Computers and Structures 89 (2011) 2460–2470

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Computers and Structures

journal homepage: www.elsevier .com/locate/compstruc

Numerical simulation of the ductile failure of mechanically and moistureloaded wooden structures

Susanne Saft, Michael Kaliske ⇑Institute for Structural Analysis, Technische Universität Dresden, D-01062 Dresden, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 October 2010Accepted 15 June 2011Available online 19 July 2011

Keywords:Wood mechanicsPlasticityMoisture diffusionHygro-mechanical coupling

0045-7949/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruc.2011.06.004

⇑ Corresponding author. Tel.: +49 351 463 34386; fE-mail addresses: susanne.saft@tu-dresden.de

tu-dresden.de (M. Kaliske).URL: http://rcswww.urz.tu-dresden.de/~statik (M.

Wooden structures are always exposed to climate changes. Since all its mechanical properties are mois-ture-dependent, the influence of moisture content on the mechanical behavior has to be considered for arealistic simulation. A hygro-mechanically coupled macroscopic material model for use within the frame-work of the FEM is introduced. Thereby, moisture diffusion, hygro-expansion and moisture-dependentlinear elastic behavior and ductile failure are covered. The tangent matrix required for the Newton pro-cedure is derived completely. The model is applied to the simulation of experiments for the determina-tion of the swelling pressure of spruce.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Besides mechanical loads, wooden constructions are always ex-posed to changes of the ambient atmosphere, such as relativehumidity (RH) and temperature. The natural biopolymer wood re-sponds to this by means of changes of moisture content and woodtemperature. Both of these quantities obviously influence themechanical behavior of wooden structures, since phenomena likeswelling, shrinkage and thermal expansion exist. Within the limitsof natural room temperature, the influence of moisture contentstrongly prevails. Since this temperature range is sufficient forthe object of research described subsequently, the model that ispresented in this paper focuses on the coupling of mechanicalbehavior and moisture transport. Furthermore, time-dependentlong-term effects like visco-elastic creep, relaxation and mechano-sorptive creep are not considered within this work. Since themodel is aimed to be applicable to wooden engineering structures,continuum macro-mechanical models are used.

Due to its specific structure, wood follows the physical laws ofcapillary porous bodies. The macro- as well as the micro-systemare able to absorb moisture when the structure is exposed towater. This is caused by the huge inner surface. Thus, the materialbehaves hygroscopic, since moisture is absorbed at increasing RH(adsorption) and emitted at decreasing RH (desorption). The woodmoisture content that arises at one RH is called equilibrium mois-

ll rights reserved.

ax: +49 351 463 37086.(S. Saft), michael.kaliske@

Kaliske).

ture content (ECM) and depends on whether the equilibrium isreached by adsorption or by desorption (sorption-hysteresis).When relative humidity is equal to 100%, the resulting wood mois-ture content is denoted as fiber saturation area (FSA). Dependingon the wood species, FSA lies between 22% and 35% for middleEuropean wood [1]. The FSA is characterised by the complete sat-uration of the cell walls’ micro-capillaries. Besides RH, which isthe main factor, amongst others temperature, density, mechanicalloading and pretreatment influence ECM and FSA. A multitude ofmodels has been developed and published that aim to describethe sorption behavior of wood. Not all of those models are physi-cally motivated, in fact, many of them are purely empirical equa-tions that approximate experimental results (e.g. [2,3]). Anoverview on typical sorption models as well as further informationon the sorption behavior of wood can amongst others be found in[4,5].

All material properties that describe the mechanical behavior ofwooden structures are more or less dependent on moisture con-tent. Since the content of this paper focuses on the modeling ofductile failure, the influence of moisture content on the stiffnessand strength properties is of interest. When moisture content in-creases, the quantities of stiffness and strength properties initiallyslightly increase and then decrease with different magnitude untilFSA is reached. Though, the influence of moisture content onstrength properties is much more distinctive then on stiffnessproperties. In many studies, experimental investigations on themoisture dependency of particular mechanical properties are pub-lished. Often, the measured data is additionally approximated byregression functions. Nevertheless, a significant lack of knowledgeis found in this field, since no publication is providing a complete

Fig. 1. Cylindrical anisotropic material directions of wood.

S. Saft, M. Kaliske / Computers and Structures 89 (2011) 2460–2470 2461

set of moisture dependent mechanical properties, not even for onewood species. A complete set of moisture dependent stiffnessproperties of spruce wood has been published by Neuhaus [6] bymeans of a moisture dependent compliance matrix. In many pub-lications, the moisture dependency of mechanical properties issimplified as being linear (e.g. [1,7,8]). When the moisture contentexceeds FSA, the dependence of mechanical properties on moisturecontent vanishes. Thus, the change of the slope of the moisturecontent-mechanical parameter-dependency is another means todefine the FSA. Since the slope does not change suddenly but grad-ually decreases, the traditional denotation ‘‘fiber saturation point’’is more and more replaced by ‘‘fiber saturation area’’ [4,5].

Besides the moisture dependency of the mechanical properties,moisture-induced volume changes have to be considered for therealistic simulation of the short-term hygro-mechanical behaviorof wood. Due to the absorption of moisture into the micro-capillar-ies, the cell walls expand until FSA is reached. This leads to swell-ing of the wood structure. When FSA is exceeded, no more swellingis observed. The amount of length-change (or strain) due to swell-ing and shrinkage significantly depends on the material directionand the wood species. In [9], the mean shrinkage value for Euro-pean wood species, which means the amount of length-changedue to change of moisture content from 0% to FSA, is given forthe longitudinal direction at 0.4%, the radial direction at 4.3% andthe tangential direction at 8.2%. Although the moisture content-swelling strain-relationship is nonlinear, for practical reasons it ismostly taken as linear for moisture contents below FSA. Thisassumption is sufficiently right, since the nonlinearity is causedby the changes of sorption behavior when FSA is reached. For thisreason, it is possible to specify differential shrinkage values whichrepresent the percentage of length-change caused by one percentchange of moisture content. Studies on free swelling and shrinkagecan e.g. be found in [10,12]. When free length-change is restrained,considerable swelling stresses occur, which can form a large part ofthe overall mechanical stresses. In e.g. [11,13,14,35], experimentalinvestigations on the swelling pressure of wood can be found.

