Fractional Derivative Viscoelasticity at Large Deformations

Nonlinear Dynamics 33: 301–321, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

Fractional Derivative Viscoelasticity at Large Deformations

KLAS ADOLFSSON and MIKAEL ENELUNDDepartment of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden;E-mail: [email protected]

(Received: 18 February 2003; accepted: 1 July 2003)

Abstract. A time domain viscoelastic model for large three-dimensional responses under isothermal conditionsis presented. Internal variables with fractional order evolution equations are used to model the time dependentpart of the response. By using fractional order rate laws, the characteristics of the time dependency of manypolymeric materials can be described using relatively few parameters. Moreover, here we take into account thatpolymeric materials are often used in applications where the small deformations approximation does not hold(e.g., suspensions, vibration isolators and rubber bushings). A numerical algorithm for the constitutive responseis developed and implemented into a finite element code for structural dynamics. The algorithm calculates thefractional derivatives by means of the Grünwald–Lubich approach. Analytical and numerical calculations of theconstitutive response in the nonlinear regime are presented and compared. The dynamic structural response of aviscoelastic bar as well as the quasi-static response of a thick walled tube are computed, including both geomet-rically and materially nonlinear effects. Moreover, it is shown that by applying relatively small load magnitudes,the responses of the linear viscoelastic model are recovered.

Keywords: Fractional derivatives, viscoelasticity, large deformations, structural dynamics.

1. Introduction

Polymers in general show inelastic behavior such as creep, relaxation and damping. Manypolymers show a weak frequency dependence of their damping properties over a broad fre-quency range. This weak frequency dependence is difficult to describe with classical vis-coelasticity based on integer derivative operators. A large number of derivative operators,resulting in many parameters, is required to obtain a reasonably accurate description of theobserved damping characteristics. By introducing fractional order derivative operators in theconstitutive relations, the number of parameters can be significantly reduced. The fractionalorder viscoelastic model has successfully been fitted to experimental data over a broad fre-quency range for several polymers using only four parameters (two ‘elastic’ constants, onerelaxation constant and the non-dimensional fractional order of differentiation) in the uniaxialcase (see, e.g., [1]). The fractional viscoelastic model has also been successfully fitted totime domain rubber data at small strains by, e.g., Welch et al. [2]. Rather few finite elementformulations of fractional derivative viscoelasticity and algorithms for calculations of transientstructural responses have been presented. Substantial contributions in this field have beenpresented by Padovan [3], Enelund et al. [4] and Schmidt and Gaul [5]. Padovan as wellas Schmidt and Gaul use a single constitutive equation that involves fractional derivativesacting on both stresses and strains. As a consequence, both the strain history and the stresshistory need to be stored and included in each time step when integrating the structural re-sponse. Enelund et al. [4] developed an internal variable formulation of fractional derivativeviscoelasticity. The rate law for the internal variable is then a differential equation of fractional

302 K. Adolfsson and M. Enelund

order. This three-dimensional general anisotropic formulation was implemented into a finiteelement framework. The advantage with this formulation is that only the history of the internalvariable needs to be stored and included thus saving computer time and memory.

Polymers are often used in applications where the small deformation approximation is notvalid and the linear small strain theory does not suffice. Linear viscoelasticity with fractionalorder derivative operators in the rate laws has been an active area of research for many years.Less attention, however, has been devoted to fractional derivative viscoelasticity in combin-ation with large deformations. Several models that extends the classical linear viscoelasticmodel (i.e., they are based on integer order derivatives) to the nonlinear domain have beenproposed. Simo [6] suggested linear rate equations where the internal variables are of stresstype while letting the elastic responses be nonlinear. Alternative formulations of this model arefound in [7–9]. These models have (due to their simplicity) been widely used in practice andare implemented in many commercial finite element codes. The starting point of the abovemodels is the multiplicative split of the deformation gradient into isochoric and volumetricparts. Lubliner [10] further decomposed the isochoric part of the deformation gradient multi-plicatively into multiple elastic and viscous components and used linear rate equations for theviscous parts. Recently, Bonet [11] proposed a model that utilizes the Lubliner decompositionof the deformation gradient while using nonlinear rate laws for the viscous components. How-ever, this formulation uses the linear model as a starting point. A fully nonlinear viscoelasticmodel is given by Reese and Govindjee [12, 13] and is, unlike the above models, proven tosatisfy the second law of thermodynamics for all admissible deformation histories, in that it isnaturally derived from the Clausius–Duhems inequality.

Here, a large deformation viscoelastic model for three-dimensional stress states under iso-thermal conditions is presented. The model is based on the models in [6, 9]. The constitutiveequations of viscoelasticity are formulated by use of internal variables of inelastic stress type,where the equations governing the evolution of the internal variables are linear differentialequations of fractional order. The elastic part of the deformation is decoupled into purely shearand bulk responses (assuming isotropic materials and nearly incompressible deformations).Further, the model is formulated in the reference configuration, i.e., the Lagrangian kinemat-ics description is used. An algorithm for the integration of the constitutive response basedon the generalized midpoint rule and convolution quadratures for fractional differentiationis developed. The present constitutive model together with the numerical algorithm for theconstitutive response are implemented into a finite element code for geometrically nonlinearstructural dynamics. Finally, some numerical examples that consider both constitutive aspectsand structural responses for fictitious rubber materials are presented.

