Viscoelastic material models in peridynamics

Applied Mathematics and Computation 219 (2013) 6039–6043

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Viscoelastic material models in peridynamics Olaf Weckner a,⇑, Nik Abdullah Nik Mohamed b a b

The Boeing Company, P.O. Box 3707, MC 42-26, Seattle, WA 98124, USA Institute of Space Science (ANGKASA), Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor D.E., Malaysia

a r t i c l e

i n f o

Keywords: Peridynamics Nonlocality

a b s t r a c t In this paper we propose a new viscoelastic material model within the framework of the nonlocal peridynamic formulation of continuum mechanics. Using Fourier- and Laplacetransforms we derive integral-representation formulas using a Green’s function approach for both local and nonlocal viscoelasticity. As an example we calculate the local and nonlocal response of an infinite viscoelastic bar impacted by a point load. We show both analytically and numerically that local viscoelasticity is recovered in the limit as the peridynamic lengthscales become small. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Materials exhibiting characteristics that are both solid- and fluid-like are categorized as viscoelastic materials, see e.g. [6,16]. Materials that exhibit this behavior include some classes of polymers but also biological tissues such as tendons and ligaments. Due to the fluid-like behavior, viscoelastic materials can be categorized as rate-type materials exhibiting a time-dependent constitutive law. Similar to plastic materials viscoelastic materials belong to the class of materials with memory, in the sense that its current state of stress does not only depends on the current deformation, as in elastic materials, but also on deformations experienced in the past. Unlike plasticity, which represent materials with permanent memory, viscoelastic material have so-called fading memory, i.e. deformations which happened in the distant past have less effect on the current state of stress. In viscoelastic materials the current state of stress is represented by an integration over all past deformations weighted by a non-negative, Lebesque-integrable, monotonic decreasing function, see e.g. [14,7,5]. Experimentally, the material characterization of viscoelastic materials consists of performing tests similar to those used for elastic solids, but slightly modified so as to enable the observation of the time dependency of the material response, see [10]. While creep and stress relaxation test are typically used to determine the long-term material characterization, the short term behavior is typically investigated using sinusoidal loading, see e.g. [6]. The viscoelastic behavior is commonly approximated by the combination of constitutive laws: while the solid-like behavior is often characterized by Hooke’s law, the fluid-like behavior or the viscous part is often modeled by Newton’s law. Consequently, one-dimensional linear viscoelastic materials are often represented by mechanical models consisting of elastic springs and viscous dash-pots, see [16]. If the spring and a dash-pot are connected in series, this model is also known as Maxwell’s model, and as a Kelvin–Voigt model when they are connected in parallel, see e.g. [9]. A slightly more general model is the so-called standard linear solid or Zener model which uses linear springs and dashpots in both serial and parallel. The Zener model is the simplest spring-dashpot model which captures both creeping and stress relaxation. Despite certain limitations in accurately capturing shear deformations spring-dashpot models are successfully used among many rheologists: in [2] the constitutive response of coagulated ⇑ Corresponding author. E-mail addresses: [email protected] (O. Weckner), [email protected] (N.A. Nik Mohamed). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.11.090

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blood is found to be accurately captured by the Zener model over the frequency range investigated using dynamic ultrasound elastography. In [3] a generalized Maxwell model is found to capture the viscoelastic effects of certain food emulsions well. In order to capture even more complicated material behavior such as size-dependent superelastic effect observed in recent experiments of single crystal Cu–Al–Ni shape memory alloys, researchers have considered going beyond local viscoelasticity by accounting for nonlocal effects, see e.g. [4]. Nonlocal viscoelastic effects are thought to be important also in the sizedependent behavior of polymers. Using a different nonlocal formulation called peridynamics [11,12] it has been shown how nonlocality can be explored to characterize wave dispersion in silicon observed at lengthscales comparable to the lattice spacing, see [15]. At the macroscale brittle-elastic material models are commonly employed in peridynamics, enabling the simulation of multiple interacting cracks. In this paper we propose an extension of this brittle–elastic peridynamic material model by accounting for viscoelastic effects. 2. Peridynamic viscoelastic materials In 1D, small deformations uðx; tÞ of the material point x 2 R with density q at time t 2 ½0; T in an infinite1 homogeneous viscoelastic body of rank N are governed by the following equation of motion2

qu€ ðx; tÞ ¼ I ½uðx; tÞ þ bðx; tÞ;

