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A hyperelastic fractional damage material model with memory
Accepted Manuscript
A Hyperelastic Fractional Damage Material Model with Memory Wojciech Sumelka, George Z. Voyiadjis PII: DOI: Reference:
S0020-7683(17)30290-1 10.1016/j.ijsolstr.2017.06.024 SAS 9631
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
3 March 2017 7 June 2017 19 June 2017
Please cite this article as: Wojciech Sumelka, George Z. Voyiadjis, A Hyperelastic Fractional Damage Material Model with Memory, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.06.024
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A Hyperelastic Fractional Damage Material Model with Memory
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Wojciech Sumelka∗ , George Z. Voyiadjis∗∗
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∗
Poznan University of Technology, Institute of Structural Engineering, Piotrowo 5 street, 60-969 Pozna´n, Poland
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[email protected]
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∗∗
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Boyd Professor, Department of Civil and Environmental Engineering, Louisiana State University,
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Baton Rouge, LA 70803, USA
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[email protected]
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Keywords: hyperelasticity; scalar damage; fractional calculus; materials; damage mechanics.
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ABSTRACT
In this paper a scalar damage model for hyperelastic materials is considered. The novelty of
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the proposed approach lies in the evolution law for the damage variable that is formulated with
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the application of fractional calculus. In this way damage evolution includes the memory, or
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in other words the current intensity of damage evolution, which is based on information from
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the past - whose length is included in the fractional operator. Based on illustrative examples,
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the flexibility of the model to mimic experimentally observed material behaviour is presented.
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1 Introduction
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Continuum damage mechanics (CDM) has reached a high level of its maturity nowadays.
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Herein on should mention the first concepts proposed by Kachanov [1] and Rabotnov [2], to1
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[4], Woo and Li [5], Krajcinovic [6], Chen and Chow [7], Voyiadjis et. al. [8, 9], Perzyna
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[10], Sumelka [11] or competetive formulations like these quantified by irreversible entropy
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and entropy generation rate by Basaran [12, 13, 14] or Sosnovskiy [15]. Within this differ-
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ent types of damage models, one of the most important aspect is the definition of damage as
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a mathematical object, namely one considers isotropic (scalar) and anisotropic (higher order
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tensors) measures for material degradation. CDM concepts were applied for different types
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of materials (e.g. metals, rocks, concrete, composites), a wide range of degradation types (e.g.
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elastic-brittle damage, elastic-plastic damage, fatigue damage, corrosion damage) and different
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processes (e.g. static, dynamic, coupled fields) - cf. Skrzypek and Ganczarski [16]. Nonethe-
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less, constant development of new materials or advancing knowledge of existing ones (e.g.
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biological) induces the necessity of further development of CDM, especially in the field of
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definition of damage initiation and evolution criteria i.e. specific evolution equations for the
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internal state variables. This crucial aspect can be formulated using different mathematical
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tools, however as presented in recent papers, the fractal theory (for definition of fractal dam-
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age variable) [17] or fractional calculus (for fatigue phenomena modeled as a change of the
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fractional derivative order) [18] can be very effective. Herein, the new insight will be posed to
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the last mentioned modelling technique for finite strains and original damage evolution law.
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Fractional calculus (FC) is a branch of mathematical analysis which considers differential
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equations of an arbitrary order [19, 20, 21]. From the point of view of mathematical modelling
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of physical phenomena, the important aspects of FC are: (i) fractional differential operators
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(FDO) are defined over an interval; (ii) there are infinitely many definitions of FDO [22]; (iii)
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each FDO can have some specific properties [23] (e.g. FDO operating on a constant function
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does not necessarily result in zero). It should be pointed out that to some extent, dependent on
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the material being considered, there should exist the most proper choice of FDO.
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It is important that the non-local action of FDO mentioned above, can operate on different
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spaces, dependent on variables on which FDO acts (cf. Fig. 1). Namely, it can be a non-local 2
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local action in a stress space (e.g. non-normal plastic flow [25]), or non-local action in space
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variable (cf. [26, 27, 28, 29, 29, 30]). Herein, the non-locality in time, should be pointed out
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as being the crux of this paper and is called memory [20]. For such class of non-local concept,
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one can distinguish two cases: (i) full memory - when time fractional derivatives are taken over
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the total time of the analysed process; and (ii) short memory - when the current state of the
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mechanical system is influenced by the closest past events, characterised by the existence of a
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characteristic time length scale `t (cf. Fig. 1).
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Figure 1: Possible non-localities in mechanics vs. FDO operation on a specific field
In a further part of this paper we explore the concept of short memory connected with the
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definition of (scalar) damage parameter evolution in terms of FC (fractional calculus) for hy-
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perelastic materials. It will be observed that both, the order of damage evolution (order of
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FC) and the applied characteristic time `t (range of FC action) control the intensity of damage
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expansion - cf. Fig. 2. Thus, in this new formulation, two additional material parameters are
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needed for practical applications, however, with a very positive and desired property of the
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overall model, namely its flexibility to map the experimental results.
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Figure 2: The concept of short memory
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2 Problem formulation
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2.1
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In this work isotropic hyperelasic materials are considered. It is assumed that there exists a
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stress threshold above which damage appears in the material. Damage is modelled through
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a scalar variable, thus the damaged range of the material behaviour is isotropic as well. The
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evolution of damage is formulated with the application of the Caputo fractional derivative,
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hence the current state of damage is influenced by past events - one says that the material has
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a memory, or that the material is non-local in time.
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Main Assumptions
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2.2
Isotropic Hyperelastic Fractional Damage Material Model with Memory
One assumes the existence of the Helmholtz free-energy function in the general form
Ψ = Ψ(F),
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(1)
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where Ψ denotes the Helmholtz free-energy function, and F stands for deformation gradient.