2. Basic Deformation and Stress Measures

Consider a continuum body whose particles initially have the positions X, referred to as thereference configuration. Also let ϕ denote the motion of the particles, then their current posi-tions x are given by the expression x = ϕ(X, t), and are thus defining a continuous sequenceof current configurations of the body. The continuum equations of motion may be formulatedwith respect to the reference configuration, such a formulation is called a Lagrangian (ormaterial) description. If instead using the variables defined in the current configuration it isknown as an Eulerian (or spatial) description. In the present study we employ a Lagrangiandescription, however, the basic measures in an Eulerian description will also be mentioned.

Fractional Derivative Viscoelasticity at Large Deformations 303

The deformation gradient tensor is denoted

F = ∂ϕ

∂X. (1)

The right and left Cauchy–Green deformation tensors are introduced as C = F TF and b =FF T, respectively, where the superscript (·)T means transpose of a second order tensor. Theright Cauchy–Green tensor appears in the Lagrangian formulation whereas the left Cauchy–Green tensor is an Eulerian measure. The displacement of the particles, initially occupyingthe positions X, under the deformation ϕ are given by

u = ϕ − X. (2)

In the theory of finite deformations there is not a unique way to define the strain of a material.For example, the Green–Lagrange strain tensor (given both in terms of the right Cauchy–Green tensor and the displacement vector) is defined as

E = 1

2(C − I ), E = 1

2

(∂u

∂X+(∂u

∂X

)T

+(∂u

∂X

)T∂u

∂X

), (3)

where I is the second order identity tensor. Note that the linear strain tensor ε is obtained byomitting the last term of the latter expression in Equation (3).

In the reference (material) configuration the second (or symmetric) Piola–Kirchhoff stresstensor S is appearing. The second Piola–Kirchhoff stress is work conjugate to the Green–Lagrange strain with respect to the reference configuration, and the hyperelastic constitutiverelation may thus be written as

S = ∂�(E)

∂E= 2

∂�(C)

∂C, (4)

where � is the strain energy function per unit undeformed volume. Its spatially versions arethe Kirchhoff stress tensor τ and the Cauchy stress tensor σ which are given by the followingpush-forward transformation

τ = Jσ = FSF T, (5)

where J = det(F ) is the Jacobian determinant.

3. Large Deformation Viscoelasticity

Isothermal and isotropic conditions are assumed throughout the present study. We start byconsidering the one-dimensional (uniaxial) case. In a viscoelastic model the stress dependsnot solely on the current strain, it also depends on the entire strain history. The standard linearviscoelastic model consists of N Maxwell chains coupled in parallel (Figure 1), each one ofwhich is associated with the elastic stiffness Ek and the elastic strain εe

k, defined as

εek = ε − εv

k , k = 1, 2, . . . , N, (6)

where εvk is the viscous strain or the non-observable internal variable. (The internal variables

may be regarded as representing irreversible processes in the material.) The linear constitutivemodel can be written as a set of coupled equations

σ = E(0)ε −N∑k=1

Ekεvk , (7)


Figure 1. Mechanical representation of the standard linear viscoelastic model.

εvk + 1

τkεvk = 1

τkε, εv

k(0) = 0, k = 1, 2, . . . , N, (8)

where a superimposed dot (˙) denotes d/dt, E(0) = ∑Nk=1 Ek is the instantaneous stiffness

and τk is the relaxation time associated with each Maxwell chain. Equations (8) are evolutionequations (or rate equations) for the internal variables. For the model to represent solid beha-vior with finite long time stiffness E(∞) at least one of the relaxation times should be infinite,e.g., let τN → ∞ when E(∞) = EN .

The generalization to large deformations is based on the approach by Simo (see, e.g., thetextbook by Hughes and Simo [9]). The idea is to use the structure of the standard linearviscoelastic model. First we introduce N internal variables of stress-type qk = Ekε

vk and the

linear strain energy �k = (1/2)Ekε2 in the kth chain. This makes it possible to decompose the

total stress in elastic and inelastic parts. The set of equations corresponding to Equations (7)and (8) then becomes

σ = ∂�(0)

∂ε−

N∑k=1

qk, (9)

qk + 1

τkqk = 1

τk

∂�k

∂ε, qk(0) = 0, k = 1, 2, . . . , N, (10)

where �(0) = ∑Nk=1 �k is the instantaneous strain-energy and �k is the strain energy in chain

k. With the linear strain energy these equations constitute an equivalent set of equations, butthe elastic part can no longer be interpreted as the stresses in the spring elements in Figure 1(the strain energy should be with respect to the elastic strain rather than the total strain).However, the chain structure is preserved. Now the linear model can be extended to accountfor a nonlinear elastic stress-strain relation by choosing suitable strain-energy functions �k.Further we see that the linear structure of the rate equation is preserved (it is only the elasticpart that is nonlinear).


