Lifetime-Oriented Structural Design Concepts- P12 potx

However, the concept of plastic effective stress first introduced mate-at a macroscopic level by [210] for smate-aturmate-ated porous media see [211] for tails, allows to overcome these di

the drained thermo-mechanical coupling tensor

A=A u − 3α t,u M B=Ced:1α t , (3.149)and the drained tensor

respectively [321]

3.3.2.1.3 Identification of Coupling Coefficients

According to [321] the poroelastic hygro-mechanical coefficients b and M can

be determined by relating differential stress and differential strain quantitiesdefined on the meso-level to respective homogenised quantities on the macro-level The so-obtained tangential Biot coefficient is determined as

which includes the expression b = S l suggested by [211] for the special case

of poroelastic materials with incompressible matrix behaviour An expression

for the Biot modulus M = ψM is obtained as

The coefficients related to damage phenomena Λ and Ξ are identified by

exploiting the symmetry relations that are connected to the existence of amacroscopic potential Using the Maxwell symmetries, the drained tensor

3.3.2.1.4 Effective Stresses

The concept of effective stress [281, 791] is a generally accepted approach insoil mechanics for the determination of stresses in the skeleton of fully satu-rated soils In addition to the original proposal of [791], several alternative sug-gestions for the definition of effective stresses exist, taking the compressibility

of the matrix material or the porosity into account (see e.g [123, 587, 128]).Based on the relevance of the concept of effective stress for the analysis offully saturated soils, this concept has also been adapted for the description ofpartially saturated soils Early formulations introduced the capillary pressure

in the (elastic) effective stress definition [127] However, difficulties to obtainsatisfactory agreements with experimental results have motivated the use oftwo independent stress fields for the constitutive modelling of unsaturatedsoils (see e.g [129, 44])

As far as the numerical modelling of partially saturated cement-based rials is concerned, the assumption of (elastic) effective stresses seems not to bewell suited for the description of shrinkage-induced cracks using stress-basedcrack-models However, the concept of plastic effective stress first introduced

mate-at a macroscopic level by [210] for smate-aturmate-ated porous media (see [211] for tails), allows to overcome these difficulties in the framework of poroplasticity– porodamage models The proposed form of the plastic effective stress is thesame as the classical Biot-type, however, a plastic effective stress coefficient

de-is used A similar form has been derived from micromechanical tions by [510] This concept has been recently extended to partially saturatedmaterials [167, 533], and is also adopted in the present formulation Fromthe coupled relations between total stresses, strains, liquid saturation andtemperature

characterises the thermodynamic force associated with the plastic strain rate[211] In contrast to the elastic effective stress tensor,σ represents the macro-

scopic counterpart to matrix-related micro-stresses with the coefficient b p as

the plastic counterpart of the Biot coefficient b By relating stress quantities

on the meso-scale to respective macroscopic quantities, a possible

identifi-cation of b p as a function of the integrity ψ, the porosity φ and the liquid saturation S l can be accomplished as

see [321] for details

3.3.2.1.5 Multisurface Damage-Plasticity Model for Partially Saturated

Concrete

According to the concept of multisurface damage-plasticity theory, nisms characterised by the degradation of stiffness and inelastic deformationsare controlled by four threshold functions defining a region of admissible stressstates in the space of plastic effective stressesσ 

asso-R) and k = 4 represents an active

hardening/softening mechanism in compression associated with the loading

In (3.163), the subscript A refers to one of the three principal directions and

qR(αR) =−∂U/∂αR denotes the softening parameter

The ductile behaviour of concrete subjected to compressive loading is scribed by a hardening/softening Drucker-Prager plasticity model

de-fDP(σ  , q

DP) =

J2− κDPI1− qDP(αDP)

with qDP(αDP) =−∂U/∂αDPas the hardening/softening parameter The

de-termination of the model parameters κDP and βDP is based on the ratio of

the biaxial and the uniaxial compressive strength of concrete f cb /f cu as [534]

κDP= √1

3



f cb /f cu − 1 2f cb /f cu − 1

whereby f cb /f cu is approximately equal to 1.16 The fracture energy concept

is employed to ensure mesh-objective results in the post-peak regime Details

of the material model are found in [534] For an efficient implementation ofthe multisurface model based on an algorithmic formulation in the principalstress space reference is made to [531]

