Plasticity of formable all-metal sandwich sheets: Virtual ...mohr.ethz.ch/papers/40.pdfPlasticity of...

International Journal of Solids and Structures 49 (2012) 2863–2880

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International Journal of Solids and Structures

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Plasticity of formable all-metal sandwich sheets: Virtual experimentsand constitutive modeling

Camille C. Besse a, Dirk Mohr a,b,⇑a Solid Mechanics Laboratory (CNRS-UMR 7649), Department of Mechanics, École Polytechnique, Palaiseau, Franceb Impact and Crashworthiness Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

a r t i c l e i n f o

Article history:Available online 2 May 2012

Keywords:Sandwich constructionPlasticityAnisotropyConstitutive modelingFinite elementsFormability

0020-7683/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijsolstr.2012.04.032

⇑ Corresponding author at: Impact and Crashworthiof Mechanical Engineering, Massachusetts Institute ofUSA.

E-mail address: [email protected] (D. M

a b s t r a c t

The mechanical behavior of a metallic sandwich sheet material composed of two flat face sheets and twobi-directionally corrugated core layers is analyzed in detail. The manufacturing of the sandwich materialis simulated to obtain a detailed unit cell model which accounts for the non-uniform thickness distribu-tion and residual stresses associated with the stamping of the core layers. Virtual experiments are per-formed by subjecting the unit cell model to various combinations of bi-axial in-plane loadingincluding the special cases of uniaxial tension, uniaxial compression, equi-biaxial tension and shear.The results demonstrate that the core structure’s contribution to the in-plane load carrying capacity ofthe sandwich sheet material is similar to that of the face sheets. The numerical results are also used toidentify the effective yield surface and hardening response of both the core layer and the face sheets.An anisotropic yield function with linear pressure dependency is proposed to approximate the equal-plastic work surfaces for the core structure and face sheets. Furthermore, a new two-surface model withnon-linear interpolation based on plastic work density is presented to describe the observed combinedisotropic-distortional hardening of the core structure.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Sandwich construction enables the design of structures of anexceptionally high bending stiffness-to-weight ratio. The underly-ing design concept is the separation of two flat sheets by a muchthicker core layer of low density. Sheet metal and fiber reinforcedplastics are typically chosen as face sheet materials, while thechoice of the low density core layer material is far more complex:in addition to basic elastic and weight properties of the core layer,multi-functionality (e.g., thermal, acoustic and energy absorptionproperties) as well as manufacturing considerations come into play(Evans et al., 1998).

To satisfy the requirement of low density (as compared to theface sheet material), lightweight bulk materials such as balsa wood(e.g., Vural and Ravichandran, 2003; Cantwell and Davies, 1996) orpolymers may be used directly in combination with steel or alumi-num skins (e.g., Palkowski and Lange, 2007). As an alternative tolow density bulk materials, man-made porous materials find widespread use. Hexagonal honeycombs are still the most widely usedconstructed sandwich core material. The elastic structure–prop-

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ness Laboratory, DepartmentTechnology, Cambridge, MA,

ohr).

erty relationships for honeycombs are known for several decades(e.g., Kelsey et al., 1958; Gibson and Ashby, 1988) and most re-search on honeycombs focused on understanding and modelingtheir large deformation behavior (McFarland, 1963; Wierzbickiand Abramowicz, 1983; Papka and Kyriakides, 1994; Chung andWaas, 2002; Mohr and Doyoyo, 2004). Extensive research has beenperformed during the past two decades on the mechanical behav-ior of polymeric and metallic foams and their use in sandwichstructures (e.g., Gibson and Ashby, 1988; Bart-Smith et al., 1998;Ashby et al., 2000; Bastawros et al., 2000; Dillard et al., 2005; Gonget al., 2005, Tan et al., 2005a,b; Demiray et al., 2007; Ridha andShim, 2008; Luxner et al., 2009). However, more recent results sug-gested that lattice materials are more efficient from a mechanicalpoint of view (e.g., Wicks and Hutchinson, 2001; Deshpande andFleck, 2001; Evans et al., 2001; Chiras et al., 2002; Liu and Lu,2004; Queheillalt and Wadley, 2005; Mohr, 2005; Hutchinsonand Fleck, 2006; Liu et al., 2006). Zupan et al. (2003) investigatedthe through-thickness compression response of egg-box structuresand reported a higher specific energy absorption as compared tometal foams. Tokura and Hagiwara (2010) analyzed the stiffnessand strength of a stamped two-layer panel material that featuredan egg-box like periodic array of domes of pyramidal shape withtriangular base. They found that it is critically important to accountfor local thickness changes and work hardening during stampingwhen estimating the bending strength of stamped core layers.

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Fig. 2. Draw bending of a prototype sandwich sheet demonstrating the formabilityof the sandwich sheet material.

2864 C.C. Besse, D. Mohr / International Journal of Solids and Structures 49 (2012) 2863–2880

Sandwich construction is mostly limited to flat panel-type ofstructures. Sandwich structures with curved mid-planes aredifficult to make. One manual manufacturing option is to formthe face sheets and core structure separately into a three-dimen-sional shape before bonding all layers together to form the stiffsandwich structure (Bitzer, 1997). As an alternative to this laborintensive manufacturing process, curved sandwich structures canbe made from flat sandwich sheets using conventional sheet metalforming processes such as stamping and deep drawing (e.g., Mohrand Straza, 2005; Parsa et al., 2010; Seong et al,, 2010; Carradoet al., 2011). In the present study, we investigate the mechanicalbehavior of a core structure that is composed of two bi-direction-ally corrugated steel layers (Fig. 1). This structure has severaladvantages in forming operations (Straza, 2007). In particular, itdelays the dimpling failure of the compressed face sheet duringbending and compression dominated stages of forming. Fig. 2shows an illustration of the successful draw bending of a prototypemade from this material.

As an alternative to physical experiments, numerical simula-tions of a representative unit cell of the sandwich material are per-formed to investigate the effective behavior of this cellularmaterial under multi-axial loading conditions. Predicting the effec-tive behavior of cellular materials based on the FE analysis of theunderlying unit cell (for periodic media) or the representative vol-ume element for statistically homogeneous microstructures hasbeen successfully used by several research groups. For example,Mohr and Doyoyo (2004) investigated the crushing response ofaluminum honeycomb using a detailed shell element model ofthe hexagonal cell structure; Youssef et al. (2005) built a finite ele-ment model of a PU foam based on X-ray tomography images;while Caty et al. (2008) developed a microstructural FE model ofa sintered stainless steel sphere assembly (similar to a closed-cell

Fig. 1. (a) Side view of the four layer sandwich structure, (b) top view of a singlecore layer.

foam) based on X-ray tomography images. It is worth mentioningthat unit cell and RVE computations are only representative for thematerial behavior if the microstructures remain mechanically sta-ble. In the case of instabilities (which are frequent in cellular mate-rials of low relative density), a careful analysis of the type ofinstabilities is needed to check the validity of the homogenizedmaterial description (see e.g., Triantafyllidis and Schraad, 1998).

