23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
c
KTH, Stockholm, 2010
CONSTITUTIVE MODELING AND VALIDATION OF CGI
Goran Ljustina, Martin Fagerstr¨om and Ragnar Larsson
where εp is the total equivalent plastic strain, ˙εp is the equivalent plastic strain rate and where ˙ε0 is a reference strain rate controlling the model rate dependence. The temperature dependence is accounted for via the so-called homologous temperature
θ =ˆ θ − θtrans
θtrans− θmelt (2)
where θtransis the transition temperature def ned as the one at or below which there is no temperature dependence (usually taken as room temperature).
3 HYPO AND HYPERELASTIC-INELASTIC MODELS
Hypoelastic-inelastic constitutive relations means constitutive relations postulated on rate-form in terms of an objective stress rate, cf. subsection 3.1 below for a number of alternatives used in the literature. Generically, for the case of isotropic hardening, these constitutive rate-relations are formulated as
´
τ = E[k] : (l − lp− lth) + h[τ , k] = E[k] : le+ h[τ , k] (3) where ´τ is the considered objective stress rate and where E is a spatial material operator tangent modulus tensor, k is an internal variable associated with the isotropic hardening and l, lpand lthare the total, the plastic and the thermal portion of the spatial velocity gradient def ned as
l= v ⊗ ∇ = ˙F · F−1, lp= λ∂Φ
∂τ, lth= α ˙θ1. (4)
In practice, this means that only the rate of deformation is necessary to determine the stress state, in contrast to hyperelastic-inelastic models where the total deformation is required, and the consequent stress rate behavior is merely a consequence of the hyper elastic-inelastic formulation. The drawbacks of the hypo-formulation are that they lack the property of being (unconditionally) thermodynamically consistent and also that no explicit expression for the mechanical dissipation can be derived.
Therefore, it is of signif cant interest to relate the proposed constitutive relations based on hypoelastic-inelastic response to the more thermodynamically consistent hyperelastic-hypoelastic-inelastic formulation for which the basic assumption is to assume the presence of the isotropic stored energy function ψ ¯C, k, θ function of the reversible part of the deformation, here represented by the reversible right Cauchy-Greenas a deformation ¯C tensor, the temperature θ and an internal hardening variables k. Within the hyperelastic-inelastic framework, the structure of the constitutive relations are based on the second law of thermo-dynamics, which may be specif ed as the dissipation inequality in terms of the second Piola-Kirchhoff stress S as
D = 1
2S : C − ∂ψ
∂ ¯C : ˙¯C−∂ψ
∂k ˙k − 1
θH· ∇Xθ ≥ 0 (5)
which directly yields the corresponding state equations for the intermediate second Piola-Kirchhoff stress
¯
Sand the heat f ux H as
S¯ = 2∂ψ
∂ ¯C, H = −K · ∇Xθ (6)
Goran Ljustina, Martin Fagerstr¨om and Ragnar Larsson
3.1 Objective stress rates for Hypo-formulation of constitutive response
The idea is to scrutinize hypo-formulations based on the spatial Green-Naghdi, Oldroyd and Zaremba-Jaumann stress rates. The two former objective stress rates are obtained based on induced, differently back–rotated stresses - either the material stress tensor T or the second Piola Kirchhoff stress tensor S associated with the Kirchhoff stress τ , respectively. These stresses and their associated objective rates are def ned as
T = Rt· τ · R ⇒ ˙T = Rt· ˆτ· R (7)
S = F−1· τ · F−t ⇒ ˙S = F−1· ˜τ · F−t (8) where ˆτ and ˜τ are the symmetric Green-Naghdi and Oldroyd stress rates, respectively, and where F is the deformation gradient and R is the rotational part of the continuum deformation gradient according to the polar decomposition F = R · U with U being the symmetric right stretch tensor. As a consequence of Eqs. (7) and (8), we f nd that ˆτ and ˜τ are obtained as
ˆ
τ = ˙τ − ω · τ + τ · ω (9)
˜
τ = ˙τ − l · τ − τ · lt (10)
where (again) l is the spatial velocity gradient and ω is a material spin according to ω = ˙R·Rt. For later comparisons, let us introduce also the convective Zaremba-Jaumann stress rate def ned with the subtle difference that ω → w from the Naghdi-Green stress rate so that
~
τ = ˙τ − w · τ + τ · w (11)
where w is the skew symmetric part of the spatial velocity gradient l.