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Assuming that the elastic range is isotropic Eq. (1) reduces to
Ψ = Ψ(U) = Ψ(C) = Ψ(E),
(2)
where U is the right stretch tensor (U2 = C), C is the right Cauchy-Green tensor and E
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denotes the Green-Lagrange strain tensor (2E = C − I).
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For convenience the following restrictions for Ψ are assumed:
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Ψ = Ψ(I) = 0,
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and in general
Ψ(F) ≥ 0, and
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Ψ is objective.
(5)
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˜ Ψ = Ψ(C, φ) = (1 − φ)Ψ(C),
(6)
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where φ is the scalar damage variable, together with the postulates:
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(4)
Next, one introduces the scalar damage concept based on the assumption that
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(3)
˜ Ψ(I) = 0,
(7)
˜ Ψ(C) ≥ 0,
(8)
˜ is objective, Ψ
(9)
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and (10)
φ ∈ [0, 1], ˜ is the effective (undamaged) Helmholtz free-energy function. where Ψ
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Based on the assumption that the Clausius-Planck inequality for purely mechanical processes
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can be written as
˙ =S:E ˙ ≥ 0, ˙ −Ψ Dint = wint − Ψ
(11)
˜ ˙ ˜ ˙ = (1 − φ) ∂ Ψ(C) : C ˙ − Ψ(C) φ, Ψ ∂C
(12)
and utilizing the chain rule
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where Dint is the density of dissipation, wint is the internal energy density, and S denotes the
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second Piola-Kirchhoff stress tensor (S = F−1 P = ST and P is the first Piola-Kirchhoff stress
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tensor). Therefore, one obtains
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thus
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!
:
˙ C ˜ + Ψ(C) φ˙ ≥ 0. 2
(13)
From Eq. (13) it is clear that:
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˜ ∂ Ψ(C) S − 2(1 − φ) ∂C
S − 2(1 − φ)
˜ ∂ Ψ(C) = 0, ∂C
(14)
˜ S = (1 − φ)S,
(15)
˜ Ψ(C) φ˙ ≥ 0,
(16)
and
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˜ ˜ = 2 ∂ Ψ(C) where S is the effective (undamaged) second Piola-Kirchhoff stress tensor. It is ∂C
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observed, that for the non-negative dissipation defined in Eq. (16), together with postulate
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Eq. (8), the following is noted φ˙ ≥ 0.
Finally, the evolution of φ is postulated utilising the fractional differential operator in the form
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C α t−`t Dt φ
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1 = αΦ T
where the bracket h·i defines the ramp function, 1 = Γ(n − α)
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a
t
C
Iφ −1 , τφ
(18)
D is the left-sided Caputo derivative
f (n) (τ ) dτ, (t − τ )α−n+1
for
t > a,
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(17)
where t denotes time variable, α is an order of derivative, T stands for characteristic time, Φ is
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the overstress function, Iφ is the stress intensity invariant, τφ is the threshold stress for damage
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evolution, Γ is the Euler gamma function, and n = bαc + 1. It should be emphasised that the
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fractional velocity of damage to be defined needs identification of the damage order α and the
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damage memory (time length scale) `t , in other words two additional material parameters are
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postulated. This is in contrast with the classical case when α = 1 and memory does not play
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any role. Moreover, the form of the evolution equation Eq. (18) induces rate dependence of the
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overall constitutive model.
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3 Examples
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3.1
Introductory Remarks
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The main aim of this section is to present the influence of the fractional damage evolution
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model on the material model defined in the preceding section. Due to this reason the effect of
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both the damage order index α and the memory (time length scale) `t is examined in detail.
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Furthermore, this parametric study is enriched by the examination of the rate effects.
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3.2
Ogden Strain-Energy Function
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One assumes that the stain-energy function postulated in Eq. (6) takes the form of the Ogden
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model, namely ˜ = Ψ(λ ˜ 1 , λ2 , λ3 ) = Ψ
N X µp p=1
αp
α
α
α
(20)
(λ1 p + λ2 p + λ3 p − 3),
where λ1 , λ2 , λ3 are the principal stretches (principal values of U), and µp , αp
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are the material parameters for Ogden model. This allows one to analyse a wide class of other
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common models like:
(p = 1, ..., N )
• Mooney-Rivilin model N = 2, α1 = 2, α2 = −2 with the constraint condition I3 (C) = λ21 λ22 λ23 = 1,
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• neo-Hookean model N = 1, α1 = 2, or
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• Varga model N = 1, α1 = 1.
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3.3
The Constitutive Equation in the Principal Stretches Directions
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Without loss of generality one limits the considerations to incompressible hyperelastic bodies.
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Therefore, the constitutive model given by Eq. (14) can be now rewritten in the principal stretch
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˜ = Ψ(λ ˜ 1 , λ2 , λ3 )) [31] directions form, namely (Ψ
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˜ 1 1 ∂Ψ Sa = (1 − φ) − 2 p + λa λa ∂λa
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,
a = 1, 2, 3,
(21)
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where Sa are the principal values of S, and p denotes the hydrostatic pressure.
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Based on Eq. (21) the transformation to the principal Cauchy stress, the most intuitive stress
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measure, is straightforward and is given by ˜ ∂Ψ σa = λ2a Sa = (1 − φ) −p + λa ∂λa 8
!
,
a = 1, 2, 3,
(22)
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where σa are the principal values of Cauchy stress tensor σ = J −1 FSFT , (J = det(F)), and
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p has to be determined from the balance of linear momentum and the boundary conditions.
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3.4
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To understand the concept of fractional damage with memory, without loss of generality, one
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considers the behaviour of the overall model for uniaxial (incompressible) tension, hence
x = χ(X, t) = (1 + tnβ )X1 e1 + √
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Solved Example
1 1 X2 e 2 + √ X3 e3 , n β 1+t 1 + tnβ
(23)
where nβ = const - thus parameter nβ controls the rate of the process Eq. (23), χ is the motion,
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e is the base vector in current configuration, and x, X are spatial and material coordinates,
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respectively.