The evolution equations of the tensor of plastic strains ˙ε p, of the reciprocal

value of the integrity (ψ −1)˙, of the plastic porosity occupied by the liquid

phase ˙φ p l and of the internal variables ˙αR and ˙αDP are obtained from thepostulate of stationarity of the dissipation functional [318] as

The parameter 0≤ β ≤ 1 contained in (3.167) and (3.168) allows a simple

partitioning of effects associated with inelastic deformations due to the induced misalignment of the asperities of the crack surfaces, resulting in anincrease of inelastic strainsε p, and deterioration of the microstructure, result-

crack-ing in a decrease of the integrity ψ An elastoplastic model ((ψ −1 )˙ = 0, ˙ ε p =0)

and a damage model ((ψ −1)˙ = 0, ˙ ε p = 0) are recovered as special cases by

setting β = 0 and β = 1, respectively.

3.3.2.1.6 Long-Term Creep

Consideration of long-term or flow creep effects is accomplished in the work of the microprestress-solidification theory [93] The evolution law of theflow strains is based on a linear relation between the rate ˙ε f and the stresstensorσas

η f (S f)= cpS

p−1

where c and p > 1 are positive constants According to [93], the microprestress

relaxation is connected to changes of the disjoining pressure Consequently,

variations of the internal pore humidity h due to drying, which entail a ing disjoining pressure, lead to a change of the microprestress S f This mech-anism partially explains the Pickett effect [631], also called drying creep

chang-3.3.2.1.7 Moisture and Heat Transport

Starting with a simplified nonlinear diffusion approach, in which the differentmoisture transport mechanisms in liquid and in vapour form are represented

by means of a single macroscopic moistudependent diffusivity [94], the lation between the moisture fluxq l and the spatial gradient of the capillarypressure∇ p c is given by

re-q l= k

In (3.174), k denotes the intrinsic liquid permeability tensor and μ l is theviscosity of water According to the hypothesis of dissipation decoupling [212],possible couplings between heat and moisture transport are disregarded in thepresent formulation In order to account for the dependence of the moisturetransport properties on the nonlinear material behaviour of concrete, k isadditively decomposed into two portions as

k = k r (S l ) [k t (T ) k φ (φ) k0+k d (αR)] , (3.175)one related to the moisture flow through the partially saturated pore spaceand one related to the flow within a crack, respectively [758] This approach

is consistent with the smeared crack concept In (3.175),k0 denotes the

ini-tial isothermal permeability tensor, k r is the relative permeability, k t counts for the dependence of the isothermal moisture transport properties

ac-on the temperature and k φdescribes the relationship between the ity and the porosity Furthermore,k dis the permeability tensor relating planePoiseuilleflow through discrete fracture zones to the degree of damage inthe continuum model, see [533, 319] for details

permeabil-Using again the hypothesis of dissipation decoupling, the relation betweenthe heat fluxq tand the gradient of the temperature∇ T can be described by

a linear heat conduction law reading

whereby D t (T , S l , φ) denotes the effective thermal conductivity.

3.3.2.1.7.1 Freeze Thaw

Authored by Max J Setzer and Jens Kruschwitz

The main reason for frost damage in porous materials is the expansion by

9 Vol.-% in the transition from water to ice, if a critical degree of saturation

in the pores is exceeded This artificial saturation, e.g observed by Auberg &Setzer [69], is as well a multi scaling as a coupled phenomenon The scalingproblem is characterised by the existence of two scales, which should be sepa-rated when modelling frost processes in hardened cement paste Most relevantfor the distinction between these scales are of course the macroscopic temper-ature changes and their typical time constants compared to the time necessary

to obtain equilibrium within a certain scale On the macroscopic scale sient conditions have to be modeled, i.e mass transport due to viscous fluidflow is slow On this scale the model deals with bigger volumes than on themicroscale In the big macroscopic volumes thermodynamic processes need alarge time span to obtain equilibrium This can be observed in practise as well

tran-as in standard experiments The second part of the theory in this contribution

is restricted to the nanoscopic CSH gel system consisting of solid CSH, porewater and air filled gel-pores with adsorbed water films The liquid water film