Within the scope of hygro-mechanical wood behavior, merelythe influence of the moisture content on the mechanical character-istics of wood is introduced so far. To realistically gather all wood-water-relationships, the influence of the loading state on moisturesorption has to be considered as well. The ECM decreases at con-stant RH and temperature, when a structure is pressure loaded atthe same time. If tensile loading occurs, ECM increases. This effectis caused by the change of pore volume and is especially distinctivein tangential direction, because of the large shrinkage value [1,4,5].In [4], this phenomenon is denoted as hygro-elastic effect. Further-more, an approach for the quantification of the effect is deduced.That has been developed by Barkas in 1949 (cited in [4]). Com-pared to the experimental results of Simpson [15], the theoreticalvalues are always too large so that the influence of the loadingstate is overestimated. Simpson also discovered, that the influenceof tensile loading on ECM is larger then the influence of compres-sive loading. Nevertheless, since the influence of loading state onthe sorption behavior is small within the range of small strainsand nearly no quantitative information is available, hygro-elasticeffects are neglected for the following considerations.

Compared to other construction materials like for examplesteel, concrete or rubber, the number of publications on materialmodels for the simulation of wood is small. Especially hygro-mechanically coupled models for the simulation of the short-termbehavior are rare. Within some investigations, the influence ofmoisture content on the elastic behavior is considered by meansof moisture-depended stiffness-properties and the considerationof shrinkage strains [16–18]. However, all of those publicationsare aimed at the simulation of the long-term behavior of wood.In some papers, irreversible deformations due to creep effects are

considered by the introduction of a plastic strain part [19,20]. In[21], a hygro-mechanically coupled nonlinear-elastic materialmodel is published which considers moisture hysteresis andstress- and moisture-dependency of the stiffness properties as wellas the stress-dependency of shrinkage and sorption. Mechanicalbehavior and moisture transfer are coupled by means of differentcoupling coefficients that have to be fitted with the help of exper-iments. At least, the definition of shrinkage as a stress-dependentproperty has to be reconsidered, since compressive loading actsas a restraint for swelling and the resulting hygro-expansionalstrains are a combination of free swelling and the ‘‘mechanical re-sponse’’ to the partly restraint. Publications on coupled numericalmodels for the simulation of the short-term ductile failure ofmechanically and moisture loaded wooden structures are notknown. The material models used as basis for the coupled formu-lation are introduced in the subsequent sections.

The goal of the paper at hand is the introduction of a macro-scopic three-dimensional constitutive model for the simulation ofductile failure of mechanically and moisture loaded wooden struc-tures. This offers the possibility to more realistically simulate thecharacteristics of wood and with that to understand better the pro-cesses within a wooden structure since experimental investiga-tions do not allow to look inside. In the subsequent sections, theconstitutive models that are used for the simulations of purelymechanical behavior, moisture transfer and hygro-mechanicalcoupling are introduced, at which the focus lies on the detaileddescription of the hygro-mechanical coupling. Thereafter, the pre-sented model is applied to the simulation of longitudinal swellingstresses according to the experimental investigations of Krauss[35].

2. Constitutive hygro-mechanical model for wood

In the following, material models for the realistic simulation ofthe material behavior of wood are introduced. All of them can beclassified as continuum-macro-mechanical models and are formu-lated for small strains. This assumption is sufficient for the bulk ofapplications in structural engineering of wood. Furthermore, themodels were developed for the simulation of spruce wood (piceaabies) but can easily be adapted to the simulation of other woodtypes. For all models, the material directions radial (r), tangential(t) and longitudinal (l) are defined (see Fig. 1). These result fromthe cylindrical anisotropy of wood. The transformation from globalto local material directions is published in detail in [22].

The material models that will be introduced below are devel-oped for the use within the framework of the Finite Element Meth-od (FEM). As degrees of freedom of every node, the displacements(u) for the three directions as well as the moisture content (m) aredefined. All subproblems of mechanics and moisture transfer aresolved at once in one system of equations. Thus, we have a mono-

1 The operator � defines the Hadamard product: (A � B)ij = Aij � Bij.

2462 S. Saft, M. Kaliske / Computers and Structures 89 (2011) 2460–2470

lithic formulation in contrast to a staggered algorithm. This proce-dure is advantageous compared to iterative solutions, which solvethe subproblems one after the other, because of minor computa-tional costs. Based on the static balance of momentum and thetransient moisture transport problem using the principle of virtualdisplacements with the test functions du and dm, we arrive at theweak forms

Gu ¼Z

BdrurdV �

ZB

duq0bdV �Z@B

dutdA ¼ 0 ð1Þ

and

Gm ¼Z

Bdmq0 _mdV �

ZB

drmqmdV �Z@B

dmqnmdA ¼ 0: ð2Þ

Herein, the mechanical stresses are denoted as r, the body forces asb and the tractions at the boundaries @B of the studied body B as t.The scalar value q0 stands for the density in absolute dry conditions.Furthermore, the time change _m of moisture content as well as thebody moisture flux qm and the moisture flux across the surface qn

m

are introduced. The Eqs. (1) and (2) are nonlinear functions of thefield variables u and m. For use within the FEM, a consistent linear-ization is required

