Model B3 is an aging viscoelastic model for concrete creep and shrinkage, developed by Prof. Bazant and coworkers. In OOFEM it is implemented in three different ways.
The first version, “B3mat”, is kept in OOFEM for compatibility. It is based on an aging Maxwell chain. The moduli of individual units in the chain are evaluated in each step using the leastsquares method.
The second, more recent version, is referred to as “B3solidmat”. Depending on the specified input it exploits either a nonaging Kelvin chain combined with the solidification theory, or an aging Kelvin chain. It is extended to the socalled MicroprestressSolidification theory (MPS), which in this implementation takes into account only the effects of variable humidity on creep; the effects of temperature on creep are not considered. The underlying rheological chain consists of four serially coupled components. The solidifying Kelvin chain represents shortterm creep; it is serially coupled with a nonaging elastic spring that reflects instantaneous deformation. Longterm creep is captured by an aging dashpot with viscosity dependent on the microprestress, the evolution of which is affected by changes of humidity. The last unit describes volumetric deformations (shrinkage and thermal strains). Drying creep is incorporated either by the “averaged crosssectional approach”, or by the “point approach”.
The latest version is denoted as “MPS” and is based on the microprestresssolidification theory [28] [29] [30]. The rheological model consists of the same four components as in “B3solidmat”, but now the implemented exponential algorithm is designed especially for the solidifying Kelvin chain, which is a special case of an aging Kelvin chain. This model takes into account both humidity and temperature effects on creep. Drying creep is incorporated exclusively by the socalled “point approach”. The model can operate in four modes, controlled by the keyword . The first mode ( ) solves only the basic creep and runs as a single problem, while the remaining three modes need to be run as a staggered problem with humidity and/or temperature analysis preceding the mechanical problem; both humidity and temperature fields are read when , only the field of relative humidity is taken into account when and finally, only temperature when .
The basic creep is in the microprestresssolidification theory influenced by the same four parameters  as in the model B3. Values of these parameters can be estimated from the composition of concrete mixture and its compressive strength using the following empirical formulae:
(105)  
(106)  
(107)  
(108) 
The nonaging spring stiffness represents the asymptotic modulus of the material; it is equal to .
The solidifying Kelvin chain is composed of Kelvin units with fixed retardation times
,
, which form a geometric progression with quotient 10. The lowest retardation time is equal to
, the highest retardation time is bigger than
. The chain also contains a spring with stiffness
(a special case of Kelvin unit with zero retardation time).
Moduli
of individual Kelvin units are determined such that the chain
provides a good approximation of the nonaging microcompliance function of the solidifying constituent,
, where
day and .
The technique based on the continuous retardation spectrum
leads to the following formulae:
where  (109)  
where  (110) 
The actual viscosities and stiffnesses of the solidifying chain change in time according to and , where
(111) 
Evolution of viscosity of the aging dashpot is governed by the differential equation
Equation (112) differs from the one presented in the original work; it replaces the differential equation for microprestress, which is not used here. The evolution of viscosity can be captured directly, without the need for microprestress. What matters is only the relative humidity and temperature and their rates. Parameters and of the original MPS theory are replaced by . The initial value of viscosity is defined as , where is age of concrete at the onset drying or when the temperature starts changing, in the present implementation it is set which corresponds to the material age when the material is cast.
As mentioned above, under variable humidity and temperature conditions the physical time in function describing evolution of the solidified volume is replaced by the equivalent time . In a similar spirit, is replaced by the solidification time in the equation describing creep of the solidifying constituent, and by the reduced time in equation
d d relating the flow strain rate to the stress.
Factors transforming the physical time into , and are defined as follows:
(117)  
(118)  
(119) 
(120)  
(121)  
(122) 
At sealed conditions (or ) the auxiliary coefficients while at room temperature (or with ) factors .
It turned out that both the size effect on drying creep as well as its delay behind drying shrinkage can be addressed through parameter in the governing equation (112). In the experiments, the average (crosssectional) drying creep is decreasing with specimen size. Unfortunately, for the standard value , the MPS model exhibits the opposite trend. For the size effect disappears, and for it corresponds to the experiments. It should be noted that for negative or infinite values of the underlying theory loses its original physical background. If the experimental data of drying creep measured on different sizes are missing, the exponent can be taken as a realistic estimate.
