Lifetime-Oriented Structural Design Concepts- P12 potx

288 3 Deterioration of Materials and Structures the drained thermo-mechanical coupling tensor A = A u − 3α t,u MB = C C C ed : 1α t , (3.149) and the drained tensor Λ = Λ u + ΞB, (3.150) 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 quantities defined on the meso-level to respective homogenised quantities on the macro- level. The so-obtained tangential Biot coefficient is determined as b = S l  1 −ψ K K s  , (3.151) 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 M =  φ  1 − S l p l K s  ∂S l ∂p l + φS l K l + S l (b −φS l ) K s  −1 (3.152) see [705, 493] for a similar formulation. For cementitious materials, expression (3.152) can be replaced by M ≈  φ ∂S l ∂p l  −1 . (3.153) In the special case of a fully saturated material (S l = 1), (3.152) yields the classical relation [211, 493] M S l =1 =  φ K l + (b − φ) K s  −1 . (3.154) The coefficients related to damage phenomena Λ and Ξ are identified by exploiting the symmetry relations that are connected to the existence of a macroscopic potential. Using the Maxwell symmetries, the drained tensor Λ can be expressed as [533] Λ = C C C :(ε −ε p − ε f )+ K K s  p l S l dp l 1 −C C C : 1α t T, (3.155) and the coupling coefficient Ξ is obtained as Ξ = MS l K K 2 s  p l S l dp l ≈ 0. (3.156) 3.3 Modelling 289 3.3.2.1.4 Effective Stresses The concept of effective stress [281, 791] is a generally accepted approach in soil 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 of fully saturated soils, this concept has also been adapted for the description of partially saturated soils. Early formulations introduced the capillary pressure in the (elastic) effective stress definition [127]. However, difficulties to obtain satisfactory agreements with experimental results have motivated the use of two independent stress fields for the constitutive modelling of unsaturated soils (see e.g. [129, 44]). As far as the numerical modelling of partially saturated cement-based mate- rials is concerned, the assumption of (elastic) effective stresses seems not to be well suited for the description of shrinkage-induced cracks using stress-based crack-models. However, the concept of plastic effective stress first introduced at a macroscopic level by [210] for saturated porous media (see [211] for de- tails), allows to overcome these difficulties in the framework of poroplasticity – porodamage models. The proposed form of the plastic effective stress is the same as the classical Biot-type, however, a plastic effective stress coefficient is used. A similar form has been derived from micromechanical considera- tions by [510]. This concept has been recently extended to partially saturated materials [167, 533], and is also adopted in the present formulation. From the coupled relations between total stresses, strains, liquid saturation and temperature σ = ψC C C :(ε −ε p − ε f ) +  1 −ψ K K s   p c S l (p c )dp c 1 −AT, (3.157) the following definition of the elastic effective stress tensor σ e = ψC C C :(ε −ε p − ε f ) − AT, (3.158) with σ = σ e +  1 −ψ K K s   p c S l (p c )dp c 1. (3.159) is obtained. The plastic effective stress tensor σ p = σ  , defined as σ  = σ − b p p c 1, (3.160) 290 3 Deterioration of Materials and Structures 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,theporosityφ and the liquid saturation S l can be accomplished as b p = ψφS l (p c ) , (3.161) 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, mecha- nisms characterised by the degradation of stiffness and inelastic deformations are controlled by four threshold functions defining a region of admissible stress states in the space of plastic effective stresses σ  E = {(σ  ,q k )| f k (σ  ,q k (α k )) ≤ 0,k=1, , 4}. (3.162) In (3.162), the index k =1, 2, 3 stands for an active cracking mechanism asso- ciated with the damage function f R,k (σ  ,q R )andk = 4 represents an active hardening/softening mechanism in compression associated with the loading function f DP (σ  ,q DP ). Cracking of concrete is accounted for by means of the Rankine criterion, employing three failure surfaces perpendicular to the axes of principal stresses f R,A (σ  ,q R )=  A − q R (α R ) ≤ 0,A=1, 2, 3. (3.163) In (3.163), the subscript A refers to one of the three principal directions and q R (α R )=−∂U/∂α R denotes the softening parameter. The ductile behaviour of concrete subjected to compressive loading is de- scribed by a hardening/softening Drucker-Prager plasticity model f DP (σ  ,q DP )=  J 2 − κ DP I 1 − q DP (α DP ) β DP ≤ 0, (3.164) with q DP (α DP )=−∂U/∂α DP as the hardening/softening parameter. The de- termination of the model parameters κ DP and β DP is based on the ratio of the biaxial and the uniaxial compressive strength of concrete f cb /f cu as [534] κ DP = 1 √ 3  f cb /f cu − 1 2f cb /f cu − 1  , (3.165) β DP = √ 3  2f cb /f cu − 1 f cb /f cu  , (3.166) 3.3 Modelling 291 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 of the multisurface model based on an algorithmic formulation in the principal stress 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 ˙α R and ˙α DP are obtained from the postulate of stationarity of the dissipation functional [318] as ˙ ε p =(1−β) 4  k=1 ˙γ k ∂f k ∂σ  , (3.167) (ψ −1 )˙ = β 4  k=1 ˙γ k ∂f k ∂σ  : C C C u : ∂f k ∂σ  ∂f k ∂σ  : σ  , (3.168) ˙ φ p l = 4  k=1 ˙γ k ∂f k ∂σ  : 1b p , (3.169) ˙α R = 3  A=1 ˙γ R,A ∂f R,A ∂q R , ˙α DP =˙γ DP ∂f DP ∂q DP , (3.170) together with the loading/unloading conditions f k (σ  ,q k ) ≤ 0; ˙γ k ≥ 0; ˙γ k f k (σ  ,q k )=0. (3.171) 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 crack- induced misalignment of the asperities of the crack surfaces, resulting in an increase of inelastic strains ε p , and deterioration of the microstructure, result- 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 =0andβ = 1, respectively. 