Thanks to the good accuracy of unit cell computations for con-structed cellular materials with stable microstructures, it may beenvisioned to use two-scale finite element models for large scalestructural analysis. In other words, a unit cell model could be as-signed to each integration point of the large scale model (e.g.,Mohr, 2006). However, in view of computational efficiency, thefeasibility of this approach appears to be limited to very simplemicrostructures. In other words, phenomenological macroscopicconstitutive models are still needed to describe the effectivebehavior of cellular materials with a complex microstructure.Deshpande and Fleck (2000) developed an isotropic yield functionfor foams where the square of the mean stress is added to thesquare of the von Mises equivalent stress. They made use of anassociated flow rule along with a stress-state dependent isotropichardening law. Xue and Hutchinson (2004) proposed an aniso-tropic constitutive model for metallic sandwich cores by addingthree square normal stress terms to the Hill’48 equivalent stressdefinition. Note that similar to the physics-based Gurson (1977)model for porous metals, the phenomenological Deshpande–Fleckand Xue–Hutchinson models incorporate the effect of the meanstress on yield through even terms.

Based on the assumption that the effect of in-plane stressesmay be neglected in sandwich structures, Mohr and Doyoyo(2004) proposed a non-associated plasticity model to describethe large deformation response of low density honeycombs. A gen-eralized anisotropic plasticity model for sandwich plate cores hasbeen presented by Xue et al. (2005). They normalized all stress ten-sor components to define an elliptical yield function (which is aneven function of the stress tensor). Due to the normalization, it iseasy to introduce yield surface shape changes (distortional harden-ing) in addition to isotropic hardening. Xue et al. (2005) also showextensive results from unit cell simulations which support theintroduction of distortional hardening. Micromechanical modelsof truss-lattice materials (Mohr, 2005) and hexagonal honeycombs(Mohr, 2006) explain distortional hardening at the macroscopic le-vel through the evolution of the unit cell geometry as the materialis subject to finite strains. In sheet metal plasticity, changing Lank-ford coefficients are seen as an indicator for texture changes (e.g.,

Fig. 3. (a) Side view and top view of the stamping experiment showing the male die (label 1) and the female die (label 2); the colored dashed lines mark the boundary of theunit cell used for selected virtual experiments; (b) side view of the flattening operation, (c) illustration of spring back.

Fig. 4. Engineering stress–strain curves for the 0.2 mm thick low carbon steel sheetfor loading along different in-plane directions.

C.C. Besse, D. Mohr / International Journal of Solids and Structures 49 (2012) 2863–2880 2865

Savoie et al., 1995) which can be related to distortional hardening.A general kinematic-distortional hardening modeling frameworkcan be found in Ortiz and Popov (1983). Aretz (2008) proposed asimple isotropic-distortional hardening model, where the shapecoefficients of a non-quadratic plane stress yield surface (Aretz,2004) are expressed as a function of the equivalent plastic strain.To account for the direction dependent strain hardening with con-stant r-values, Stoughton and Yoon (2009) made use of a non-asso-ciated flow rule and integrated four stress–strain functions tocontrol the evolution of the shape and size of a Hill’48 yieldcriterion.

The present work is organized as follows. In Section 2, a detailedunit cell model of the bi-directionally corrugated sandwich sheetmaterial is constructed accounting for thickness non-uniformitiesand residual stresses associated with the manufacturing of the corestructure. This model is then used in Section 3 to determine themacroscopic stress–strain response of the sandwich material forout-of-plane compression and shear loading and uni-axial and bi-axial in-plane loading. A macroscopic constitutive model is thenproposed in Section 4 which includes an anisotropic pressuredependent yield surface along with an isotropic-distortional hard-ening model. In Section 5, the macroscopic model predictions arecompared with the unit cell simulations and discussed in detail,before presenting the conclusions in Section 6.

2. Computational models for virtual experiments

The term ‘‘virtual experiments’’ is used to refer to the estima-tion of the effective properties of the proposed sandwich materialbased on detailed numerical simulations. Based on the results fromvirtual experiments, a homogeneous equivalent ‘‘macroscopicmodel’’ of the unit cell will be developed for large scale structuralanalysis. Note that the detailed modeling approach is only feasiblewith reasonable computational effort at the unit cell level, whereasa macroscopic model is required for the design of structures madefrom sandwich materials. The mechanical behavior of sheet mate-rials in forming and crash simulations can be predicted with

remarkably high accuracy using state-of-the-art computationalmodels. Since the proposed sandwich material corresponds to asheet metal assembly, it is expected that the virtual experimentswill provide representative estimates of its effective behavior.

2.1. Description of the sandwich material

The material coordinate system (eL, eW,eT) as shown in Fig. 1 isintroduced to describe the microstructure. The coordinate axis eT isaligned with the thickness direction of the sandwich sheet material(out-of-plane direction) whereas eW and eL denote the so-called in-plane directions. The L-direction is parallel to the connecting lineof two neighboring domes while the W-direction is defined aseW = eT � eL. The core structure features five symmetry planes;the bi-layer assembly is symmetric with respect to the central

Table 1Yield stress ratios.

RLL RWW RTT RLW RLT RWT

1.00 1.03 0.94 0.90 1.00 1.00


(W, L)-plane (Fig. 1b). Furthermore, each layer is symmetric withrespect to the (W, T)- and (L, T)-planes.

In the manufacturing process, we can control the geometry ofthe stamping tools, while the final geometry of a core layer afterstamping depends also on the plastic properties of the basis sheetmaterial. The stamping tool consists of a set of male and femaledies. The male die comprises a periodic array of pins that are posi-tioned on a triangular pattern of spacing D = 2.06 mm (Fig. 3a). Allpins have the same diameter dm = 1.23 mm and feature a corner ra-dius rm = 0.31 mm (Fig. 3a). The receiving female die features thecorresponding periodic array of holes of a diameter df = 1.44 mmalong with a corner radius rf = 0.31 mm (Fig. 3a). Note that all tooldimensions have been chosen in accordance with the results froman optimization of the effective out-of-plane stiffness of the sand-wich structure (Besse and Mohr, in press). The relative density q⁄

of the core structure describes the ratio of the overall mass densityof the core structure to the density of the basis sheet material. Inthe case of incompressible sheet materials, the relative density ofthe core structure is given by

q� ¼ 2tc

ð1Þ

where t denotes the initial sheet thickness and C/2 is the effec-tive height of a single core layer after stamping. Here we havet = 2 mm and C = 1.31 mm; hence, the sandwich structure heightis H = 1.71 mm and the relative density q⁄ = 31.

