A new framework for large strain electromechanics based on convex multi-variable strain energies: Finite Element discretisation and computational implementation

TLDR: In this article, the Finite Element implementation of two mixed variational principles is presented, as an example, for the simulation of EAPs in the presence of extreme deformations and electric fields.

About: This article is published in Computer Methods in Applied Mechanics and Engineering.The article was published on 2016-04-15 and is currently open access. It has received 51 citations till now. The article focuses on the topics: Finite element method & Piecewise.

Figures

Table 4: Material properties for example 5.3. Ratios c1 and c2 defined as in equation (87).

Figure 9: Torsional actuator. Fiber reinforced dielectric elastomer. Contour plot of the hydrostatic pressure p for (a) ∆ϕ = 5.49MV , (b) ∆ϕ = 7.27MV , (c) ∆ϕ = 8.43MV , (d) ∆ϕ = 10.01MV , (e) ∆ϕ = 10.43MV , (f) ∆ϕ = 10.71MV , (g) ∆ϕ = 11.07MV , (h) ∆ϕ = 11.48MV and (i) ∆ϕ = 11.67MV . Fibers arrangement: θ = ψ = π/4 in X2 > 0 and θ = 5π/4 and ψ = π/4 in X2 < 0. Results obtained with the mixed formulations. Constitutive model defined in (96) and (97) with material parameters given in Table 7. Results shown for a discretisation of (2× 8× 30)× 6 tetrahedral elements (5, 185 × 3 and 5, 185 degrees of freedom associated to the spatial coordinates x and electric potential, respectively).

Table 5: Comparison of performance betweenDPF andMWF formulations. Mesh discretisation details. Column 2: number of tetrahedral elements (Elems.). Column 3: number of degrees of freedom (Dofs.) associated to the spatial coordinates x. Colummn 4: number of degrees of freedom (Dofs.) associated to the electric potential. Column 5: number of degrees of freedom (Dofs.) associated to the fields U ≡ {F ,H ,D0,ΣF ,ΣH ,ΣD0 ,Σd}. Column 6: number of degrees of freedom (Dofs.) associated to the strain/stress fields J,ΣJ .

Figure 2: Three dimensional patch test. (a) View of half undistorted mesh in the reference configuration. (b) View of half distorted mesh in the reference configuration. (c) Example of deformed geometry after applying a voltage difference of ∆ϕ = 30MV in the OX3 direction for the constitutive model defined in (85).

Table 6: Comparison of performance betweenDPF andMWF formulations. Stress components (kPa), electric displacement components (10−4N/mV ) and displacements (m) at points A, B and C. Results obtained using the DPF formulation (columns 2 to 4) and mixed formulations (columns 5 to 7). Prescribed potential difference ∆ϕ = 62.5MV/m. Coarse, medium and fine discretisations of (8× 8× 6)×6, (12× 12× 6)×6 and (16× 16× 6)×6 tetrahedral elements, respectively.

Figure 8: Torsional actuator. Fiber reinforced dielectric elastomer. Contour plot of hydrostatic pressure p and Eulerian electric field E1 for the following spherical parametrisation of the vector N in the direction of the fibers: (a) p and (d) E1 for θ = π/2 and ψ = π/4 in X2 > 0 and θ = −π/2 and ψ = π/4 in X2 < 0, (b) p and (e) E1 for θ = ψ = π/4 in X2 > 0 and θ = 5π/4 and ψ = π/4 in X2 < 0 and (c) p and (f) E1 for θ = ψ = 0. Results obtained with the MWF formulation for ∆ϕ = 11.53MV/m. Constitutive model defined in (96) and (97) with material parameters given in Table 7. Results shown for a discretisation of (2× 8× 30) × 6 tetrahedral elements (5, 185 × 3 and 5, 185 degrees of freedom associated to the spatial coordinates x and electric potential, respectively).

Frequently asked questions

Q1. What are the contributions mentioned in the paper "A new framework for large strain electromechanics based on convex multi-variable strain energies: finite element discretisation and computational implementation" ?

The aim of this paper is to present, as an example, the Finite Element implementation of two of these mixed variational principles. These types of enhanced methodologies are known to be necessary in scenarios in which the simpler displacement-potential based formulation yields non-physical results, such as volumetric locking, bending and shear locking, pressure oscillations and electro-mechanical locking, to name but a few. This results in mixed formulations with a computational cost comparable to that of the displacement-potential based approach, yet far more accurate. 

Q2. What are the future works in "A new framework for large strain electromechanics based on convex multi-variable strain energies: finite element discretisation and computational implementation" ?

The new framework presented opens very interesting possibilities to similar multi-physics problems within the field of smart materials, which are the current focus of the authors. 

Q3. What is the well accepted constitutive inequality?

The most well accepted constitutive inequality, namely ellipticity, is automatically satisfied if the internal energy density functional e per unit of undeformed volume e = e(∇0x,D0) is defined as [1]e (∇0x,D0) = W (F ,H , J,D0,d) ; d = FD0. 

Q4. How does the extended set of variables enables the introduction of work conjugates?

the extended set of variables {F ,H , J,D0,d} enables the introduction of work conjugates {ΣF ,ΣH ,ΣJ ,ΣD0 ,Σd}, where the satisfaction of multi-variable convexity guarantees that the relationship between both sets is one to one and invertible. 

Q5. What is the importance of the new definition of multivariable convexity?

It should be emphasised that the new definition of multivariable convexity ensures [1] the material stability and well posedness of the equations. 

Q6. What are the discrete stationary conditions of the variational principle W?

