materials based design of structures computational modeling of the mechanical behavior of gold polymer nanocomposites

Materials based design of structures: computational modeling of themechanical behavior of gold-polymer nanocomposites Swantje Bargmann, Celal Soyarslan, Edgar Husser, Natalia Konchakova

Materials based design of structures: computational modeling of the

mechanical behavior of gold-polymer nanocomposites

Swantje Bargmann, Celal Soyarslan, Edgar Husser,

Natalia Konchakova

DOI: 10.1016/j.mechmat.2015.11.008

To appear in: Mechanics of Materials

Received date: 22 May 2015

Revised date: 2 November 2015

Accepted date: 9 November 2015

Please cite this article as: Swantje Bargmann, Celal Soyarslan, Edgar Husser, Natalia Konchakova,Materials based design of structures: computational modeling of the mechanical behavior of gold-

polymer nanocomposites, Mechanics of Materials (2015), doi:10.1016/j.mechmat.2015.11.008

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• Separate parameter identification for gold and epoxy resin constituents

• Captured characteristics of the response curves, gained stability withimpregnation

b Institute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht,

Max-Planck Straße 1, 21502 Geesthacht, Germany

Abstract

The impact of small scale geometric confinement on deformation mechanisms

is subject of intense research in materials sciences nowadays Nanoporousmetals have a microstructure with an extremely high volume-specific surfacecontent Due to a very high local strength and a relatively regular intercon-nection of the nanoconstituents as well as a low mass density, nanoporousmetals are very good candidates for strong and light-weight structural ma-terials

The modeling of a modern nanocomposite material of gold-polymer is inthe focus of the contribution A gradient extended crystal plasticity theory

is applied to the computation of the mechanical response of the metal part

of the composite and an elastic-viscoplastic continuum model is used for thesimulation of the polymer material The gradient hardening contribution isincluded into the crystal plasticity model in order to study the influence ofthe ligament size Numerical results of the deformation of the gold-polymernanocomposite under compression are presented Simulation results are com-pared to the corresponding experimental data

Keywords: nanocomposite, metal-polymer composite, extended crystal

plasticity, gold, size effect

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1 Introduction

Due to increasing significance of newly developed materials and/or terials with improved properties, material’s characterization and simulationbecome more and more powerful tools Recently, research on nanoporousmetals significantly increased [11, 12, 25, 38, 49, 50] due to their relativelyhigh ductility and high specific strength It has been shown that nanoporousmetals exhibit size effects such as increasing strength with decreasing di-mensions [21, 50] In case of nanoporous materials, it is the (ligament)size which matters By ligament, we refer to the regularly interconnectednano-/microstructural constituents Of course, the material’s response fur-ther depends on its nanoporous architecture The gold-polymer compositeconsidered in this contribution is hierarchical To date, no established un-derstanding of how to design a microstructured composite exists As a con-sequence, there is a strong need for experimental as well as numerical studies

ma-at all scales

This work aims at contributing to designing integrated nanostructuredmultiphase materials systems in which microstructure design enables strength.What happens at the small scale during deformation is amazingly complex

as dislocation nucleation, dislocation starvation, capillary effects, lattice fects, size effects, among others, occur As a consequence, the composite’smechanical properties differ greatly from those of the corresponding bulkmaterials

de-This paper focuses on material modeling of a metal-polymer ite under large deformations, considering non-local effects via introduction

nanocompos-of the plastic strain gradient into a crystal plasticity theory 1 The modelingand analysis extends the experimental investigation of Wang and Weissm¨uller[50] and this modelling approach allows to account for a bending dominateddeformation behavior which, at this scale, results into a pronounced devel-opment of strain gradients Extensive plastic localization is indeed expectedfor NPG as well as for polymer infiltrated composites For the time being,

∗ Corresponding author

URL: http://www.tuhh.de/icm (Swantje Bargmann)

1 Although the here considered ligament sizes are within the nano scale ( 50-150 nm), the application of a continuum-based crystal plasticity model is justified by the fact that dislocation-starvation scenarios are not supported by corresponding results of Ngˆ o et al [31] and, hence, do not apply to nanoporous gold.

is latent hardening, i.e., an active slip system influences the hardening of aneighboring inactive slip system It controls the shape of the single crys-tal yield surface Among others, slip interactions and the development ofgeometrically necessary dislocations (GNDs) play a role in latent harden-ing While there exist some experimental works on nanoporous metals (seeabove), there hardly are any with respect to nanoporous metal-polymer com-posites [50], and even fewer deal with the computational modeling of theirbehavior.