3.2 Hypoelastic-inelastic formulation
Traditionally, for hypoelastic-inelastic consitutive models the stress rate response is postulated for the objective stress rate in terms of the elastic material operator Eeas
´
τ = Ee: le (12)
where Eeis taken as the constant isotropic spatial material tensor
Ee= 2GIdev+ K1 ⊗ 1 with Idev= Isym−1
31⊗ 1 (13)
where Isym is the fourth order symmetric unit tensor. Moreover, G and K are the elastic constants pertinent to shear and volumetric response, respectively.
3.3 Hyperelastic-inelastic formulation
To compare the hypo and hyper formulations let us from Eq. (61) express the rate of the second Piola-Kirchhoff stress tensor as
˙¯
S = 1 2Le
2 : ¯C with Le2 = 4 ∂2ψ
∂ ¯C⊗ ∂ ¯C (14)
Goran Ljustina, Martin Fagerstr¨om and Ragnar Larsson
where Le2 is the elastic second Lagrangian material tangent operator. Push-forward transformation of this relation yields (after some elaborations under the assumption of elastic and plastic isotropy) to the Oldroyd stress rate as
˜
τ = Ee2 : le− 2lp· τ (15)
where Ee2 = ¯F⊗ ¯F : Le2 : ¯Ft⊗ ¯Ft is the elastic second Eulerian material tangent operator induced via Le2. Please carefully note that it was used that (a⊗b)ijkl = aikbjl. Thus, in order to compare with the postulation proposed in Eq. (3), choosing E = Ee2 and h = 2lp · τ implies a thermodynamically consistent formulation in the sense that the postulated Oldroyd rate behavior is in line with a hyperelasto-viscoplastic formulation based on multiplicative split of the deformation gradient.
4 NUMERICAL EXAMPLE
Let us consider next the response at simple shear deformation and uniaxial stress compression. We thus emphasize that different responses are generally obtained depending on which stress rate the rate be-havior Ee: leis postulated with respect to. We thus consider the shear response with respect to the stress rates Green-Naghdi (SGN), Zaremba-Jaumann (SJM) and two version of the Oldroyd stress rate, the f rst one is the ad–hoc model (SOR) where linear elastic response in the Oldroyd stress rate is specif ed, whereas for the second one (SORa) the Oldroyd stress rate is consistent with the hyperelastic–inelastic model as outlined in Subsection 3.3. For comparison, we also consider two hyperelastic-inelastic Neo-Hooke models (NH) and (NHL) where the latter is formulated in logarithmic strains. The results for the simple shear test are shown in Figure 1a, where it is observed that all stress rate formulations yields more or less the same response. The same tendency is observed for the uniaxial compressive test, cf. Figure 1b. However, the results differ slightly at large strains between the consistent neo–Hookean NHL–model and the SGN, SJM, and SORa–models.
a) b)
Figure 1: Stress response in f nite shear deformation a) and uniaxial compression b) REFERENCES
[1] Johnson, G. R. & Cook, W. H. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. Proc. 7th Int. Symp. On Ballistics 541–547 (1983).
[2] Nagtegaal, J. C. & Dejong, J. E. Some aspects on non-isotropic work-hardening in f nite strain plasticity. Proc. of the Workhop on Plasticity of Metals at Finite Strain 65–107 (1981).
[3] Cescotto, S. & Habraken, A.-M. A note on the response to simple shear of elasto-plastic materials with isotropic hardening. European Journal of Mechanics, A/Solids 10, 1–13 (1991).
23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
c
KTH, Stockholm, 2010