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Next, the deformation gradient for the motion of Eq. (23) in the matrix representation can be
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expressed as
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(1 + tnβ ) 0 ∂χ(X, t) F = F(X, t) = = √ 1n 0 ∂X 1+t β 0 0
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1
√
nβ
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1+t
.
(24)
Finally, the corresponding right-stretch tensor needed for stress calculation (cf. Eqs (21-22)) is given by
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λ1 0 0 (1 + tnβ ) 0 U = 0 λ2 0 = √ 1n 0 1+t β 0 0 λ3 0 0
0 0 √
1 nβ
1+t
.
(25)
Based on the calculated stretches (Eq. (25)) the associated stress state resulting from the Ogden 9
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˜ = Ψ(λ ˜ 1 )): model is given by (Ψ ˜ 1 ∂Ψ 1 S1 = (1−φ) − 2 p + λ1 λ1 ∂λ1
!
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# N α 1 1 X αp −1 − 2p −1 = (1−φ) − 2 p + µp λ1 − λ1 , (26) λ1 λ1 p=1
S2 = S3 = −(1 − φ)λ1 p, 149
and finally applying the boundary conditions
⇒
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thus one obtains
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# N α 1 X αp −1 − 2p −1 S1 = (1 − φ) µp λ1 − λ1 . λ1 p=1
(27)
(28)
(29)
Next, according to the formula Eq. (22), and simultaneously assuming that in the remaining
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part of this analysis, the direction 1 will be considered only, the Cauchy stresses for the Ogden
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model and other related models are as follows:
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• Ogden model (N=3)
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σ1 = (1 − φ)
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(1 − φ) µ1
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λα1 1
−
− λ1
α1 2
"
3 X
µp
p=1
+ µ2
λα1 2
α λ1 p
−
− λ1
α2 2
−
− λ1
αp 2
+ µ3
#
= λα1 3
−
− λ1
α3 2
i
;
(30)
• Mooney-Rivilin model σ1 = (1 − φ) µ1 λ21 − λ−1 + µ2 λ−2 ; 1 1 − λ1
• neo-Hookean model
σ1 = (1 − φ) µ1 λ21 − λ−1 ; 1 10
(31)
(32)
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• Varga model
h i −1 σ1 = (1 − φ) µ1 λ1 − λ1 2 .
(33)
Finally, in the numerical procedure, for the subsequent time points ti (t ∈ [0, tf ] where tf is a
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time for which λ1 = 10) the following flow chart is applied:
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• calculate current the right-stretch tensor U|t=ti (Eq. (25))
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• calculate current stress intensity invariant Iφ |t=ti
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• test Iφ > τφ ◦ NO - φ|t=ti = φ|t=ti−1
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◦ YES - update damage variable according to Eq. (18) • calculate stresses (Eqs (30)-(33)).
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It is important that for damage update that the approximation of Caputo operator proposed in
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[32] was used. Therefore the LHS of Eq. (18), for t = tm , can be written as
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a = t0 < t1 < ... < tk < ... < tm = t,
C α a Dt φ(t)|t=tm m−1 X
∼ =
t−a tm − t0 = , m m
m ≥ 2,
(34)
hn−α [(m − 1)n−α+1 − (m − n + α − 1)man−α ]φ(n) (t0 ) + Γ(n − α + 2)
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h=
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[(m − k + 1)n−α+1 − 2(m − k)n−α+1 + (m − k − 1)n−α+1 ]φ(n) (tk ) + φ(n) (tm ) , (35)
k=1
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where φ(n) (tk ) denotes the classical n-th derivative at t = tk . It should be pointed out that
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approximation Eq. (35) is a weighted sum of classical n-th derivatives at points from the past
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(first two terms) and at current time point (t = tm - third term). In the following examples it is
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assumed that α ∈ (0, 1], therefore n = 1. Thus, in the computational scheme, two first terms
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of Eq. (35) are considered on the RHS of Eq. (18) (and approximated utilising the backward 11
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difference), whereas the third term after the approximation of φ(1) (tm ), utilising the backward
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difference also, allowed one finally to calculate the needed updated value of the damage vari-
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able φ(tm ) = φ|t=ti .
3.5
Damage evolution
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One assumes that the overstress function for fractional damage evolution takes the form of the
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power function Φ
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nφ Iφ Iφ −1 = −1 , τφ τφ
(36)
where nφ is a material parameter.
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One assumes additionally that I˜φ has an analogous form to the strain-energy function under
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consideration (by analogy to Huber-Mises-Hencky plasticity where the equivalent stress is
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proportional to the deviatoric part of the strain energy), thus e.g. for the Ogden model
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I˜φ =
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N X µ∗p
where µ∗p , αp∗ , nφ
α∗
αp
(37)
(p = 1, ..., N ) are material parameters for stress intensity invariant.
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α∗
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p=1
α∗
(λ1 p + λ2 p + λ3 p − 3), ∗
4 Results and Application
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4.1
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Numerical study
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The material parameters used in this study, assumed by analogy to [33], are summarised in
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Table 1. It should be emphasised that parameters nβ , α and `t will vary dependent on the case
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considered.
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Tab. 1 - Benchmark material parameters α1 = α1∗ = 1.3
µ1 = µ∗1 = 1.0 N m−2
α2 = α2∗ = 5.0
µ2 = µ∗2 = 0.0019 N m−2
T = 1.0 s
τφ = 20 N m−2
α ∈ (0, 1)
nφ = 2.0
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α3 = α3∗ = −2.0 µ3 = µ∗3 = 0.0159 N m−2
In the first part of the analysis the material behaviour, for the stretch range λ1 ∈ [1, 10], without
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damage effects, was considered (cf. Figs 3-4). In Fig. 3 the stress intensity invariant (Eq. (37))
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vs. principal stretch, for different Ogden type models is presented. The discrepancy between
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the solutions is clear, and the classical Ogden model shows the most intensive non-linearity.