The extension to a three-dimensional formulation is motivated by Equations (9) and (10)and is carried out by introducing tensors to represent the physical quantities. By using theLagrangian description the set of equations becomes

S = 2∂�(0)

∂C−

N∑k=1

Qk, (11)

Qk + 1

τkQk = 1

τk

(2∂�k

∂C

), Qk(0) = 0, k = 1, 2, . . . , N, (12)

where S is the symmetric (or second) Piola–Kirchhoff stress tensor, C is the right Cauchy–Green deformation tensor and Qk is the internal variable corresponding to chain k. Con-sequently, they are all appearing in the reference configuration. For the instantaneous elasticstress and the stress in chain k, we will occasionally use the following short notations

S(0) = 2∂�(0)

∂Cand Sk = 2

∂�k

∂C, (13)

respectively.

Remark. The model has not been proven to satisfy the second law of thermodynamics in theClausius–Duhem form. However, for small deformations it reduces to the linear viscoelasticmodel, which is well-known to satisfy the Clausius–Duhem inequality.

3.1. FRACTIONAL ORDER RATE LAWS

By simply replacing the integer derivative operators in the rate equations for the internalvariables in Equations (8), (10) and (12) by fractional derivative operators of orders α ∈(0, 1) (which is the interesting interval for viscoelastic applications), we obtain the fractionalderivative model of viscoelasticity (cf. [14] in the linear case). First we define fractionaldifferentiation of order α of a function y(t) as [15]

Dαy(t) ≡ 1

�(1 − α)

d

dt

t∫0

y(t)

(t − t )αdt , 0 < α < 1, (14)

where � is the gamma function. Note that the fractional derivative is not a local operator, i.e.,the derivative is not only dependent of the value at the point but the value of the function onthe whole interval.

In the case of large deformations we use a stress-like internal variable and the rate equationsyield, cf. Equation (12),

DαkQk + 1

ταkk

Qk = 1

ταkk

(2∂�k

∂C

), Qk(0) = 0, k = 1, 2, . . . , N, (15)

where τk can now be interpreted as the most probable relaxation time out of a continuousdistribution of relaxation times. The fractional order of differentiation αk then plays the role ofa distribution parameter for the corresponding distribution of relaxation times [16]. It should


be mentioned that the formal initial conditions to Equation (15) in fact involve fractionalintegrations D−(1−αk)Qk, but it can easily be shown that

Qk(0) = 0 ⇒ D−(1−αk)Qk(0) = 0. (16)

However, from a physical point of view the first condition is the relevant one. The initialconditions Qk(0) = 0 are consistent with an initial response following the instantaneousstress response S(0) in Equation (13). This means that for small strains the initial conditionsimply an initial response following Hooke’s law.

Remark. Again, the model has not been proven to satisfy the second law of thermodynam-ics in the Clausius–Duhem form. However, for small deformations it reduces to the linearfractional viscoelastic model. It is natural to ask if the linear model satisfy the second lawof thermodynamics for αk ∈ (0, 1). Lion [17] shows that the Clausius–Duhem inequality issatisfied for arbitrarily strain histories.

3.2. ISOCHORIC-VOLUMETRIC SPLIT

The multiplicative decomposition of the deformation gradient into isochoric and volumetricparts has become a standard procedure in large deformation viscoelasticity. The decomposeddeformation gradient tensor is written as

F = J 1/3F with J = det(F ), (17)

where F is the isochoric part and J is the volumetric (or dilatational) part known as theJacobian determinant. The isochoric part of the right Cauchy–Green deformation tensor is,

now according to the definition, given by C = FTF . We may also note that det(F ) = 1,

which means that it represents volume preserving deformations.Rubber materials are significantly stiffer in bulk than in shear deformations, and are there-

fore often modeled as nearly incompressible materials. Because of that, it is reasonable toassume uncoupled strain energy functions in shear and bulk. By also regarding isotropicmodels, we may split the strain energy function into an isochoric part and a volumetric partaccording to

�k(C) = �k,iso(C)+�k,vol(J ). (18)

These assumptions give us the possibility to decompose the second Piola–Kirchhoff stresstensor into pure shear and bulk responses. The calculation is straightforward and is performedby using the chain rule for differentiation and general tensor properties. For the elastic stressin chain k, we obtain

Sk = Sk,iso + Sk,vol = J−2/3Dev

[2∂�k,iso

∂C

]+ JpkC

−1, (19)

where pk = ∂�k,vol/∂J is the hydrostatic pressure, Dev[·] = (·) − (1/3)[(·) : C]C−1 is thedeviator operator in the Lagrangian description. Further, (:) denotes double scalar product.For the exact incompressible case pk are unknowns, regarded as Lagrange multipliers, and arein general determined from the equilibrium equations of the corresponding boundary valueproblem. For nearly incompressible materials a reasonable and often used assumption is thatthe material is only viscoelastic in shear. For such a material the volumetric response is time


independent, and we may use the decomposed elastic stress in Equation (19) to simplify theviscoelastic model. The total stress together with the evolution equations then become

S = J−2/3Dev

[2∂�(0),iso

∂C

]+ Jp(0)C

−1 −N∑k=1

Qk,

DαkQk + 1

ταkk

Qk = 1

ταkk

J−2/3Dev

[2∂�k,iso

∂C

], Qk(0) = 0, k = 1, 2, . . . , N, (20)

where Qk are the deviatoric internal variables and p(0) = ∑Nk=1 pk.