A tin mill product of the type ‘‘black plate’’ is chosen for oursimulations. This light gage low-carbon, cold-reduced material ismodeled as elasto-plastic material. The measured engineeringstress–strain curves for this sheet material are presented inFig. 4. The assumed isotropic elastic properties are a Young’s mod-ulus of 210 GPa and an elastic Poisson’s ratio of 0.3. An anisotropicHill’48 yield surface is used along with an associated flow rule andan isotropic hardening model assuming the Lankford ratiosr0 = 0.69, r45 = 1.15, and r90 = 0.76 (obtained from experiments).The corresponding yield stress ratios are given in Table 1. Thestress–strain curve for strains of up to 0.25 has been identifiedfrom experiments, while a modified Swift law has been used toextrapolate the data for large strains. We omit a more realistic iter-ative extrapolation based on experimental data in the post-neckingrange (e.g., Dunand and Mohr, 2010; Tardif and Kyriakides, inpress) since the overall mechanical response of the sandwichmaterial is expected to be insensitive with regards to this choice.

2.2. Important modeling assumptions

The key simplifications with respect to representing the realsandwich structure are:

(1) Assumption of perfect alignment of the two core layers; itcan be seen from the photograph shown in Fig. 1a that smallmisalignment errors are present in the prototype material;perfect alignment will be assumed in the virtualexperiments.

(2) Negligence of property changes due to brazing; in reality,the four constituent layers of the sandwich material arebrazed together; the temperature history throughout braz-ing may change the steel properties (preliminary experi-ments have shown this); this effect will be neglected in

the present study; for example, Cu-brazing at a temperatureof 1100 �C for 1 h is expected to introduce a phase change(Fe–C eutectic temperature is 725 �C).

(3) Negligence of stress relaxation and thermal residual stres-ses; these two complex phenomena would also need to betaken into account when modeling a real brazed material.

(4) Assumption of rigid braze joints; it is assumed that the brazejoints are very thin (and strong), such that the deformationin these joints is negligibly small with respect to the defor-mation of the core layers.

An attempt was made to confront the results from virtualexperiments with experiments on real prototypes. It was foundthat point (2) presents a first order effect which makes it almostimpossible to achieve good agreement. Note that the stress–strainresponse for the brazed material is expected to be different fromthat assumed in our models because of the possible effect of phasechanges, recrystallization, stress relaxation and thermal residualstresses. Furthermore, we could only make small coupon size sam-ples which were not suitable for biaxial testing. Here, virtualexperiments will be performed for in-plane and out-of-planeloading.

2.3. Manufacturing simulations

A finite element model of a unit cell of the core structure is ob-tained after simulating three manufacturing steps: (1) the stamp-ing of flat sheets to create the dimpled shaped layers, (2) theforming of flat bonding lands on each layer, and (3) elastic spring-back. The residual plastic strain fields are imported from one stepto the next one. Once the core structure is created, flat face sheetsare added.

Step #1: Stamping. The colored dashed rectangles in Fig. 3a indi-cate the size of the unit cell models which are used toperform the virtual simulations. The green lines definethe smallest model; the model defined by the red rect-angle is twice as long as the green model, while the bluemodel is twice as wide. The different dimensions areneeded to facilitate the definition of periodic boundaryconditions (which depend on the specific loading caseto be studied). In the case of the small green model,two punches (male die) along with their receivingfemale dies are needed for the stamping of this unit cell.All forming tools are modeled as analytical rigid sur-faces. A mesh with five first-order solid elements (typeC3D8R from the Abaqus element library) in the thick-ness direction is chosen to account for high through–thickness stresses as well as through-thickness necking.The receiving dies are fixed in space while the punchesmove along the T-direction (Fig. 5). To guaranteequasi-static conditions throughout the stamping simu-lations, the punch velocity increases linearly from 0 to1 m/s over a time interval of 40 ls. Subsequently, it iskept constant until the maximum stamping depth isreached. A kinematic contact formulation with a frictioncoefficient of 0.1 is employed to model the contactbetween the tools and the sheet surfaces. Throughoutstamping, the in-plane displacement component uL isset to zero for all nodes on the boundary surface of nor-mal eL, while uW = 0 on all boundaries of normal vectorew.

Step #2: Forming of the bonding land and joining of the core layersand face sheets. After completing the previous stampingsimulations (step #1, explicit time integration), an addi-tional forming step is introduced (step #2, explicit time

Fig. 5. Sequence of computed configurations during the stamping of a unit cell of asingle core layer.


integration) where flat rigid plates are used to flattenthe bonding lands (Fig. 2b). A first rigid plate applies apressure to the bottom surface of the corrugated sheetuntil the resulting material thickness below the centersof the punches equals about 80% of the initial sheetthickness. Similarly, a second rigid plate is used to applya pressure to the top surface of the corrugated sheet. Thesimulation is stopped as the initial sheet thicknessabove the dies equals about 90% of the initial sheetthickness. A tie contact model (Abaqus, 2008) is usedto join the core layers to each other as well as to therespective top and bottom face sheets. The flatness ofthe bonding lands (contact areas) is important to avoidan artificial mesh distortion when using the tie contactmodel. In reality, the flatness of the bonding lands is alsoimportant as it enhances braze joint strength.

Step #3: Springback analysis. After joining all layers together withthe tie contact, a spring back analysis is performed (step#3, implicit time integration). The final shape anddimensional changes associated with spring back arenegligibly small for the present design, but it is stillimportant to compute a macroscopically stress-free con-figuration before starting any virtual experiments onthis unit cell model.

2.4. Models for virtual experiments

The outcome of the manufacturing simulations is a unit cellmodel which includes the residual stress and plastic strain fieldsdue to manufacturing. Here, we briefly describe the boundary con-ditions and output variables that have been used to characterizethe effective mechanical behavior of the sandwich material. Allsimulation results will be presented in Section 3.

2.4.1. Out-of-plane compressionThe red unit cell model is positioned between two flat rigid

plates (of normal eT). The upper plate moves along the T-axis, whilethe lower plate is fixed in space. To guarantee quasi-static condi-tions throughout the simulation (with explicit time integration),the loading velocity increases linearly from 0 to 0.003 m/s maxi-mum over a time interval of 5 ms and is kept constant until theend of the virtual experiment. A kinematic contact formulationwith a friction coefficient of 0.1 is employed to model the contactbetween the tools and the sheet surfaces. Periodic boundary condi-tions are defined for all nodes positioned on the lateral boundaries(of normal ew and eL).

