17th U.S. National Congress on Theoretical and Applied Mechanics Michigan State University, 15-20 June 2014
Abstract: S-12-453
Modeling Of Soft Materials Via Multiplicative Decomposition of Deformation Gradient Ashraf HADOUSH, Carnegie Mellon University in Qatar,
[email protected] Hasan DEMIRKOPARAN, Carnegie Mellon University in Qatar,
[email protected] Thomas J. PENCE, Dept. Mechanical Eng., Michigan State University,
[email protected]
Finite elasticity is widely used to investigate the response of highly deformable engineering materials. In general for large deformations, the deformation gradient F, and it’s derived quantities, is the basic asset to express stored energy of hyperelastic material model. If other mechanical effects are present in addition to finite elasticity then the decomposition of F = Fˆ F∗ is commonly used. This serves to distinguish and relate particular portions of F to specific parts of the material response. The usual scheme is then that Fˆ models elastic response and it is associated to the rules of variational calculus. The F∗ portion then models none lastic response, usually by means of a time dependent evolution law. Recently, the arguments of variational calculus have been applied to both portions of the deformation gradient decomposition for hyperelastic material [1, 2]. The decomposition itself is then determined by an additional internal balance equation that is generated by such a variational treatment. To demonstrate let us recall a special case of the generalized Blatz–Ko material model from hyperelasticity, see [3], that has a stored energy function W (F) in the form 1/2
W (I2 , I3 ) = µ (I2 /I3 + 2I3
− 5)/2
(1)
where µ > 0 is a material parameter that can be interpreted as a shear modulus. The variables I2 and I3 denote the second and the third principal scalar invariants of C = FT F. In analogy to (1) the internally ˆ in the form balanced material treatment considers a W (F, F) 1/2 ∗ 1/2 W (Iˆ2 , Iˆ3 , I2∗ , I3∗ ) = µˆ (Iˆ2 /Iˆ3 + 2Iˆ3 − 5)/2 + µ ∗ (I2∗ /I3∗ + 2I3 − 5)/2
(2)
where µˆ > 0 and µ ∗ > 0 are material parameters that generalize the role of µ in (1). The variables Iˆ2 ˆ = Fˆ T F. ˆ The variables I ∗ and I ∗ are the and Iˆ3 are the corresponding principal scalar invariants of C 2 3 ∗ ∗T ∗ corresponding principal scalar invariants of C = F F . Uniaxial Loading Consider a cube of such a material with unit length sides in the reference configuration. The principal stretching λ1 then stores a total energy w(λ1 , λ2 ) × 1 where λ2 is lateral contraction. The stored energy function w(λ1 , λ2 ) expresses a reduced form of (1) as
w(λ1 , λ2 ) = µ (2 λ2−2 + λ1−2 + 2 λ1 λ22 − 5)/2
(3)
Let P be the applied longitudinal force that is uniformly distributed and it follows then P = Txx λ22 where Txx is longitudinal Cauchy stress component. Equilibrium under the applied load requires minimization of the E (λ1 , λ2 ) = w(λ1 , λ2 ) − P(λ1 − 1) with respect to λ1 and λ2 and that gives −1/4
λ2 = λ1
−5/2
Txx = µ (1 − λ1
,
).
(4)
Now consider the internally balanced material with W given by (2). Then F is decomposed as λ1 = λˆ 1 ∗ λ1∗
,
λ2 = λˆ 2 ∗ λ2∗
(5) 1
Modeling Of Soft Materials Via Multiplicative Decomposition of Deformation Gradient — 2/2
1 0.9 0.8
Txx / µ hat
0.7 0.6 Conventional and also κ = 0 k = 0.05 k = 0.5 k=1 k = 10 κ=∞
0.5 0.4 0.3 0.2 0.1 0 1
1.5
2
2.5
3
3.5
4
4.5
5
λ1
Figure 1. The evolution of normalized standard Cauchy stress component Txx /µ as expressed in (4)2 and ˆ ∗. the internally balanced normalized Cauchy stress component Txx /µˆ as expressed in (8), where κ = µ/µ whereupon (2) gives w(λ1 , λ2 , λˆ 1 , λˆ 2 ) = µˆ (2 λˆ 2−2 + λˆ 1−2 + 2 λˆ 1 λˆ 22 − 5)/2 + µ ∗ (2 λ2∗ −2 + λ1∗ −2 + 2 λ1∗ λ2∗ 2 − 5)/2
(6)
The same loading scenario is applied to a unit cube, so that now E (λ1 , λ2 , λˆ 1 , λˆ 2 ) = w(λ1 , λ2 , λˆ 1 , λˆ 2 ) − P(λ1 − 1) is to be minimized with respect to λ1 , λ2 , λˆ 1 and λˆ 2 and solving the system leads to −1/4
λ2 = λ1
−1/2
Txx = µˆ λ1
−1/4 λˆ 2 = λˆ 1 ,
, 1/2
(λˆ 1
− λˆ 1−2 ) ,
1/2 −1/2 1/2 µˆ (λˆ 1−2 − λˆ 1 ) + µ ∗ (λˆ 12 λ1−2 − λˆ 1 λ1 ) = 0 .
(7) (8)
Equation (8)1 is the stress-stretch relation. However, unlike the corresponding relation (4)2 for the conventional Blatz-Ko material, the internally balanced model expresses Txx not only in terms of λ1 but also in terms of λˆ 1 . For a given λ1 the associated λˆ 1 is found from the internal balance equation (8)2 , which is the internal balance equation for the material model (2) under uniaxial loading. The evolution of Cauchy stress component Txx is shown in Figure 1. In this figure the conventional hyperelastic Cauchy stress Txx is normalized by µ. For the internally balanced material, the Cauchy stress component Txx is ˆ The hyperelastic stress response Txx /µ coincides with the internally balanced stress normalized by µ. ˆ ∗ → 0. More generally the normalized Cauchy stress Txx /µˆ response Txx /µˆ in the special limit µ/µ ˆ ∗ increases. response shows a progressive softening behavior as the ratio µ/µ
Acknowledgments This work is made possible by NPRP grant # 4-1333-1-214 from the Qatar National Research Fund (a member of the Qatar Foundation). The statements made herein are soley the responsibility of the authors.
References [1]
[2]
[3]
H. Demirkoparan, T. J. Pence, and H. Tsai. Hyperelastic internal balance by multiplicative decomposition of the deformation gradient. Manuscript under Journal Review, 2013. A. Hadoush, H. Demirkoparan, and T.J. Pence. Internally balanced solid response in compressible hyperelasticity described by a deformation gradient product decomposition. In ICMM3, 2013. P. J. Blatz and W. L. Ko. Application of finite elasticity to the deformation of rubbery materials. Trans. Soc. Rheol., 6:223–251, 1962.
