Optimization of fiber geometry for fiber reinforced composites considering damage J. Kato , E. Ramm Institute of Structural Mechanics, University of Stuttgart, Pfaffenwaldring 7, D-70550 Stuttgart, Germany article info Article history: Received 19 May 2009 Accepted 4 January 2010 Available online 18 February 2010 Keywords: Shape optimization Damage Fiber reinforced composites abstract The present contribution deals with an optimization strategy of fiber reinforced composites. Although the methodical concept is very general we concentrate on Fiber Reinforced Concrete with a complex failure mechanism resulting from material brittleness of both constituents matrix and fibers. Because of these unfavorable characteristics the interface between fiber and matrix plays a particularly important role in the structural response. A prominent objective for this kind of composite is the improvement of ductility. The influential factors on the entire structural response of this composite are (i) material parameters involved in the interface, (ii) the material layout at the small scale level, and (iii) the fiber geometry on the macroscopic structural level. The purpose of the present paper is to improve the structural ductility of the fiber reinforced composites applying an optimization method with respect to the geometrical layout of continuous long textile fibers. The method proposed is achieved by applying a so-called embedded reinforcement formulation. This methodology is extended to a damage formulation in order to represent a realistic structural behavior. For the optimization problem a gradient-based optimization scheme is assumed. An optimality criteria method is applied because of its numerically high efficiency and robustness. The performance of the method is demonstrated by a series of numerical examples; it is verified that the ductility can be substantially improved. & 2010 Elsevier B.V. All rights reserved. 1. Introduction 1.1. Overview The most widely used light-weight fiber reinforced composites are fiber reinforced polymers (FRP) where often long glass, carbon or aramid fibers are placed into a polymer matrix leading to a good-natured ductile material. The present study addresses a new composite material, namely Fiber Reinforced Concrete (FRC), sometimes also called Textile Reinforced Concrete. It differs from FRP that the fibers are placed in a fine grained concrete or mortar matrix (Fig. 1), often as a reinforcement mesh with a relatively low fiber content, making it economically attractive. Unlike conventional steel reinforcement, this kind of textile fiber is corrosion free; this property allows to manufacture light-weight thin-walled composite structures. However, FRC structures show very complex failure mechan- isms resulting on the one hand from the material brittleness of both constituents, fibers and matrix, and on the other hand from their interface behavior introducing the necessary ductility. The specific characteristic of FRC is an ideal target for optimization applying the overall structural ductility as objective which ought to be maximized for a prescribed fiber volume. In this context the ‘structural ductility’ means ‘energy absorp- tion capacity’ which is measured by the internal energy summed over the entire structure up to a prescribed displacement of a dominant control point. For this task geometrical parameters on the small scale level like fiber location, orientation, size, length, spacing as well as combinations of different fiber materials can be considered as the representative design variables, see Kato et al. [15]. As an example for FRC Karihaloo and Lange-Kornbak [14] introduce a multi-objective optimization problem to maximize tensile strength, deformability and toughness of high perfor- mance fiber reinforced concrete mixes (HPFRC) in which chopped ‘short’ fibers are utilized. In their investigation the fiber diameter, length, fracture toughness and fiber volume fraction are chosen as the design variables and an analytical solution instead of the finite element method is applied for the optimization problem. Kato et al. [16] introduce a multiphase material optimization to improve the ductility of FRC with respect to fiber size, length and the combination of different fiber materials using the finite ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/finel Finite Elements in Analysis and Design 0168-874X/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2010.01.001 Corresponding author. Tel.: + 49 711 685 66123; fax: + 49 711 685 66130. E-mail addresses: [email protected], [email protected](J. Kato). Finite Elements in Analysis and Design 46 (2010) 401–415
Transcript
ARTICLE IN PRESS
Finite Elements in Analysis and Design 46 (2010) 401–415
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design
0168-87
doi:10.1
� Corr
E-m
(J. Kato)
journal homepage: www.elsevier.com/locate/finel
Optimization of fiber geometry for fiber reinforced compositesconsidering damage
J. Kato �, E. Ramm
Institute of Structural Mechanics, University of Stuttgart, Pfaffenwaldring 7, D-70550 Stuttgart, Germany
a r t i c l e i n f o
Article history:
Received 19 May 2009
Accepted 4 January 2010Available online 18 February 2010
The present contribution deals with an optimization strategy of fiber reinforced composites. Although
the methodical concept is very general we concentrate on Fiber Reinforced Concrete with a complex
failure mechanism resulting from material brittleness of both constituents matrix and fibers. Because of
these unfavorable characteristics the interface between fiber and matrix plays a particularly important
role in the structural response. A prominent objective for this kind of composite is the improvement of
ductility. The influential factors on the entire structural response of this composite are (i) material
parameters involved in the interface, (ii) the material layout at the small scale level, and (iii) the fiber
geometry on the macroscopic structural level.
The purpose of the present paper is to improve the structural ductility of the fiber reinforced
composites applying an optimization method with respect to the geometrical layout of continuous long
textile fibers. The method proposed is achieved by applying a so-called embedded reinforcement
formulation. This methodology is extended to a damage formulation in order to represent a realistic
structural behavior.
For the optimization problem a gradient-based optimization scheme is assumed. An optimality
criteria method is applied because of its numerically high efficiency and robustness. The performance of
the method is demonstrated by a series of numerical examples; it is verified that the ductility can be
substantially improved.
& 2010 Elsevier B.V. All rights reserved.
1. Introduction
1.1. Overview
The most widely used light-weight fiber reinforced compositesare fiber reinforced polymers (FRP) where often long glass, carbonor aramid fibers are placed into a polymer matrix leading to agood-natured ductile material.
The present study addresses a new composite material,namely Fiber Reinforced Concrete (FRC), sometimes also calledTextile Reinforced Concrete. It differs from FRP that the fibersare placed in a fine grained concrete or mortar matrix (Fig. 1),often as a reinforcement mesh with a relatively low fiber content,making it economically attractive. Unlike conventional steelreinforcement, this kind of textile fiber is corrosion free; thisproperty allows to manufacture light-weight thin-walledcomposite structures.
However, FRC structures show very complex failure mechan-isms resulting on the one hand from the material brittleness of
both constituents, fibers and matrix, and on the other hand fromtheir interface behavior introducing the necessary ductility. Thespecific characteristic of FRC is an ideal target for optimizationapplying the overall structural ductility as objective which oughtto be maximized for a prescribed fiber volume.
In this context the ‘structural ductility’ means ‘energy absorp-tion capacity’ which is measured by the internal energy summedover the entire structure up to a prescribed displacement of adominant control point. For this task geometrical parameters onthe small scale level like fiber location, orientation, size, length,spacing as well as combinations of different fiber materialscan be considered as the representative design variables, see Katoet al. [15].
As an example for FRC Karihaloo and Lange-Kornbak [14]introduce a multi-objective optimization problem to maximizetensile strength, deformability and toughness of high perfor-mance fiber reinforced concrete mixes (HPFRC) in which chopped‘short’ fibers are utilized. In their investigation the fiber diameter,length, fracture toughness and fiber volume fraction are chosen asthe design variables and an analytical solution instead of the finiteelement method is applied for the optimization problem.