ð1Þ

where the linear operator I captures internal forces while bðx; tÞ captures external forces. The equation of motion (1) has the same form for both local and nonlocal formulation but with different operators L

N Z X

I ½uðx; tÞ ¼

n¼0 NL

I ½uðx; tÞ ¼

1

0

N Z X n¼0

an ðt  sÞC n @ ns @ 2x uðx; sÞds;

1 0

an ðt  sÞ@ ns

Z

ð2Þ

þ1

C n ðx0  xÞðuðx0; sÞ  uðx; sÞÞdx0ds:

ð3Þ

1

The initial conditions are given by

u0 ðxÞ ¼ uðx; t ¼ 0Þ;

v

0

ð4Þ

_ t ¼ 0Þ: ðxÞ ¼ uðx;

Eqs. (2) and (3) describe the generalization of the local and nonlocal behavior of viscoelastic material respectively, characterized by the nth time-derivative of displacement vector. The idea of using nth time derivative goes back to the concept of rate-type material proposed by Truesdell in [1]. We use this general form to cater the viscoelastic material behavior as well as viscoplasticity under large deformation [8]. Applying the Fourier-transform3 with respect to the spatial variable x $ k we obtain

€ ðk; tÞ þ u

N X

x2n ðkÞ

n¼0

x2n ðk; tÞ ¼

8 2 < k Cn q

Z 0

1

an ðt  sÞ@ ns u ðk; sÞds ¼

 tÞ bðk;

with;

q

ð5Þ

local;

: C n ð0ÞC n ðkÞ nonlocal: q

Applying the Laplace-transform4 with respect to the time variable t $ s we find the transformed solution

~ ðk; sÞ ¼ u

PN 2 n1 ~ sÞ=q þ su ~  0 ðkÞ þ u  ð1Þ   0ðn1Þ ðkÞ bðk; þ  þ u n¼1 xn ðkÞan ðsÞ½u0 ðkÞs 0 ðkÞ þ ; PN 2 2 n ~ n ðsÞs s þ n¼0 xn ðkÞa

ð6Þ

n  ðnÞ  with u 0 ðkÞ :¼ @ s uðk; 0Þ. In the following we limit our consideration to the simplest peridynamic viscoelastic model with out memory: N ¼ 1; a0 ðtÞ ¼ a1 ðtÞ ¼ DðtÞ:

_ tÞ; L½uðx; tÞ ¼ C 0 @ 2x uðx; tÞ þ C 1 @ 2x uðx; Z þ1 Z 0 NL L½uðx; tÞ ¼ C 0 ðx0  xÞ½uðx0 ; tÞ  uðx; tÞdx þ L

1

ð7Þ þ1

0

_ 0 ; tÞ  uðx; _ tÞdx : C 1 ðx0  xÞ½uðx

ð8Þ

1

C 0 ðnÞ is commonly referred to as the micromodulus function whereas C 0 is the Young’s modulus of local elasticity. C 1 ðnÞ and C 1 capture the nonlocal and local viscous effects. In Fourier-space we have

 tÞ=q; € ðk; tÞ þ 2DðkÞu ðk; tÞ ¼ bðk; _ ðk; tÞ þ x20 ðkÞu u

ð9Þ

1 It is physically intuitive that as the distance between a pair of particles gets very large, the interaction between them becomes negligible. In what follows we shall assume that this happens fast enough to ensure the convergence of the various infinite integrals encountered. 2 0 € ¼ @ 2 ðÞ etc. with @ n ðÞ ¼ @ n ðÞ We use the following notations: ðÞ_ ¼ @ t ðÞ; ðÞ t R t @tn and @ t ððÞÞ ¼ ðÞ. R 3  f ðkÞ ¼ F ff ðxÞg ¼ R eikx f ðxÞdx; f ðxÞ ¼ F 1 ff ðkÞg ¼ 21p R eikx f ðkÞdk. R cþı1 R1 4 ~ f ðsÞ ¼ Lff ðtÞg ¼ 0 expðstÞf ðtÞdt; f ðtÞ ¼ L1 f~f ðsÞg ¼ 1ı cı1 expðstÞ~f ðsÞds.