The effective engineering out-of-plane normal stress is definedas

RcT ¼

FcT

ATð2Þ

with the initial cross-sectional area AT ¼ffiffi3p

2 D2. The correspond-ing out-of-plane engineering normal strain is defined as

ET ¼uT

Cð3Þ

where uT and FT define the displacement and force applied bythe moving upper plane.

2.4.2. Out-of-plane shearWe limit our attention to shear loading along the W- and L-

directions. We expect a similar shear response for intermediatedirection because of the hexagonal symmetry of the core structure.A possible direction dependency of the shear response could onlyarise from the small anisotropy in the basis material and possiblemicrostructural instabilities that could result in change of defor-mation mechanism (unlikely for this thick-walled core structure).

The red unit cell model is used for shear in the (L,T)-plane, whilethe blue unit cell model is used for shear in the (W-T)-plane. Theboundary conditions for shear loading in the (L,T)-plane are:

Periodicity of the structure along the L-direction: the displace-ments of a node on a first (W,T)-boundary plane are identical tothe displacements of the corresponding node with the same xw

and xT coordinate on the second (W,T)-boundary plane.Symmetry of the mechanical problem along the W-direction:

the in-plane displacement uwof all nodes on the (L,T)-boundaryplanes is set to zero.

Analogously, for virtual shear testing in the (W,T)-plane, theboundary conditions are:

Periodicity of the structure along the W-direction: the displace-ments of a node on a first (L,T)-boundary plane are identical to thedisplacements of the corresponding node with the same xL and xT

coordinate on the second (L,T)-boundary plane.Symmetry of the mechanical problem along the L-direction: the

in-plane displacement uL of all nodes on the (W,T)-boundaryplanes is set to zero.

The shear load is introduced by a displacement uL along the L-direction applied to all nodes on the (W,T)-boundary planes ofthe top face or by displacement uw along the W-direction appliedto all nodes on the (L,T)-boundary planes of the top face for shear-ing on the other direction. To guarantee quasi-static conditionsthroughout the shearing simulations, the loading velocity increaseslinearly from 0 to 0.003 m/s maximum over a time interval of 5 msand is kept constant until the end of the step. Denoting the corre-sponding reaction forces as Fc

L and FcW, we define the out-of-plane

engineering shear stresses and strains as

TcLT ¼

FcL

Asand Tc

WT ¼Fc

W

Asð4Þ

and

CLT ¼uL

Cand CWT ¼

uW

Cð5Þ

2.4.3. Uniaxial in-plane loadingThe green model is used for in-plane loading. Due to the sym-

metry of the mechanical problem with respect to the L-W-plane,a green model with one core layer and one face sheet is used forin-plane simulations. The specific boundary conditions are:

The in-plane displacement along the L-direction of all nodes onthe first (W,T)-boundary plane is set to zero. A kinematic con-straint is imposed on all nodes on the second (W,T)-boundary


plane to guarantee that the plane remains flat. The displacement ofthese nodes is denoted as uL.

Analogously, the in-plane displacement along the W-directionof all nodes on the first (L,T)-boundary plane is set to zero. A kine-matic constraint is imposed on all nodes on the second (L,T)-boundary plane to guarantee that the plane remains flat. The dis-placement of these nodes is denoted as uW.

The out-of-plane displacement uT of a set of nodes located ontop of the core dimples (i.e., the center of the sandwich core) isset to zero.

For uniaxial tension and compression along the L-direction, weprescribe uL while uW is free (Fig. 6a). Conversely, we prescribe uW

and leave uL free for uniaxial loading along the W-direction(Fig. 6b).

In the case of in-plane loading, the engineering stresses for theface sheets, core layer and the entire sandwich material are

RcL ¼

FcL

AcL

; RfL ¼

FfL

AfL

and RsL ¼

FcL þ Ff

L

AcL þ Af

L

ð6Þ

RcW ¼

FcW

AcW

; RfW ¼

FfW

AfW

and RsW ¼

FcW þ Ff

W

AcW þ Af

W

ð7Þ

with the initial cross-sectional areas AfL ¼

ffiffi3p

2 Dt, AcL ¼

ffiffi3p

4 DC, AfW ¼ D

2 t,and Ac

W ¼ DC4 . The corresponding macroscopic engineering strains

read

EL ¼uLD2

� � and EW ¼uWffiffi

3p

D2

� � ð8Þ

2.4.4. Combined in-plane loadingThe same unit cell and boundary conditions as for uniaxial

in-plane loading are used to perform virtual experiments for com-bined in-plane loading. We introduce the biaxial loading angle todescribe the ratio of in-plane strains,

Fig. 6. Illustration of the displacement boundary conditions for un

tan b ¼ dEW

dEL¼ duwffiffiffi

3p

duLð9Þ

The virtual experiments are then carried out for radial loading(i.e., monotonic loading with constant b).

3. Results from virtual experiments

The program of virtual experiments includes both out-of-planeand in-plane loading. The emphasis of the present work is on in-plane loading. Selected results for out-of-plane loading are in-cluded to shed some light on the overall mechanical behavior ofthe bi-directionally corrugated core material.

3.1. Uniaxial out-of plane compression

The macroscopic response for out-of-plane loading is shown inFig. 7. The curve starts with a linear elastic regime followed by amonotonically hardening plastic response as the stress exceeds30 MPa. Observe from the deformation snapshots taken throughoutdifferent stages of loading that the dome height is progressively re-duced during loading. This induces a state of compression in the flatbonding land area in the center of the core structure where regionsof very high plastic strains develop. It is important to note that theout-of-plane compressive response of the present material is verydifferent from that of traditional cellular materials. We observe

no peak stress (which indicates the absence of plastic collapseof the cellular microstructure);

no plateau regime (which indicates the absence of progressivefolding of the cellular microstructure).

However, as for traditional cellular materials, densification isexpected to occur for the present material. The simulations werestopped too early to see the effect of densification on the stress–strain curve. The careful comparison of snapshots #2 and #3 re-veals that a contact zone develops between the upper and lower

iaxial tension (a) in the L-direction and (b) in the W-direction.

Fig. 7. Out-of-plane compression: (a) macroscopic engineering stress–strain curve; (b) deformed configurations corresponding to the points labeled in the stress–straincurve. Note that the plastic strains at stage 1 are primarily residual strains from the stamping of the core structure.


domes which corresponds to an increase of the apparent size of thebonding lands.