3.3.2.1.6 Long-Term Creep Consideration of long-term or flow creep effects is accomplished in the frame- work of the microprestress-solidification theory [93]. The evolution law of the flow strains is based on a linear relation between the rate ˙ ε f and the stress tensor σ as ˙ ε f = 1 η f (S f ) G G G ed : σ, (3.172) with the fourth-order tensor G G G ed = E  C C C ed  −1 and Young’s modulus E.The viscosity η f is a decreasing function of the microprestress S f and can be written as [93] 292 3 Deterioration of Materials and Structures 1 η f (S f ) = cpS p−1 f , (3.173) 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 chang- 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. 3.3.2.1.7 Moisture and Heat Transport Starting with a simplified nonlinear diffusion approach, in which the different moisture transport mechanisms in liquid and in vapour form are represented by means of a single macroscopic moisture-dependent diffusivity [94], the re- lation between the moisture flux q l and the spatial gradient of the capillary pressure ∇p c is given by q l = k μ l ·∇p c . (3.174) In (3.174), k denotes the intrinsic liquid permeability tensor and μ l is the viscosity of water. According to the hypothesis of dissipation decoupling [212], possible couplings between heat and moisture transport are disregarded in the present formulation. In order to account for the dependence of the moisture transport properties on the nonlinear material behaviour of concrete, k is additively decomposed into two portions as k = k r (S l )[k t (T )k φ (φ) k 0 + k d (α R )] , (3.175) one related to the moisture flow through the partially saturated pore space and one related to the flow within a crack, respectively [758]. This approach is consistent with the smeared crack concept. In (3.175), k 0 denotes the ini- tial isothermal permeability tensor, k r is the relative permeability, k t ac- counts for the dependence of the isothermal moisture transport properties on the temperature and k φ describes the relationship between the permeabil- ity and the porosity. Furthermore, k d is the permeability tensor relating plane Poiseuille flow through discrete fracture zones to the degree of damage in the continuum model, see [533, 319] for details. Using again the hypothesis of dissipation decoupling, the relation between the heat flux q t and the gradient of the temperature ∇T can be described by a linear heat conduction law reading q t = −D t ∇T, (3.176) whereby D t (T,S l ,φ) denotes the effective thermal conductivity. 3.3 Modelling 293 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 scaling problem is characterised by the existence of two scales, which should be sepa- rated when modelling frost processes in hardened cement paste. Most relevant for 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 tran- sient conditions have to be modeled, i.e. mass transport due to viscous fluid flow is slow. On this scale the model deals with bigger volumes than on the microscale. In the big macroscopic volumes thermodynamic processes need a large time span to obtain equilibrium. This can be observed in practise as well 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, pore water 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 surface thermodynamics and the theory of disjoining pressure. At least thermal or thermodynamic 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 the ice is highly dynamic. However, the response time for movement from gel to ice 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 the- 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 & Setzer [450] and Kruschwitz & Bluhm [449] respectively. Last describe the frost heave of a critical filled cementitious matrix. In the mentioned com- bination the macroscopic, thermodynamic aspects of the model base on the Theory of Porous Media. This theory is a combination of the mixture theory and the concept of volume fractions. The interactions of the nanostructure of the hardened cement paste are modelled by a smeared micromechanical model. This part of the model is characterised by the properties of the two phase system solid and pore liquid. The transport on the micro structure and the unfrozen, adsorbed water film between matrix and ice are included. 