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 0 ðkÞ ¼ F fu0 ðxÞg; v  0 ðkÞ ¼ F fv 0 ðxÞg. Using the individual Laplacewith x21 ðkÞ ¼ 2DðkÞ. The transformed initial conditions are u transforms

 1 eDðkÞt sinðxd ðkÞtÞ ¼ ¼: gðk; tÞ; 2 xd ðkÞ þ 2DðkÞs þ x0 ðkÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xd ðkÞ ¼ x20 ðkÞ  D2 ðkÞ; L1



ð10Þ

s2

ð11Þ

L fv ðkÞ þ u ðkÞðs þ 2DðkÞÞg ¼ D_ ðtÞu ðkÞ þ DðtÞðv ðkÞ þ 2DðkÞu ðkÞÞ; 1

0

0

0

0

0

ð12Þ

we can use the convolution theorem of Laplace-transforms to obtain the solution in Fourier ðk; tÞ space:

 ðk; tÞ ¼ u

Z

t

gðk; sÞ

 t  sÞ bðk;

0

q

 0 ðkÞgðk; tÞ_: ds þ v 0 ðkÞgðk; tÞ þ u

ð13Þ

Finally we use the convolution theorem of Fourier-transforms to obtain the following integral representation of the solution of Eqs. (7, 8) in ðx; tÞ space.

Z

Z

Z þ1 Z þ1 0 0 _ 0 ; tÞdx0 with gðx0 ; t 0 Þdx dt þ v 0 ðx  x0 Þgðx0 ; tÞdx0 þ u0 ðx  x0 Þgðx q 0 1 1 1  DðkÞt  Z 1 DðkÞt e sinðxd ðkÞtÞ 1 e sinðxd ðkÞtÞ ikx ¼ e dk: gðx; tÞ ¼ F 1 2p 1 xd ðkÞ xd ðkÞ

uðx; tÞ ¼

t

þ1

bðx  x0 ; t  t 0 Þ

ð14Þ ð15Þ

gðx; tÞ is a Green’s function for this problem. This integral representation of the solution holds for both local and nonlocal formulation by choosing the corresponding dispersion relation L;NL xd ðkÞ and attenuation coefficient L;NL DðkÞ. As an example we assume that the nonlocal stiffness (i ¼ 0) and attenuation (i ¼ 1) distribution is as follows: 2 4C i 4C i 1 2 2 C i ðnÞ ¼ pffiffiffiffi 3 eðn=‘i Þ $ Ci ðkÞ ¼ 2 e4k ‘i : p‘i ‘i

ð16Þ

The amplitudes of the stiffness and damping distribution are chosen such that in the large wavelength limit (jkj  1) waves have the same phase velocity and the same attenuation in both local and nonlocal formulation. 3. Impact on a viscoelastic bar Next we concentrate on an impact loading at x ¼ 0; t ¼ 0 on a bar initially at rest

^DðxÞDðtÞ; bðx; tÞ ¼ b

ð17Þ

u0 ðxÞ ¼ v 0 ðxÞ  0;

ð18Þ

where the solution is proportional to the Green’s function

uðx; tÞ ¼

^ b

q

gðx; tÞ:

ð19Þ

^ ^ 1 We normalize with respect to the local parameters: n ¼ ^xx ; ki ¼ ‘^xi with ^ x ¼ pCffiffiffiffiffiffi ; s ¼ ^tt with ^t ¼ CC 10 and gðn; sÞ ¼ uðx¼nuxc;t¼stÞ C0 q ^ b with uc ¼ pffiffiffiffiffiffi and obtain C0 q

gðn; sÞ ¼

1 2p

Z

1

1

ef1 ðjÞs

sinðf2 ðjÞsÞ ijn e dj; f2 ðjÞ

ð20Þ

8   8 1j2 k2 > < ! k22 2 1e 4 1 > j!1 1 > NL > > ; f1 ðjÞ ¼ ¼ > k21 > : ! j2 þ Oðj4 Þ <   j j!0 2 ^ 8 f1 ðjÞ ¼ tD k ¼ ¼ ^x > 1 < j! > > !1 2 > L j > f ð j Þ ¼ ¼ ; > 1 2 > : ! j2 þ Oðj4 Þ : j!0 2

8 qffiffiffiffiffiffiffiffiffiffiffiffiffi 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < ! 4 4 > 1j2 k2 2  4 >  k1 > j!1 k0 NL > > ; f2 ðjÞ ¼ 4 1e k42 0  NL f12 ðjÞ ¼ > : ! jjj þ Oðj3 Þ 0 <   > j j !0 ¼ f2 ðjÞ ¼ ^txd k ¼ 8 ^x > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < ! 1 > > > 2 > L f2 ðjÞ ¼ j2 ð1  j Þ ¼ j!1 > : > 4 : : ! jjj þ Oðj3 Þ j!0

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O. Weckner, N.A. Nik Mohamed / Applied Mathematics and Computation 219 (2013) 6039–6043 u 0.5

0.4

0.3

0.2

0.1

1

2

3

4

5

t

Fig. 1. Normalized displacement history in local and nonlocal viscoelasticity.

