A Formulation for Hyperelastic Damaged Materials

Transcription

1 Paper Civil-Comp Press, 2015 Proceeings of the Fifteenth International Conference on Civil, Structural an Environmental Engineering Computing, J. Kruis, Y. Tsompanakis an B.H.V. Topping, (Eitors), Civil-Comp Press, Stirlingshire, Scotlan A Formulation for Hyperelastic Damage Materials M. Miñano an F.J. Montáns Escuela Técnica Superior e Ingeniería Aeronáutica y el Espacio Universia Politécnica e Mari, Spain Abstract Hyperelastic materials are characterize by an energy-preserving behavior which results in an ientical path for loaing, unloaing an reloaing. Rubbery materials consist frequently on a cross-linke elastomeric substance containing some percentage of particles of carbon as fillers. Due to such fillers, for example, the virgin loaing path iffers substantially from the unloaing-reloaing one. The behavior may then be consiere as a result of amage in the material. This effect is known as part of the Mullins effect an is present not only in carbon-fille rubbers but also in biological materials. In this paper we present a novel formulation of continuum amage mechanics in hyperelastic materials an an efficient computational proceure for moelling the Mullins effect in isochoric, isotropic materials. The formulation is base on the iea that unamage hyperelastic behavior cannot be measure, but the unloaingreloaing behavior of a amage material can be reaily obtaine an on that only the unloaing-reloaing curve presents a real hyperelastic behavior. The unloaingreloaing curve may be escribe by any appropriate constitutive moel but using spline-base functions both the virgin loaing an the unloaing-reloaing curves are exactly capture. The moel is efficient for finite element implementation. Keywors: hyperelasticity, logarithmic strains, Mullins effect, amage mechanics, living tissues, polymers. 1 Introuction Many materials like biological tissues an rubbers are usually moelle as hyperelastic isochoric materials [1, 2, 3, 4, 5]. Hyperelastic materials preserve the energy uring close cycles. Nevertheless many of them isplay a issipative behavior known 1

2 as Mullins effect [6, 7, 8, 9, 10, 11, 12] which is frequently moelle as without permanent strains. There is as yet no wiesprea agreement on the physical phenomena involve whose consequence is a reuction of stiffness [12, 13, 14, 15, 16, 17, 18] from a phenomenological point of view. In practice, several other inelastic effects arise uner loaing an unloaing cycles as resiual strains [19, 20], epenence of the shape of the stress-strain curves with rate an temperature, inuce anisotropy, among others. The simplest approach is to moel the effect as an isotropic softening effect through continuum-base amage moels [2, 5, 12, 14, 17, 21, 22, 23, 24]. The most wiely use approach for the case of isotropic amage is to suppose an unamage store energy function an then apply the reuction factor (1 D) where D [0,1) is the scalar amage variable, first propose by Kachanov [25]. Actually, an this is a main point of this paper, the unamage store energy function can not be measure, but only the amage one. Aitionally a amage criterion an an evolution function [2, 26, 27] are usually establishe for the amage variable. Finally, a parameter-fitting proceure is use to obtain the material constants of the amage hyperelastic moel. We propose a ifferent approach in which using spline-base hyperelastic moels we are able to capture exactly the experimental stress-strain curves without using any fitting material parameter [28, 29]. Then a new formulation is evelope in which the unloaing-reloaing hyperelastic curves are irectly obtaine from the experimental ata. In the following sections, we first present the continuum constitutive equations for isochoric, isotropic amage following a new approach similar to the split operator typical of computational plasticity. Then we introuce the amage scalar variable. Finally we give some illustrative examples. 2 Theoretical backgroun Let E be the logarithmic strains in the reference configuration an T the generalize Kirchoff stress, work-conjugate of the logarithmic stress [30]. Let w D be an internal variable an W(E,w D ) the store energy per unit reference volume. The conservation of energy states that then, Ẇ = T : Ė = W(E,w D) E T = W(E) E = W(E,w D) E : Ė + W (E,w D) w D w D E : Ė (1) ẇd =0 + W (E,w D) w D E Ė=0 w D The ifferentiation in Equation (2) can be construe as a trial-hyperelastic stress preictor an a amage-stress-corrector parts which is a typical approach employe in computational elastoplasticity an viscoleasticity. However, as a remarkable ifference with computational plasticity an viscoelasticity, the trial an final states correspon to the same strain. (2) 2