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The same conclusions can be stated for Fig. 4 where stress ratios (Eqs (30)-(33)) vs. principal
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stretch are shown. Because of this observations the classical Ogden model was chosen for
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further analysis.
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The second part of the study includes the analysis of damage evolution for the Ogden model,
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for a motion defined in Eq. (23) assuming different rates of deformation (nβ ∈ {1, 5, 10}),
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different orders of evolution (α ∈ {0.3, 0.5, 0.7, 0.9, 0.95, 1.0}) and constant memory (`t =
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of order of damage evolution is more pronounced and intensifies as α 0.3.
In the third part of the study the same configuration as for the second was repeated for constant rate of deformation (nβ = 1) and different orders of evolution (α ∈ {0.3, 0.5, 0.7, 0.9, 0.95, 1.0}) and different memories (`t ∈ {0.001, 0.01, 0.1}s) - cf. Fig. 6.
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0.01s) - cf Fig. 5. It is observed that as the rate of deformation is higher (nβ 10) the influence
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It is observed that damage evolution intensifies as α 0.3 for all cases, whereas for growing
memory (`t 0.1s) the answer of the material is stabilizing.
Finally, the fourth part of the study covers the analysis of stresses, for analogous conditions as in the second part also - cf. Figs 7-8. For clarity on all plots the undamaged answer of the material is presented. It is observed that for lower rates (nβ 0.5) damage is slightly 13
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influenced by the order of evolution α. This situation is different for higher rates (nβ 5)
where the order of evolution dominates the behaviour. Nonetheless, in all cases the lower damage evolution order (α 0.3), the most severe softening of the material is observed.
As a concluding remark of this parametric analysis one can state the advantages of fractional model compared to the classical one (curves for α = 1 in Figs 5-8). It is clear that fractional ef-
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fects are observed only when α ∈ (0, 1), then damage evolution is controlled by two additional
216
material parameters α ([α] = [−]) and time length scale (range of memory) `t ([`t ] = [s]). In
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fractional range one obtains a variety of solutions that considerably differs both quantitatively
218
and qualitatively. This states the strength of fractional model (of crucial importance from the
219
point of view of modelling), namely for a limited set of new material parameters one covers a
220
broad range of possible material behaviours keeping simultaneously clear physical interpreta-
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tion.
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Figure 3: Stress intensity invariant vs. principal stretch - without damage
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Figure 4: Stress ratios vs. principal stretch - without damage
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Figure 5: Evolution of damage parameter for different rates of deformation (nβ ∈ {1, 5, 10}) 16 and `t = 0.01s)
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Figure 6: Evolution of damage parameter for different values of the memory parameter `t ∈ 17 {0.001, 0.01, 0.1}s and nβ = 1
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Figure 7: (Top) Ogden stresses for different evolutions of damage parameter; (Bottom) magnification - (nβ ∈ {0.5} and constant memory parameter `t = 0.01s)
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Figure 8: Ogden stresses for different evolutions of damage parameter (for different rates of deformation nβ ∈ {1, 5} and constant memory parameter `t = 0.01s)
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4.2
Identification for failure of the abdominal aortic aneurysm
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To show the applicability of the presented model one considers the analysis of failure of the
224
abdominal aortic aneurysm (AAA) [34]. For this purpose the Ogden model (N = 3) was used 19
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Material parameters for AAA are shown in Table 2 and they were calibrated in Matlab software
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based on soft computing methods (quasi static uniaxial tension test) to obtain best fitting of the
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strain-stress curves (see Fig. 9). It is observed that up to λ1 = 1.65 AAA follows the path of
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an undamaged hyperelastic material model, while for λ1 > 1.65 softening appears. It is clear
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from Fig. 9 that the numerical results using the proposed hyperelastic fractional damage model
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conform with the observed experimental results. Tab. 2 - AAA material parameters
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µ1 = µ∗1 = 75.0 N cm−2
α2 = α2∗ = 11.0
µ2 = µ∗2 = 0.1425 N cm−2
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α1 = α1∗ = 1.3
α3 = α3∗ = −1.0 µ3 = µ∗3 = 1.1925 N cm−2 τφ = 14.5 N cm−2
T = 0.5 s α = 0.75
nφ = 1.7
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`t = 0.065s
Figure 9: Numerical vs. experimental [34] results
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5 Conclusions
233
In this paper a new isotropic hyperelasic material model accounting for isotropic damage,
234
where damage evolution is described in terms of fractional calculus, is formulated in the frame-
235
work of thermodynamics. Two additional material parameters are introduced in this new for-
236
mulation compared with classical formulations: (i) order of damage evolution; and (ii) memory
237
(time length scale) of damage evolution. Both parameters allow flexible modelling of material
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softening as presented based on benchmark solutions and especially applied in final analysis
239
of failure of the abdominal aortic aneurysm.
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The obtained results allow to formulate the flowing conclusions:
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• because of damage evolution definition which utilises the concept of overstress function, in the softening range of material operation rate effects are observed, • for lower orders of damage velocity, intensification of damage evolution is shown,
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• the dependence of fractional damage velocity on time length scale (memory) shows that
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for long memory (in relation to characteristic time) changes in softening become limited.
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[1] L.M. Kachanov. Time of the rupture process under creep conditions. Izv. AN SSR, Otd. Tekh. Nauk, 8:26–31, 1958.
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55:280–286, 1988. [5] C.W. Woo and D.L. Li. A universal physically consistent definition of material damage. International Journal of Solids and Structures, 30(15):2097–2108, 1993. [6] D. Krajcinovic. Continuum damage mechanics: when and how? International Journal of Damage Mechanics, 4:217–229, 1995.