Note that the functions NL f1;2 are bounded for j ! 1. Therefore the above integral does not converge in the classical, Riemann sense. The reason is that there is a Dirac function D ‘‘hidden’’ in the solution since in peridynamics the smoothness of an elastic solution corresponds to the smoothness of the loading and initial conditions, see e.g. [13]. By rewriting the solution

 Z 1 1 sinðNL f2 ðjÞsÞ NL e f1 ðjÞs ð s Þ eijn dj;  f 1 NL f ðjÞ 2p 1 2 sinðNL f2 ð1ÞsÞ NL f1 ðsÞ ¼ e f1 ð1Þs ; NL f ð1Þ 2

NL

gðn; sÞ ¼ DðnÞf1 ðsÞ þ

ð21Þ ð22Þ

the integral now converges. Note that while the normalized local solution does not have any more material parameters, the nonlocal solution still has two normalized lengthscales. The asymptotical behavior shown suggests that the nonlocal solution will behave more and more similar to the local solution when these lengthscales are small. This will also be demonstrated by numerically evaluating the integral in (20) using Mathematica.5 Setting k0 ¼ k1 2 f0:5; 1; 1:5g we plot both local (dashed red6 line) and nonlocal solutions (blue lines) at the point n ¼ 1:0 for the time interval s 2 ½0; 5 together with the steady-state solution (black line) in Fig. 1. As expected the nonlocal solution converges to the local solution with the same elastic and viscous properties when the nonlocal lengthscales are small. 4. Summary In this paper we propose a new viscoelastic material model within the framework of the nonlocal peridynamic formulation of continuum mechanics. We show both analytically and numerically that local viscoelasticity is recovered in the limit as the peridynamic lengthscales become small. While previous research in peridynamics has focussed mainly on brittle-elastic material behavior, we hope to extend the class of problems to materials exhibiting significant viscoelastic behavior. One of the advantages of the nonlocal peridynamic formulation is that it utilizes integral operators rather than differential operators to describe internal forces, thus reducing the smoothness requirements in the spatial domain. Adding a bond failure criterion to the model proposed in this paper would then enable the simulation of crack formation and propagation in viscoelastic materials. References [1] W. Noll, S. Antman, C. Truesdell, The Non-linear Field Theory of Mechanics, Springer Verlag, Berlin, Heidelberg, New York, 2004, ISBN 3-540-02779-3. [2] Guy Cloutier, Ce’dric Schmitt, Anis Hadj Henni, Characterization of blood clot viscoelasticity by dynamic ultrasound elastography and modeling of the rheological behavior, J. Biomech. 44 (2011) 622–629. [3] C. Bengoechea et al, Linear and non-linear viscoelasticity of emulsions containing carob protein as emulsier, J. Food Eng. 87 (2008) 124–135. [4] Lei Qiao et al, Nonlocal superelastic model of size-dependent hardening and dissipation in single crystal Cu–Al–Ni shape memory alloys, Phys. Rev. Lett. 106 (2011) 085504. [5] M.J. Leitmann, G.M. Fisher, The linear theory of viscoelasticity, Encyclopedia of Physics, vol. 6(a3), Springer Verlag, Berlin, Heidelberg, New York, 1973. [6] J. Ferry, Viscoelastic Properties of Polymers, John Wiley & Sons, New York, 1980. [7] J.C. Saut, D.D. Joseph, Fading memory, Arch. Ration. Mech. Anal. 81 (1) (1983). [8] E. Mitsoulis, The non-linear field theory of mechanics, Rheol. Rev. (2007) 135–178. [9] A.C. Pipkin, Lectures on Viscoelasticity Theory, Springer Verlag, Heidelberg, Berlin, New York, 1972. pp. 1092–1094. 5 6

The finite limit of integration was chosen such that doubling the length of this interval did not change the solution in the ‘eye-ball’ norm. For interpretation of color in Fig. 1, the reader is referred to the web version of this article.

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