is an essential part of the Setzers model [726], which was determined by [812]experimentally By going down in length scales it adopts primarily surfacethermodynamics and the theory of disjoining pressure At least thermal orthermodynamic equilibrium is established under normal conditions This can

be assumed for cubes of length up to 120 μm [731] At constant temperature,

the non-freezing interlayers and films are in equilibrium with ice and vapour.The temperature of the bulk ice governs the pressure and by this the equilib-rium Experiments have shown that the ice freezes in situ, referring to [778].That means on the submicroscopic scale the motion of the pore water to theice is highly dynamic However, the response time for movement from gel toice and the flow distance is rather small Nevertheless, the pressure gradient

is extremely high

By a combination of the Theory of porous Media (TPM), mainly influenced

by de Boer [135], Ehlers [252], Bluhm [130], etc., and a micromechanical ory of surface forces developed by Setzer [723] the artificial saturation phe-nomenon can be described [448] Basis of this model is the work of Kruschwitz

the-& Setzer [450] and Kruschwitz the-& Bluhm [449] respectively Last describe thefrost heave of a critical filled cementitious matrix In the mentioned com-bination the macroscopic, thermodynamic aspects of the model base on theTheory of Porous Media This theory is a combination of the mixture theoryand the concept of volume fractions The interactions of the nanostructure

of the hardened cement paste are modelled by a smeared micromechanicalmodel This part of the model is characterised by the properties of the twophase system solid and pore liquid The transport on the micro structure andthe unfrozen, adsorbed water film between matrix and ice are included

3.3.2.2 Chemo-Mechanical Modelling of Cementitious Materials

It has been shown in Subsections 3.1.2.3, 3.1.2.3.2, 3.1.2.3.3, 3.1.2.2.2 thatthe main microstructural mechanisms of environamentally induced corrosionand deterioration processes are by now fairly well understood There seems toexist, however, a gap between research focused on the material level and dura-bility oriented computational analysis of concrete structures Although consid-erable progress has been achieved in the modeling of the mechanical behavior

of concrete subjected to various loading conditions (see Subsection 3.1.1.1),environmental influences affecting the durability of concrete structures arestilc l accounted for by more or less heuristic evaluations of the degradationprocess and its influence on the residual structural safety Recent progress

in computational durability mechanics (see e.g [75, 211, 800, 798, 214]), gether with appropriate numerical discretization methods in space and time[460, 453] (see also Chapter 4) open the perspective of a more fundamentalapproach to obtain not only estimates for the life-time, but also to provideinsight into the degradation mechanisms as a result of the interaction betweenmechanical and environmental loading

to-Using a continuum mechanics-based mode of description, concrete jected to mechanical and non-mechanical loading is generally described as amulti-phase material whose behaviour is influenced by the interaction of thesolid skeleton containing the cementititious matrix and the aggregates and theliquid and gaseous pore fluids To this end, the scale of observation may eithertake the micro-scale or macro-scale as a point of departure In the framework

sub-of a micro-scale approach the individual constituents are described by means

of classical continuum mechanics for one-phase materials, formulating priately the interactions between the constituents and the contact conditions,respectively To this end, the exact knowledge of the morphology of the ma-terial, in particular of the geometry of the pore space, is required This is,however, not available in general This difficulty motivates the description

appro-of porous materials on the basis appro-of a macroscopic approach The Theory appro-ofMixtures (see e.g [254] for more details) has been established as a suitablehomogenisation procedure, which allows to treat multi-phase materials as acontinuum while each constituent may be described by its own kinematics andbalance equations The interactions between the constituents are included byproduction terms within the balance equations

Since the Theory of Mixtures contains no microscopic information of themixture it need to be complemented by the concept of volume This leads tothe well established concept of the Theory of Porous Media (TPM) It defines

the volume fraction of each constituent dv α and the volume of the mixture

dv, which provides a representation of the local microscopic composition of multi-phase materials: φ α = dv α /dv The sum of the volume fractions of all

constituents has to be equal to one

α φ α= 1 The TPM provides a generalcontinuum mechanically and thermo dynamically established concept for themacroscopic description of multi-phase materials like concrete

virgin material → mech damage mech damage ← virgin material

Fig 3.143 Chemo-mechanical damage of porous materials within the Theory of

Mixtures Three types of deterioration are illustrated: virgin material, mechanicallydamaged material, chemically damaged material and chemo-mechanically damagedmaterial