A stress measure that has a more obvious physical meaning than the second Piola–Kirchhoff stress is the Cauchy stress, which is defined in the current configuration. Byapplying a push-forward transformation to Equation (19) and among other things make use ofEquation (17) the Cauchy stress in chain k is obtained from

σ k = J−1FSkFT = J−1dev

[2F

∂�k,iso

∂CF

T]

+ pkI , (21)

where dev[·] = (·)− (1/3)[(·) : I ]I is the deviator operator in the Eulerian description and I

is the second order identity tensor.

4. Numerical Integration of Constitutive Response

First consider the definition of the fractional derivative of a function y(t), see Equation (14).Fractional derivatives can be numerically approximated by using convolution quadrature [18].We restrict ourselves to functions with homogeneous initial conditions (as is the case forthe internal variables Qk). The fractional derivative at time n+1t = (n + 1)�t can then beapproximated by [18]

n+1(Dαy) = 1

(�t)α

n∑j=0

bj (α)n+1−jy, n = 0, 1, 2, . . . , (22)

where �t is a uniformly distributed time step and bj (α) are the convolution quadratureweights. The weights bj (α) can be obtained by making a binomial expansion of a generatingfunction b(ζ ). A first order scheme (the discretization error is of first order in the time step) isobtained from the following generating function according to

b(ζ ) = (1 − ζ )α =∞∑j=0

bj (α)ζj

= 1 − αζ + α(α − 1)

2! ζ 2 + · · · + (−1)nα(α − 1) · · · (α − n + 1)

n! ζ n + · · · , (23)

where the weights can be identified and calculated by the recursion relation below

b0(α) = 1, b1(α) = −α, . . . , bj (α) = (j − 1 − α)

jbj−1(α), . . . . (24)


The same convolution quadrature may also be found by a suitable truncation of Grünwald’sdefinition of fractional differentiation [15]

n+1(Dαy) = 1

(�t)α

n∑j=0

bj (α)n+1−jy with bj (α) = �(j − α)

�(−α)�(j + 1). (25)

The calculation of the weights can be simplified by using the following recursion formula forgamma functions

�(j − α)

�(j + 1)= (j − 1 − α)

j

�(j − 1 − α)

�(j). (26)

We rewrite the approximation of the fractional derivatives as

n+1(Dαy) = 1

(�t)α(n+1y + ny) with ny =

n∑j=1

bj (α)n+1−jy, (27)

where ny plays the role of −(ny) when α = 1. It has been used that b0(α) = 1.Next we start to develop an algorithm for the integration of the constitutive response. First

we take the time step to be uniformly distributed, i.e., n+1Qk = Qk((n + 1)�t). Then byapplying the generalized midpoint rule to the rate equations Equation (15), we obtain

n+1(DαkQk) = 1

ταkk

[θ(n+1Sk − n+1Qk) + (1 − θ)(nSk − nQk)

],

with Sk = 2∂�k

∂C, (28)

where θ ∈ [0, 1] serves as a parameter to control the implicitness of the algorithm. For ex-ample, with θ = 0 we get the explicit forward Euler method and with θ = 1 we get the implicitbackward Euler method. Now, combine the recently formulated expression for numericalfractional differentiation in Equation (27) with Equation (28), and after some rearrangementswe obtain the following expression for the update of the internal variables

n+1Qk = (�t/τk)αk

1 + θ(�t/τk)αk

[θ n+1Sk + (1 − θ)(nSk − nQk)

]

− 1

1 + θ(�t/τk)αknQk with nQk =

n∑j=1

bj (αk)n+1−jQk. (29)

Finally, the total second Piola–Kirchhoff stress Equation (11) is given by

n+1S = n+1S(0) −N∑k=1

n+1Qk (30)

Unlike classical viscoelasticity, fractional viscoelasticity requires the whole history of theinternal variables to be saved and included in the calculation of a new time step. This is dueto the non-local character of fractional derivatives, cf. Equation (14). The classical model isextracted by letting αk = 1 and the sum in Equation (29) simply becomes −(nQk).

By differentiating the update of the second Piola–Kirchhoff stress tensor with respect to theright Cauchy–Green deformation tensor at the end of the time increment, we obtain the fourth


order material tangent stiffness tensor consistent with the developed algorithm for viscoelasticresponses

n+1C = 2∂ n+1S

∂ n+1C= n+1C(0) −

N∑k=1

θ(�t/τk)αk

1 + θ(�t/τk)αkn+1Ck with Ck = 2

∂Sk

∂C, (31)

where Ck is the elastic tangent stiffness tensor, (see, e.g., [9]). The tangent stiffness is crucialthen performing Newton iterations in order to maintain the quadratic convergence rate of themethod.