Kato et al. [16] introduce a multiphase material optimization toimprove the ductility of FRC with respect to fiber size, length andthe combination of different fiber materials using the finite
J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415402
element method for the discretization. In that study the conceptof ‘volume fraction’ of fiber material(s) is applied describing thedesign variables mentioned. This approach is considered as amaterial distribution problem derived from conventional topologyoptimization.
An optimal fiber orientation of composites has been investi-gated in several contributions. Most of them focus on the optimalfiber orientation using laminated FRP structures. In those casesthe fiber angle in individual plies is chosen as design variable, seeStegmann and Lund [34] and Stolpe and Stegmann [35], just tomention only a few references.
The design variables in both mentioned approaches, i.e.optimizing material or fiber angle distribution, are defined locallywithin certain plies or even in a patch of finite element; thisrestriction limits to achieve final optimal fiber layouts. Forexample in the investigation of Kato et al. [16] fiber materialsare defined only in prescribed design elements, furthermore onlystraight fibers are allowed. In Stegmann and Lund [34] and Stolpeand Stegmann [35] the fibers are discontinuous between adjacentelements, leading to a so-called discrete fiber distribution.
The purpose of the present study is to introduce a methodol-ogy improving the structural ductility of FRC with respect to thefiber geometry which is independent of the fixed finite elementmesh. The mechanical model of FRC is briefly summarized in thenext section. The optimization problem is solved by a gradient-based optimization scheme. An optimality criteria method, seePatnaik et al. [28], is used because of its numerically highefficiency and robustness. For the sensitivity analysis a variationalsemi-analytical direct method is used. The present method is notrestricted to FRC but may as well be applied to other fiberreinforced composites, for example fiber reinforced glass (FRG).
1.2. Present numerical model
We introduce so-called embedded reinforcement elementswhere the fiber geometry is globally defined. These embeddedreinforcement elements have been originally introduced byPhillips and Zienkiewicz [32]. Chang et al. [8] modified theconcept allowing for straight reinforcement segments to beplaced at any angle with respect to the local axes of isoparametricconcrete elements. Balakrishnan and Murray [1] introduce anembedded formulation with bond–slip relation between concreteand reinforcement. Further improvements by Elwi and Hrudey[10] allow for a general curved reinforcement formulation in theembedded element. Hofstetter and Mang [11] apply the em-bedded reinforcement formulation for a thin-walled prestressconcrete shell structure where geometry of curved tendons areintroduced by an analytical expression. The extension to a three-dimensional formulation is discussed by Barzegar and Maddipudi[2]. Recently Huber [12] applies the bond–slip relation by
Balakrishnan and Murray [1] for a 3D model with straightreinforcement bars considering nonlinear material models forboth concrete and steel reinforcement.
In the present study we apply the embedded reinforcementelement including the bond–slip relation by Balakrishnan andMurray [1] for a two-dimensional model in which a curved fibergeometry is allowed. The fiber geometry is defined globally byBezier-splines. The materials for both concrete and fibers aremodeled by a gradient-enhanced damage formulation, seePeerlings et al. [29,30], Peerlings [31]. For the interface betweenconcrete and fiber a discrete bond model is applied, see Krugeret al. [18–20], Xu et al. [37]. The assumption introduced byBalakrishnan and Murray [1] is adopted for the interfacialkinematical relation.
1.3. Structure of paper
The paper is organized as follows. Firstly the applied materialmodels are briefly presented in Section 2. Secondly the concept ofthe globally defined fiber geometry is introduced in Section 3.Then the embedded reinforcement element is described inSection 4. The finite element formulation of FRC is introduced inSection 5, which is composed of three individual materialformulations, namely the gradient enhanced damage for bothconcrete and fibers and the interface model. Some details areshifted to the Appendix, e.g. the transformation matrices relevantfor the fiber orientations and the linearization of the model.Finally the optimization problems and the derivation of thesensitivity analysis are discussed in Sections 6 and 7.
Note that in the present paper the following superscripts ð�Þc,ð�Þ
f , and ð�Þi denote the terms for ‘concrete’, ‘fiber’, and ‘interface,respectively. In some cases we utilize a compact expression, e.g.ð�Þ
cþ f¼ ð�Þ
cþð�Þ
f . A subscript ð�ÞL or ð�ÞG indicates that the valueð�Þ is measured in the local or global coordinate system,respectively. However the notion ð�ÞG is introduced only whenwe need to emphasize it, otherwise we skip it for simplicity.
2. Applied material models
2.1. Material model for constituents concrete and fiber
In this study the nonlinear material behavior of both concreteand fiber is described by an isotropic continuum damage model.Firstly an equivalent strain measure is defined. For the concretematrix de Vree’s definition (de Vree et al. [36]) of equivalentstrains ec
J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415 403
where I1 denotes the first invariant of the strain tensor and J2 thesecond invariant of the deviatoric strain tensor. k indicates theratio of compression relative to the tension strength and n isPoisson’s ratio. For the fiber we follow Mazars’s definition (Mazarsand Pijaudier-Cabot [24]) since the fiber is assumed to be a one-dimensional model
efv ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi/ef
LS2
q; ð2Þ
where efL is the fiber strain and /�S denotes the Macauley bracket
/xS¼ ðxþjxjÞ=2. For the damage evolution of both concrete andfiber we use an exponential damage law introduced by Mazarsand Pijaudier-Cabot [24] as
DðkÞ ¼ 1�k0
k ð1�aþae�bðk�k0ÞÞ if kZk0; ð3Þ
where D stands for the damage parameter ð0rDr1Þ, a definesthe final softening stage and b governs the rate of damage growth.k0 is a threshold variable which determines damage initiation andk represents the most severe deformation the material hasexperienced during loading. In a conventional local damagemodel, k is related to the local equivalent strain ev and thehistory variable k is defined by the Kuhn–Tucker relations, i.e._kZ0, ev�kr0, _kðev�kÞ ¼ 0. For non-local damage, k is related toa weighted volume average of the local equivalent strain ev,denoted as non-local equivalent strain ~ev. In the gradient-enhanced damage model (Peerlings et al. [29,30], Peerlings [31])~ev is approximated implicitly as follows:
~ev�cr2 ~ev ¼ ev; ð4Þ
where r2 denotes the Laplacean operator and c is a positiveparameter of the dimension length squared regularizing thelocalization of the deformation. Thus in the Kuhn–Tuckerequations ev is replaced by the non-local equivalent strain ~ev
which is discretized in the finite element sense. Elastic unloadingis included in the traditional way.