Unlike crystalline materials, epoxy resin is an amorphous solid which pears in disordered form Its stress response when subjected to large mono-tonic plastic deformations is rate dependent and highly nonlinear composed

ap-of a yield-peak, post-yield strain sap-oftening and rapid strain hardening wherethe post–yield stress softening is linked to the evolution of the local free-volume Upon subsequent unloading, a Bauschinger-type reverse-yieldingphenomenon is observed A mathematical model which accounts for theseobserved phenomena, which is also applied in the current contribution ispresented in [1, 16] Regarding continuum elasto-viscoplastic theory usesthe local free-volume associated with certain metastable states and the resis-tance to plastic flow as internal state variables Using the dependence of theHelmholtz free energy on the plastic deformation gives rise to rapid strain-hardening subsequent to post-peak strain softening as well as a backstress inthe flow rule

To gain further insight into the model, we take a detailed look at the

2 Continuum-mechanically based contributions on the simulation of size-effects in archical structures can be found in, e.g., [8, 39].

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predicted mechanical response of a two-dimensional representative volumeelement The numerical implementation of the strongly coupled and nonlin-ear system of equations is realized with the finite element method in spaceand a mixed implicit-explicit integration procedure in time

2 Material properties

2.1 Gold

Although gold is a rather exclusive material, it is of great interest incase of nanoporous metals as under compression it deforms up to large plas-tic strains (e.g., up to 120 % true strain [50]), whereas usually nanomate-rials only allow for small deformations [26] As observed in experiments,the strength of nanoporous gold crystals increases if crystal dimensions arereduced [24] Moreover, experiments show an intensive microstructure evo-lution in nanoporous gold during deformation and the interaction of disloca-tions with the surface [50] The mechanical tests and the microstructure char-acteristics inferred from the electron backscatter diffraction (EBSD) suggestlattice dislocations to be the carriers of plastic deformation in nanoporousgold

The experiments of Volkert and Lilleodden [46] on single crystal micron gold columns show the dependence of yield stress and apparent strainhardening rate on the column diameter EBSD analyses revealed that nanoporousmetals are polycrystalline with a grain size of 10μm - 100 μm [50] Further,nanoporous gold exhibits a strong size effect at constant porosity, see theexperimental data of Wang and Weissm¨uller [50] These types of size ef-fects motivate the design of nanocomposites - in order to take advantage andfine-tune their mechanical behavior A recent study on the elastic and plas-tic Poisson’s ratios of nanoporous gold during compression to large plasticstrains is given in [30]

sub-A Young’s modulus of 79 GPa and Poisson’s ratio of 0.44 are stated in,e.g., [4, 24, 47] The yield stress of sub-micron gold is in the range between

154 MPa and 560 MPa The magnitude of the Burgers vector is mately 0.2878 nm [22]

approxi-2.2 Epoxy resin

Epoxy resin is a class of important engineering materials which is used

in many areas of material design such as the production of different ponents of material systems featuring high static and dynamic loadability

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Unlike gold, which is crystalline, epoxy resin is an amorphous solid whichappear in a disordered form The viscosity is very low, which is especiallyadvantageous for infusion, protrusion and injection processes at temperaturesbetween 10◦C and 50◦C Pure epoxy specimens show elasto-plastic mechan-ical behavior during compressive loading Their stress-strain behavior ishighly nonlinear and typically involves three consecutive steps composed of

a yield-peak, post-yield strain softening (especially at high temperatures)and rapid strain hardening Moreover, upon unloading after monotonic plas-tic deformation a Bauschinger-type reverse-yielding phenomena is observed.The plastic response of the material is also sensitive to the applied loadingrate The elasticity parameters of epoxy resin are as follows: a Young’s mod-ulus of 1.038 GPa, Poisson’s ratio of 0.35, as well as the material density of1.2 g/cm3, the tensile strength of 60− 75 MPa and the compressive strength

of 80− 90 MPa, see [34, 42, 50] The initial yield stress is 28 ± 3 MPa [50].2.3 Gold-polymer composite

The metal-polymer composite consists of solid phase gold (ca 30− 40%volume fraction) and solid phase polymer, where nanoporous gold forms thebasis Despite its exclusiveness, gold is one of the preferred model materialsfor the design of nanocomposites as its ligament size is easily controlled dur-ing processing Due to their extremely small structural size, the ligaments ofnanoporous metals possess an extraordinary high local strength The porousmaterial is infiltrated with polymer For details on the composite prepara-tion, the reader is referred to [50] During the infiltration with the poly-mer the metallic network is not deformed, however, it leads to ductilization.Quoting [50] ‘The interpenetrating nanocomposite material [ ] exhibits anumber of unusual and technologically attractive properties, specifically duc-tility in tension, the option of cold-forming, high electric conductivity and astrength significantly exceeding that of each of the constituent phases.’