3.3.2.2.1 Models for Ion Transport and Dissolution Processes

Authored by Detlef Kuhl and G¨ unther Meschke

3.3.2.2.1.1 Introductory Remarks

Based on insights and data obtained from experimental investigations oncalcium dissolution and coupled chemo-mechanical damage processes (seeSubesection 3.1.2.3.2) constitutive models formulated on a macroscopic level

of observation have been developed for the analysis of the time dependentdissolution process of concrete and concrete structures One class of mod-els is based on a phenomenological chemical equilibrium model relating the

calcium concentration of the skeleton and the pore solution s(c) in

conjunc-tion with the concept of isotropic damage mechanics [422], as proposed byG´erard[307] and subsequent publications (G´erard[308], G´erard et al.[311], Pijaudier-Cabot et al [635, 634, 636] and Le Bell´ego et al.[477, 479, 478])

Ulm et al.[801] and Ulm et al [799] have proposed a chemo-plasticitymodel formulated within the Biot-Coussy-Theory of porous media [211].This model is also based on a chemical equilibrium model, using empiricalrelations for the conductivity and aging In both models, the irreversible char-acter of skeleton dissolution is not accounted for Hence, chemical unloading

or cyclic chemical loading processes cannot be described

From the experiments the key-role of the porosity for the changing rial and transport properties of chemo-mechanically loaded cementitious ma-terials becomes obvious Based on this observation and in order to consider

mate-the interaction phenomena of chemical and mechanical material degradationdescribed in Subsection 3.1.2.3.2 a fully coupled chemo-mechanical damagemodel has been developed in [454, 455] within the framework of the Theory ofPorous Media The material is described as ideal mixture of the fully saturatedpore space and the matrix In this model, the pore fluid acts as a transportmedium for calcium ions The pore pressure, however, is not accounted for inthe present version of the model.

The changing mechanical and transport properties are related to the tal porosity defined as the sum of the initial porosity, the chemically in-duced porosity and the apparent mechanical porosity Together with theassumptions of chemical and mechanical potentials the need for further as-sumptions or empirical models is circumvented Micro-cracks are interpretedaccording to Kachanov [422] as equivalent pores affecting, on a macroscopiclevel, the conductivity and stiffness but not the mass balance The evolution

to-of the mechanically and chemically induced porosities are both controlled

by internal parameters This enables the modeling of cyclic loading tions and allows a consistent thermodynamic formulation of the coupled fieldproblems [454]

condi-The link between the mechanical and the chemical field equations is

ac-complished by the definition of the total porosity φ as the sum of the initial porosity φ0, the porosity due to matrix dissolution φ c and the apparent me-

chanical porosity φ m:

The chemically induced porosity φ c can be calculated by multiplying the

amount of dissolved calcium of the skeleton s0− s by the averaged molar

volume of the skeleton constituentsM/ρ

φ c= M

where s0 denotes the initial skeleton concentration The apparent

mechan-ically induced porosity φ m considers the influence of mechanically inducedmicro pores and micro cracks on the macroscopic material properties of theporous material It is obtained by multiplying the scalar damage parameter

d m by the current volume fraction of the skeleton 1− φ0− φ c

This definition of the mechanical porosity φ m takes into account that cracking is restricted to the solid matrix material

micro-3.3.2.2.1.2 Initial Boundary Value Problem

The coupled system of calcium diffusion-dissolution, mechanical

deforma-tion and damage is characterized by the concentradeforma-tion field c of calcium ions

in the pore solution and the displacement field u as external variables and

a set of internal variables concerning the irreversible material behavior Themacroscopic balance of linear momentum is given by:

The matrix dissolution-diffusion problem is governed by the macroscopic ance of the calcium ion mass in the representative elementary volumedivqc + [[φ0+ φ c ] c ] · + ˙s = 0 , (3.181)

bal-wherebyqc is the mass flux of the solute The term [[φ0+ φ c ] c ] ·accounts for

the change of the calcium mass due to the temporal change of the porosityand the concentration, which is up to one dimension smaller compared to the

calcium mass production resulting from the dissolution of the skeleton ˙s [452].