2.2. Material model for interface
In this study nonlinear interfacial behavior between fiber andmatrix is expressed by a discrete bond model, see Kruger et al.[18]. This model was obtained by experiments for different textilefiber materials and leads to a realistic interface response of FRC.The significant factors governing interfacial response are the bondstrength and the debonding behavior. The influence of materialproperties at a small scale level and the stresses perpendicular tothe fiber direction are included in the material formulation asimportant parameters. The bond stress–slip ðsi
L�uiLÞ relation is
expressed as
siL ¼ ~w � bþð1�bÞ �
1
1þ ~wR
� �1=R( )
� s0 for uiLrw1; ð5Þ
where ~w ¼ uiL=w0 denotes the normalized slip. ui
L is the slip lengthwhich will be introduced in the following section. w0 is a factordefined by the initial tangent k1. k2 is the tangent at slip w1
(see Fig. 2) where the bond stress achieves the maximum bondstrength. b¼ k2=k1 and s0 ¼ k1 �w0 are parameters to calculatethe stresses and R defines the radius of curvature at slip w1. Thestress–slip relation for the range ui
L4w1 is simply describedby the adhesion strength sm and the friction bond strength sf
(see Fig. 2),
sm ¼ sm;0c; sf ¼ sf ;0c ð6Þ
with
c¼ 1þtanh arsR
0:1fc�afnes 1�
r2
ðrþhÞ2
!�124
35; ð7Þ
where c denotes an additional parameter ð1oco2Þ whichconsiders the influence of the kind of fiber material, the loadingcondition and the stresses perpendicular to a fiber direction. sm;0
and sf ;0 denote the initial adhesion strength and sliding frictionstrength, respectively. r describes a fiber (roving) radius, n isPoisson’s ratio of a fiber and h is the surface roughness of a fiber.ar and af are constants assuming the lateral deformation of afiber. These properties are given depending on the kind of fibermaterial used. fc is the uniaxial compressive strength of concrete,es the uniaxial strain and sR defines the stress perpendicular to afiber. For a detailed description of this model it is referred toKruger et al. [18–20]. In this model loading and unloadingconditions are also considered.
This one-dimensional interface model is originally formulatedfor a fiber in a three-dimensional setting. If this model is utilizedin a two-dimensional space, the interface has to be approximatedto hold the original total interface area.
3. Global layout of fiber geometry
3.1. Concept and design variables
The geometry of a continuous long fiber is defined in the globalcoordinate system. One of the big advantages of FRC structures isthat they do not need thick concrete covers. Furthermore hooks oftextile fibers are not used. Due to these characteristics the layoutof textile fibers in FRC can be rather simple often parallel fibers ora mesh of straight fibers are used, see Fig. 1. Slightly curved fibersare advantageous if an optimal structural response is looked for.
We approximate the fiber geometry by Bezier-splines, defined asparametric curves by control points. A quadratic Bezier-spline andits mathematical formulation are introduced in Fig. 3, where rstands for a position vector of the spline; W ð0rWr1Þ is the localcoordinate system of the spline. pj indicates the j-th control point. Ofcourse we could apply another parameterization allowing moregeneral geometrical definitions such as a level set function (Fig. 4).
The fiber approximated by Bezier-splines is embedded in thestructure and the control points are moved in order to obtain theoptimal fiber layout. The entire domain of structure is defined in aparametric space s ð0rsr1Þ, see Fig. 3. Thus the normalizedcoordinates of control points turn out to be the design variablesdefining the global fiber layout in the physical space. According tothis the j-th position vector of control point pj can be expressed asfollows
rjðsxj ; s
yj Þ ¼ Oðx; yÞþðsx
j Lx; syj LyÞ; ð8Þ
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Fig. 3. (a) Quadratic Bezier-spline and (b) concept of global layout of fiber geometry.
Fig. 4. Element patch describing intersections and related Newton–Raphson algorithm.
J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415404
where O stands for the coordinate origin of the structure; x, yare the corresponding global coordinates of O. L denotes thecontour lengths of the structure and the superscripts x, y on L aswell as s indicate the direction. Inserting Eq. (8) into the generalmathematical formulation of Bezier-splines leads to the geo-metric formulation of a fiber including the design variables s asfollows
rðW; sx; syÞ ¼Xnb
j ¼ 0
FjðWÞrjðsxj ; s
yj Þ with Fj ¼
nb!
ðnb�jÞ!j!Wjð1�WÞnb�j;
ð9Þ
where nb is the order of the Bezier-spline. Note that thecoefficients F are independent of the design variables s. Oncethe fiber geometry is defined by Eq. (9) the intersections between
fiber and fixed finite element mesh can be calculated. It isnecessary to determine the global coordinates for intersections offibers and mesh in order to establish the stiffness matrix andafterwards the internal forces of embedded fiber elements. Thisprocedure is described in the sequel.
3.2. Determination of intersections between finite element mesh
and fiber
The basic procedure determining the intersections betweenthe mesh and a fiber is explained in Box 1. This procedure iscontinued until the end of the fiber. Note that the local coordinatesystem W and additional parameter $ have been introduced
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J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415 405
only for the determination of intersections and are no longernecessary thereafter.
Step 1:
(1.1) start with control point p0 and obtain its position vector r0(W), where W=0
(1.2) calculate the descent along the fiber at p0, i.e. rWrðWÞ at W=0(1.3) find intersection of element boundary with descent line
Step 2:
(2.1) parameterize element boundary by normalized parameter $ð0r$r1Þ using position vectors ra and rb of pointsa and b(2.2) determine global coordinates of intersection (x, y)p by Newton – Raphson scheme, see box in Fig 4, where R is the
residual vector between points rðWÞ and rð$ÞStep 3:
(3.1) if converged at (2.2), store the global coordinates of intersection p; ðx; yÞp ¼ rðWÞ(3.2) otherwise choose other element boundary of element and restart from (2.1)
Step 4:(4.1) replace reference point p0 by the determined intersection p and repeat the same procedure from (1.2) with the
new descent rWrðWÞ
3.3. Inverse mapping for local coordinate of fiber
In the previous section we introduced the procedure todetermine the global coordinates of the intersections ðx; yÞp. It isalso necessary to determine the associated parametric (natural)coordinates ðx;ZÞp of the corresponding finite element in order toperform the integration for the internal virtual work of the fiber.
This so-called nonlinear inverse mapping is described in detailin Elwi and Hrudey [10] and Barzegar and Maddipudi [2]. For thegeneral isoparametric mapping the global coordinate ðx; yÞ of anarbitrary point p in an element is expressed by the shape functionN and the global coordinates of the element nodes ðxk; ykÞ,
x
y
" #p
¼Nðx;ZÞ 0
0 Nðx;ZÞ
" #p
xk
yk
" #; ð10Þ
where the expression ð�Þ emphasizes a known value. k is thenumber of nodes of an element. In case of the inverse mappingEq. (10) is transformed to the following equation such that theresidual vector function R vanishes
Rðx;ZÞ ¼x
y
" #p
�Nðx;ZÞ 0
0 Nðx;ZÞ
" #p
xk
yk
" #¼
0
0
� �: ð11Þ
Finally, this nonlinear equation is solved by a Newton–Raphson scheme as shown in the box of Fig. 5 obtaining theassociated natural coordinates ðx;ZÞp of point p. Once thesecoordinates of the intersections are determined the integration ofthe embedded reinforcement can be performed. For furtherprocessing of the fiber mechanics the curved fiber is for
Fig. 5. Determination of natural
simplicity approximated by a straight line leading to apolygonal layout as already indicated in Fig. 5.