4.1. CONSTITUTIVE RESPONSE

In this section is the numerical algorithm for the integration of the constitutive responsedemonstrated and somewhat verified through some examples. We use the backward Eulerintegration rule (θ = 1) in all examples. Due to the flexibility in fitting the fractional vis-coelastic model to experimental data, it is often enough to use a single internal variable tomodel the viscous part of the response. Therefore we use the simplest viscoelastic model withinstantaneous and long time elastic responses, i.e., only two Maxwell chains (N = 2) areincluded in the model and one of the relaxation constants is taken to be infinite. The elasticpart is modeled as nearly incompressible and isotropic by using a modified Neo–Hooke model,particularly well suited for rubber-like materials. The strain energy function of such a modelis given by

�k = 1

2Gk(I1 − 3) + 1

2Kk(J − 1)2, k = 1, 2, (32)

where I1 = tr(C) is the first invariant of C. The second Piola–Kirchhoff stress tensor maynow be determined as

Sk = GkJ−2/3

(I − 1

3tr(C)C−1

)+KkJ (J − 1)C−1, k = 1, 2, (33)

where the material parameters Gk and Kk can be identified as the elastic shear and elastic bulkmodulus in chain k appearing in the small strain theory, respectively.

Stress relaxation of a fictitious rubber material is analyzed for two different deformationmodes with the following material parameters:

G(0) = G1 +G2 = 10 MN/m2, G(∞) = G2 = 2/3G(0) MN/m2,

K(0) = K1 +K2 = 2000 MN/m2, K(∞) = K2 = 2/3K(0) MN/m2,

τ1 = τ = 0.02 s, τ2 = ∞ and α ∈ (0, 1].First we consider stress relaxation in simple shear. It is obtained from the followingdeformation gradient

F = 1 0 0

0 1 κ

0 0 1

(34)

applied as a step with magnitude κ at time t = 0. The fractional viscoelastic response,Equations (11) and (15), can be solved analytically for step loads in terms of Mittag–Leffler


Figure 2. Normalized stress relaxation (the shear component) versus non-dimensional time in simple shear. Theinfluence of different orders of differentiation α is shown.

functions (or generalized exponential functions). In [14] an extensive derivation is given foruniaxial stress states in the case of small deformations. The total Cauchy stress becomes

σ =[(G(0) − G(∞))Eα[−(t/τ)α] +G(∞)

] −κ2/3 0 00 2κ2/3 κ

0 κ −κ2/3

, t ≥ 0, (35)

where

Eα(u) =∞∑k=0

uk

�(1 + αk)(36)

is the α:order Mittag–Leffler function. Note that, for the Neo–Hooke model, we end up withthe same shear stress components as in the linear theory. Next we consider stress relaxationin pure dilatation. The stress state arises from the deformation F = δI applied as a step withmagnitude δ at time t = 0. The total Cauchy stress then takes the form

σ = [(K(0) −K(∞))Eα[−(t/τ)α] +K(∞)

](δ3 − 1)I , t ≥ 0. (37)

In the following examples the stress response is given as a function of time for deform-ations large enough so that nonlinear effects become important. Figure 2 shows the stressrelaxation according to Equation (35) for three different values of the fractional derivative ex-ponent α in the rate equation. In the case of α = 1 the stress has reach its long time asymptoticresponse, but so is not the case for lower values of α. The reason for this is that the time forthe stress to relax depends heavily on the derivative exponent. In Figure 3 the expression inEquation (37) is compared with the stress response of the linear fractional viscoelastic modelfor different deformation magnitudes. The stress is normalized with the linear instantaneouselastic response. Both lines will eventually coincide when the deformation is small enough


Figure 3. Normalized stress relaxation versus non-dimensional time in pure dilatation. Comparison of the linearand the large deformation fractional viscoelastic models at different magnitudes of the deformation. The derivativeexponent used is α = 0.5.

(δ → 1). In Figure 4 the stresses are also calculated by using the numerical algorithm witha time step �t smaller than the relaxation constant τ to ensure sufficiently accuracy. Theshear stress component and the bulk stress component are displayed in Figures 4a and 4b,respectively. We conclude that the numerical results agree very well with the analytical results.

5. Structural Dynamics

A way to link the viscoelastic model to structural dynamics is achieved by using the developednumerical algorithm for the update of the stresses in the equations of motion. For this purposewe employ a displacement based finite element discretization that accounts for geometricallynonlinear effects. We use the total Lagrangian formulation, i.e., the element quantities areintegrated over the original mesh of the structure. Therefore, the following finite elementmatrices are with respect to the reference configuration. The procedure is outlined in detail byBathe [19].