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[11] W. Sumelka and T. Łodygowski. Reduction of the number of material parameters by ann approximation. Computational Mechanics, 52:287–300, 2013. [12] C. Basaran and C.Y. Yan. A thermodynamic framework for damage mechanics of solder
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[30] K. A. Lazopoulos and A. K. Lazopoulos. On fractional bending of beams. Archive of Applied Mechanics, 86(6):1133–1145, 2016. [31] P. Haupt. Continuum Mechanics and Theory of Materials. Springer, second edition, 2002.
[32] Z. Odibat. Approximations of fractional integrals and Caputo fractional derivatives. Applied Mathematics and Computation, 178:527–533, 2006. [33] G.A. Holzapfel. Nonlinear Solid Mechanics - A Continuum Approach for Engineering.
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A Hyperelastic Fractional Damage Material Model with Memory Wojciech Sumelka, George Z. Voyiadjis PII: DOI: Reference:
S0020-7683(17)30290-1 10.1016/j.ijsolstr.2017.06.024 SAS 9631
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
3 March 2017 7 June 2017 19 June 2017
Please cite this article as: Wojciech Sumelka, George Z. Voyiadjis, A Hyperelastic Fractional Damage Material Model with Memory, International Journal of Solids and Structures (2017), doi: 10.1016/j.ijsolstr.2017.06.024
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1
A Hyperelastic Fractional Damage Material Model with Memory
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Wojciech Sumelka∗ , George Z. Voyiadjis∗∗
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∗
Poznan University of Technology, Institute of Structural Engineering, Piotrowo 5 street, 60-969 Pozna´n, Poland
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[email protected]
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7
∗∗
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Boyd Professor, Department of Civil and Environmental Engineering, Louisiana State University,
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Baton Rouge, LA 70803, USA
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[email protected]
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Keywords: hyperelasticity; scalar damage; fractional calculus; materials; damage mechanics.
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ABSTRACT
In this paper a scalar damage model for hyperelastic materials is considered. The novelty of
14
the proposed approach lies in the evolution law for the damage variable that is formulated with
15
the application of fractional calculus. In this way damage evolution includes the memory, or
16
in other words the current intensity of damage evolution, which is based on information from
17
the past - whose length is included in the fractional operator. Based on illustrative examples,
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the flexibility of the model to mimic experimentally observed material behaviour is presented.
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1 Introduction
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Continuum damage mechanics (CDM) has reached a high level of its maturity nowadays.
21
Herein on should mention the first concepts proposed by Kachanov [1] and Rabotnov [2], to1
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[4], Woo and Li [5], Krajcinovic [6], Chen and Chow [7], Voyiadjis et. al. [8, 9], Perzyna
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[10], Sumelka [11] or competetive formulations like these quantified by irreversible entropy
25
and entropy generation rate by Basaran [12, 13, 14] or Sosnovskiy [15]. Within this differ-
26
ent types of damage models, one of the most important aspect is the definition of damage as
27
a mathematical object, namely one considers isotropic (scalar) and anisotropic (higher order
28
tensors) measures for material degradation. CDM concepts were applied for different types
29
of materials (e.g. metals, rocks, concrete, composites), a wide range of degradation types (e.g.
30
elastic-brittle damage, elastic-plastic damage, fatigue damage, corrosion damage) and different
31
processes (e.g. static, dynamic, coupled fields) - cf. Skrzypek and Ganczarski [16]. Nonethe-
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less, constant development of new materials or advancing knowledge of existing ones (e.g.
33
biological) induces the necessity of further development of CDM, especially in the field of
34
definition of damage initiation and evolution criteria i.e. specific evolution equations for the
35
internal state variables. This crucial aspect can be formulated using different mathematical
36
tools, however as presented in recent papers, the fractal theory (for definition of fractal dam-
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age variable) [17] or fractional calculus (for fatigue phenomena modeled as a change of the
38
fractional derivative order) [18] can be very effective. Herein, the new insight will be posed to
39
the last mentioned modelling technique for finite strains and original damage evolution law.
40
Fractional calculus (FC) is a branch of mathematical analysis which considers differential
41
equations of an arbitrary order [19, 20, 21]. From the point of view of mathematical modelling
42
of physical phenomena, the important aspects of FC are: (i) fractional differential operators
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(FDO) are defined over an interval; (ii) there are infinitely many definitions of FDO [22]; (iii)
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each FDO can have some specific properties [23] (e.g. FDO operating on a constant function
45
does not necessarily result in zero). It should be pointed out that to some extent, dependent on
46
the material being considered, there should exist the most proper choice of FDO.
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It is important that the non-local action of FDO mentioned above, can operate on different
48
spaces, dependent on variables on which FDO acts (cf. Fig. 1). Namely, it can be a non-local 2
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local action in a stress space (e.g. non-normal plastic flow [25]), or non-local action in space
51
variable (cf. [26, 27, 28, 29, 29, 30]). Herein, the non-locality in time, should be pointed out
52
as being the crux of this paper and is called memory [20]. For such class of non-local concept,
53
one can distinguish two cases: (i) full memory - when time fractional derivatives are taken over
54
the total time of the analysed process; and (ii) short memory - when the current state of the
55
mechanical system is influenced by the closest past events, characterised by the existence of a
56
characteristic time length scale `t (cf. Fig. 1).
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Figure 1: Possible non-localities in mechanics vs. FDO operation on a specific field
In a further part of this paper we explore the concept of short memory connected with the
58
definition of (scalar) damage parameter evolution in terms of FC (fractional calculus) for hy-
59
perelastic materials. It will be observed that both, the order of damage evolution (order of
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FC) and the applied characteristic time `t (range of FC action) control the intensity of damage
61
expansion - cf. Fig. 2. Thus, in this new formulation, two additional material parameters are
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needed for practical applications, however, with a very positive and desired property of the
63
overall model, namely its flexibility to map the experimental results.