294 3 Deterioration of Materials and Structures 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 that the main microstructural mechanisms of environamentally induced corrosion and deterioration processes are by now fairly well understood. There seems to exist, 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 are stilc l accounted for by more or less heuristic evaluations of the degradation process and its influence on the residual structural safety. Recent progress in computational durability mechanics (see e.g. [75, 211, 800, 798, 214]), to- gether with appropriate numerical discretization methods in space and time [460, 453] (see also Chapter 4) open the perspective of a more fundamental approach to obtain not only estimates for the life-time, but also to provide insight into the degradation mechanisms as a result of the interaction between mechanical and environmental loading. Using a continuum mechanics-based mode of description, concrete sub- jected to mechanical and non-mechanical loading is generally described as a multi-phase material whose behaviour is influenced by the interaction of the solid skeleton containing the cementititious matrix and the aggregates and the liquid and gaseous pore fluids. To this end, the scale of observation may either take the micro-scale or macro-scale as a point of departure. In the framework of a micro-scale approach the individual constituents are described by means of classical continuum mechanics for one-phase materials, formulating appro- 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 of porous materials on the basis of a macroscopic approach. The Theory of Mixtures (see e.g. [254] for more details) has been established as a suitable homogenisation procedure, which allows to treat multi-phase materials as a continuum while each constituent may be describedbyitsownkinematicsand balance equations. The interactions between the constituents are included by production terms within the balance equations. Since the Theory of Mixtures contains no microscopic information of the mixture it need to be complemented by the concept of volume. This leads to the 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 general continuum mechanically and thermo dynamically established concept for the macroscopic description of multi-phase materials like concrete. 3.3 Modelling 295 virgin material → mech. damage mech. damage ← virgin material chem. dissolution ← virgin material chem. dissolution ← virgin material Representative Elementary Volume (REV) Theory of Mixture - material point φ 0 1 − φ 0 d m φ m ˙s φ c φ m φ 0 φ c 1 − φ Fig. 3.143. Chemo-mechanical damage of porous materials within the Theory of Mixtures. Three types of deterioration are illustrated: virgin material, mechanically damaged material, chemically damaged material and chemo-mechanically damaged material 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 on calcium dissolution and coupled chemo-mechanical damage processes (see Subesection 3.1.2.3.2) constitutive models formulated on a macroscopic level of observation have been developed for the analysis of the time dependent dissolution 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 by G ´ erard [307] and subsequent publications (G ´ erard [308], G ´ erard et al. [311], Pijaudier-Cabotetal.[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-plasticity model formulated within the Biot-Coussy-Theory of porous media [211]. This model is also based on a chemical equilibrium model, using empirical relations 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 mate- rial and transport properties of chemo-mechanically loaded cementitious ma- terials becomes obvious. Based on this observation and in order to consider 296 3 Deterioration of Materials and Structures the interaction phenomena of chemical and mechanical material degradation described in Subsection 3.1.2.3.2 a fully coupled chemo-mechanical damage model has been developed in [454, 455] within the framework of the Theory of Porous Media. The material is described as ideal mixture of the fully saturated pore space and the matrix. In this model, the pore fluid acts as a transport medium for calcium ions. The pore pressure, however, is not accounted for in the present version of the model. The changing mechanical and transport properties are related to the to- tal porosity defined as the sum of the initial porosity, the chemically in- duced porosity and the apparent mechanical porosity. Together with the assumptions of chemical and mechanical potentials the need for further as- sumptions or empirical models is circumvented. Micro-cracks are interpreted according to Kachanov [422] as equivalent pores affecting, on a macroscopic level, the conductivity and stiffness but not the mass balance. The evolution of the mechanically and chemically induced porosities are both controlled by internal parameters. This enables the modeling of cyclic loading condi- tions and allows a consistent thermodynamic formulation of the coupled field problems [454]. 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 φ 0 , the porosity due to matrix dissolution φ c and the apparent me- chanical porosity φ m : φ = φ 0 + φ c + φ m . (3.177) The chemically induced porosity φ c can be calculated by multiplying the amount of dissolved calcium of the skeleton s 0 − s by the averaged molar volume of the skeleton constituents M/ρ φ c = M ρ [s 0 − s] , (3.178) where s 0 denotes the initial skeleton concentration. The apparent mechan- ically induced porosity φ m considers the influence of mechanically induced micro pores and micro cracks on the macroscopic material properties of the porous material. It is obtained by multiplying the scalar damage parameter d m by the current volume fraction of the skeleton 1 −φ 0 − φ c φ m =[1−φ 0 − φ c ] d m . (3.179) This definition of the mechanical porosity φ m takes into account that micro- cracking is restricted to the solid matrix material. 