The spatially discretized structural equations takes the form (in matrix notation)

Mu + f int = f ext (38)

with corresponding initial conditions

u(0) = 0u, u(0) = 0v, (39)

where M is the mass matrix, f int is the internal nodal force vector and f ext is the externalnodal force vector (corresponding to applied load). The internal nodal force vector f int iscalculated as in [19] from the element spatial distribution of stress S using Equation (30).Both f int and f ext depend nonlinearly on the displacement, however, for simplicity f ext iscalculated without taking care of the change in configuration of the structure.


Figure 4. Normalized stress relaxation versus non-dimensional time. The numerical result is compared with theanalytical result for α = 0.5.


Many different ways exist to numerically integrate the equations above, for example, theNewmark family of methods [20], where the update of the nodal displacement vector u andthe nodal velocity vector u are approximated by

n+1u = nu +�t nu + (�t)2

2

[(1 − 2β)nu + 2β n+1u

], (40)

n+1u = nu +�t[(1 − γ )nu + γ n+1u

], (41)

while �t is the time step. The parameters β and γ can be chosen to control stability, accuracyand implicitness.

5.1. IMPLICIT ALGORITHM

When using implicit integration we try to find the equilibrium solution to the linearized struc-tural equations in an iterative procedure. By choosing the trapezoidal version of Newmark’smethod (β = 1/4 and γ = 1/2) we get an implicit algorithm, which with a sufficient numberof iterations within each time step is unconditionally stable and second order accurate. Thediscretized linearized structural equations at time n+1t = (n+ 1)�t are given by

M n+1u(i) + n+1K(i−1)

�u(i) = n+1f ext − n+1f(i−1)int , (42)

n+1u(i) = n+1u(i−1) +�u(i), (43)

where i is the iteration number, while K is the structural tangent stiffness matrix, whichdepends on the current displacement state of the structure [19]. For the calculation of K thematerial tangent stiffness tensor C in Equation (31) and the stress tensor S need to be providedin accordance with Bathe [19]. Note that the external nodal force vector is not updated duringthe iterations.

Assume that the solution is known at time nt = n�t (taking n+1u0 = nu). By combin-ing Equations (40) and (41) (with the present choice of β and γ ) with Equation (42) whileusing Equation (43), we obtain the following expression for the correction �u(i) to the nodaldisplacement vector(

n+1K(i−1) + 4

(�t)2M

)�u(i)

= n+1f ext − n+1f(i−1)int − M

[4

(�t)2(n+1u

(i−1) − nu)− 4

�t

nu − nu

]. (44)

The improved (if the solution converges) nodal displacement vector n+1u(i)

is then updatedaccording to Equation (43) and the element spatial distribution of the right Cauchy–Green de-formation n+1C

(i)may be updated. Thereby the stress n+1S(i) and the material tangent stiffness

n+1C(i)

within each element can be obtained by using the scheme described in Section 4. Withthis solution n+1f

(i)

int and n+1K(i)

can be calculated. The iterations are repeated until (for thepurpose) sufficiently accuracy is reached. Finally, the nodal acceleration vector and velocityvector are updated from the Newmark equations, i.e.,

n+1u = 4

(�t)2(n+1u − nu) − 4

�t

nu − nu (45)


and

n+1u = nu + �t

2(nu + n+1u) (46)

and a new time step can be taken.

5.2. EXPLICIT ALGORITHM

By choosing β = 0 and γ = 1/2 in the Newmark method equations (40) and (41), we obtainthe nearly explicit Newmark method, and the nodal displacement vector and the nodal velocityvector become

n+1u = nu +�t nu + (�t)2

2nu, (47)

n+1u = nu + �t

2(nu + n+1u). (48)

This method is second order accurate and only conditionally stable [20]. The stability con-dition is discussed in Section 6. In return this algorithm does not need expensive iterationsand calculations of the structural tangent stiffness within each time step. As in the implicitalgorithm we assume that the solution of the structural equations of motion is known at timent and we try to solve it at the end of the time interval n+1t

M n+1u = n+1f ext − n+1f int. (49)

The update of the nodal displacement vector n+1u is given from Equation (47). Thereby, itis possible to calculate n+1C and the updated element spatial distribution of stress n+1S isobtained from the numerical algorithm of the constitutive response described in Section 4.With this solution n+1f int is calculated and Equation (49) is solved for the nodal accelerationvector n+1u. The nodal velocity vector n+1u may now be updated, and a new time step can betaken.