4. Embedded reinforcement element
4.1. Kinematical assumption for interface between concrete
and fiber
The embedded reinforcement element applied in this studyconsiders a bond–slip relation between concrete and fiber, seeFig. 6(a). We introduce the kinematical relation of the interfacebased on the assumption by Balakrishnan and Murray [1]. Inthe kinematical assumption the slip at an arbitrary point isconsidered as the relative displacement between concrete andfiber measured along the axis of the fiber. The components of thedisplacements can be written as
ufL ¼ uc
LþuiL; ð12Þ
where uiL is the slip length or relative displacement introduced in
Eq. (5). ufL and uc
L are the displacements of fiber and concrete atthe considered point, respectively, see Fig. 6(b). The slips of theoriginal curved fiber between two adjacent elements have to beequal; this is not the case for the polygonal geometry assumedabove. In order to satisfy the compatibility at least in an averagesense the slip length ui
L is projected onto the global x-axis
d ¼ cosy � uiL-ui
L ¼ td with t ¼ ðcosyÞ�1; ð13Þ
where y is the angle between fiber axis and x-axis, see Fig. 6. Thusthe compatibility of the slip-length is enforced for d. From ui
L the
coordinates of intersections.
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Fig. 6. (a) Embedded reinforcement element patch and (b) notion for displacements of slip.
J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415406
local bond strain eiL is obtained which in turn leads to the local
fiber strain efL
efL ¼ ec
L|{z}Te1e
cG
þeiL: ð14Þ
Matrix Te transforms the global strain eG of a two-dimensionalcontinuum into the local one eL under plane stress condition(see Appendix A). Te1 represents the first row of Te extractingthe local strain ec
L in fiber direction from the global concretestrain ec
G.
5. Finite element formulation of FRC
5.1. Virtual work
Since we apply a gradient enhanced damage model for bothconcrete and fibers and use a nonlinear interface model betweenfibers and matrix the virtual work is decomposed into
dW¼ dWint�dWext ¼ dWcintþdWf
intþdWiint�dWext ¼ 0; ð15Þ
where dWcint, dWf
int, and dWiint stand for the internal virtual work
of concrete, fibers and interfaces, respectively, and dWext denotesthe external virtual work.
The gradient enhanced damage model leads to a two-fieldformulation ðO¼Oc
[OfÞ and its formulation at the actual time
tþ1 is
dWuðu; duÞ ¼ZOde : r dO�
ZOdu � b dO�
ZGdu � t dG¼ 0; ð16Þ
dWeð~ev; d~evÞ ¼
ZOdr ~ev � sdOþ
ZOd~evð~ev�evðeÞÞdO¼ 0 ð17Þ
where ‘time’ t does not mean the ‘real time’ but simply the‘loading step number’ for a nonlinear static problem. du and d~ev
are the virtual displacement and non-local equivalent strainfields, respectively. r is the stress tensor, b and t are theprescribed body and traction forces. Eq. (16) is the usual virtualwork expression, whereas Eq. (17) defines the weak form of anadditional equilibrium equation for the non-local equivalentstrain where s¼ cr ~ev is a work equivalent stress vector (seePeerlings et al. [29,30], Peerlings [31]). Without loss of generalitywe omit the body force in this study.
Both equations of the virtual work can be split up into theparts of concrete and fibers, dWc
u=e, dWfu=e, the latter one being
reduced to a one-dimensional expression referring to the local
stress and strain field in the fibers,
dWfu;int ¼
ZOfdef
LsfLdOf
¼
ZOfðdec
LþdeiLÞs
fLdOf ; ð18Þ
dWfe ¼
ZOfdr ~ef
v;LtfLdOf
þ
ZOfd~ef
v;Lð~efv;L�e
fv;Lðe
fLÞÞdO
f : ð19Þ
According to Eq. (14) the fiber strain efL can be decomposed
into a contribution of the local concrete strain ecL and that of the
interface eiL, as indicated in Eq. (18). The work of the latter part
together with the virtual work inside the interface due to the sliplength ui
LZOidui
LsiLdOi
defines the total virtual work of the interface slip:
dWiint ¼
ZOfdei
LsfLdOf
þ
ZOidui
LsiLdOi
¼ 0 8 duiL: ð20Þ
5.2. Discretization
The virtual work expressions (16)/(17) and (18)/(19) togetherwith (20) contain three independent variables, namely thedisplacement field in the concrete element u, the non-localequivalent strain ~ev and the slip length ui
L which are discretized inthe finite element sense. In the present study a two-dimensional8-node quadratic plane stress element is applied for the concretematrix. The non-local strain is discretized by bilinear shapefunctions within this element. Please note, that the interface slipis discretized as a 3-node quadratic one-dimensional elementðni ¼ 3Þ, whereas the non-local strain enhancement of the fibersare only linearly interpolated based on the two values at the fiberbeginning and end obtained from the non-local strain values inthe concrete element
u¼Xnc
k ¼ 1
Nkdk or u¼Nd; ð21Þ
~ev ¼Xne
k ¼ 1
~Nkek or ~ev ¼~Ne; ð22Þ
uiL ¼
Xni
k ¼ 1
Nikðu
iLÞ
k¼Xni
k ¼ 1
NikðtdÞk ¼
Xni
k ¼ 1
Nkdk
or uiL ¼Nd; ð23Þ
where d is a vector with eight nodal displacements, e with fournodal values and d contains three nodal slip values.
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J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415 407
The two nodal values of the projected slip lengths are summedin the vector d
d ¼ ½d1
d2
d3�T: ð24Þ
N contain the shape function for the interface defined in theglobal coordinate system. Analogously the local bond strain ei
L ofEq. (14) in one element can be expressed as
eiL ¼
Xni
k ¼ 1
BikðtdÞk ¼ tBid ¼ Bd; ð25Þ
where Bi and B stand for the so-called B-operator matrices for theinterface defined in local and global coordinate systems, respec-tively. The local fiber strain ef
L can be written according to Eq. (14)
efL ¼ Te1e
cGþe
iL ¼ Te1Bf dþBd: ð26Þ
Introducing Eqs. (21)–(23) into the virtual work expressionsleads to
where KT, Du and R stand for the tangential stiffness matrix, theincremental displacement/strain vector and the residual forcevector, respectively. The superscripts n and nþ1 on the matrixand vectors indicate the iteration number in the increment. Forthe derivation of the corresponding stiffness matrices in KT andthe forces in R it is referred to Appendices B to D.