Gu;lin ¼ Guðdu;u;mÞ þ DGuðdu;u;m; Du;DmÞ; ð3Þ

Gm;lin ¼ Gmðdm;u;mÞ þ DGmðdm;u;m; Du;DmÞ: ð4Þ

After an additional spatial discretization of the body into particularFinite Elements, the following system of equations is achieved

KF;u KF;m

KQ ;u KQ ;m

� ��

um

� �¼ Fapl

0

" #� Fnr

Q

� �; ð5Þ

whereas the components Ki,j are the parts of the tangent matrix andare obtained from the second terms of Eqs. (3) and (4). Fnr describesthe internal nodal forces, Q the nodal moisture flow and Fapl theexternal mechanical loadings (no correspondent in moisture trans-port). For each element, the significant terms are defined as

Fnr ¼Z

VBT

u : rdV ; ð6Þ

Q ¼Z

VBT

m � qmdV ð7Þ

and

KF;u ¼@Fnr

@u¼Z

VBT

u :@r@u

dV ¼Z

VBT

u : C : BudV ; ð8Þ

KQ ;m ¼@Q@m¼Z

VBT

m �@qm

@mdV þ

ZA

BTm;A �

@qnm

@mdA; ð9Þ

KF;m ¼@Fnr

@m¼Z

VBT

u :@r@m

dV : ð10Þ

Since in the following constitutive models, the influence of the load-ing state on moisture transport is not considered, the remainingcomponent of the tangent matrix is defined as

KQ ;u ¼@Q@u¼ 0: ð11Þ

In Eqs. (6)–(10) Bu, Bm and BmA are the operator matrices for struc-tural and transport problems and the fourth order tensor C is themechanical material tangent.

The constitutive models for the description of the behavior ofwood, that are introduced below, aim to provide a physical contentto the previous equations. In the following, Voigt’s notation is usedfor the presentation of tensors.

2.1. Modeling of mechanical behavior

2.1.1. ElasticityThe material model that is used for the simulation of linear elas-

tic anisotropic behavior has frequently been published (e.g.[23,24]) and thus, is only described briefly in the following. Theformulation is based on a St. Venant–Kirchhoff-material, wherethe potential of the deformation-energy–density is defined as

W ¼ 12e : Ce : e: ð12Þ

The outcome of this is the stress–strain-relationship in form ofHooke’s law for small strains

r ¼ Ce : e: ð13Þ

Assuming the symmetry of the elastic material tensor Ce, it can bedefined using the three Young’s moduli (Er,Et,El), shear moduli(Grt,Gtl,Grl) and Poisson’s ratios (mrt,mtl,mrl)

ðCeÞ�1 ¼

1Er

� vrtEt� vrl

El0 0 0

� vrtEt

1Et

� vtlEl

0 0 0

� vrlEl� v tl

El

1El

0 0 0

0 0 0 1Grt

0 0

0 0 0 0 1Gtl

0

0 0 0 0 0 1Grl

2666666666664

3777777777775: ð14Þ

2.1.2. Ductile failureFor the simulation of ductile failure, a multi-surface plasticity

model with a C1-continuous yield criterion is used, which has beenpublished by Resch, Kaliske [24] in the recent past. In contrast to aprevious model of Schmidt, Kaliske [23,25], solely ductile failurewith hardening post-collapse behavior is considered. Due to thediscretization dependency of the solution when softening behavioroccurs, the use of cohesive elements is more appropriate in case ofbrittle failure due to tensile or shear loading [23]. Furthermore, theC1-continuous transition of the yield surfaces and the exclusion ofsoftening in the post-collapse range results in a stable incrementaliterative solution procedure. As input quantities, only engineeringparameters are used.

The formulation is defined for the three material directionsintroduced above and separates a linear elastic and a plastic range.The limit is defined by a yield condition that bases on the Tsai/Wuyield criterion

f ¼ r : b : rþ q� 1 6 0; ð15Þ

whereas the tensor of strength

b ¼ bs � bf ð16Þ

is determined by the two parts1

bs ¼12� 1� 1

2� sgnðr � 1Þ; ð17Þ

and

bf ¼

1f 2cr

1f 2ct

1f 2cl

000

266666666664

377777777775: ð18Þ

S. Saft, M. Kaliske / Computers and Structures 89 (2011) 2460–2470 2463

Depending on the stress state, bs activates the octants of the stressdomain, the yield surface has to be defined for. bf contains the com-pression strength properties for the three material directions. Incontrast to the formulation of Schmidt, Kaliske, every part of themulti-surface represents a particular octant of the stress domain,related to one or the coupling of two or three failure modes. To at-tain a C1-continuous transition of the yield surfaces, the condition

dfdriðri ¼ 0Þ ¼ 0 ð19Þ

has to be fulfilled, which is assured by the absence of a linear termwithin the yield condition (Eq. (15)). Post-yield damage is consid-ered in the yield condition due to the scalar term q, which dependson the inner variable a. At the point of failure (f = 0 and a = 0), theparameter is defined as q = 0. In the post-yield domain (f = 0 anda > 0), softening behavior is excluded by the condition @q

@a ðaÞ 6 0(hardening may be understood as negative damage). Fig. 2 showsthe yield surface of the presented model at q(a) = 0. At this, the lightgrey area stands for the octant of the stress domain with three cou-pled failure modes, the medium grey surfaces represent the combi-nation of two failure modes and the dark grey areas stand for onesingle failure mode.