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Figure 2: The concept of short memory
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2 Problem formulation
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2.1
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In this work isotropic hyperelasic materials are considered. It is assumed that there exists a
67
stress threshold above which damage appears in the material. Damage is modelled through
68
a scalar variable, thus the damaged range of the material behaviour is isotropic as well. The
69
evolution of damage is formulated with the application of the Caputo fractional derivative,
70
hence the current state of damage is influenced by past events - one says that the material has
71
a memory, or that the material is non-local in time.
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Main Assumptions
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73
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2.2
Isotropic Hyperelastic Fractional Damage Material Model with Memory
One assumes the existence of the Helmholtz free-energy function in the general form
Ψ = Ψ(F),
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(1)
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where Ψ denotes the Helmholtz free-energy function, and F stands for deformation gradient.
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Assuming that the elastic range is isotropic Eq. (1) reduces to
Ψ = Ψ(U) = Ψ(C) = Ψ(E),
(2)
where U is the right stretch tensor (U2 = C), C is the right Cauchy-Green tensor and E
78
denotes the Green-Lagrange strain tensor (2E = C − I).
79
For convenience the following restrictions for Ψ are assumed:
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Ψ = Ψ(I) = 0,
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and in general
Ψ(F) ≥ 0, and
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Ψ is objective.
(5)
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˜ Ψ = Ψ(C, φ) = (1 − φ)Ψ(C),
(6)
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where φ is the scalar damage variable, together with the postulates:
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(4)
Next, one introduces the scalar damage concept based on the assumption that
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(3)
˜ Ψ(I) = 0,
(7)
˜ Ψ(C) ≥ 0,
(8)
˜ is objective, Ψ
(9)
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and (10)
φ ∈ [0, 1], ˜ is the effective (undamaged) Helmholtz free-energy function. where Ψ
88
Based on the assumption that the Clausius-Planck inequality for purely mechanical processes
89
can be written as
˙ =S:E ˙ ≥ 0, ˙ −Ψ Dint = wint − Ψ
(11)
˜ ˙ ˜ ˙ = (1 − φ) ∂ Ψ(C) : C ˙ − Ψ(C) φ, Ψ ∂C
(12)
and utilizing the chain rule
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where Dint is the density of dissipation, wint is the internal energy density, and S denotes the
92
second Piola-Kirchhoff stress tensor (S = F−1 P = ST and P is the first Piola-Kirchhoff stress
93
tensor). Therefore, one obtains
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thus
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!
:
˙ C ˜ + Ψ(C) φ˙ ≥ 0. 2
(13)
From Eq. (13) it is clear that:
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˜ ∂ Ψ(C) S − 2(1 − φ) ∂C
S − 2(1 − φ)
˜ ∂ Ψ(C) = 0, ∂C
(14)
˜ S = (1 − φ)S,
(15)
˜ Ψ(C) φ˙ ≥ 0,
(16)
and
97
˜ ˜ = 2 ∂ Ψ(C) where S is the effective (undamaged) second Piola-Kirchhoff stress tensor. It is ∂C
98
observed, that for the non-negative dissipation defined in Eq. (16), together with postulate
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Eq. (8), the following is noted φ˙ ≥ 0.
Finally, the evolution of φ is postulated utilising the fractional differential operator in the form
101
C α t−`t Dt φ
102
1 = αΦ T
where the bracket h·i defines the ramp function, 1 = Γ(n − α)
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a
t
C
Iφ −1 , τφ
(18)
D is the left-sided Caputo derivative
f (n) (τ ) dτ, (t − τ )α−n+1
for
t > a,
(19)
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C α a Dt f (t)
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(17)
where t denotes time variable, α is an order of derivative, T stands for characteristic time, Φ is
104
the overstress function, Iφ is the stress intensity invariant, τφ is the threshold stress for damage
105
evolution, Γ is the Euler gamma function, and n = bαc + 1. It should be emphasised that the
106
fractional velocity of damage to be defined needs identification of the damage order α and the
107
damage memory (time length scale) `t , in other words two additional material parameters are
108
postulated. This is in contrast with the classical case when α = 1 and memory does not play
109
any role. Moreover, the form of the evolution equation Eq. (18) induces rate dependence of the
110
overall constitutive model.
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3 Examples
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3.1
Introductory Remarks
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The main aim of this section is to present the influence of the fractional damage evolution
114
model on the material model defined in the preceding section. Due to this reason the effect of
115
both the damage order index α and the memory (time length scale) `t is examined in detail.
116
Furthermore, this parametric study is enriched by the examination of the rate effects.
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3.2
Ogden Strain-Energy Function
118
One assumes that the stain-energy function postulated in Eq. (6) takes the form of the Ogden
119
model, namely ˜ = Ψ(λ ˜ 1 , λ2 , λ3 ) = Ψ
N X µp p=1
αp
α
α
α
(20)
(λ1 p + λ2 p + λ3 p − 3),
where λ1 , λ2 , λ3 are the principal stretches (principal values of U), and µp , αp
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are the material parameters for Ogden model. This allows one to analyse a wide class of other
122
common models like:
(p = 1, ..., N )
• Mooney-Rivilin model N = 2, α1 = 2, α2 = −2 with the constraint condition I3 (C) = λ21 λ22 λ23 = 1,
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• neo-Hookean model N = 1, α1 = 2, or
126
• Varga model N = 1, α1 = 1.
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3.3
The Constitutive Equation in the Principal Stretches Directions
128
Without loss of generality one limits the considerations to incompressible hyperelastic bodies.
129
Therefore, the constitutive model given by Eq. (14) can be now rewritten in the principal stretch
130
˜ = Ψ(λ ˜ 1 , λ2 , λ3 )) [31] directions form, namely (Ψ
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˜ 1 1 ∂Ψ Sa = (1 − φ) − 2 p + λa λa ∂λa
!