6. Structural optimization of FRC
6.1. Optimization problem
In general an optimization problem is defined by an objectivefðsÞ, equality constraints hðsÞ and inequality constraints gðsÞ. Inthis study the objective is to maximize the structural ductility fora prescribed fiber volume. As the ductility is defined by theinternal energy summed up over the entire structure with aprescribed nodal displacement u (Maute [22] and Maute et al.[23]), the mathematical formulation of the optimization problemof FRC can be written as follows:
minimize fðsÞ ¼�
ZOc
Zec
rcdecdOcþ
ZOf
Ze f
L
sfLdef
LdOfþ
ZOi
Zu i
L
siLdui
LdOi
" #
ð33Þ
subject to hðsÞ ¼ZOf
ltf dOf�V ¼ 0 ð34Þ
sLrsirsU i¼ 1; . . . ;ns ð35Þ
where V denotes the prescribed fiber volume, sL and sU the lowerand upper bounds of the design variables, and ns the number ofdesign variables. tf represents the thickness of a fiber, which isassumed to be constant along the entire fiber. l is the length of asingle fiber within an embedded reinforcement element anddepends on the design variables si.
6.2. Equilibrium conditions and total derivative of design function
The sensitivities of the design functions (objective, constraintsetc.) depend on the gradients of the state variables uðd;e;dÞ.These are derived from the three equilibrium conditions (16),(17), and (20) at position nþ1.
The total derivative of the design functions with respect tothe design variables can be decomposed into an explicit andan implicit part. As indicated the design functions dependon the structural response which in turn is implicitly related tothe optimization variables, for example the objective f ¼ fðs;uðd; e;dÞÞ. This leads to
rsð�Þ ¼rexs ð�Þþruð�Þrsu¼rex
s ð�Þþrdð�Þrsdþreð�Þrseþrdð�Þrsd;
ð36Þ
where rexs ð�Þ describes the explicit derivative with respect to the
design variables. An optimality criteria method (see Patnaik et al.[28]) is applied to solve the optimization problem. For thesensitivity analysis a variational semi-analytical direct methodis adopted and described in the sequel.
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7. Sensitivity analysis
7.1. Overview of sensitivity analysis
The main effort of sensitivity analysis is the calculation of implicitpart rsu. For a direct sensitivity analysis this part is obtained byexploiting the stiffness expression containing the tangent stiffnessmatrix and the so-called pseudo-load vector, see Eq. (64).
The accuracy of the sensitivities strongly depends on that of thepseudo-load vector. This pseudo-load vector is obtained throughthe derivatives of the equilibrium conditions, Eqs. (16), (17), and(20) with respect to the design variables and by assembling theindividual pseudo-load vectors for each equilibrium condition.
For a linear elastic analysis the pseudo-load vector can beeasily obtained formulating a discrete sensitivity approach. It ismore demanding for materially or geometrically nonlinearproblems. Therefore the derivation of the pseudo-load vector forthe damage formulation used in the present study is detailed inSections 7.4 to 7.6. For this the gradients of constitutive equationsand also the explicit part of the derivative of objective functionare described first in the next two sections.
7.2. Gradients of constitutive equations
The gradients of the variables in the constitutive equations atposition nþ1 are calculated for the total sensitivity analyses.
The derivatives with respect to a design variable s aredecomposed into explicit and implicit parts:
�
concrete and fiber strains ecL, ef
L,
rsecðdÞ ¼rex
s ecþrims ec ¼rsðB
cÞ|fflfflffl{zfflfflffl}
¼ 0
dþBcrsd; ð37Þ
rsefL ¼rsec
LþrseiL : e
cL ¼ e
cLðB
fðsÞ;Te1ðsÞ;dÞ and ei
L ¼ eiLðBðsÞ;dÞ;
ð38Þ
with
rsecL ¼r
exs ec
Lþrims ec
L ¼rsðTe1ÞB
f dþTe1rsðBfÞd|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Note that the explicit terms for concrete in Eqs. (37), (43)–(45)are zero because Nc, Bc, ~N
c, and ~B
cdo not depend on the design
variables s; in other words all explicit terms of derivatives forconcrete vanish.
Utilizing the above equations the stress derivatives of concretematrix rsrc, fiber rssf
L, and interface rssiL at position nþ1 are
rsrc ¼
@rc
@ec
@ec
@sþ@rc
@~ecv
@~ecv
@s¼Cc
ed rexs ec|fflffl{zfflffl}¼ 0
þrims ec
1CAþE
cr
exs~ec
v|fflfflffl{zfflfflffl}¼ 0
þrims~ec
v
0@
1A
0B@
¼Ccedr
ims ecþE
cr
ims~ec
v; ð49Þ
rssfL ¼
@sfL
@efL
@efL
@sþ@sf
L
@~efv;L
@~efv;L
@s
¼Cfed;Lðr
exs ef
Lþrims ef
LÞþEf
Lðr
exs~ef
v;Lþrims~ef
v;LÞ; ð50Þ
rssiL ¼
@siL
@uiL
@uiL
@s¼ kLðr
exs ui
Lþrims ui
LÞ; ð51Þ
with the abbreviations
Ec�@rc
@~ecv
and Ef
L �@sf
L
@~efv;L
: ð52Þ
Ec
and Ef
L are detailed in Eqs. (80) and (89) in Appendices B andC. Ced is the so-called elasto-damage ‘secant’ material tensor, seeEq. (83). kL in Eq. (51) denotes the tangent modulus of theinterface which is explicitly obtained from Eq. (5).
In the present case the design variables s are defined on theglobal level and not related directly to variables on the elementlevel needed in the present case. Therefore a semi-analyticalapproach is most appropriate calculating the above introducedderivatives by a finite difference method.
7.3. Sensitivity for explicit part of objective function
The explicit part of sensitivity of the objective function given inEq. (33) is expressed as follows,
rexs f ¼rex
s ðfcþ ffþf iÞ ð53Þ
with
rexs fc¼ 0; ð54Þ
rexs ff¼�
ZOf
Ze f
L
ðrexs ðs
fLÞde
fLþs
fLr
exs def
LÞdOf�
ZOf
x
Ze f
L
sfLdef
LrsjJfjdOf
x;
ð55Þ
rexs f i¼�
ZOi
Zu
iL
ðrexs ðs
iLÞdui
LþsiLr
exs dui
LÞdOi�
ZOi
x
Zu
iL
siLdui
LrsjJijdOi
x:
ð56Þ
The second terms are integrated in the parametric space x.Eq. (54) is satisfied because the functions of concrete, e.g.
shape functions and B-operator, are independent of the designvariables as mentioned above. The determinants of Jacobianmatrices jJf
j and jJij for fiber and interface elements map the
parametric element domains onto their real space. The stress
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derivatives rexs sf
L and rexs si
L are related to the explicit parts asshown in Eqs. (50) and (51).
In the following sections the implicit part of sensitivity of theobjective function is discussed.
7.4. Sensitivity for first equilibrium equation
The derivative of equilibrium condition Eq. (16) with res-pect to a design variable s is obtained considering Eq. (27)as followsZOcrsðB
cÞT|fflfflfflffl{zfflfflfflffl}
¼ 0
rcdOcþ
ZOc
BcTrsðr
cÞdOcþ
ZOc
x
BcTrcrsjJ
cj|fflffl{zfflffl}
¼ 0
dOcx
þ
ZOf½rsðB
fÞTðTe1Þ
TþBf TrsðT
e1Þ
T�sf
LdOfþ
ZOf
Bf TðTe1Þ
TrsðsfLÞdO
f
þ
ZOf
x
Bf TðTe1Þ
TsfLrsjJ
fjdOf
x�rsltþ1
ZGx
NcTt0j~JjdGx ¼ 0; ð57Þ
where the virtual displacement field du in (16) is assumed to bearbitrary so that its derivative rsdu vanishes. jJc
j is thedeterminant of Jacobian matrix of the concrete element. Themetric j~Jj maps a line differential on the boundary.