The algorithmic implementation of the model is described indetail in [23,24]. Nevertheless, a brief summery of the basic princi-ples is given, since it will be relevant again within the scope ofhygro-mechanical coupling. The mathematical description of themulti-surface plasticity model bases on the Kuhn-Tucker comple-mentary conditions (c: consistency parameter)

f ðr; qÞ 6 0; c P 0; f ðr; qÞ � c ¼ 0: ð20Þ

The strains are composed of the elastic part �e and the plastic part�p

� ¼ �e þ �p: ð21Þ

A general return mapping algorithm combined with an associatedflow rule [26]

_�p ¼ c � @f ðr; qÞ@r

ð22Þ

is used to iteratively determine the strain parts. Within the numer-ical implementation, which bases on the approach of [26], the givenvalues �p

n and an of the last converged substep are used as initial val-ues for the current substep. When the iteration is terminated andappropriate values �p

nþ1 and an+1 are found for the current step,the elastic–plastic material tangent

Fig. 2. Yield surface of the multi-surface plasticity model for fcr = fct = �6 N/mm2,fcl = �43 N/mm2 and q = 0 and rr = rt = [�6; 10] N/mm2 and rl = [�43; 10] N/mm2

(see [24]).

Cepnþ1 ¼

@rnþ1

@�nþ1ð23Þ

can be determined according to [23,26].

2.2. Modeling of moisture transfer

Moisture transport in wood below FSA is controlled by diffu-sion. For modeling, the description of moisture diffusion in woodis mostly regarded as being of the same mathematical type as heattransport. Thus, Fick’s Law of diffusion

qm ¼ �q0 � D � rm ð24Þ

is used in analogy to Fourier’s Law of heat conduction (see e.g.[16,27]). At this, moisture flow qm is proportional to the gradientof moisture content, the tensor of diffusion coefficients D and thedensity in absolute dry conditions q0 specify the material. The factthat moisture flows towards decreasing concentrations is indicatedby the negative sign. The assumption of Fick’s Law is applicable inthe case of steady-state moisture transport simulation. In contrast,using the time-dependent form of Fick’s Law

@m@t¼ r � ðD � rmÞ ð25Þ

for transient simulations of moisture diffusion merely gives anapproximation of the real transport behavior. Discrepancies of thesimulation in comparison to experimental results are often denotedas ‘‘non-Fickian behavior’’ [16,28]. Large gradients of moisture con-tent lead to higher discrepancies. When the transient form of Fick’sLaw is used, moisture diffusion is merged to one single transportprocess. In fact, transient moisture diffusion has to be subdividedinto three phenomena: (1) moisture diffusion in wood pores, (2)bound water diffusion in cell walls and (3) sorption. To properly de-scribe this complex process, a coupled system of differential equa-tions has to be solved. The depicted formulation amongst othersrequires the moisture dependent diffusion coefficients of water va-por and bound water. It is published e.g. in [29,30] and currently ob-ject of research. In the scope of multi-scale modeling of wood,Eitelberger, Hofstetter [31] use this formulation as basis for thedetermination of steady-state moisture diffusion coefficients.

In the present work, nevertheless, Fick’s Law is used because ofthe insufficient availability of the input parameters required for thecoupled transport model. Furthermore, the focus lies in thedescription of the mechanical behavior of wood in considerationof influence of moisture content. Due to the significant natural var-iation of all (mechanical and transport) material properties, thedeterministic numerical simulation of wooden structures can al-ways merely provide an approximative result. The natural varia-tion of input properties can e.g. be captured by the considerationof uncertain data [32].

The steady-state diffusion coefficient, which is needed as inputquantity for Fick’s Law, depends on the material directions as wellas on moisture content. Thus, D is defined as

DðmÞ ¼DrðmÞDtðmÞDlðmÞ

264

375: ð26Þ

The derivative @qm@m needed to calculate the first part of KQ,m (see Eq.

(9)) is determined by

@qm

@m¼ @

@m½�q0 � D � rm� ¼ �q0

@D@m� rmþ D � @rm

@m

� �ð27Þ

with

@rm@m

¼ Bm: ð28Þ

2464 S. Saft, M. Kaliske / Computers and Structures 89 (2011) 2460–2470

To ensure the stability of the incremental iterative Newton solutionprocedure, C1-continuous approximation functions for the compo-nents of D should be used. Within the present work, the approachof Hanhijärvi [17] is chosen

DiðmÞ ¼ ai � ebi �m: ð29Þ

The influence of changes of ambient atmosphere is considered withthe help of the boundary conditions

qnm ¼ Sm � q0 � ðECM �mÞ; ð30Þ

since, as mentioned above, ECM depends on RH and ambient tem-perature. For the determination of ECM, a sorption isotherm is used,which was found to fit good for wood sorption data [33]

ECM ¼ 0:01 � �T � lnð1� RHÞ0:13 � 1� T

647:1

� ��6:46

" # 1110�T�0:75

; RH½��; T½K�: ð31Þ

The moisture flux qnm across the surface furthermore depends on the

surface emissivity Sm and the density in absolute dry conditions. Forthe moisture dependent surface emissivity, again the approach ofHanhijärvi [17] is chosen

SmðmÞ ¼ a � eb�m: ð32Þ

Due to the moisture-dependency of qnm, the derivative of Eq. (30)

with respect to m also influences the tangent matrix (see Eq. (9))

@qnm

@m¼ @SmðmÞ

@m: ð33Þ

Fig. 3. Stress–strain relationship for moisture dependent ductile failure.