,
a = 1, 2, 3,
(21)
131
where Sa are the principal values of S, and p denotes the hydrostatic pressure.
132
Based on Eq. (21) the transformation to the principal Cauchy stress, the most intuitive stress
133
measure, is straightforward and is given by ˜ ∂Ψ σa = λ2a Sa = (1 − φ) −p + λa ∂λa 8
!
,
a = 1, 2, 3,
(22)
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where σa are the principal values of Cauchy stress tensor σ = J −1 FSFT , (J = det(F)), and
135
p has to be determined from the balance of linear momentum and the boundary conditions.
136
3.4
137
To understand the concept of fractional damage with memory, without loss of generality, one
138
considers the behaviour of the overall model for uniaxial (incompressible) tension, hence
x = χ(X, t) = (1 + tnβ )X1 e1 + √
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1 1 X2 e 2 + √ X3 e3 , n β 1+t 1 + tnβ
(23)
where nβ = const - thus parameter nβ controls the rate of the process Eq. (23), χ is the motion,
140
e is the base vector in current configuration, and x, X are spatial and material coordinates,
141
respectively.
142
Next, the deformation gradient for the motion of Eq. (23) in the matrix representation can be
143
expressed as
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0 0
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(1 + tnβ ) 0 ∂χ(X, t) F = F(X, t) = = √ 1n 0 ∂X 1+t β 0 0
145
146
1
√
nβ
CE
1+t
.
(24)
Finally, the corresponding right-stretch tensor needed for stress calculation (cf. Eqs (21-22)) is given by
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144
λ1 0 0 (1 + tnβ ) 0 U = 0 λ2 0 = √ 1n 0 1+t β 0 0 λ3 0 0
0 0 √
1 nβ
1+t
.
(25)
Based on the calculated stretches (Eq. (25)) the associated stress state resulting from the Ogden 9
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˜ = Ψ(λ ˜ 1 )): model is given by (Ψ ˜ 1 ∂Ψ 1 S1 = (1−φ) − 2 p + λ1 λ1 ∂λ1
!
"
# N α 1 1 X αp −1 − 2p −1 = (1−φ) − 2 p + µp λ1 − λ1 , (26) λ1 λ1 p=1
S2 = S3 = −(1 − φ)λ1 p, 149
and finally applying the boundary conditions
⇒
150
thus one obtains
"
p = 0,
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# N α 1 X αp −1 − 2p −1 S1 = (1 − φ) µp λ1 − λ1 . λ1 p=1
(27)
(28)
(29)
Next, according to the formula Eq. (22), and simultaneously assuming that in the remaining
152
part of this analysis, the direction 1 will be considered only, the Cauchy stresses for the Ogden
153
model and other related models are as follows:
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• Ogden model (N=3)
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151
σ1 = (1 − φ)
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h
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(1 − φ) µ1
155
156
λα1 1
−
− λ1
α1 2
"
3 X
µp
p=1
+ µ2
λα1 2
α λ1 p
−
− λ1
α2 2
−
− λ1
αp 2
+ µ3
#
= λα1 3
−
− λ1
α3 2
i
;
(30)
• Mooney-Rivilin model σ1 = (1 − φ) µ1 λ21 − λ−1 + µ2 λ−2 ; 1 1 − λ1
• neo-Hookean model
σ1 = (1 − φ) µ1 λ21 − λ−1 ; 1 10
(31)
(32)
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• Varga model
h i −1 σ1 = (1 − φ) µ1 λ1 − λ1 2 .
(33)
Finally, in the numerical procedure, for the subsequent time points ti (t ∈ [0, tf ] where tf is a
159
time for which λ1 = 10) the following flow chart is applied:
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160
• calculate current the right-stretch tensor U|t=ti (Eq. (25))
161
• calculate current stress intensity invariant Iφ |t=ti
162
• test Iφ > τφ ◦ NO - φ|t=ti = φ|t=ti−1
164
◦ YES - update damage variable according to Eq. (18) • calculate stresses (Eqs (30)-(33)).
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It is important that for damage update that the approximation of Caputo operator proposed in
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[32] was used. Therefore the LHS of Eq. (18), for t = tm , can be written as
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C α a Dt φ(t)|t=tm m−1 X
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hn−α [(m − 1)n−α+1 − (m − n + α − 1)man−α ]φ(n) (t0 ) + Γ(n − α + 2)
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h=
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k=1
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where φ(n) (tk ) denotes the classical n-th derivative at t = tk . It should be pointed out that
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approximation Eq. (35) is a weighted sum of classical n-th derivatives at points from the past
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(first two terms) and at current time point (t = tm - third term). In the following examples it is
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assumed that α ∈ (0, 1], therefore n = 1. Thus, in the computational scheme, two first terms
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of Eq. (35) are considered on the RHS of Eq. (18) (and approximated utilising the backward 11
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difference), whereas the third term after the approximation of φ(1) (tm ), utilising the backward
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difference also, allowed one finally to calculate the needed updated value of the damage vari-
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able φ(tm ) = φ|t=ti .
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Damage evolution
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One assumes that the overstress function for fractional damage evolution takes the form of the
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power function Φ
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nφ Iφ Iφ −1 = −1 , τφ τφ
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where nφ is a material parameter.
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One assumes additionally that I˜φ has an analogous form to the strain-energy function under
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consideration (by analogy to Huber-Mises-Hencky plasticity where the equivalent stress is
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proportional to the deviatoric part of the strain energy), thus e.g. for the Ogden model
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N X µ∗p
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αp
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(p = 1, ..., N ) are material parameters for stress intensity invariant.
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α∗
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α∗
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4 Results and Application
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4.1
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Numerical study
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The material parameters used in this study, assumed by analogy to [33], are summarised in
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Table 1. It should be emphasised that parameters nβ , α and `t will vary dependent on the case
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considered.