Substituting Eqs. (49) and (50) into Eq. (57) results inZOc
In Eq. (58) all implicit and explicit terms are assembledto the left and right hand side, respectively. The right handside of Eq. (58) leads to the pseudo-load vector ~P
d. Note
that the stiffness matrices on the left hand side of Eq. (58)correspond to those in the tangent stiffness matrix KT introducedin (32).
The same procedure is adopted for the derivatives of thesecond and third equilibrium conditions.
7.5. Sensitivity for the second equilibrium equation
Analogously the derivative of equilibrium condition Eq. (17)with respect to a design variable s is obtained considering Eq. (28).
Inserting Eqs. (43)–(48) into the obtained derivative of theequilibrium condition Eq. (17) and arranging it as in the previoussection yieldsZOc½cð ~B
detailed in Eqs. (81) and (90) in Appendices B and C. The virtualnon-local equivalent strain d~ev in (17) is assumed to be arbitrary,thus its derivative rsd~ev vanishes.
7.6. Sensitivity for the third equilibrium equation
Similarly the derivative of equilibrium condition Eq. (20) isobtained considering Eq. (29)ZOfrsðBÞ
TsfLdOf
þ
ZOf
BTrssf
LdOfþ
ZOf
x
BTsf
LrsjJfjdOf
x
þ
ZOirsðNÞ
TsiLdOiþ
ZOi
NTrssi
LdOiþ
ZOi
x
NTsi
LrsjJijdOi
x ¼ 0:
ð61ÞAgain the derivative rsdui
L vanishes because duiL in (20) is an
arbitrary test function.The pseudo-load vector is derived by inserting Eqs. (50) and (51)
into Eq. (61). However the bond–slip relation Eq. (5) does notinclude any term related to the non-local equivalent strain oppositeto Eq. (50). Excluding the non-local term from Eq. (50), i.e.
rssfL ¼Cf
ed;Lðrexs ef
Lþrims ef
LÞ; ð62Þ
and substituting Eqs. (51) and (62) into Eq. (61) results in thefollowing expressionZ
J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415410
7.7. Total sensitivity analysis
Assembling Eqs. (58), (59) and (63) leads to the followingcompact matrix expression
Kcþ fdd Kcþ f
de Kfdd
Kcþ fed Kcþ f
ee 0
Kfdd
0 Kidd
26664
3777526664rsd
rse
rsd
37775¼rsltþ1
26664
P
0
0
37775�
X5
l ¼ 1
~Pd
l
X3
l ¼ 1
~Pe
l
X6
l ¼ 1
~Pd
l
266666666664
377777777775: ð64Þ
which has the format of the typical stiffness equation adding upall terms on the right hand side to a new pseudo-load vector
Ppseudo :
KTrsu ¼ Ppseudo ¼rsltþ1Pþ ~Ppseudo: ð65Þ
KT denotes the tangent stiffness matrix at the time step tþ1 asmentioned before. The next question is how to deal with thederivative of load factor rsl. Note that the derivatives based on aload-controlled algorithm differ from those based on a displace-ment-controlled algorithm controlling a certain nodal displacement‘component’ uj ¼ uj of the structure which is more suitable for theoptimization of ductility. For a detailed description it is referred toSchwarz et al. [33], Lipka et al. [21]. A load-controlled algorithmrenders rsl¼ 0 while for a displacement controlled algorithm onlythe sensitivity of the nodal displacement for the controlled degree offreedom uj is equal to zero. The sensitivity of the load factor basedon a discretized formulation is derived subsequently,
rsuj ¼rsltþ1uj
ltþ1þðrsujÞpseudo ¼ 0; ð66Þ
νκ
Fig 7. Deep
where uj and ðrsujÞpseudo are the j-th component of vectors u andðrsuÞpseudo expressed as
u ¼K�1T ltþ1P; ð67Þ
ðrsuÞpseudo ¼K�1T~Ppseudo: ð68Þ
Substituting uj and ðrsujÞpseudo into (66) yields
rsltþ1 ¼�ðrsujÞpseudo
ujltþ1: ð69Þ
According to the above equations the derivative of the totalnodal displacement vector rsu is
rsu ¼ ursltþ1
ltþ1þðrsuÞpseudo: ð70Þ
Finally, the total sensitivity of the objective function can beobtained by inserting Eq. (70) into Eq. (36) and accumulating eachsensitivity over the load increment step number nstep as
rsf ¼Xnstep
t ¼ 1
ðrexs ftþru f trsutÞ ð71Þ
where ft indicates the ductility increment in the t-th loadincrement. Note that in the damage model the stress r is anexplicit function of the total strain e even if un-/re-loadingsituations occur. Thus an incremental iterative procedure for theupdate of the stress r, mandatory for example in plasticity, is notnecessary.
This allows to use directly the derivative of the total strain rseor displacement rsu at position nþ1 in the sensitivity analysis,i.e. both derivatives do not have to be accumulated over loadincrements. This also avoids that errors of the sensitivities areaccumulated during load incrementation.
νκ
beam.
ARTICLE IN PRESS
0
0.125
0.250
0
0
4
8
12
0
0.005
0.2 0.4
Fig 8. Results of optimization for (a) a linear elastic and (b) a materially nonlinear response.
J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415 411
As said above this approach is restricted to material models inwhich the stress r is a function of the total strain e. Details of thisdirect sensitivity approach using a total strain or displacement isdescribed in Bugeda et al. [7].
Fig 9. Hanging deep beam.
8. Numerical examples
8.1. Optimization for a deep beam
In the first numerical simulation a FRC beam reinforced withfour carbon fibers is investigated as displayed in Fig. 7 where alsothe material properties are given. For the properties of theinterface it is referred to Kruger et al. [18–20], Xu et al. [37].Plane-stress conditions are assumed. Due to symmetry only onehalf of the system is analyzed, the FE mesh is given in Fig. 7(c).The beam thickness is assumed to be only 1 mm, since no out-of-plane actions are considered. The 200 elements are used forconcrete and 68 elements for the interface, respectively.
The number and location of slip nodes may change duringoptimization depending on the actual fiber geometries. Thus,mesh adaptation for slip nodes is carried out after each structuralanalysis. The fiber geometry is approximated by a symmetricbiquadratic Bezier-spline, see Fig. 7(b). As shown in Fig. 7(a) theorigin O of the parametric element is defined at the edge ofthe beam. Due to symmetry the number of design variables isreduced, i.e. the location of the control points p3 and p4 is coupledto p1 and p0, respectively. Further simplification of the fibergeometry can reduce the number of design variables. Firstly, they-coordinate of p1 is set equal to that of p2. Secondly, thex-coordinate of p1 is placed always at the center betweenthe x-coordinates of p0 and p2. Thus the number of designvariables for a single fiber is three, i.e. s1, s2, and s3, see Fig. 7(b).The initial set of the design variables is (i) s1 ¼ 0:075 (i.e. thex-coordinate of p0 is 0:075� 400 mm) for all fibers and (ii) s2 ands3 are assumed to be 0.15, 0.38, 0.62, and 0.85 for each fiber.