3.2. Out-of-plane shear

Virtual experiments for out-of-plane shear loading are per-formed in the L-T- and W-T-planes. The corresponding engineeringshear stress–strain curves (Fig. 8) are almost the same (stress levelis about 3% higher for L-direction). We observe an initial yield pointat around 15 MPa. Thereafter, the stress continues to increasemonotonically. Careful inspection of the deformed shapes showsa distortion on the dome structure due to out-of-plane shear. Ob-serve the apparent jump in the displacement field near the centerof the vertical unit cell boundaries. This is a three-dimensional ef-fect. For example, for shear along the W-direction, the uW-displace-ment field is continuous and satisfied the periodicity conditionsalong the boundaries, but it varies along the L-direction whichgives the impression of a jump when looking at the projection onthe W-T-plane. The highest strains are observed near the brazejoints which is expected as the net cross-section is the smallestin that area.

3.3. Uniaxial in-plane tension

Fig. 9 shows the engineering stress–strain curves for uniaxialin-plane tension. The red curves show the results for tension alongthe L-direction, while the blue curves correspond to tension alongthe W-direction. The effective stress–strain curves are monotoni-

cally increasing and their shapes resemble that of a conventionalmetal. For tension in the L-direction, the initial yield stress is about130 MPa for the entire sandwich material and reaches a value ofabout 160 MPa at an engineering strain of 0.15. Fig. 10a elucidatesthe contribution of the face sheets and the core layers to the overallaxial force of the sandwich material. For tension along the L-direc-tion, the face sheets contribute about 57% to the overall force level,while the core layers contribute the other 43%. This strong contri-bution of the core layer to the in-plane deformation resistance ofthe sandwich material is a very special feature of the bi-direction-ally corrugated core structure. Note that for most traditional sand-wich material it is assumed that the contribution of the core layerto the in-plane stiffness and strength is negligible.

It is worth noting that the same basis material (alloy and thick-ness) is used for the face sheets and the core structure. This alsoimplies that the overall weight of the sandwich material is equallysplit between the face sheets and the core structure. Therefore, theweight-specific response of the face sheets is more effective thanthat of the core structure (for uniaxial tension), but nonethelessthe latter may still be seen as very high for a cellular material. InFig. 9c, we also plotted the stress–strain response of the basismaterial for reference (dashed lines). The comparison of thedashed and solid curves reveals that a higher effort is needed to de-form the face sheets in the sandwich material as compared to test-ing these independently from the core structure. The coupling withthe core structure results in non-uniform deformation fields (seecontour plots in Fig. 9d) which increases the plastic work requiredfor axial straining of the face sheets.

Fig. 8. Out-of-plane shear: (a) macroscopic engineering shear stress–strain curves; (b) side views of deformed configurations corresponding to the points labeled in thestress–strain curves.


The small differences between the two curves shown in Fig. 9aindicate some anisotropy in the sandwich material response. Thebreakdown into the contributions of the faces and core layers(Fig. 9b and c) demonstrates that this anisotropy may be attributedto the face sheet response. However, as shown by the dashed linesin Fig. 9c, this anisotropy is not only due to the original (texture re-lated) anisotropy in the basis material. It is also due to the interac-tion with the core structure.

The core structure is compressible (from a macroscopic point ofview) and hence the definition of an r-value is not very meaningfulto describe the anisotropy. Instead, we determine an apparentplastic Poisson’s ratio from a plot of the width versus axial strain(Fig. 10b). For the present material, we obtain vp

LW ¼ �0:33 andvp

LW ¼ �0:28 for uniaxial tension along the L- and W-directions,respectively.

3.4. Uniaxial in-plane compression

The effective engineering stress–strain curves for uniaxial com-pression are shown in Fig. 11a. They both exhibit a maximum instress followed by a slightly decreasing stress level. The initialsmall strain response is very similar to that for uniaxial tensionand we observe an initial yield stress of about 130 MPa. The shal-

low peak in stress is associated with the out-of-plane deformationof the face and core sheets which may be considered as a local col-lapse mode of the sandwich microstructure. This deformationmode is local in the sense that the sandwich mid-plane remainsflat (as imposed by the symmetry conditions). The local bendingstiffness of the core sheet is determined by the dimple pattern.In the case of compressive loading along the W-direction, less ef-fort (as compared to the L-direction) is required to initiate anout-of-plane deformation mode as a plastic hinge can easily formperpendicularly to the loading direction (which corresponds tothe expected orientation of a plastic hinge). This is due to the factthat the domes are positioned such that this hinge line can formbetween the domes i.e., in the area where the corrugated sheetsexhibit the lowest plastic bending resistance. In the case of com-pression along the L-direction, the dome positioning prohibitsthe formation of a hinge line perpendicularly to the loading direc-tion. This explains as to why the unit cell is more distorted for com-pression along the L-direction as compared to the W-direction.Fig. 11 indicates that the decrease in stress level is mostly due tothe folding of the core structure for W-compression. In the caseof L-compression, the macroscopic deformation resistance of thecore structure remains more or less constant which is consistentwith the described plastic hinge mechanism.

Fig. 9. Uniaxial in-plane tension: engineering stress–strain curves for (a) entire sandwich cross-section, (b) the core structure, and (c) the face sheets; (d) 3D views of thedeformed configurations corresponding to the points labeled in the stress–strain curves.


3.5. Biaxial in-plane behavior

We limit our attention to states of loading that are of interest tosheet metal forming. Moreover, we assume that the effect of in-plane anisotropy on the mechanical properties is small and focuson bi-axial in-plane loadings of positive strain along the W-direc-tion (0� 6 b 6 180�). In particular, we consider

� b = 0� (transverse plane strain tension along L-direction)� b = 180� (transverse plane strain compression along L-direction)� b = 45� (equi-biaxial stretching)� b = 135� (in-plane shear, i.e., equal L-compression and W-

stretching)

In addition, in view of constructing an isotropic macroscopicyield surface, three intermediate loading angles are considered:

� b = 11.3�� b = 101.3�, i.e., dEL

dEW¼ 1

tanð101:3Þ ¼ �0:2 which is close to uniaxial

tension along the W-direction� b = 168.7�, dEW

dEL¼ �0:2

Fig. 12a summarizes all measured engineering stress–straincurves for the different loading cases of the biaxial experimentsfor the sandwich structure, the core structure and the face sheet.The red dotted lines recall the results for uniaxial tension. Allcurves for the L-direction are in hierarchical order with respectto b as expected for a conventional engineering material. The stresslevel for transverse plane strain tension (b ¼ 0�) is the highestwhich is a common feature of materials with a convex yield surface

and an associated flow rule. Similarly, the stress level for trans-verse plane strain compression (b ¼ 180�) is the lowest. The stressalong the L-direction is almost zero for b ¼ 101:3� which is consis-tent with the observation that the corresponding stress–straincurve for the W-direction coincides with that for uniaxial tension(Fig. 12a).