6. Numerical Examples

The purpose of the numerical algorithm will be demonstrated through some examples. Forthis purpose the simplest viscoelastic model that describe solid behavior is used, i.e., onlytwo Maxwell chains are included and one of the relaxation constants is taken to be infinite.We restrict ourselves to polymeric rubber materials, therefore the deformations are assumedto be nearly incompressible. By choosing the strain energy function for a modified Neo–Hooke model equation (32), the elastic Piola–Kirchhoff stress tensor in the Maxwell chainsis given by the expression in Equation (33). For uniaxial stress and plane stress states it isconvenient to treat the deformations as exactly incompressible. The hydrostatic pressure pk,see Equation (19), is then determined by imposing the incompressibility condition det(C) = 1rather than using the last part of Equation (33). In the uniaxial case the elastic stress becomes

Sk = Gk

(1 − 1

C√C

), k = 1, 2, (50)


where Gk = Ek/3, and Ek being the elastic modulus in chain k. In case of plane stressconditions the elastic stress becomes

Sk = Gk

[1 − (C33)

2C22 (C33)2C12

(C33)2C12 1 − (C33)

2C11

], k = 1, 2, (51)

where

C33 = 1

C11C22 − (C12)2. (52)

Note that the bulk modulus Kk has lost its role as a material parameter, this is of course dueto the incompressibility condition. When using implicit time integration the elastic tangentstiffness tensor is also needed, see Equation (31), which is easily calculated by differentiatethe expression for the stresses above. The backward Euler method (θ = 1) will be used in allexamples for the integration of the total stress response that give f int.

6.1. QUASI-STATIC RESPONSE OF A VISCOELASTIC THICK WALLED TUBE

The viscoelastic correspondence principle is often used to find analytical solutions to vis-coelastic boundary value problems (e.g., [21]). If fractional operators appear in the evolutionlaws for the internal variables, we may utilize the following Laplace transform pair [22]:

L(Eα[−(t/τ)α])(s) = sα−1

sα + τ−α. (53)

One solution that may be obtained is the quasi-static solution of a thick walled circular tubewith an internal pressure distribution applied as a step with magnitude p at time t = 0. Theradial displacement for plane stress conditions and incompressibility (i.e., letting Kk → ∞)is found to be

ulin = pb2

6(a2 − b2)

[(1

G(0)− 1

G(∞)

)Eα[−(t/τ∗)α] + 1

G(∞)

](r + 3

a2

r

)

with1

τα∗= G(∞)

G(0)

1

τα, t ≥ 0, (54)

where r is the radius while a and b are the outer and inner radii, respectively. (The solutionabove in Equation (54) is valid for linear viscoelasticity and linearized kinematics.) The finiteelement solution of the same problem will be compared to this expression, and thereby givean indication of its accuracy. The tube is discretized using four-node plane stress elementsthat account for nonlinear kinematics. Figure 5a shows the finite element discretization of thetube. In the quasi-static analysis we use the implicit algorithm from Section 5 (without themass matrix M). The original geometrical data and material parameters of the tube are asfollows:

a = 0.3 m, b = 0.15 m, G(0) = G1 +G2 = 2 MN/m2,

G(∞) = G2 = 1 MN/m2, τ1 = τ = 0.02 s, τ2 = ∞ and α1 = α = 0.67.

In Figure 6 the radial displacement at the inner radius is shown for different magnitudes of thepressure. The pressure is here with respect to the undeformed mesh. Further, the displacement


Figure 5. The element mesh of the tube. The deformed mesh is shown for the pressure magnitude p = 500 kN/m2

at the end of the time interval.

is normalized with the analytical expression evaluated at t = 0 in Equation (54). The finiteelement solution agrees very well with the linear analytical solution when the pressure issmall. For responses in the nonlinear regime we have no analytical expression, but as can beseen the finite element results follows the same qualitative trend as the linear solution. Finally,Figure 5b shows the deformed mesh for the largest pressure distribution at the end of the timeinterval.

6.2. DYNAMIC RESPONSE OF A VISCOELASTIC BAR

Another example is the dynamic response of a viscoelastic bar of a fictitious rubber material.The bar is held fix at one end and is subjected to an axial step load P at time t = 0 at the otherend. The load magnitudes will be chosen large enough so that geometrical effects becomeimportant. The bar and its material have the following data:

length L = 0.5 m, cross sectional area A = 0.0025 m2, density = 1,000 kg/m3,

E(0) = E1 + E2 = 10 MN/m2, E(∞) = E2 = 7 MN/m2,

τ1 = τ = 0.02 s, τ2 = ∞ and α ∈ (0, 1],where the length and area are the original ones (when the bar is in rest). The stress state isassumed to be uniaxial and we make use of Equation (50). The long time elastic tip displace-ment u(∞) can be found analytically and is obtained by using long time asymptotic materialproperties.

P = S(∞)A√C with S(∞) = S2 and C =

(1 + u(∞)

L

)2. (55)


Figure 6. Normalized inner radial displacement u/ulin(0) versus non-dimensional time t/τ for a viscoelastic tube

(α = 0.67) with an internal step pressure. The response is displayed for different pressure magnitudes.

Figure 7. Tip displacement u versus non-dimensional time t/τ for a viscoelastic bar subjected to a step load withmagnitude P = 1000 N is shown for α = 0.5. We see that the explicit and implicit time integration schemes givethe same result.


Figure 8. Tip displacement u versus non-dimensional time t/τ for a viscoelastic bar subjected to a step load withmagnitude P = 1000 N. The influence of different values of α is displayed.