2.3. Hygro-mechanical coupling

As mentioned above, in the scope of hygro-mechanical couplingfor the elastic–plastic short-time behavior of wood, merely theinfluence of the moisture content on the mechanical behavior isconsidered. The influence of loading state on moisture transportis neglected. For the consideration of moisture-induced strains aswell as resulting deformations and stresses, the mechanical strains�(u) have to be expanded by the part �(m)

� ¼ �ðuÞ � �ðmÞ ð34Þ

that depends on the shrinkage values for the different materialdirections and the change of moisture content (current moisturecontent referred to a reference value)

�ðmÞ ¼ b � ðm�mref Þ; b ¼

br

bt

bl

000

2666666664

3777777775: ð35Þ

Moisture dependency of the mechanical properties is directly con-sidered by analytical relations (cf. Eqs. (14) and (18))

Ce ¼ CeðmÞ; ð36Þ

bf ¼ bf ðmÞ: ð37Þ

Analogously to the components of the tensor of diffusion coeffi-cients, C1-continuous approximation functions should be chosento describe the moisture dependency of the mechanical propertiesto ensure the stability of the incremental iterative solution proce-dure. Within the present work, the moisture-dependent stiffnessproperties are expressed by sine functions that approximate theexperimental results of Neuhaus [6] for spruce wood (see [32,34])

ðCeÞ�1ij ðmÞ ¼ aij � sinðbij �mþ cijÞ þ dij: ð38Þ

Moisture-dependency of the compression strength values as a firstapproach is modeled by a simple linear approach basing on the gen-eral information of [7,8] (fci,12: compression strength of materialdirection i at a moisture content of m = 0.12)

fciðmÞ ¼ fci;12 � ð1þ ða �mþ bÞÞ: ð39Þ

In Fig. 3, an exemplary stress–strain-relationship for ductile failurecaused by homogeneous pressure and decreasing moisture contentis shown.

In the course of consistent linearization, the derivative @r@m has to

be determined for the calculation of KF,m (see Eq. (10)). Therefore, adistinction has to be made between the linear elastic

� ¼ �e; �p ¼ 0 ð40Þ

and the plastic domain

� ¼ �e þ �p; �e–0; �p–0: ð41Þ

First, the case of linear elasticity is regarded. Due to the linearity,the separation of the total internal nodal forces Fnr into a mechani-cal and a moisture-induced part is possible

Fnr ¼ Fnru � Fnr

m ð42Þ

with

Fnru ¼

ZV

BTurðu;mÞdV ¼

ZV

BTu : CeðmÞ : �ðuÞdV ð43Þ

and

Fnrm ¼

ZV

BTurðmÞdV ¼

ZV

BTu : CeðmÞ : �ðmÞdV : ð44Þ

The derivative of Fnr with respect to m is then determined by

@Fnr

@m¼Z

VBT

u@

@mðrðu;mÞ � rðmÞÞdV ð45Þ

with

@rðu;mÞ@m

¼ @

@mðCeðmÞ : �ðuÞÞ; @

@mð�ðuÞÞ ¼ 0; ð46Þ

@rðu;mÞ@m

¼@CeðmÞ@m

: �ðuÞ� �

ð47Þ

and

S. Saft, M. Kaliske / Computers and Structures 89 (2011) 2460–2470 2465

@rðmÞ@m

¼ @

@mðCeðmÞ : �ðmÞÞ

¼@CeðmÞ@m

: �ðmÞ þ CeðmÞ :@�ðmÞ@m

� �: ð48Þ

Since the derivative of the moisture-induced strains with respect tomoisture content is the tensor of shrinkage values (see Eq. (35)), therequired term is achieved as

@r@m¼@CeðmÞ@m

: ð�ðuÞ � �ðmÞÞ � CeðmÞ : b: ð49Þ

For the plastic domain, the determination of @r@m is much more com-

plicated, since the distribution of stresses and strains into mechan-ical and moisture-induced parts is not longer possible due to thenonlinear mechanical behavior. Starting point is the multi-surfaceplasticity formulation of Resch, Kaliske [24] introduced in Sec-tion 2.1.2. At this, the following variables depend on moisturecontent:

� elasticity tensor Ce (see Eq. (36)),� tensor of strength b (see Eq. (16)),� tensor of compressive strength bf (see Eqs. (18), (37)),� tensor of activation bs (see Eq. (17)),� strains � and stresses r.

Since moisture-dependency of the material properties is de-scribed by analytical functions as mentioned above, the derivatives

of Ce and bf with respect to m may be taken as known. The deriv-

ative of bs with respect to m has to be determined numerically, be-cause the activation function is not continuously differentiable.Furthermore, the following assumptions are made:

� Linear hardening is assumed. Unlike the inner variable a, whichis attended by the plastic strain part �p, the hardening factor Kdoes not depend on moisture content

qðmÞ ¼ �K � aðmÞ: ð50Þ

� The tensor of strength b is always symmetric.� Moisture content does not change during the local plastic iter-

ation. Thus, the determination of the plastic strain part andthe stresses is independent of moisture content. Those valuesmay be regarded as known for each time step.

The derivative of r with respect to m is determined with thehelp of the combination of two approaches.

2.3.1. Approach 1The equilibrium state at the end of the plastic iteration satisfies

the yield criterion (see Eq. (15)) for every time step

f ðrðu;mÞ;mÞ ¼ rðu;mÞ : bðmÞ : rðu;mÞ þ qðmÞ � 1 ¼ 0: ð51Þ

Since b(m) is always symmetric, it is found that

@

@mf ðrðu;mÞ;mÞ ¼ @

@mðrðu;mÞ : bðmÞ : rðu;mÞ þ qðmÞÞ

¼ 0: ð52Þ

Differentiation leads to

rðu;mÞ :@bðmÞ@m

: rðu;mÞ þ 2 rðu;mÞ : bðmÞ :@rðu;mÞ@m

� �þ @qðmÞ

@m

¼ 0: ð53Þ

Eq. (53) still contains 2 unknown terms, @rðu;mÞ@m and @qðmÞ

@m . A secondconditional equation has to be developed.

2.3.2. Approach 2, part 1Concerning the split of strains, it it known that

� ¼ �ðuÞ � �ðmÞ; � ¼ �e þ �p: ð54Þ

After differentiation with respect to m, we arrive at

@�@m¼ @�ðuÞ

@m� @�ðmÞ

@m¼ �b: ð55Þ

The derivative of the total strain with respect to m is constant andwith this

@

@t@�@m

� �¼ @2�@m@t

¼ 0; ð56Þ

@2�@m@t

¼ @

@t@�e

@mþ @�

p

@m

� �¼ 0: ð57Þ

With ð_Þ ¼ @ðÞ@t , one obtains

@ _�@m¼ @

_�e

@mþ @

_�p

@m¼ 0: ð58Þ

For a better understanding and clear equations, the declaration ofthe moisture-dependency of the parameters is omitted in thefollowing.