The contributions of the core structure and face sheet to theoverall material response are also shown in Fig. 12a. A hierarchicalorder of the stress–strain curves for the L-direction is observed atthis level. However, we note that the W-stress–strain curves forb ¼ 45� (tension–tension) and b ¼ 135� (compression–tension) dointersect when considering the core structure only. The snapshotsof various stages of loading in Fig. 12b reveal that the sandwichstructure is distorted for b ¼ 135� which might explain the lowerapparent strain hardening of the core layer as compared tob ¼ 45� where the core layer appears to be stretched (and flat-tened) in a more uniform manner. It is worth noting that the facesheets remain approximately flat (i.e., the waviness is smaller thanthe face sheet thickness) when both in-plane normal stresses arepositive. As the second principal stress becomes negative (i.e.,compression along the L-direction), we observe local out-of-planedeformation modes (see deformed configurations for b > 101:3�).

3.6. Volume change of core structure

The volume change is determined from the displacement uT

along the T-direction of the nodes located one the inner surfaceof the face sheets. At a given time step, the average surface dis-placement uT is determined and used to compute the engineeringthickness strain. The corresponding plastic volume change is ex-pressed through the volumetric strain

Fig. 10. (a) Decomposition of the section force (per unit width) for uniaxial tensionalong the L-direction into the contributions of the core structure (black) and theface sheets (red); (b) Engineering strain along the width direction as a function ofthe axial engineering strain for uniaxial tension along the L- and W-directions. (Forinterpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)


EpV ¼ ð1þ Ep

LÞð1þ EpWÞð1þ Ep

TÞ � 1 ð10Þ

Fig. 13 shows a plot of the plastic volumetric strain as a functionof the plastic work for all virtual in-plane experiments performed.Except for the initial phase of uniaxial compression and ¼ 180�, weobserve a volume reduction (compaction). This is expected for ten-sile loading conditions, but negative volumetric strains are also ob-served for compression-dominated loading such as b ¼ 135� andb ¼ 168:7�. It is tentatively explained by the distortion of the com-pressed face sheet which can accommodate a local increase in corethickness, while the increase of the average core thickness is muchsmaller.

4. Phenomenological macroscopic constitutive model

4.1. Modeling approach

The goal is to describe the macroscopic behavior of the sand-wich sheet material using a composite shell model. A composite

shell model assumes that the effect of the out-of-plane normalstress is negligible which is a strong but typical assumption madein the context of thin-walled structures. Different constitutivemodels are assigned to the thickness integration points of the com-posite shell element. We therefore need to provide constitutivemodels that describe the effective behavior of the face sheets andthe core structure when built into a sandwich structure. As analternative, one could consider the entire sandwich sheet as ahomogeneous medium and develop a single constitutive modelonly. However, such a description would be suitable for membraneloading only whereas it is expected to break down in the case ofbending loading. The constitutive model for the face sheet basismaterial is known, but it provides only a poor approximation ofthe effective behavior of the face sheets when these are integratedinto a sandwich structure.

4.2. Notation and kinematics

The constitutive equations are written in the material coordi-nate system which is defined through the longitudinal in-planedirection (L), the width in-plane direction (W) and the thicknessdirection (T). The Cauchy stress vector summarizes the non-zerostress components in that coordinate frame, r = {rLL rWW sLW}T,while a standard co-rotational formulation is used to update theorientation of the material coordinate frame as the shell elementis subject to large rotations and distortions (Abaqus, 2008). Thework-conjugate logarithmic strain components are summarizedby the strain vector, e = {eLL eWW cLW}T. A superscript ‘p’ is usedto denote the corresponding plastic strains, ep. Bold lower case let-ters are used to denote vectors, while second-order tensors are de-noted by bold letters. Square brackets are used exclusively toindicate the argument of a function, e.g., f = f[x].

4.3. Elastic constitutive equation

The core structure features hexagonal in-plane symmetry.Neglecting possible elastic anisotropy in the basis material, wecan therefore use an isotropic elasticity model to describe theeffective in-plane behavior, for the core structure as well as forthe face sheets,

r ¼ Cðe� epÞ ð11Þ

with

C ¼ Em

1� v2e1 � e1 þ e2 � e2 þ ve1 � e2 þ ve2 � e1 þ

1� v2

e3 � e3

� �

ð12Þ

Em and v denote the elastic modulus and Poisson’s ratio for uni-axial in-plane loading, while e1 = {1 0 0}T, e2 = {0 1 0}T, e3 = {0 0 1}T

and .

4.4. Macroscopic yield surface

The yield function will be chosen such that it defines the enve-lopes of equal plastic work (Table 2). We thus computed the plasticwork per initial unit volume for each virtual experiment and plot-ted the corresponding true stress data points (rW, rL) for selectedamounts of plastic work in Fig. 14 (face sheets) and Fig. 15 (corestructure). Note that we assumed plastic incompressibility forthe face sheets (ev = 0), while the volumetric strains reported inFig. 13 are used when calculating the true effective stresses forthe core structure. Colored dots in Fig. 12a show the engineeringstress–strain couples corresponding to the four level of plasticwork densities represented in Figs. 14 and 15.

Fig. 11. Uniaxial in-plane compression: engineering stress–strain curves for (a) entire sandwich cross-section, (b) the core structure, and (c) the face sheets; (d) 3D views ofthe deformed configurations corresponding to the points labeled in the stress–strain curves.


Both the core structure and the face sheets are made from aHill’48 material and it is hence natural to choose the Hill’48 yieldfunction as a starting point for the construction of a yield functionfor the cellular material. However, the data for the core structureshows a pronounced tension/compression asymmetry which can-not be represented by the Hill’48 function. As a first approxima-tion, the tension/compression difference in our study isattributed to a linear pressure dependency of the effective inelasticmaterial behavior. We therefore define the yield condition as,

f ¼ �r� K ¼ 0 ð13Þ

where the equivalent stress depends both on the deviatoric anddiagonal terms of the Cauchy stress tensor,

�r ¼ �rHill þ arm ð14Þ

with

�rHill¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFðrL�rTÞ2þGðrW�rTÞ2þHðrL�rWÞ2þ2Ls2

Wsþ2Ms2TLþ2Ns2

LW

q

ð15Þ

and

rm ¼rL þ rW þ rT

3ð16Þ

Note that the above yield function preserves the convexity ofthe original Hill’48 criterion as the associated Hessian matrix isnot affected by the linear pressure term. In the case of plane stress,the yield function reduces to

f ½r� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

L þ Gr2W þ HðrL � rWÞ2 þ 2Ns2

LW

qþ a

3ðrL � rWÞ � k ¼ 0

ð17Þ

We fitted the above expression to our virtual experimental data.The solid envelopes in Figs. 14a, d and 15a, d show the fit to thedata using the parameters listed in Table 3.