3 The uncouple form of the store energy function is given by W = U (J)+W ( E,w D ) (3) For the isotropic case, the Valanis-Lanel hypothesis states for a fixe value ofw D W ( λ 1,λ 2,λ 3,w ) ( ) ( ) ( ) D = ω λ 1,w D + ω λ 2,w D + ω λ 3,w D (4) Usually amage is consiere to affect only the isochoric contribution an to not affect incompressibility, so W (lnλ 1,lnλ 2,lnλ 3,w D ) = ω(e 1,w D )+ω(e 2,w D )+ω(e 3,w D ) (5) We efine the amage variable as the maximum isochoric energy reache up to time t ( w D = max W (τ) ) (6) τ (,t) A similar amage variable has been introuce by several authors, see for example [22, 27, 31]. For ifferent level of amage we have W ( E,w D1 ), W ( E,w D2 ),... (7) an uring unloaing-reloaing without further amage W D1( E ), W D2( E ),... (8) The functions of the Equation (8) are true hyperelastic functions, however the Equation (4) is a potential function which epens on amage (issipative variable). Each energy function of Equation (8) can be written using the Valanis-Lanel ecomposition as ( ) ( ) ( ) ( ) E = ω Dn E 1 +ωdn E 2 +ωdn E 3 (9) W Dn Experimental eviences show that amage ecreases the store energy. We efine the energy release rate tensor as Y = W( ) E,w D w D = γd (10) w D E whereγ = W ( E,w D ) / wd is the scalar amage power loss factor. The store energy rate results Ẇ= U (J)+Ẇ( ) E,w D ( U (J) ) ( ) = E : Ė + W D E : E E E E :,w D w D Ė+ W w }{{ D E : Ė }}{{} 0 0 (11) (12) = T v : Ė +T : Ė (13) 3

4 strain energy release rate trial state Virgin loaing curve release energy Master loaing-unloaing curve ψ =1 store energy stress Figure 1: Representation of the the virgin loaing an master loaing-unloaing curves, of the store an release energies an efinition of the trial state. where tr T is efine as the trial isochoric Kirchhoff stress tensor, obtaine with the amage frozen,y is the correction ue to amage an T is the resulting isochoric Kirchhoff stress tensor. Finally this leas to T = tr T for no amage increase; ẇ D = 0 T = tr T Y for amage increase; ẇ D > 0 A representation of all these terms is given in Figure 1. Accoring to the amage variable introuce, we establish the following amage criterion f D = W w D 0 W w D (14) The consistency conition uring amage process states that ( f D = tr T Y w ) D : Ė = 0 forẇ D > 0 (15) E Consiering that the evolution of amage can only occur when there is increasing eformation Ė 0 From Equation (10) D := w D E =tr T Y = T (16) Y = γ 1+γ tr T T = 1 1+γ tr T (17) Compare with computational elastoplasticity, the Equation (17) is basically similar to the raial return algorithm of Wilkins [32]. 4

5 It can be shown [33] that the Valanis-Lanel ecomposition results into a expression of the type W (E 1,E 2,E 3,w D ) = ψ(w D )W 0 (E 1,E 2,E 3 ) (18) wherew 0 is a reference, largely amage, store energy ( ( ) ( ) ( ) E 1,E2 3),E = ω0 E 1 +ω0 E 2 +ω0 E 3 W 0 (19) The factor of proportionalityψ(w D ) is obviously ifferent for each level of amage. In summary, only two scalar functions, which can be obtaine from a tensile loaingunloaing test, are require. The energyw0 can be obtaine from the user-prescribe spline-base store energy function for the maximum expecte strain. The factor of proportionality ψ(w D ) is etermine as the quotient between the stress given by the uniaxial virgin testt an the stress given by the master hyperelastic curvet 0. ψ(w D (E)) = T T 0 (20) Equation (18) is similar to that of classical amage formulations [2], nevertheless it shoul be note that in our formulation it is a result of isotropy an Valanis-Lanel ecomposition. It is noteworthy that the unamage energy use in most of the moels [2, 22, 27, 34] cannot be obtaine experimentally whereas the amage one W 0 can be reaily etermine. In aition, the virgin loaing curve cannot be obtaine from a hyperelastic curve because itself involves amage which is a issipative process. 3 Examples We give in this section some illustrative examples of the moel. The examples are selecte, with assume test ata, simply to show the capabilities of the moel. The material ata neee for the moel is merely an experimental virgin loaing curve an the master amage unloaing-reloaing curve. This master unloaing-reloaing curve must be given up to the maximum expecte strain. We have interpolate both experimental curves using piece-wise splines in orer to obtain analytical, continuous functions. The original ata employe an the resulting uniform spline interpolation are shown in Figure 2. The material is moele as quasi-incompressible. Therefore, we introuce a penalty volumetric function with a large equivalent bulk moulus in orer to numerically enforce incompressibility when require. No aitional information is neee for the moel. As a first simulation we have impose the uniaxial strain historye 1 shown in Figure 3a. As a result of incompressibility E 2 = E 3 = 0.5E 1. As shown in Figure 3c, the preicte stress-strain ata exactly simulate the prescribe ata. Note that we represent T 1 T 2 to remove the ineterminate pressure. In Figure 3b it can be seen the resulting amage variable evolution versus the strain. Finally in Figure 3 we illustrate the evolution of the reference isochoric store energy uring the simulation. 5