2.3.3. Approach 2, part 2From the linear elastic stress–strain-relationship we know

r ¼ Ce : �e; _r ¼ Ce : _�e ð59Þ

and with that

_�e ¼ S : _r; S ¼ ðCeÞ�1: ð60Þ

Differentiation with respect to m leads to

@ _�e

@m¼@S@m

: _rþ S :@ _r@m

: ð61Þ

In the course of algorithmic implementation, the derivative with re-spect to time is given by

ð_Þ ¼ 1Dt� ððÞnþ1 � ðÞnÞ: ð62Þ

For Dt – 0 and independent of m, we arrive at

@ _�e

@m¼@S@m

: ðrnþ1 � rnÞ þ S :@rnþ1

@m� S :

@rn

@m: ð63Þ

Since the total equation is analyzed at time n + 1, the current values

of S;@S

@m and m have to be inserted into Eq. (63).

2.3.4. Approach 2, part 3When an associated flow rule is used, which is done in the mod-

el of Resch, the plastic strain rate is determined by Eq. (22) and rateof the hardening value is obtained by

_q ¼ �K � _a ð64Þ

with

_a ¼ c � @f@q: ð65Þ

Insertion of Eq. (65) into Eq. (64), an rearrangement (which is pos-sible when @f

@q is a scalar value as in our case 1) leads to

c ¼ � 1K� 1@f@q

� _q ð66Þ

2466 S. Saft, M. Kaliske / Computers and Structures 89 (2011) 2460–2470

and after insertion into Eq. (22)

_�p ¼ � 1K� 1@f@q

� _q � @f@r

: ð67Þ

After differentiation with respect to m, we arrive at

@ _�p

@m¼ � 1

K� 1@f@q

@ _q@m� @f@rþ _q � @2f

@r@m

!ð68Þ

with

@f@r¼ 2 � b : r; ð69Þ

@2f@r@m

¼ 2@b@m

: rþ b :@r@m

� �: ð70Þ

Insertion of Eq. (70) into (68) leads to

@ _�p

@m¼ � 1

K� 1@f@q

@ _q@m� @f@rþ _q � 2

@b@m

: rþ b :@r@m

� �� �: ð71Þ

Analogously to (63), the derivative with respect to time is split forthe algorithmic implementation

@ _�p

@m¼ � 1

K� 1@f@q

@qnþ1@m �

@f@rnþ1

� @qn@m �

@f@rnþ1

� þðqnþ1 � qnÞ � 2 �

@b

@m : rnþ1

þðqnþ1 � qnÞ � 2 � b :@rnþ1@m

0BBBB@

1CCCCA: ð72Þ

Again, b;@b

@m and m have to be taken for time step n + 1.

2.3.5. Approach 2, assembly of parts 1, 2 and 3Insertion of Eqs. (63) and (72) into (58) leads to

0 ¼@S@m

: ðrnþ1 � rnÞ þ S :@rnþ1

@m� S :

@rn

@m

� 1K� 1@f@q

@qnþ1

@m� @f@rnþ1

� @qn

@m� @f@rnþ1

� �

� 1K� 1@f@q

� ðqnþ1 � qnÞ � 2 �@b@m

: rnþ1

� 1K� 1@f@q

� ðqnþ1 � qnÞ � 2 � b :@rnþ1

@m:

Herein, the following moisture dependent parameters are knownfrom the plastic iteration:

� for the current step n + 1

S;@S

@m ;@f

@rnþ1; qnþ1;rnþ1;b and

@b

@m

� for the previous step nqn and rn.One has to consider, that for Eq. (73) the derivatives of qn and rn

with respect to moisture content m at time step n + 1 are required.Since it is not possible to determine those values and the deriva-tives with respect to mn are known from the previous step, it is as-sumed that

@mn

@mnþ1� 1 ð74Þ

and with this

@rn

@mnþ1¼ @rn

@mn� @mn

@mnþ1¼ @rn

@mnð75Þ

and

@qn

@mnþ1¼ @qn

@mn� @mn

@mnþ1¼ @qn

@mn: ð76Þ

The assumption is maintainable because it has no influence on thesolution of the coupled hygro-mechanical problem, but as worstslightly impairs the convergence of the incremental iterative solu-tion procedure. Hence, analogous to approach 1, the residual un-known terms are @rnþ1

@m and @qnþ1@m and the second conditional

equation is found.

2.3.6. Assembly of approaches 1 and 2Finally, approaches 1 and 2 have to be brought together to

determine the derivative @rnþ1@m , needed for the tangent matrix. Since

no time derivative is involved in approach 1, all parameters have tobe analyzed at time n + 1

0 ¼ rnþ1 :@b@m

: rnþ1 þ 2ðrnþ1 : b :@rnþ1

@mÞ þ @qnþ1

@m: ð77Þ

Eqs. (73) and (77) are rearranged as system of equations of the form

A B

C D

� ��

@rnþ1@m@qnþ1@m

" #¼

R1

R2

� �ð78Þ

with

R1 ¼ �@S@m

: ðrnþ1 � rnÞ þ S

:@rn

@m� 1

K� 1@f@q

@qn

@m� @f@rnþ1

þ 1K� 1@f@q

� ðqnþ1 � qnÞ � 2 �@b@m

: rnþ1; ð79Þ

R2 ¼ �rnþ1 :@b@m

: rnþ1; ð80Þ

A ¼ S � 1K� 1@f@q

� ðqnþ1 � qnÞ � 2 � b; ð81Þ

B ¼ � 1K� 1@f@q

@f@rnþ1

; ð82Þ

C ¼ 2 � rnþ1 : b; ð83Þ

D ¼ 1: ð84Þ

The solution of the equation system leads to the coupled plasticmaterial tangent KF,m.