4.5. Distortional-isotropic hardening

Figs. 14d and 15d shows the yield envelopes for two distinctplastic work densities in a single plot. The comparison of the cali-brated yield envelopes demonstrates that the elastic domain is notincreasing in a self-similar manner. Instead, the shape of the yieldsurfaces changes substantially (distortional hardening). In Eq. (14),changes of k represent isotropic hardening, while changes in thecoefficients F, G, H, N and a would represent distortional hardening.As an alternative to modeling the evolution of the yield surfacecoefficients (e.g., Aretz (2005)), we describe the yield surface evo-lution through a linear combination of two distinct yield functionsf1[r] and f2[r],

f ½r� ¼ ð1� dÞf1½r� þ df2½r� ð18Þ

where the weighting factor d = d[Wpl] is defined as a function ofthe plastic work density. Denoting the plastic work density associ-ated with the yield functions f1 and f2 as W1

pl and W2pl, respectively,

we impose the order W2pl > W1

pl

Fig. 12. Biaxial in-plane loading: (a) Engineering normal stress–strain curves for the L-direction (left column) and W-direction (right column) for the full sandwich cross-section (first row), the core structure (second row), and the face sheets (third row); the label b indicates the bi-axial loading angle, colored dots indicate stress–strain couplescorresponding to level of plastic work densities; (b) 3D views of the deformed configurations corresponding to the points labeled in the stress–strain curves.


Note that the linear combination (with positive weights) of twoconvex functions is still convex. Furthermore, the correspondingweighted equivalent stress,

d ¼ ð1� dÞ�r1 þ d�r2 ð19Þ

is still a homogeneous function of degree one. For plastic workdensities smaller than W1

pl and greater than W2pl, we assume isotro-

pic hardening only. The yield function evolution law may thus bewritten as

f ½r� ¼

�r1½r� � ð1þ dÞk1 for Wpl 6W1pl

ð1� dÞð�r1½r� � k1Þ þ dð�r2½r� � k2Þ for W1pl < Wpl < W2

pl

�r2½r� � dk2 for W2pl 6Wpl

8>><>>:

ð20Þ

where the weighting function d > �1 defines a monotonicallyincreasing function of the plastic work density which fulfills theconstraints d½W1

pl� ¼ 0 and d½W2pl� ¼ 1.

Fig. 12 (continued)

Fig. 13. Plastic volume change during in-plane loading for all virtual experimentsperformed as a function of the plastic work per initial volume. The red dashed lineshows the model approximation according to Eq. (24). (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version ofthis article.)


The boundary values W1pl and W2

pl are chosen such that the mod-el provides the best overall approximation of the stress–strain re-sponse. The distortional hardening is attributed to changes is thegeometry of the core structure. Geometrical changes are negligiblefor small macroscopic strains and hence the assumption of isotro-pic hardening is made in that range. The value W1

pl represents thusthe work density at which the changes in the core structure geom-etry are no longer negligible as far as their effect on the shape ofthe macroscopic yield surface is concerned. The upper value W2

pl

indicates another mechanism change. For Wpl > W2pl, necking at

the microstructural level becomes important and a macroscopicdescription of the material response may no longer be adequate.Here, the assumption of isotropic hardening for Wpl > W2

pl is justmade for computational robustness.

4.6. Flow rule and volume change

An associated plastic flow rule is adopted to describe the evolu-tion of the plastic in-plane strains. Formally, we write

dep ¼ dkofor

ð21Þ

with the plastic multiplier dk P 0. The increment in plasticwork density (per initial volume) can be written as

Fig. 14. Envelopes of equal plastic work (per unit initial volume) for the face sheets in the true stress plane (rW, rL). The open dots present the results from virtualexperiments, the black solid lines in (a) and (d) represent the least square fit of the yield function given by Eq. (17). The solid envelopes in (b) and (c) have been computedbased on the isotropic-distortional hardening model given by Eq. (25).

Table 2Piece-wise linear approximation of the stress–strain curve of the basis material.

Plastic strain 0 0.02 0.05 0.10 0.20 0.30 0.40 0.60 0.80 1.00True stress (MPa) 310 352 388 421 465 489 508 535 555 571


dWpi ¼ ð1þ EpvÞr dep ð22Þ

where a constitutive equation needs to be specified to deter-mine the evolution of the plastic (engineering) volumetric strainEp

v . Plastic incompressibility is assumed as a first approximationfor the face sheets, i.e., Ep

v ¼ 0. The core layer on the other handis considered as compressible. The change in volumetric strain isdirectly defined as a function of the plastic work density,

dEpv ¼ h½WPl�ðdWplÞ ð23Þ

In particular, the linear function

h½WPl� ¼ �kWPl ð24Þ

with k = 0.0008 MPa�1 provides a reasonable approximation of thepresent experimental data for the core structure (see red dashedcurve in Fig. 13). Note that the above expression is only valid upto the theoretical densification strain of Ep

v ¼ q� � 1.

4.7. Summary of material model parameters

The proposed material model is specified through the followingparameters:

– The elastic parameters, Young’s modulus Em and Poisson’s ratiov, that describe the planar isotropic elastic behavior;

Fig. 15. Envelopes of equal plastic work (per unit initial volume) for the core structure in the true stress plane (rW, rL). The open dots present the results from virtualexperiments, the black solid lines in (a) and (d) represent the least square fit of the yield function given by Eq. (17). The solid envelopes in (b) and (c) have been computedbased on the isotropic-distortional hardening model given by Eq. (26).

Table 3Yield function parameters.

W1Pl (N/mm2) G1 (–) H1 (–) N1 (–) a1 (–) k1 (MPa)

Core 1 1 0.6 1.5 0.03 105Face 0.5 0.77 0.53 1.5 �0.1 373

W2Pl (N/mm2) G2 (–) H2 (–) N2 (–) a2 (–) k2 (MPa)

Core 12 1.2 0.22 1.5 �0.43 114Face 50 0.85 0.3 1.5 �0.33 459

Table 4Weighting function parameters.

d10 d1

1 d20 d2

1

Core �0.09 1.09 0.74 0.26Face 0.91 9.09 0.91 0.09


– The yield functions f1 and f2; each function fi is specifiedthrough five parameters: Gi, Hi, Ni, ai and Ki.

– The weighting function d[Wpl] which describes combined iso-tropic-distortional hardening.