The system of differential equations (3.180)-(3.181) is completed by

bound-ary conditions on the boundbound-ary Γ given by

Herein,εdenotes the linearized strain tensor andCsis the fourth order

elas-ticity tensor of the the skeleton The derivative of the free energy function Ψ m

with respect to the strain tensorεyields the stress tensorσ:

whereγ=−∇c is the negative gradient of the concentration field The tive of the dissipation potential Ψ cwith respect to the negative concentrationgradientγ results in the calcium ion mass flux vectorqc

deriva-qc= ∂Ψ c

of the pore fluid, which is discussed in the next subsection In eqs (3.186) and(3.187) D0 denotes the second order conductivity tensor of the pore fluid.Consequently, the macroscopic conductivity of the porous material is given

byD = φ D0 φ = φ0 defines the chemical and mechanical sound macroscopic

material (φ c = φ m = 0), characterized by the subscript s Hence, the

macro-scopic conductivity of the virgin material is given byD s = φ0D0 In contrast

to existing reaction-diffusion models describing calcium leaching, the dence ofD0on the square root of the calcium concentration within the porefluid is considered This dependency follows from Kohlrausch’s law, describ-ing the molar conductivity of strong electrolytes (see Atkins [66] and Section3.3.2.2.1.4), using Nernst-Einstein’s relation In the isotropic case, the con-ductivity tensorD0 is given in terms of the second order identity tensor 1,

depen-the calcium ion conductivity for depen-the infinitely diluted solution D00≥ 0 and the constant D 0c ≥0.

3.3.2.2.1.4 Migration of Calcium Ions in Water and Electrolyte Solutions The molar conductivity Λ of a strongly electrolyte solution is given as function of the calcium ion concentration in the pore fluid c by the empirical

Kohlrauschlaw, see Kohlrausch [436]

to-charges of a cation Ca2+, F = 9.64853 ·104C/mol is the Faraday constant,

Λ0 = 11.9 · 10 −3 Sm2/mol is the molar conductivity at infinite dilution and

Λ c is the Kohlrausch constant of the molar conductivity Based upon themodel of ionic clouds, the Debye-H¨uckel-Onsager theory (Debye &H¨uckel[230] and Onsager [602]) verifies Kohlrausch’s law and allows to

determine the Kohlrausch constant Λ c,

fluid conductivity D0 macroscopic conductivity φD0

c

φD0

02

68

κ c

c0

0.0 0.2 0.8 1.0

Fig 3.144 Conductivity of the pore fluidD0 [10−10m2/s] as function of the

cal-cium concentration c [mol/m3] and the total temperature T [K] Macroscopic ductivity of non-reactive porous media φD0[10−10m2/s] as function of the calcium

con-concentration c [mol/m3] and the porosity φ [ −] with φ=φ(κ c , d m= 0)

 R T

(3.190)

where the constants A and B account for electrophoretic and

relax-ation effects associated with the ion-ion interactions These constants aregiven in terms of the universal gas constant, the total temperature, the

elementary charge e = 1.602177 ·10 −19 C, the constant q = 0.586, the electric

permittivity  = 6.954 ·10 −10 C2/J m and the viscosity η = 0.891 ·10 −3 kg/ms of

water (see e.g Atkins [66]) From comparing equations (3.188) and (3.189)

the macroscopic diffusion constants D00 and D 0c can be determined

deter-history variables κ c and κ m and for any concentration c.

Figure 3.144 contains plots of the conductivity D0 in the pore fluid and

the macroscopic conductivity φD0 vs the calcium concentration within the

range c = 0 corresponding to pure water and c = c0 corresponding to theconcentrated pore solution of the virgin material The diagrams on the lefthand side of Figure 3.144 underline the relevance of using higher order iontransport models, considering electrophoretic and relaxation effects, as a ba-sis for realistic calcium leaching models Standard and higher order transport

models are characterized by D 0c = 0 and D 0c =0, respectively A pronounced change of the conductivity D0 proportional to the square root of the con-

centration c can be observed within the considered concentration range The ratio of the conductivities related to the fully degraded material D0(0) and

the virgin material D0(c0) is approximately 9 : 4 The average value of the

conductivity D0 is approximately D0≈ 4·10 −10 m/s2 This is in accordancewith the suggestion by Delagrave et al [232], that in numerical analyses