6 150 Uniaxial stress [MPa] Sample ata from virgin loaing Sample ata from unloaing reloaing Uniform master amage spline Uniform hyperelastic spline Uniaxial logarithmic strain Figure 2: Original stress-strain ata an spline fit. As a secon simulation, we prescribe a simple shear history which eformation graient is 1 γ(t) t 0 X = 1 (21) 1 where γ(t) is the shearing parameter. Notice that the impose eformations are isochoric. In Figure 4a is shown the shear history. The resulting strains are shown in Figure 4b. The preictions using the moel are shown in Figure 4c an Figure 4. We note that the shear stress reaches a maximum epening on the nonlinearity of the store energy with the logarithmic shear strain. This is an observation inherent to the simple shear test at large strains an which is physically soun [30]. Logarithmic strain measures simply reveal this fact. We can observe that for large strains the componentst 11 ant 22 reach values much larger than shear stresses [30]. The evolution uring the simulation of the amage multiplierψ an of the amage variablew D are given in Figure 5. As final simulation, in orer to evaluate the capabilities of the moel we have prescribe a combine nonproportional axial-torsion loa test. This test has been stressriven. Figure 6a shows the prescribe stresses where T 22 = T 33 = T 13 = T 23 = 0. The resulting strains are given in Figure 6b. Figure 7 represents the evolution of the reference isochoric energy W0 an of the amage variablew D. In the table of Figure 8 we represent the numerical performance of the algorithm. We have use plain Newton algorithms both locally an globally. We have selecte two global step, one in which there is amage evolution an another step where no amage evolution occurs. The local column shows the number of local iterations use in obtaining the energy multiplier parameter ψ until the consistency conition is sat- 6

7 a) b) c) ) Figure 3: Preictions for the uniaxial test. a)prescribe uniaxial strain. b) Evolution of the amage variable uring the simulation. c) Preictions an uniaxial prescribe ata. ) Evolution of the reference isochoric store energy uring the simulation. isfie with a relative tolerance of the amage function of For this example, the typical number of iterations employe per global iteration is 3. The global column shows the number of global equilibrium iterations require in each step using a consistent tangent [33]. Usually it employs 4-5 iterations per step for a relative tolerance of the loas of We note that the computational algorithm may be consistently linearize an if so, secon orer convergence is achieve. In fact, note that iterations are ue to the hyperelastic response except uring amage evolution [33]. 4 Conclusions In this work we present a new formulation an computational proceure for amage in isotropic, isochoric hyperelastic materials. The approach is base on the iea that is not possible to measure the unamage store energy function, but only the amage one. The continuum formulation is base on an operator split which can be interprete as a trial-hyperelastic stress preictor an a amage-stress-corrector. Employing spline-base store energy functions, we are capable of replicating both the virgin loaing an the unloaing-reloaing behavior. We have shown that the algorithm is very efficient per finite element simulations. 7

8 a) b) c) ) Figure 4: Simple shear test preiction. a) The time history for the shear parameter. b) Resulting strains compute from the eformation graient. c) Preictions for the T 12 stress versus the shear parameter. ) Preictions for the componentest 11 an T 22. Acknowlegements Partial financial support for this work has been given by grant DPI from the Dirección General e Proyectos e Investigación of the Ministerio e Economía y Competitivia of Spain. References [1] R.W. Ogen, Non-linear elastic eformations, Dover, New York, [2] G.A. Holzapfel, Nonlinear Soli Mechanics: A Continuum Approach for Engineering, Wiley, Chichester, [3] M.T. Shaw, W.J. MacKnigth, Introuction to Polymer Viscoelasticity, John Wiley & Sons., [4] A.S. Argon, The Physics of Deformation an Fracture of Polymers, Cambrige University Press, [5] J.D. Humphrey, Cariovascular Soli Mechanics: Cells, Tissues, an Organs, Springer Science & Business Meia,

9 a) b) Figure 5: a) Evolution of the amage multiplier uring the simple shear test. b) Evolution of the amage variable uring the simulation. Kirchhoff stress [MPa] T 11 T Step 4 E 11 E 22 Log. strain E 33 E Step Figure 6: Axial-torsion test. a) History of the impose axial stresst 11 an shear stress T 12. b) Preictions for the strain componentse 11, E 22, E 33 an E 12. 9

10 20 W wd 50 ψ Step Figure 7: Axial-torsion test. a) Evolution of the reference isochoric energy. b) Evolution of the amage variable. Figure 8: Axial-torsion test. Number of local an global iterations employe when there is amage evolution an when no amage evolution occurs. 10

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