CMM-2009 – Computer Methods in Mechanics 18–21 May 2009, Zielona Góra, Poland

Theoretical and computational aspects of implementation of anisotropic constitute model formetals with microstructural defects∗

Wojciech Sumelka and Adam Glema

Poznan University of Technology, Faculty of Civil and Environmental EngineeringInstitute of Structural Engineering, Division of Computer Aided Design

ul. Piotrowo 5, 60-965 Poznan, Polande-mail: Wojciech.Sumelka@put.poznan.pl

Abstract

The material model is stated in terms of the continuum mechanics, in the framework of thermodynamics. The aim of the paper is theimplementation of the anisotropic material model as a user subroutine in Abaqus/Explicit environment. The rate form of the anisotropicmaterial model and the applied Eulerian description have enforced the use of the, so called, objective rate, to fulfil the frame indifferencerequirements. The Lie derivative is accepted to obtain the covariant description of the constitutive structure while in Abaqus/Explicitthe Green-Naghdi rate is implemented. The fundamental results of the mathematical anisotropic material model description is given,detailed discussion of the implementation of the Lie derivative in Abaqus/Explicit program is presented and the numerical results foran adiabatic process for anisotropic body with strain rate reaching 104s−1 are shown.

Keywords: anisotropy, objective rate, frame indifference, constitutive relation

1. Introduction

One of the most important requirement, concerning eachphysical theory, is that it must be objective (frame indifferent)[4, 5, 7]. It means that each theory can not depend on the observeror in terms of mathematics, on the selected coordinate system inwhich governing equations are written.

Objectivity, can be obtained in physical theory, if the math-ematical model, used to describe the analysed phenomenon, isbased on the tensorial calculus [5]. Unfortunately, even if we usetensorial calculus to reach objectivity, the obtained theory willnot be unique. The objective rate of defined tensorial fields isnot uniquely defined. There is infinite set of objective rates to beconsidered.

Definition of the objective rates is of the crucial importancein rate type constitutive structures. The objective tensor field,regarding constitutive theories, e.g. stress field, do not have ob-jective material time rate (or direct flux). To reach objectivity onehave to choose, so called “objective rate”.

It will be shown that the consequent use of the objective Liederivative gives us the covariant anisotropic material model de-scription. The anisotropic rate type constitutive structure is ap-plied to high strain rate thermo-mechanical (adiabatic) processby taking into account the user subroutine capability of Abaqusfinite element software. During numerical analysis the explicittime integration scheme is accepted as a most efficient solutionfor analysis where wave effect plays crucial role [6, 3].

2. Anisotropic constitutive structure

2.1. The concept of microdamage tensor

The discussed rate type anisotropic material model structurebases on the phenomenological approach and the fundamentalvariables are introduced as a state variable. Among others wellestablished in literature e.g. equivalent plastic strain, back stress,

the new state variable that governs the local anisotropic proper-ties of microdamage is proposed [2, 3, 6]. This variable is calledmicrodamage tensor and is symmetric second order tensor field- denoted in the paper by ξ. The physical interpretation of thisvariable is that its norm (defined as Euclidean norm)

ξ =√

ξ : ξ, (1)

defines the scalar ξ called volume fraction porosity which is theratio of the void volume to the volume of a material element.

The evolution equation (rate type based on Lie derivative) formicrodamage is proposed in an additive form where the nucle-ation and growth are considered (the growth term is inactive be-fore the nucleation)

Lυξ =∂h∗

∂τ

1

Tm

〈Φ[In

τn(ξ, ϑ,∈p)− 1]〉

︸ ︷︷ ︸

nucleation

+∂g∗

∂τ

1

Tm

〈Φ[Ig

τeq(ξ, ϑ,∈p)− 1]〉

︸ ︷︷ ︸

growth

, (2)

where the scalar functions of tensorial arguments h∗ and g∗ de-scribe the microcrack interaction for nucleation and the microc-rack interaction for growth process, respectively while Tm de-notes the relaxation time for mechanical disturbances, In, Ig

are the stress intensity invariants, τn, τeq denote two thresholdstresses and τ is the Kirchhoff stress.

2.2. Fundamental relations

The fundamental results of the presented theory for adiabaticprocess are governed by the following relations describing theevolution of stress tensor τ and temperature ϑ

Lυτ = Le : d − Lthϑ − (Le + gτ + τg) : dp

, (3)

∗The support of Polish Ministry of Higher Education and Science under grant N N519 419435 “The evolution of properties and failure criteria of materials and structuresunder fast dynamic loadings” is kindly acknowledged.