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Tab. 1 - Benchmark material parameters α1 = α1∗ = 1.3
µ1 = µ∗1 = 1.0 N m−2
α2 = α2∗ = 5.0
µ2 = µ∗2 = 0.0019 N m−2
T = 1.0 s
τφ = 20 N m−2
α ∈ (0, 1)
nφ = 2.0
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In the first part of the analysis the material behaviour, for the stretch range λ1 ∈ [1, 10], without
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damage effects, was considered (cf. Figs 3-4). In Fig. 3 the stress intensity invariant (Eq. (37))
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vs. principal stretch, for different Ogden type models is presented. The discrepancy between
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the solutions is clear, and the classical Ogden model shows the most intensive non-linearity.
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The same conclusions can be stated for Fig. 4 where stress ratios (Eqs (30)-(33)) vs. principal
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stretch are shown. Because of this observations the classical Ogden model was chosen for
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further analysis.
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The second part of the study includes the analysis of damage evolution for the Ogden model,
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for a motion defined in Eq. (23) assuming different rates of deformation (nβ ∈ {1, 5, 10}),
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different orders of evolution (α ∈ {0.3, 0.5, 0.7, 0.9, 0.95, 1.0}) and constant memory (`t =
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of order of damage evolution is more pronounced and intensifies as α 0.3.
In the third part of the study the same configuration as for the second was repeated for constant rate of deformation (nβ = 1) and different orders of evolution (α ∈ {0.3, 0.5, 0.7, 0.9, 0.95, 1.0}) and different memories (`t ∈ {0.001, 0.01, 0.1}s) - cf. Fig. 6.
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0.01s) - cf Fig. 5. It is observed that as the rate of deformation is higher (nβ 10) the influence
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It is observed that damage evolution intensifies as α 0.3 for all cases, whereas for growing
memory (`t 0.1s) the answer of the material is stabilizing.
Finally, the fourth part of the study covers the analysis of stresses, for analogous conditions as in the second part also - cf. Figs 7-8. For clarity on all plots the undamaged answer of the material is presented. It is observed that for lower rates (nβ 0.5) damage is slightly 13
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influenced by the order of evolution α. This situation is different for higher rates (nβ 5)
where the order of evolution dominates the behaviour. Nonetheless, in all cases the lower damage evolution order (α 0.3), the most severe softening of the material is observed.
As a concluding remark of this parametric analysis one can state the advantages of fractional model compared to the classical one (curves for α = 1 in Figs 5-8). It is clear that fractional ef-
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fects are observed only when α ∈ (0, 1), then damage evolution is controlled by two additional
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material parameters α ([α] = [−]) and time length scale (range of memory) `t ([`t ] = [s]). In
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fractional range one obtains a variety of solutions that considerably differs both quantitatively
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and qualitatively. This states the strength of fractional model (of crucial importance from the
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point of view of modelling), namely for a limited set of new material parameters one covers a
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broad range of possible material behaviours keeping simultaneously clear physical interpreta-
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tion.
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Figure 3: Stress intensity invariant vs. principal stretch - without damage
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Figure 4: Stress ratios vs. principal stretch - without damage
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Figure 5: Evolution of damage parameter for different rates of deformation (nβ ∈ {1, 5, 10}) 16 and `t = 0.01s)
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Figure 6: Evolution of damage parameter for different values of the memory parameter `t ∈ 17 {0.001, 0.01, 0.1}s and nβ = 1
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Figure 7: (Top) Ogden stresses for different evolutions of damage parameter; (Bottom) magnification - (nβ ∈ {0.5} and constant memory parameter `t = 0.01s)
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Figure 8: Ogden stresses for different evolutions of damage parameter (for different rates of deformation nβ ∈ {1, 5} and constant memory parameter `t = 0.01s)
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4.2
Identification for failure of the abdominal aortic aneurysm
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To show the applicability of the presented model one considers the analysis of failure of the
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abdominal aortic aneurysm (AAA) [34]. For this purpose the Ogden model (N = 3) was used 19
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Material parameters for AAA are shown in Table 2 and they were calibrated in Matlab software
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based on soft computing methods (quasi static uniaxial tension test) to obtain best fitting of the
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strain-stress curves (see Fig. 9). It is observed that up to λ1 = 1.65 AAA follows the path of
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an undamaged hyperelastic material model, while for λ1 > 1.65 softening appears. It is clear
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from Fig. 9 that the numerical results using the proposed hyperelastic fractional damage model
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conform with the observed experimental results. Tab. 2 - AAA material parameters
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µ1 = µ∗1 = 75.0 N cm−2
α2 = α2∗ = 11.0
µ2 = µ∗2 = 0.1425 N cm−2
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α3 = α3∗ = −1.0 µ3 = µ∗3 = 1.1925 N cm−2 τφ = 14.5 N cm−2
T = 0.5 s α = 0.75
nφ = 1.7
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Figure 9: Numerical vs. experimental [34] results
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5 Conclusions
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In this paper a new isotropic hyperelasic material model accounting for isotropic damage,
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where damage evolution is described in terms of fractional calculus, is formulated in the frame-
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work of thermodynamics. Two additional material parameters are introduced in this new for-
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mulation compared with classical formulations: (i) order of damage evolution; and (ii) memory
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(time length scale) of damage evolution. Both parameters allow flexible modelling of material
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softening as presented based on benchmark solutions and especially applied in final analysis
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of failure of the abdominal aortic aneurysm.
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The obtained results allow to formulate the flowing conclusions:
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• because of damage evolution definition which utilises the concept of overstress function, in the softening range of material operation rate effects are observed, • for lower orders of damage velocity, intensification of damage evolution is shown,
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• the dependence of fractional damage velocity on time length scale (memory) shows that
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for long memory (in relation to characteristic time) changes in softening become limited.
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[1] L.M. Kachanov. Time of the rupture process under creep conditions. Izv. AN SSR, Otd. Tekh. Nauk, 8:26–31, 1958.
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