The total number of design variables is 12 (3� 4 fibers). Takinginto account that thick concrete covers for textile fibers areobsolete, we adopt the lower bound sL ¼ 0:01 and the upper onesU ¼ 0:99 for s2 and s3 of all fibers. For the design variable s1,the lower and upper bounds are set to sL ¼ 0:01 and sU ¼ 0:4,respectively. The analysis is carried out with a displacementcontrolled method; the control point c is at the lower center of thebeam. For comparison the structure is optimized based on either alinear elastic or the damage model. The prescribed nodaldisplacement u (y-direction) at the control point is either 0.005or 0.4 mm. The fiber volume is kept constant (1.4%) during theoptimization leading to a fiber thickness tf ¼ 0:4 mm.
Firstly we optimize the ductility of the structure for a linearelastic response, which means maximizing the overall stiffnessof the structure. Fig. 8(a) shows the optimized fiber layout.The figure on the right side of Fig. 8(a) introduces the stressdistribution of fibers. After optimization the two fibers are shiftedto the upper part in compression and the two others to the lower
ARTICLE IN PRESS
0
1
2
00
4
8
12
0 0.02 0.04 0.060.005
Fig 10. Results of optimization for (a) a linear elastic and (b) a materially nonlinear response.
Fig 11. Problem description of the third example and parametric element.
0
5
10
15
20
25
0 0.1 0.2
Fig 12. Results of optimization for (a) a linear elastic and (b) a materially nonlinear response.
J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415412
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J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415 413
one carrying the tension force, respectively. The two upper fiberswind up with almost the same location. As a result an increase of14% of ductility could be obtained.
Fig. 8(b) shows the optimized fiber layout in the materiallynonlinear situation applying the damage model. Also the damagedistribution of concrete is displayed; fibers are not yet damaged atthis stage. After optimization one fiber is shifted to the upper partand the three others to the lower one. These three fibers preventthe concrete from a premature damage propagation. Compared tothe elastic case the fibers are more curved which is structurallyreasonable. As a result an increase of 44% of ductility could beobtained.
One could expect that also the fourth fiber moves to the lowerpart. This suggests that the achieved solution does not representthe global minimum, a consequence of the underlying nonconvexoptimization problem.
8.2. Optimization for a hanging deep beam
As second numerical example a hanging deep beam is chosenas displayed in Fig. 9. The material properties of concrete, carbonfibers and interface as well as loading condition and mesh followthe previous example. Due to symmetry only one half of thesystem is analyzed under plane-stress conditions. Again asymmetric biquadratic Bezier-spline is adopted to approximateeach fiber geometry. 180 elements are used for concrete and 50elements for the interface.
The initial set of design variables for all three fibers inthe parametric space is (i) s1 ¼ 0:025 and (ii) s2 and s3 equal 0.25,0.5 or 0.75. The fiber volume is kept constant (1.1%) during theoptimization.
Fig. 10(a) and (b) show the optimized fiber layouts based on alinear elastic and a materially nonlinear response, respectively.The prescribed displacement u at the control point c under theload is either 0.005 or 0.06 mm. For the linear elastic casethe upper straight fiber reduces the compressive deformation andthe middle curved fiber reflects the cable effect between the fixedsupports. For the damage case, the upper and middle fibers areutilized to resist the damage propagation of concrete in thevicinity of the supports. As a result an increase of 13% of ductilitywas obtained.
8.3. Optimization for a splitting plate
The third example is a splitting plate shown in Fig. 11. Againthe same material properties, loading condition, mesh and initialassumption of fiber geometry are used as in the previous twoexamples. The 124 elements are used for concrete and 24elements for the interface. For the present example theparametric element is restricted to the area below the cutoutsection, see Fig. 11; this means that fibers cannot be located in thenon-design space.
The initial set of the design variables is: (i) s1 ¼ 0:025 and (ii) s2
and s3 are 0.25, 0.50 or 0.75. The fiber volume is kept constant(0.74%) during the optimization.
Fig. 12(a) and (b) show the optimized fiber layouts based on alinear elastic and a damage response, respectively. The prescribeddisplacement at control point c is either 0.002 or 0.2 mm. For thelinear elastic case, the upper straight fiber controls the tensiledeformation around the reentrant corners. The middle and lowerfibers reduce the compressive deformation. As a result an increaseof 5% of ductility was obtained.
In case of damage the middle fiber is also shifted to theupper part to resist the damage propagation of concrete togetherwith the upper fiber. These two fibers are damaged at the
prescribed displacement. The location of the lower fiber stays inthe lower part of the plate although we could expect that it alsomoves as well to the cutout area (see comment on local minimumfor first example). Anyhow an increase of 99% of ductility could beobtained.
9. Conclusions
Shape optimization is applied to fiber geometry for textile fiberreinforced composites. The two brittle materials, namely concretematrix and fibers, get the necessary ductility from the interfacebehavior of the two constituents. It could be shown that the mainpurpose of the present study, namely to increase the structuralductility of FRC with respect to the geometrical layout ofcontinuous fibers, was successful. For this objective, it is of coursenot sufficient to base the optimization process on a linear materialmodel; thus it is mandatory to consider material nonlinearities inthe optimization process. As an example, we applied a damagemodel to both constituents together with a nonlinear interfacemodel. The formulation could be extended to other materialmodels and other composites. It can also be enhanced by materialoptimization varying the fiber content in a design element as itwas introduced in Kato et al. [16].
For the sensitivity analysis a variational semi-analytical direct
method was applied. The strategy of sensitivity analysis using a‘total’ strain or displacement was discussed. This approach doesnot accumulate errors as it is the case when an ‘incremental’sensitivity approach is applied, which is for example necessary forplasticity (e.g. Lipka et al. [21], Maute et al. [23]).
In semi-analytical methods errors of sensitivities tend toincrease when distinct ‘rigid body rotations’ appear, see forexample Olhoff and Rasmussen [26], Olhoff et al. [27], Cheng andOlhoff [9], Mlejnek [25], Boer and Keulen [4,5], Keulen and Boer[17], and Bletzinger et al. [3]. This tendency was also observed inthe present study when the structural response reaches thepostpeak range, where several elements are severely damagedand other elements may already be in the unloading phase,similar to a ‘plastic hinge deformation’. The unloaded elementseventually will encounter rigid body rotations which in turnmay lead to inaccurate sensitivities. However in the presentstudy the structural damage is not driven so far into completefailure.
Acknowledgments
The present study is supported by Grants of the ‘‘DeutscheForschungsgemeinschaft’’ DFG (German Research Foundation)within the Research Projects Ra 218/19 and Ra 218/21. Thissupport is gratefully acknowledged.