3. Example

The material models introduced above are applied to the simu-lation of the swelling pressure of a wood sample in longitudinaldirection. The simulations are carried out on the basis of experi-mental investigations that have been presented by Krauss [35]for different wood types and published in 1988. Since the materialmodels are validated for spruce wood, this wood type is chosen forthe simulation. The example does not aim for the recalculation ofexperimental results. In fact, the numerical model is used to gaininsight into the processes within the wood sample, such as time-dependent moisture and stress distribution and plastification,which is not possible by means of experiments. Furthermore,experimental results on the swelling pressure of wood significantlydepend on the test set-up, at which no reference method exists.Thus, ‘‘swelling pressure’’ is no material property, as it is denotedin some publications and the comparison of results is hardly

S. Saft, M. Kaliske / Computers and Structures 89 (2011) 2460–2470 2467

possible. Nevertheless, this kind of experiments is chosen as exam-ple within this work, because moisture-dependent plasticity playsan important role and numerical simulations provide the means tolearn more about the process of restricted swelling.

3.1. Experimental investigations of Krauss

In order to achieve a high accuracy of the results for the longi-tudinal swelling stress, a specimen geometry with a high length inlongitudinal direction compared to the other directions is chosen.The dimensions of the used dog bone specimen can be found inFig. 4. Measurement of swelling pressure is based on the assump-tion, that during the experiment no permanent but rather tempo-rary deformations are possible. Hence, the swelling pressure isdefined as the force that is needed to redeform the temporarydeformations. Swelling pressure then is calculated as the ratio ofswelling force and cross-section area of the tapered middle part.Those definitions enable the comparison of the maximum mea-sured swelling stress with the simulated one, which matches withthe average of longitudinal stresses in the studied cross-section. Toobtain an increase of moisture content, the absolute dry specimensare put into distilled water until the maximum of swelling stress isreached. The lower and upper surfaces are restricted from defor-mations apart from temporary deformations of maximum 0.5 lmand the swelling forces are measured. The results of the experi-mental investigations are presented as regression functions ofthe longitudinal stresses subject to time

r ¼ a � e�bt � tc; t½min� ð85Þ

with a = 0.444, b = 0.00268 and c = 0.656 for spruce wood. Thestress-time-dependency is shown in Fig. 5 for a moistening-timeof 3000 min. Since experimental data is merely recorded until themaximum of swelling stress is reached, the regression function issolely validated for this time range.

Fig. 4. Dimensions of the dog bone specimen used in [35] and discretization.

3.2. Finite element simulation

3.2.1. DiscretizationThe dog bone specimen is discretized by 20-node brick ele-

ments with displacement and moisture degrees of freedom (seeFig. 4). At the upper and lower surfaces, deflections in z-directionare restricted. For the simulation of moisture absorption, addi-tional plane surface elements are generated for all surfaces.

3.2.2. Material models and material propertiesThe material models introduced above are used for the simula-

tion of the hygro-mechanical behavior of the specimens. The de-tailed material properties of each model part are listed in Table 1.

3.2.3. Loading processSince the transport of free water (water logging) is not captured

by the material models, the moistening process is modeled in analternative way by the change of RH from RH = 0.001 to RH = 0.99at a temperature of T = 293.15 K (T = 20 �C). RH thereby is in-creased during a time of 10 s. After that the transient simulationis continued for further 5000 min. This assumption seems reason-able, since swelling occurs due to the adsorption of moisture intothe micro capillaries of the cell wall and is not influenced by thefree water in the macro pores which exists due to water logging.Thus, the magnitude of swelling strain is the same, independentof the way of moistening. Furthermore, swelling strain dependson the change of moisture content (see Eq. (35)) and not on itsabsolute value. Besides, maximum swelling stress is affected andlimited by the compressive strength of the material. However,due to the moisture-dependence of mechanical properties, theabsolute value of moisture content influences the maximum swell-ing stress. Differing from those theoretical considerations, an influ-ence of the moistening procedure on the measured swelling stressis detected in [36]. Unfortunately, the question why swelling stressdue to air moistening is always higher then swelling stress due towater logging stays unanswered. Thus, the simulated swellingstresses are expected to be higher then those investigated byKrauss.

Fig. 5. Stress-time-dependency, regression function of [35].

Table 1Material properties for spruce wood.

Moisture-dependent compliance matrix (Section 2.3, Eq. (38))

aij [mm2/N] bij [�] cij [�] dij [N/mm2]

S11 1/1350 7.8 �1.7 0.00174S12 �1/1820 12 �1.75 �0.0009S13 �1/49200 13.2 �1.7 �0.0000535S21 a12 b12 c12 d12

S22 1/550 7.38 �1.79 0.0038S23 �1/23500 11 �1.6 �0.000076S31 a13 b13 c13 d13

S32 a23 b23 c23 d23

S33 1/98000 9.8 �1.7 0.0000885S44 1/27 5 �1.56 0.054S55 1/650 5 �1.65 0.00268S66 1/1630 11 �1.9 0.00193

Moisture-dependent plasticity (Section 2.3, Eq. (39))

fci,12 [N/mm2] ai [�] bi [�]

fcr �6 �4.5 0.54fct �6 �4.5 0.54fcl �43 �5 0.6

Shrinkage values (Section 2.3, Eq. (35))

br [�] bt [�] bl [�]

0.19 0.36 0.015

Moisture transfer (Section 2.2, Eqs. (24), (29))

ai [mm2/s] bi [�] q0 [kg/mm3]

Dr 8 � 10�5 4 450 � 10�9

Dt 8 � 10�5 4Dl 2 � 10�4 4

Surface emissivity (Section 2.2, Eq. (32))

a [mm/s] b [�]

Sm 3.2 � 10�5 4

Fig. 6. Convergence rate of the incremental iterative solution procedure.

