considered. In the numerical model, the cylinders