Figure 9. Normalized tip displacement u/ulin(∞)

versus non-dimensional time t/τ for a step loaded viscoelasticbar for α = 0.5. Comparison of the linear and nonlinear model for different load magnitudes. Note that for P = 1N the models coincide.


Figure 10. Normalized tip displacement u/u(∞) versus non-dimensional time t/τ for a viscoelastic bar subjectedto a step load with magnitude P = 1000 N. For α = 1 the long time asymptotic value of the response is reachedfor t/τ = 50.

We may note that the displacement is not linear in the applied force. A finite element discret-ization with bar elements that include for geometrically nonlinear effects is employed. Thebar is discretized into five equal elements. The FE equations will be solved by using both theimplicit algorithm and the explicit algorithm outlined in Section 5. The explicit scheme is onlyconditionally stable, and the critical time step in the linear elastic case is �t = 2/ωmax, whereωmax is the highest eigen frequency of the finite element model. Numerical experiments in[23] indicate that this is also a good estimation of the critical time step in the linear fractionalviscoelastic case (and it seems to be a good estimate of the critical time step even for thepresent model). The implicit scheme, however, is unconditionally stable provided a sufficientnumber of iterations within each time step is used. Based on accuracy considerations the timestep is chosen as �t = 0.005τ . The free edge tip displacement is calculated using the twodifferent time stepping algorithms and is displayed in Figure 7. The results of the algorithmsare in very close agreement. Due to the fact that the time step is limited by accuracy con-ditions rather than stability conditions, the explicit code will be used henceforth. (Note thatthe explicit code is significantly faster in that no iterations in each time step are needed.)In Figure 8 the tip displacement versus time is displayed for different values of α and wemay observe that the value of α clearly effects the results. For example the higher frequencycomponents (frequencies considerable higher than ωτ = 1) seem to decay faster with lowervalue of α. A comparison between the present nonlinear model and a FE code using linearkinematics together with the linear fractional viscoelastic model is presented in Figure 9 fordifferent magnitudes of the applied load P . Note that for small load magnitudes (resulting indeformations in the linear regime) the models coincide. The response is normalized with thelinear long time response at the free bar end ulin

(∞) = PL/AE(∞). The last example, Figure 10,shows the long time elastic tip displacement. We observe that the long time asymptote value isonly reached for α = 1, this is in total agreement with the result in Figure 2. The time for the


response to reach its long time value increases substantially with decreasing value of α. Alsonote that the FE code indeed gives the same long time value as the exact value determinedfrom Equation (55).

7. Concluding Remarks

A three-dimensional fractional derivative viscoelastic model for large deformations has beenformulated by using Lagrangian kinematics. The elastic part of the response is split into purelyisochoric and volumetric parts by assuming isotropy and nearly incompressible deformations,which is a reasonable assumptions for rubber materials. The modeling of the inelastic responseis motivated by linear viscoelasticity and is carried out by using internal variables of stresstype. The equations governing the evolution of the internal variables are differential equationsof fractional order. A numerical algorithm for the calculation of the constitutive response isdeveloped by employing the generalized midpoint rule to the evolution equations and usingconvolution quadrature to numerically approximate the fractional derivatives. This algorithmis incorporated into finite element environments for calculations of structural response.

The elastic stress is modeled by a modified Neo–Hooke model, which is useful for describ-ing the responses of nearly incompressible rubber materials. Further, expressions for uniaxialand plane stress states are derived from this model by considering exactly incompressible con-ditions. Examples of constitutive response for two deformation modes (shear and bulk) showthat the numerical results are in very close agreement with results from analytical expressions.It also shows the effect of imposing different values of the derivative exponent α in the rateequations, which indicates the capability of the model to capture the behavior of real material.

The quasi-static response of a tube subjected to a pressure distribution at its inner radius iscomputed using the present viscoelastic model together with the present algorithm for the con-stitutive response. In the case of small pressure magnitudes, the results are in close agreementwith the analytical results. This somewhat verifies the present algorithm. A second exampleconsiders the transient response of a viscoelastic bar (inertia effects are included). The resultsare in close agreement with numerical results obtained by the linear theory in the case ofsmall load magnitude. It is well known that by using a displacement based finite elementformulation for three dimensional and plane strain conditions, problems with the convergencein the incompressible limit occur. In these cases, a mixed finite element formulation is usuallypreferable. However, in the present examples plane stress conditions are assumed and therewill be no difficulties in the incompressible limit. In the dynamic analyses, it is possibleto use either the implicit or the explicit time stepping algorithms. The explicit algorithm isonly conditionally stable, however, numerical experiments show that the explicit algorithm ispreferable since the time step for obtaining reasonable accuracy is smaller than the criticaltime step of the explicit code (the algorithms are of the same accuracy order in the time step).Also note that the explicit code is faster in that no iterations within each time step is needed.

Acknowledgments

The work has been supported by The Swedish Research Council for Engineering Science(TFR). Fruitful discussions with Professor Kenneth Runesson at the Department of AppliedMechanics, Chalmers University of Technology, are gratefully acknowledged.


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Fractional Derivative Viscoelasticity at Large Deformations

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