Whereas the composite’s mechanical response is isotropic at the terscale, it is anisotropic at the nanoscale Microstructural investigationsshow that the ligaments are of single crystal type and possess a disorderedstructure Experimental data of [50] clearly shows a ligament-size-dependentstress-strain behavior of nanoporous gold for ligament sizes from 15 nm to

millime-150 nm All samples demonstrate an elastic-plastic deformation behavior.Composite specimens with the smallest ligament size fail at less strain thanthose with larger ligaments, and also exhibit a somewhat larger sample-to-sample scatter in yield stress

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The resulting composite has a high strength as well as ductile ior under tension One interesting property is that the (Vickers) hard-ness of the composite is considerably higher than those of its components:hardnesscomposite(387±14 MPa) > hardnessepoxy(129±6 MPa) > hardnessnpg

behav-(33± 4 MPa) [50] Further, for nanoporous gold as well as for the composite,the hardness increases with decreasing ligament size, cf [50]

3 Material Modeling

The nanocomposite is modeled as a nanoporous single crystal filled withpolymer The kinematics are based on the multiplicative decomposition ofthe deformation gradient F into elastic and plastic parts as F = Fe · Fp

[28, 37] The elastic Green–Lagrange strain tensor is defined as Ee = 1/2[Ft

C = Ft· F = Ft

the elastic right Cauchy–Green tensor Ce = Ft

e · Fe, and the plastic leftCauchy–Green tensor Bp = Fp· Ft

p The relevant stress measures are thefirst Piola–Kirchhoff stress P

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3.1 Metal part: sub-micron sized gold

The deformation problem is linked to the microscopic behavior via the plasticvelocity gradient tensor Lp

where Me = Ce· Se is the Mandel stress

Further, the free energy Ψ is introduced and additively split into an elastic(Ψe), a local hardening (Ψlh) and a gradient hardening (Ψgh) contribution

αβ and the gradient hardening modulus by Hg l is

an internal length scale Further, the following evolution equation for thegeometrically necessary dislocations (GND) densities gi is introduced [7]

˙giα =X

β

να[nα· sβ] giβ− 1b∇iνα· sα (7)Here and below, the index “i” refers to the intermediate configuration b isthe length of the Burgers vector bα= b sα The first term accounts for latenthardening effects, i.e., the hardening of inactive slip systems due to activeslip systems

The viscoplastic (rate-dependent) flow rule is assumed as follows

t?, the drag stress C0 and the rate sensitivity parameter m are the same forall slip systems, but can easily be extended to slip system dependence.

3.2 Polymer constituent: epoxy resin

The continuum theory for the elastic-viscoplastic deformation of epoxyresin used in the current work is due to3 [1, 16] Besides the plastic deforma-tion gradient, the internal state variables are the local free-volume η which

is associated with certain metastable states and resistance to plastic flow s.Analogously to the extended crystal plasticity theory, the Helmholtz freeenergy Ψ is additively decoupled into elastic Ψe and plastic Ψp parts Ψe isformulated as

1

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where λp = p

trBp/3 denotes the effective distortional plastic stretch Γ−1

is the inverse of the Langevin function Γ (z) = coth (z)− 1/z µR and λL arethe rubbery modulus and network locking stretch, respectively

Using above forms, the equation for the stress is derived as follows

Sback= 2 dev

sym

one has ˙Fp= Dp·Fp

with Fp(X, 0) = 1 Noting that trTe =trT , the flow rule defined for Dpreads

Dp= νp

devTe− Sback

2¯τ

, where νp = ν0



¯τ

¯

τ =

r1

2[devTe− Sback] : [devTe− Sback] (17)For n→ 0 rate-independent formulation is handled, whereas for n → 1 linearviscosity is

4 Numerical parameter identification

4.1 Microcompression of gold single crystal

Single crystalline gold is under consideration in order to calibrate localand non-local hardening parameters based on microcompression investiga-tions performed by Volkert & Lilleodden [46] For this purpose, the numer-ical example is focused on two small micropillars with nominal diameters