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 concentration field c of calcium ions 3.3 Modelling 297 in the pore solution and the displacement field u as external variables and a set of internal variables concerning the irreversible material behavior. The macroscopic balance of linear momentum is given by: div σ =0. (3.180) The matrix dissolution-diffusion problem is governed by the macroscopic bal- ance of the calcium ion mass in the representative elementary volume div q c +[[φ 0 + φ c ] c ] · +˙s =0, (3.181) whereby q c is the mass flux of the solute. The term [[φ 0 + φ c ] c ] · accounts for the change of the calcium mass due to the temporal change of the porosity and 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 boundary Γ given by σ · n = t  , q c · n = q  c , u = u  ,c= c  (3.182) and initial conditions in the domain Ω given by u(t =0)=u 0 ,c(t =0)=c 0 , (3.183) where q  c is the calcium ion mass flux across the boundary and c  is the prescribed concentration. 3.3.2.2.1.3 Constitutive Laws The elasto-damage constitutive law is characterized by the free energy func- tion Ψ m : Ψ m = 1 −φ 2 ε : C C C s : ε . (3.184) Herein, ε denotes the linearized strain tensor and C C C s is the fourth order elas- ticity tensor of the the skeleton. The derivative of the free energy function Ψ m with respect to the strain tensor ε yields the stress tensor σ: σ = ∂Ψ m ∂ε =[1− φ] C C C s : ε . (3.185) The diffusion-dissolution problem is defined by the dissipation potential Ψ c of the calcium ions in the representative elementary volume Ψ c = φD l 2 γ · γ , (3.186) [...]... alkali-silica s reaction This assumption, which is used for most model formulations in the literature [632, 798, 772], implies, that the degradation of concrete caused by ASR is mainly induced by structural effects These structural effects may result from hindered deformations due to geometrical constraints or from gradients in the ASR expansion following from a non-uniform moisture distribution It should be... 3.3.2.2.2.1 Introductory Remarks Several numerical models have been developed in order to characterize the observed behavior of concrete affected by the Alkali-Silica Reaction (ASR) on a material level or even a structural level Depending on the level of observation these models follow either a mesoscopic or a macroscopic approach A mesoscopic approach involves the analysis of a single representative aggregate... mentioned, that even under stress-free conditions (σ s = 0) microcracks can develop in the vicinity of the aggregate particles e.g due to geometrical incompatibilities However, on the macroscopic level the structural effects have a much more severe influence on the deterioration of concrete structures than these microcracks on the level of the aggregate particles Although the fluid frictional stresses σ β are... are well-defined and can be easily determined by means of macroscopic strain measurements on reactive concrete specimens Both chemical material parameters ( u / r − 1 and k) depend on the concrete mix design, the type of aggregates, the temperature and the moisture content In particular, the moisture dependence plays a dominant role in the ASR deterioration The role of moisture within the alkali-silica... u / r − 1 and of their dependence on the liquid saturation sl according to experimental results by Larive [469] and to model results by Steffens et al [772] 3.3 Modelling 313 specimens of a certain mix design were stored under different hygral conditions (immersed in water, exposed to different relative humidities, wrapped in aluminum foil), whereby the temperature was kept constant at 38◦ C [471] For... ) These correlations are based on approx 200 cyclic triaxial tests 3.3.4 Models for the Fatigue Resistance of Composite Structures Authored by Gerhard Hanswille and Markus Porsch 3.3.4.1 General Most design codes consider the static and fatigue resistance of composite steelconcrete structures with separate verifications for the ultimate limit state and the limit state of fatigue For headed shear studs... Studs without Any Pre-damage Figure 3.148 shows the semi-empirical model for the prediction of the mean value of the ultimate shear resistance of headed studs [684], which was taken as the basis for the design rules in current national and international codes It was derived for headed studs with a diameter of 16 mm to 22 mm embedded in solid slabs of normal weight concrete on the basis of the results . 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 that the main microstructural mechanisms of environamentally induced corrosion and deterioration processes are by now. more or less heuristic evaluations of the degradation process and its influence on the residual structural safety. Recent progress in computational durability mechanics (see e.g. [75, 211, 800,. behavior of concrete affected by the Alkali-Silica Reaction (ASR) on a material level or even a structural level. Depending on the level of ob- servation these models follow either a mesoscopic