D0/2 should be used as the macroscopic conductivity in order to fit

exper-imental results Standard transport models are not capable to capture thesignificant increase of the conductivity with a decreasing calcium ion con-centration corresponding to propagating chemical damage in reactive porousmedia As expected, the results of the standard model and the present model

are identical in the case of infinitely diluted solutions (c = 0) The

sensitiv-ity of the ion transport with regards to temperature changes is studied by

including plots of the conductivity for T = 273 K, corresponding to the

freez-ing point of water (no calcium ion transport occurs below this temperature),

and for T = 323 K, representing approximately a desert climate, in Figure

3.144 According to equations (3.188), (3.191) and (3.192), the conductivity

D0 depends linearly on the total temperature T Within the considered perature interval D0is only changed by approximately 16% Compared to theinfluence of the concentration the influence of the temperature plays a minorrole in the transport process of ions within the pore water of cementitiousmaterials

tem-On the right hand side of Figure 3.144, the macroscopic conductivity is

plotted for various values of the threshold calcium concentration κ c and thecorresponding values of the porosity, respectively, assuming a non-reactiveporous material

3.3.2.2.1.5 Evolution Laws

According to Simo & Ju [744] the evolution of the damage parameter

d m (κ m) is described by the damage criterion

where η and κ m are the equivalent strain function and the internal able defining the current damage threshold From the Kuhn-Tucker load-ing/unloading conditions and the consistency condition

vari-Φ m ≤ 0 , ˙κ m ≥ 0 , Φ m ˙κ m = 0 , Φ˙m ˙κ m = 0 , (3.194)

follows, that κ m is unchanged for Φ m < 0 and calculated as κ m = η otherwise.

The description of the elasto-damage material model is completed by the

definition of the equivalent strain η and the damage function d m Here theequivalent strain measure proposed by de Vree et al [814] is used

in which I1 = tr[ε ], I2 = [tr2[ε]− ε : ε ]/2 and J2 = [ε dev : ε dev ]/2 are the

first and the second invariant of the strain tensorεand the second invariant

of the strain deviatorε dev , respectively The parameter k s denotes the ratio

of tensile to compressive strength and ν sthe Poisson’s ratio of the skeleton.The exponential damage function is given by

characterized by the chemical porosity φ c (s) Starting from a chemical

equi-librium state between the calcium solved in the pore fluid and the calciumbound in the skeleton, the dissolution process requires a decreasing concen-

tration c in the pore fluid Otherwise, if c is increased, the structure of the

skeleton is unchanged In order to describe chemically induced degradation

similarly to the elasto-damage problem, an internal variable κ c is introduced,which corresponds to the current chemical equilibrium state Based on this

internal variable κ c , the chemical reaction criterion Φ c is formulated as

According to the Kuhn-Tucker conditions and the consistency condition

Φ c ≤ 0 , ˙κ c ≤ 0 , Φ c ˙κ c = 0 , Φ˙c ˙κ c = 0 , (3.198)the process of matrix dissolution is associated with a decreasing chemical

equilibrium calcium concentration ( ˙κ c ≤ 0) The dissolution threshold κ c is

unchanged for Φ c < 0 and equal to the current calcium concentration of the pore fluid (κ c = c) otherwise The conditions (3.197) and (3.198) for the

occurence of chemical reactions are identical to those given by Mainguy &Coussy[512] This identity is shown in Kuhl et al [454]

As already mentioned, the current state of the calcium concentration in

the skeleton s is controlled by the spontaneous calcium dissolution It can

be described as a function of the chemical equilibrium threshold κ c given byG´erard[307, 308] and Delagrave et al [232])

15 10

5 0

c cshare material constants related to the averaged fluid calcium concentration

of the progressive dissolution of the portlandite and the CSH phases, s his thesolid calcium concentration related to the portlandite-free cement matrix Aplot of function (3.199) and an illustration of the material parameters aregiven in Figure 3.145

3.3.2.2.2 Models for Expansive Processes

Authored by Falko Bangert and G¨ unther Meschke