D = 1μm and D = 8 μm The internal length scale parameter for puregold is taken to be l = 1.5 μm, in accordance with the argumentation ofBegley & Hutchinson [10] with respect to annealed and soft metals Thematerial parameters for gold are stated in Section 2.1, whereas the modelparameters for the microcompression example are summarized in Table 1.For Young’s modulus a value of 48 GPa is taken according to measurements

in [46] This value differs from the expected value which is due to the factthat only grains with low symmetry orientation have been selected for pillarfabrication In this orientation, a double slip configuration as illustrated inFig 1 was observed with slip directions sα oriented with 37◦ and 31◦ wrt

to the compression axis leading to Schmid factors f1 = 0.48 and f2 = 0.44respectively, cf also [46] In this regard, latent hardening effects betweenboth slip systems are addressed via the dislocation interaction term in Eq.(7)

In the experimental range of investigated sample sizes, the size dence of the macroscopic yield stress ¯Y = ¯Y (D) is approximated very well

The resolved microscopic yield stress is then obtained as Yα = fα Y where¯

fα is the Schmid factor of the corresponding slip system

The pillar geometry is approximated by a tapered cylinder having anaspect ratio of 2.8 and a taper angle of 2◦ Furthermore, a base is con-sidered in the model where the radius of curvature between the pillar andthe base is set to be D/4 for both sample sizes The complete FE-modelconsists of 11240 8-node hexahedral elements The unknown field variables,i.e., the displacements and the GND densities, are interpolated using linearshape functions Uniaxial compression load is applied via a rigid surface(not depicted in Fig 1) under consideration of Coulomb friction In analogy

to Zhang et al [51], a friction coefficient of µC = 0.03 is assumed at theinterface of the indenter and the pillar

Due to the mechanical boundary conditions in the experiments, microhardconditions are assumed for the plastic slip rates at the horizontal bound-aries, i.e., να N(b) · sα = 0 on ∂Bν

i All other boundaries are allowed todeform in the experiment which is captured by microfree conditions, i.e.,

˙giα = 0 on ∂Bgi The two types of boundaries are indicated by a colormap

in Fig 1

Table 1: Model parameters for gold.

Local hardening modulus Hl 10.2 [MPa]

Gradient hardening modulus Hg 150.7 [MPa]

Rate sensitivity parameter m 20 [−]

The stress-strain response for the compression test is presented in Fig

2 and the deformation behavior is illustrated in Fig 3 As expected and

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Figure 1: FE-model of the representative gold micropillar including indication of different boundaries and illustration of the double slip configuration as considered in the numerical compression test.

observed in corresponding experiments, the smaller sample shows a cantly higher gradient effect This is reflected by the formation of a stronglypronounced double slip band, in accordance to experiments [46] Both slipbands are differently stressed: one slip system is more pronounced and henceaccommodates more deformation

signifi-4.2 Compression of bulk polymer

As a next step, a compression test of pure bulk polymer is modeled,

in particular the one done by [50] The sample has initial dimensions of

1 mm x 1 mm and is compressed from above with a constant engineeringstrain rate of 10−4s−1 Since the test represents uniform field distributions,

a single element is used in the evaluation of the stress response under metric compression where contraction free boundary conditions are devised

strain-of the converged parameter set are given in Table 2 Fig 4 displays thestress-strain response of the corresponding numerical example compared tothe experimental data The behavior is well-captured throughout the wholerange of the analyzed monotonic deformation.

5 Numerical study of gold-polymer composite

5.1 Mechanical behavior of the nanocomposite

The nanoporous gold-epoxy composite has a very complex structure, seeFig 5 a) Numerical reconstruction of the real microstructure is too complexfor such a detailed material model as presented in this contribution As a

consequence, we study a two-dimensional honeycomb structure as an tion of the gold-polymer composite structure, cf Fig 5 b) Microstructuralinvestigations show that in nanoporous gold the nanoscale pore network iscontiguous, see, e.g., Fig 5 a) Hence, the gold skeleton is referred to asopen cell In the selected plane strain configuration with its tunnel like mi-crostructure, there exists a continuity in the out of plane direction, but thegold hexagonal skeleton divides the domain into regular and isolated epoxyislands This set-up of the honeycomb structure accounts for the influence

idealiza-of one gold ligament on the neighboring ones as well as the mutual influence

of gold on polymer (and vice versa) during deformation

Microstructural investigations show that the ligaments consist of singlecrystalline gold at the nanoscale Thus, in all simulations, the gold phase

of the representative volume element (RVE) consists of a single crystal gether with applied periodic boundary conditions this amounts for a singlecrystal gold skeleton extending indefinitely in the plane This assumption

To-is reasonable due to observed crystallographic coherence of the ligaments atlong range distances, see, e.g., [50] The remaining material parameters ofthe constituents gold and polymer are the same as before

Mimicking experiments, a compression test in vertical direction is lated where with the selected configuration certain gold ligaments, henceforth