The weighting function may be presented as a parametric ornon-parametric function. For the present sandwich material, theparametric function

df ½Wpl� ¼logðd1

1ðWpl

W2plÞ þ d1

0Þ for Wpl < W2pl

d12ð

Wpl

W2plÞ þ d2

0 for pl26Wpl

8><>: ð25Þ

provides a good approximation of the face sheet response. Theyield envelopes for Wpl = 15 N/mm2 and Wpl = 30 N/mm2 shownin Fig. 14 have been computed using the above expression forthe weighting function.

For the core structure, we propose the function

d½Wpl� ¼d1

1Wpl

W2pl

� �þ d1

0 for Wpl < W2pl

d21

Wpl

W2pl

� �þ d2

0 for W2pl 6Wpl

8>>><>>>:

ð26Þ

Fig. 16. Comparison of the force (per unit width) versus engineering strain curves for all virtual experiments. Different colors show the force for the entire sandwich section(blue), the face sheets (red) and the core structure (black). Dashed lines depict the results from virtual experiments, while the solid lines correspond to the macroscopicmodels. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)


to describe the apparent strain hardening. Fig. 15 includes theintermediate yield envelopes for Wpl = 4 N/mm2 and Wpl = 8 N/mm2 that have been determined using Eq. (18) in combinationwith Eq. (26). All model parameters as calibrated for the presentface sheet and core materials are summarized in Tables 3 and 4.

5. Validation and discussion

The constitutive model is implemented into the finite elementsoftware Abaqus/explicit through its VUMAT user material subrou-tine interface. In the VUMAT code, we adopt a standard return


mapping algorithm with a backward-Euler time integrationscheme (Simo and Hughes, 1998). It is subsequently used in con-junction with a composite shell element. The cross-section of thecomposite shell element is composed of three layers representingthe top and bottom face sheets (each 0.2 mm thick) along with a1.2 mm thick core layer. Three thickness integration points (fornumerical integration with the Simpson rule) are employed perlayer.

5.1. Comparison: macroscopic model versus virtual experiments

All in-plane experiments are simulated using the compositeshell model. The results are reported in terms of the section normalforces FL and FW as a function of the corresponding engineeringnormal strains EL and EW. The solid blue lines in Fig. 16 depictthe results for the composite shell element while the dashed bluelines show the corresponding results from the virtual experiments.In addition, we also computed the individual contributions of theface sheets (red curves) and the core structure (black curves).

We observe good overall agreement of the force-strain curvesfor most loading cases. The best agreement of model and virtualexperiments is observed for transverse plane strain loading(b ¼ 0� and b ¼ 180�). For uniaxial tension and compression, thepredicted force agrees well with that of the virtual experiment,but it is underestimated by up to 15% at large strains. The modelpredictions are less accurate for combined loading. However, theforce level predictions are still reasonable when quantifying the er-ror in absolute terms. For example, the relative error in FL exceeds100% for b ¼ 101:3�, but the absolute difference is less than 20 N/mm. The comparison of the force-strain curves for the face sheetsdemonstrates a good agreement for almost all experiments. Theobserved differences in force level for the sandwich may thus beattributed to deficiencies in the model predictions of the effectivebehavior of the core structure. The plots of the yield envelopes inFig. 15 demonstrate that the error in the core model predictionsare not due to the yield functions. Instead, it is speculated thatthe flow rule is not very accurate. Note that all biaxial experimentsare strain-driven and the flow rule therefore determines the load-ing path in stress space.

5.2. Discussion

An attempt was made to come up with a simple micromechan-ics-based two-scale finite element model of the core structure. Forexample, a simplified three-dimensional shell element model ofthe unit cell could be assigned to each thickness integration pointof macroscopic composite shell model (see for instance Mohr(2006)). However, our preliminary results have shown that athree-dimensional shell element model (at the micro-scale) pro-vides only a poor quantitative prediction of the effective stress–strain response obtained from our virtual experiments (that makeuse of fine solid elements). Similarly, analytical solutions ofstrongly simplified mechanical models of the core structure (e.g.,a truncated cone of uniform thickness) turned out to be inadequatefrom both a qualitative and quantitative point of view. Here, weproposed a simple phenomenological modeling framework to de-scribe the effective behavior of the face sheets and core layers,respectively. Such models are only of little value as far as their pre-dictive capabilities outside the range of calibration are concerned.However, at this stage, where the sandwich material itself is stillunder development, it appears to be reasonable to propose a phe-nomenological model to evaluate the mechanical performance ofthree-dimensional structures made from this sandwich sheetmaterial.

The modeling of the effective behavior of constructed cellularmaterials is particularly challenging due to the evolution of the

material microstructure. This evolution causes the distortion ofthe macroscopic yield surface which is described through a phe-nomenological isotropic-distortional hardening model in the pres-ent work. The introduction of two fixed yield surfaces f1 and f2

results in a rather simple model which can be easily calibratedbased on experiments. Note that the plastic work density is theonly internal state variable of the model. This is a very strong sim-plifying assumption which is expected to break down in case ofnon-radial loading paths. The final deformed configurations forb ¼ 11:3� and 101:3� shown in Fig. 12b have been subject toapproximately the same amount of plastic work density(Wpl = 25 N/mm2). Clearly, the state of the material is very differentamong these configurations. Further improvements of the abovemodel would therefore not only require a modified flow rule, butalso the introduction of additional state variables to provide a moreaccurate description of the microstructural evolution.

6. Conclusions

The mechanical behavior of a newly-developed all-metal sand-wich sheet material composed of two flat face sheets and two bi-directionally corrugated core layers is investigated. The mass perunit area of this sandwich material is equally split between thecore structure and the face sheets. Virtual experiments are per-formed using a detailed finite element model of the unit cell ofthe periodic material microstructure. The model accounts for thethickness variations and residual stresses due to the manufactur-ing of the core layers. Various combinations of in-plane loadinghave been applied to the unit cell, including uniaxial tension,equi-biaxial tension, in-plane shear and uniaxial compression. Itis found that the core structure is used very efficiently, contribut-ing up to 43% of the effective yield strength of the sandwich sheetmaterial for in-plane loading.

The experimental data is used to determine the macroscopicyield surfaces based on an equal plastic work definition for boththe core structure and the face sheets. An anisotropic yield surfacewith linear pressure dependency is proposed to approximate theexperimental data. Furthermore, a new isotropic-distortional hard-ening modeling framework is proposed to provide a first approxi-mation of the stress–strain response for radial loading paths. Theconstitutive model is implemented into a commercial finite ele-ment software and used in conjunction with a composite shell ele-ment model to describe the effective in-plane behavior of thesandwich sheet material.

Acknowledgement

The financial support of CellTech Metals (San Diego, CA) isgratefully acknowledged.

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