(37)Virtual and incremental variations of the elements in the sets Y and ΣY in above equation (37) are denoted asδY = {δF , δH , δJ, δΣD0}; δΣY = {δΣF , δΣH , δΣJ , δΣd}; ∆Y = {∆F ,∆H ,∆J,∆Σd}; ∆ΣY = {∆ΣF ,∆ΣH ,∆ΣJ ,∆Σd}. (38) The discrete stationary conditions of the variational principle ΠW (31) with respect to virtual changes in the geometry and in the electric potential (i.e. equilibrium and Gauss law) areDΠW [δu] = nx ∑a=1Ra x · δua; Ra x =∫VP W ∇0N x a dV − ∫Vf 0N x a dV − ∫∂tVt0N x a dA;DΠW [δϕ] =nϕ ∑a=1Raϕ · δϕa; Raϕ = ∫VD0 ·∇0Nϕa dV − ∫Vρ0N ϕ a dV −∫∂ωVω0N ϕ a dA,(39) where the first Piola-Kirchhoff stress tensor P W is evaluated asP W = ΣF + ΣH Fx + ΣJHx + Σd ⊗D0. (40)The discrete stationary conditions of the variational principle ΠW (31) with respect to virtual changes of the elements of the set Y and D0 (i.e. constitutive and Faraday laws) isDΠW [δY , δD0] = nF ∑a=1Ra F : δF a +nH ∑a=1Ra H : δHa +nJ ∑a=1RaJδJ a+nd ∑a=1Ra d · δda +nD0 ∑a=1Ra D0 · δDa0,(41)17where the different residuals emerging in above equation (41) areRa F =∫V(∂W ∂F −ΣF)NFa dV ; R a H =∫V(∂W ∂H −ΣH)NHa dV ;RaJ =∫V(∂W∂J − ΣJ)NJa dV ; R a d =∫V(∂W∂d −Σd)Nda dV ;Ra D0 =∫V(∂W ∂D0 + ∇0ϕ+ Fx TΣd)Nda dV .(42) The discrete stationary conditions with respect to the elements of the setΣY (37) (i.e. compatibility) yieldsDΠW [δΣY ] =nΣF ∑a=1Ra ΣF : δΣF a +nΣH ∑a=1Ra ΣH : δΣH a +nΣJ ∑a=1RaΣJ δΣ a J+nΣd ∑a=1Ra 

Q7. What are the discrete stationary conditions of the variational principle in (67)?

The discrete stationary conditions with respect to the elements in the set D (67) and Σd (i.e. constitutive laws) areDΠΦ[δD, δΣd] = nF ∑a=1Ra F : δF a +nH ∑a=1Ra H : δHa +nJ ∑a=1RaJδJ a+nΣD0 ∑a=1Ra ΣD0 · δΣa D0 +nΣd ∑a=1Ra 

Q8. What is the definition of the stiffness matrix?

The stiffness matrix contribution KabDΣD arising from the linearisation of R a D with respect to incremental changes in the elements of the set ΣD is defined asKabDΣD = − ∫V NFa N F b The author0 0 00 NHa N H b The author0 0 0 0 NJa N J b 0 0 0 0 −ND0a ND0b The author dV . (81)Finally, the expression for the matrix KabDΣd emerges asKabDΣd =∫V[NFa N d b ΦFΣd N H a N d b ΦHΣd N J a N d b ΦJΣd N D0 a N d b ΦΣD0Σd ] dV .(82) The stiffness matrix contribution KabΣDΣd yieldsKabΣDΣd = [ 0 0 0 Kab D0Σd ] ; Kab D0Σd =∫VND0a N d b F T x dV . (83)The last stiffness matrix involved in the variational principle ΠΦ (33), namely, KabΣdΣd is obtained asKab ΣdΣd =∫VND0a N Σd b ΦΣdΣd dV . (84)25The objective of this section is to present a series of numerical examples in order to prove the robustness, accuracy and applicability of the computational framework presented above. 

Q9. What is the order of convergence of the Jacobian J and its work conjugate?

As expected, the constant interpolation of the Jacobian J and its work conjugate ΣJ affects the optimal convergence of the different variables, specially those purely mechanical and the pair {d,Σd}, for which a decrease of one is observed in the order of convergence. 

Q10. What is the stiffness matrix contribution associated to the residual RaY?

The stiffness matrix contribution associated to the residual RaY , namely K ab YD0 is defined asKabYD0 =∫V[NFa N D0 b WFD0 N H a N D0 b WHD0 N J a N D0 b WJD0 N d aN D0 b WdD0 ] dV .(53) 

Q11. What is the advantage of using a constitutive model defined by the energy functional in (10)?

An advantage of employing a constitutive model defined by the energy functional in (10) is that it ensures the existence of the Helmholtz’s energy density ab initio, which cannot be necessarily guaranteed otherwise. 

Q12. What is the significance of the Baker-Ericksen inequality?

Bustamante and Merodio [16] studied under what ranges of deformation and magnetic field the Baker-Ericksen inequality [15] would be compromised, specifically considering smart materials belonging to the class of magneto-sensitive elastomers2. 

Q13. What is the difference between the two mixed formulations?

For the two mixed formulations included in this paper, the use of interpolation spaces in which the sets {F ,H , J,D0,d} and its dual counterpart {ΣF ,ΣH ,ΣJ ,ΣD0 ,Σd} are described as piecewise discontinuous across elements enables these fields to be resolved locally, following a standard static condensation procedure, thus leading to a computational cost comparable to that of the displacement-potential based approach, yet with far more accuracy. 

Q14. What is the procedure to obtain the stiffness matrix contribution of Kabxx?

The procedure to obtain the stiffness matrix contribution Kabxx has beenalready presented in Reference [24] in the context of nonlinear elasticity[Kab xx]ij = Eijk[kab xx]k ; kab xx =∫V(ΣH + ΣJF x) (∇0N x a ×∇0Nxb ) dV .(46)