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CMM-2009 – Computer Methods in Mechanics 18–21 May 2009, Zielona Góra, Poland

and

ρcpϑ = −divq + ϑρ

ρRef

∂τ

∂ϑ: d

+ ρχ∗

τ : dp + ρχ∗∗

K : Lυξ (4)

where Le is linear elastic operator, Lth is thermal expansion op-erator, d denotes the spatial rate of deformation, dp is the rateof viscoplastic deformation, ρ is density, cp is specific heat andχ∗, χ∗∗ are irreversibility coefficients.

3. Material model implementation

The material model structure has been implemented as a usersubroutine VUMAT in Abaqus/Explicit program. As stated pre-viously, in the presented material model, for stress rate (and forall other rates), the fundamental variable in VUMAT subroutine,Lie derivative has been taken into account, thus

Lυτ = τ − lT · τ − τ · l, (5)

while in contrast, in Abaqus/Explicit VUMAT user subroutine,the Green-Naghdi rate is calculated, through the following for-mula [1]

τ(G−N) = τ + τ · Ω − Ω · τ , (6)

where Ω = Ω(G−N) = R · RT represents the angular velocityof the material and R denotes the rotation tensor.

It should be pointed out that material model inAbaqus/Explicit VUMAT user subroutine must be defined inso called corotational coordinate system, which is defined bythe spin tensor Ω (see Fig. 1). To give the physical meaningof the corotational coordinate system, one can say that in thiscoordinate system the stress tensor τ becomes τ = RT τR andwhat is very important that the material time derivative of the

corotational stress tensor τ = RT τ (G−N)R [8].

Figure 1: Initial (X Y Z) and corotational (X Y Z) coordinatesystems

If we assume, that in the iterative procedure forward differ-ence scheme is taken to calculate the material derivative of thesecond order tensor, from Eqns (5) and (6) we obtain

τ |i+1= RT |i+1 [τ |i +∆tLυτ |i

+ ∆t(lT |i ·τ |i +τ |i ·l |i)]

R |i+1 (7)

and

τ |i+1= RT |i+1

[

τ |i +∆tτ(G−N) |i

+ ∆t(Ω |i ·τ |i −τ |i ·Ω |i)]R |i+1, (8)

in corotational coordinate system respectively. Thus, it is clearthat the Green-Naghdi rate, produces an additional term

∆t (Ω |i ·τ |i −τ |i ·Ω |i) . (9)

That is why, one have to subtract term (9) to compute Lie deriva-tive. In the procedure we have formulated the stress update asfollows

τ |i+1= RT |i+1 [τ |i +∆t (2τ |i ·d |i

+Lυτ |i) + Υ |i]R |i+1, (10)

where Υ |i= −∆t |i (Ω |i ·τ |i −τ |i ·Ω |i) and τ |i= R |iτ |i RT |i.

4. Numerical results for adiabatic process

Figure 2: The evolution of microdamage tensor in subsequenttime steps

The numerical results of IBVP for an adiabatic process, withstrain rates reaching 104s−1 will be presented. The evolution ofthe microdamage tensor will be shown to predict the potentialdegradation path before fracture appearance (see Fig. (2)).

References

[1] Abaqus Version 6.8 Documentation Collection, 2008.

[2] Glema, A., Łodygowski, T., Perzyna, P., Sumelka, W. Con-stitutive Anisotropy Induced by Plastic Strain Localiza-tion, 35th SOLID MECHANICS CONFERENCE, Kraków,Poland, September 4-8, pp. 139-140, 2006.

[3] Glema, A., Łodygowski, T., Sumelka, W., Perzyna, P., TheNumerical Analysis of the Intrinsic Anisotropic Microdam-age Evolution in Elasto-Viscoplastic Solids, InternationalJournal of Damage Mechanics, first published on Novem-ber 21, 2008 as doi:10.1177/1056789508097543.

[4] G.A. Holzapfel. Nonlinear Solid Mechanics - A ContinuumApproach for Engineering. Wiley, 2000.

[5] J.E. Marsden and T.J.H Hughes. Mathematical Foundationsof Elasticity. Prentice-Hall, New Jersey, 1983.

[6] Perzyna, P., The Thermodynamical Theory of Elasto-Viscoplasticity Accounting for Microshear Banding and In-duced Anisotropy Effects, 35th SOLID MECHANICS CON-FERENCE, Kraków, Poland, September 4-8, pp. 35-36,2006.

[7] C. Truesdell and W. Noll. The non-linear field theoriesof mechanics, volume in: Handbuch der Physik III/3.Springer-Verlag, Berlin, S: Fluugge, edition, 1965.

[8] H. Xiao, O.T. Bruhns, and A. Meyers. Logarithmic strain,logarithmic spin and logarithmic rare. Acta Mechanica,124:89–105, 1997.

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