Appendix A. Transformation matrices
The transformation matrices introduced in the text are listedhere. First the usual rotation matrix Td is given
Td¼
cosðxG; xLÞ cosðyG; xLÞ
cosðxG; yLÞ cosðyG; yLÞ
" #¼
l1 m1
l2 m2
" #; ð72Þ
where lk and mk ðk¼ 1;2Þ are introduced as abbreviations.The strain transformation matrix Te transforms the ‘vector’ eG
with the components of the global strain tensor to the local one eL
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so that
eL ¼
e11
e22
2e12
264
375¼ TeeG: ð73Þ
Incidentally, the stress transformation matrix Ts which trans-forms the global stress ‘vector’ rG to the local one rL follows as
rL ¼
s11
s22
s12
264
375¼ TsrG; ð74Þ
where
Te ¼
l21 m2
1 l1m1
l22 m2
2 l2m2
2l1l2 2m1m2 l1m2þ l2m1
2664
3775 and
Ts ¼
l21 m2
1 2l1m1
l22 m2
2 2l2m2
l1l2 m1m2 l1m2þ l2m1
2664
3775 ð75Þ
The stress transformation matrix Ts can be replaced by therelationship ðTsÞ�1
¼ ðTeÞT.
Appendix B. Linearization of gradient enhanced damagemodel for concrete
The derivatives of Eqs. (27) and (28) with respect to nodaldisplacements d and nodal non-local strains e lead to thefollowing four stiffness matrices for the gradient enhanceddamage model for concrete
Kcdd ¼
@fcint;u
@d¼[nc
ele
e ¼ 1
ZOc
BcTCc
edBcdOc; ð76Þ
Kcde ¼
@fcint;u
@e¼[nc
ele
e ¼ 1
ZOc
BcTE
c ~NcdOc; ð77Þ
Kced ¼
@fcint;e
@d¼�
[ncele
e ¼ 1
ZOcð ~N
cÞTF
cBcdOc; ð78Þ
Kcee ¼
@fcint;e
@e¼[nc
ele
e ¼ 1
ZOc½cð ~B
cÞT ~B
cþð ~N
cÞT ~N
c�dOc; ð79Þ
with
Ec�@rc
@~ecv
¼@rc
@Dc
@Dc
@kc
@kc
@~ecv
; ð80Þ
Fc�@ec
v
@ec¼@ec
v
@I1
@I1
@ecþ@ec
v
@J2
@J2
@ec; ð81Þ
and
@I1
@ec¼@ec
ii
@eckl
¼ dikdil ¼ dkl;@J2
@ec¼
1
6
@ecaaec
bb
@eckl
�1
2
@ecabe
cab
@eckl
¼1
3ec
aadkl�eckl;
ð82Þ
where d is the Kronecker symbol. Cced is the matrix of the ‘secant’
material tensor for isotropic elasto-damage
Cced ¼
@rc
@ec¼ ð1�Dc
ÞCcel with rc ¼ ð1�Dc
ÞCcele
c; ð83Þ
where Ccel denotes the matrix of the elastic material tensor, and D
is the damage parameter. The second term on the right handside of Eq. (80) is zero unless damage is initiated in an element.The third term is equal to either unity for loading or zero for
un/reloading. Thus Ec
is a term which controls damage and theloading condition simultaneously. The same situation also holdsfor the damage formulation of fibers.
In the above linearization procedure the following tworelations are derived from Eqs. (80), (22) and from Eq. (81)
@rc
@e¼@rc
@~ecv
@~ecv
@e¼ E
c ~Nc;
@ecvðeÞ
@d¼@ec
v
@ec
@ec
@d¼ F
cBc; ð84Þ
Appendix C. Linearization of gradient enhanced damagemodel for fiber
The derivatives of Eqs. (27) and (28) with respect to nodaldisplacements d and nodal non-local strains e lead to fourstiffness matrices for the gradient enhanced damage model forfiber
Kfdd ¼
@ffint;u
@d¼[nf
ele
e ¼ 1
ZOf
Bf TðTe1Þ
TCfed;LTe1Bf dOf
¼[nf
ele
e ¼ 1
ZOf
Bf TCf
ed;GBf dOf ; ð85Þ
Kfde ¼
@ffint;u
@e¼[nf
ele
e ¼ 1
ZOf
Bf TðTe1Þ
TEf
L~N
fdOf ; ð86Þ
Kfed ¼
@ffint;e
@d¼�
[nfele
e ¼ 1
ZOfð ~N
fÞTF
f
LTe1Bf dOf ; ð87Þ
Kfee ¼
@ffint;e
@e¼[nf
ele
e ¼ 1
ZOf½cð ~B
fÞTðTd
1ÞTTd
1~B
fþð ~N
fÞT ~N
f�dOf
¼[nf
ele
e ¼ 1
ZOf½cð ~B
fÞT ~B
fþð ~N
fÞT ~N
f�dOf ; ð88Þ
with
Ef
L �@sf
L
@~efv;L
¼@sf
L
@Df
@Df
@kf
@kf
@~efv;L
; ð89Þ
Ff
L �@ef
v;L
@efL
: ð90Þ
In the linearization procedure the following two relations arederived from Eq. (26) and from Eqs. (89) and (22),
@sfL
@d¼@sf
L
@~efL
@~efL
@d¼Cf
ed;LTe1Bf ;@sf
L
@e¼
@sfL
@efv;L
@efv;L
@e¼ E
f
LNf ; ð91Þ
The two remaining relations are derived from Eqs. (90), (26)and from Eq. (31), respectively,
@efv;Lðef
LÞ
@d¼@ef
v;L
@efL
@efL
@d¼ F
f
LTe1Bf ;@tf
L
@e¼@ðcr ~ef
v;LÞ
@e¼ cTd
1~B
f: ð92Þ
Appendix D. Linearization of interface element
The derivatives of Eq. (29) with respect to nodal displacementsd and nodal slip parameters d lead to the stiffness matrices for
ARTICLE IN PRESS
J. Kato, E. Ramm / Finite Elements in Analysis and Design 46 (2010) 401–415 415
the interface
Kfdd¼@ff
int;u
@d¼[nf
ele
e ¼ 1
ZOf
Bf TðTe1Þ
TCfed;LBdOf ; ð93Þ
Kfdd¼@fi
int;i
@d¼[nf
ele
e ¼ 1
ZOf
BTCf
ed;LTe1Bf dOf ; ð94Þ
Kidd¼@fi
int;i
@d¼[ni
ele
e ¼ 1
ZOf
BTCf
ed;LBdOfþ
ZOi
NTkLNdOi
� �; ð95Þ
where the following two relations are derived from Eq. (26) andfrom Eq. (23), respectively,
@sfL
@d¼@sf
L
@efL
@efL
@d¼Cf
ed;LB;@si
L
@d¼@si
L
@uiL
@uiL
@d¼ kLN: ð96Þ
Stiffness matrix Kfdd has already been introduced in Eq. (85).
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