Fig. 7. Simulated stress-time-dependency with plastified parts.

2468 S. Saft, M. Kaliske / Computers and Structures 89 (2011) 2460–2470

3.3. Convergence

Due to the consistent linearization of the weak forms (see Eqs.(1)–(5)), which is described within this work, a quadratic conver-gence rate of the incremental iterative solution procedure isachieved. As an example, the convergence of the substeps 34 to38 (time range between 100 and 400 min) is shown in Fig. 6 bymeans of the reductions of the residual norm. The dotted line de-fines the tolerance. At first glance, it may seem unusual, that theresidual norm increases after the application of load (iteration stepzero), but this can easily be explained. Since the structure is exclu-sively loaded by the change of RH and with that by moisture con-tent changes and no mechanical loading is applied, the residualnorm, which is defined by the norm of the residual forces, merelyresults from the internal nodal forces (at this, ‘‘forces’’ describesmechanical forces and moisture flows, see Eq. (5)). At iteration stepzero, only the surface elements contribute to the internal nodalforces by means of moisture flows across the surfaces. Since forthe solid elements, the difference of moisture content and refer-ence moisture content is zero, no mechanical strains and stressesoccur. Only at iteration step one, the solid elements notice thechange of moisture content and contribute to the internal nodalforces by means of mechanical swelling-stresses. Thus, the residualnorm increases from iteration step zero to iteration step one.

3.4. Results

The results of the described simulation are presented in Figs. 7and 8. The values of longitudinal stress and moisture content aredetermined as the average value of the black bordered cross-sec-tion (Fig. 4). Since the distribution of stresses and moisture content

plays an important role, this is shown as well by means of the plas-tified parts in Fig. 7 and the moisture content within the specimenin Fig. 8. In each case, one fourth of the specimen is shown (seeFig. 4).

Moisture absorption as well as moisture transfer in wood are aslow process. Thus, at the beginning of the simulation, moisturecontent slowly increases at the outer regions. After 200 min, mois-ture content has gone up to 0.2 at the edges. At this time, plastifi-cation has already started at regions of higher moisture contentand more-dimensional stress-state (the darker the color, the morepronounced is the plastification). Swelling stress rapidly increasesuntil about 550 min, the maximum value of �26.7 N/mm2 isreached. Afterwards, redistribution of stresses combined with thedecrease of compressive strength with increasing moisture contentleads to a decrease of the average stress. The first range of fast de-crease (550–1600 min) is finished when the moisture content ofthe studied cross-section has reached ECM and does not change

Fig. 8. Simulated moisture content-time-dependency with distribution of moisturecontent.

S. Saft, M. Kaliske / Computers and Structures 89 (2011) 2460–2470 2469

anymore. At the time of 1600 min, the whole middle part of thespecimen has reached ECM and the middle part as well as the outerregions are plastified. Afterwards, the average stress decreasesmore slowly until the whole specimen has reached ECM.

As expected, the maximum simulated swelling stress of�26.7 N/mm2 is much higher then the one investigated by Krauss(�8.5 N/mm2). Besides the different moistening procedure as men-tioned above, the difference may be explained by the followingreasons:

� the time-dependent decrease of stresses at constant strains(relaxation) as well as the time-dependent increase of deforma-tions at constant stresses are not considered within the numer-ical model,� the mechanical properties of the wood used in [35] may differ

from those that were used in the simulation (no fitting of mate-rial properties),� as mentioned above, measured swelling pressure strongly

depends on the experimental set-up, whereas no referencemethod and with that no reference value exists,� the compliance of the experimental set-up as well as possible

friction effects significantly diminish the measured swellingpressure, whereas within the simulation an ideally rigid andfrictionless support is assumed.

4. Summary and outlook

Wooden structures show a complex mechanical behavior,which is distinctively moisture-dependent. For the simulation ofthis behavior, appropriate macroscopic, short-term material mod-els are introduced. The coupling of moisture transfer and mechan-ical behavior allows for the simultaneous determination ofmoisture distribution and deformation state, respectively strainsand stresses, under consideration of coupling effects. For the firsttime, a moisture-dependent multi-surface plasticity model forthe description of ductile failure in wood is shown and the materialtangent is derived. This coupled hygro-elastic–plastic formulationis used for the simulation of the swelling pressure of spruce wood,

according to experiments carried out by Krauss [35]. As expected,since the described simulations are not aimed at the recalculationof experimental results, the simulated swelling pressure is largerthen the experimentally investigated one. The simulated swellingpressure may be seen as the one that is at maximum possible forthe studied specimen type. However, the numerical models intro-duced in this work provide an approach that helps to gain a deeperunderstanding of the processes within the specimen.

The coupled hygro-mechanical formulation is aimed to beapplicable to all kinds of wooden structures as well as all woodspecies. Therefore, the model has to be validated with the help ofexperimental results for other wood types than spruce. As men-tioned at the beginning, a significant lack of knowledge is foundconcerning experimental investigations on moisture dependentmechanical properties as well as diffusion properties. Further re-search work is imperatively required in this field. To fully coverthe hygro-mechanical behavior of wood with the help of simula-tions, moisture dependent brittle failure as well as time-dependentand mechano-sorptive effects have to be captured. Furthermore,large strains have to be considered to cover the full range of engi-neering applications.

Acknowledgments

The authors gratefully acknowledge the support of the workby the German Federal Cultural Foundation (KUR). The work issupported within the framework of the KUR-Programme for theConservation of Moveable Cultural Assets.

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