When parameter is changed from its recommended value it is advantageous to rewrite the governing differential equation to the following form
With the “standard value” (reverted size effect) the new parameter is also 2, for (correct size effect) and finally with the first parameter and the second parameter becomes dimensionless. The other advantage of is that the governing differential equation becomes linear and can be solved directly.
The rate of thermal strain is expressed as and the rate of drying shrinkage strain as , where both and are assumed to be constant in time and independent of temperature and humidity.
There are two options to simulate the autogenous shrinkage in the MPS material model. The first one is according to the B4 model
For the normally hardening cement, the ultimate value of the autogenous shrinkage can be estimated from the composition using the empirical formula of the B4 model
Similarly, in the Model Code, the ultimate shrinkage strain can be estimated from the mean concrete strength at the age of 28 days and from the cement grade
The model description and parameters are summarized in Tab. 35 for “B3mat”, in Tab. 36 for “B3solidmat”, and in Tab. 37 for “MPS”. Since some model parameters are determined from the composition and strength using empirical formulae, it is necessary to use the specified units (e.g. compressive strength always in MPa, irrespectively of the units used in the simulation for stress). For “B3mat” and “B3solidmat” it is strictly required to use the specified units in the material input record (stress always in MPa, time in days etc.). The “MPS” model is almost unitindependent, except for in MPa and in kg/m, which are used in empirical formulae.
For illustration, sample input records for the material considered in Example 3.1 of the creep book by Bazant and Jirásek is presented. The concrete mix is composed of 170 kg/m of water, 450 kg/m of typeI cement and 1800 kg/m of aggregates, which corresponds to ratios and . The compressive strength is MPa. The concrete slab of thickness 200 mm is cured in air with initial protection against drying until the age of 7 days. Subsequently, the slab is exposed to an environment with relative humidity of 70%. The following input record can be used for the first version of the model (B3mat):
B3mat 1 n 0.2 d 0. talpha 1.2e5 relMatAge 28. fc 45.4 cc 450. w/c 0.3778 a/c 4. t0 7. timefactor 1. alpha1 1. alpha2 1.2 ks 1. hum 0.7 vs 0.1 shmode 1
Parameter corresponds to typeI cement, parameter to curing in air, parameter to an infinite slab. The volumetosurface ratio is in this case equal to one half of the slab thickness and must be specified in meters, independently of the length units that are used in the finite element analysis (e.g., for nodal coordinates). The value of relMatAge must be specified in days. Parameter relMatAge 28. means that time 0 of the analysis corresponds to concrete age 28 days. If material B3mat is used, the finite element analysis must use days as the units of time (not only for relMatAge, but in general, e.g. for the time increments).
If only the basic creep (without shrinkage) should be computed, then the material input record reduces to following: B3mat 1 n 0.2 d 0. talpha 1.2e5 relMatAge 28. fc 45.4 cc 450. w/c 0.3778 a/c 4. t0 7. timefactor 1. shmode 0
Description  B3solid material model for concrete creep 
Record Format  B3solidmat d(rn) # n(rn) # talpha(rn) # mode(in) # [ EmoduliMode(in) #] Microprestress(in) # shm(in) # [ begoftimeofinterest(rn) #] [ endoftimeofinterest(rn) #] timefactor(rn) # relMatAge(rn) # fc(rn) # cc(rn) # w/c(rn) # a/c(rn) # t0(rn) # q1(rn) # q2(rn) # q3(rn) # q4(rn) # c0(rn) # c1(rn) # tS0(rn) # w_h(rn) # ncoeff(rn) # a(rn) # ks(rn) # [ alpha1(rn) #] [ alpha2(rn) #] hum(rn) # vs(rn) # q5(rn) # kt(rn) # EpsSinf(rn) # es0(rn) # r(rn) # rprime(rn) # at(rn) # kSh(rn) # inithum(rn) # finalhum(rn) # 
Parameters   num material model number 
 d material density  
 n Poisson ratio  
 talpha coefficient of thermal expansion  
 mode optional parameter; if (default), parameters are predicted from composition of the concrete mixture (parameters fc, cc, w/c, a/c and t0 need to be specified). Otherwise values of parameters are expected.  
 EmoduliMode optional parameter; analysis of retardation spectrum (, default value) or leastsquares method () is used for evaluation of Kelvin units moduli  
 Microprestress basic creep; drying creep (must be run as a staggered problem with preceding analysis of humidity diffusion. Parameter shm must be equal to 3. The following parameters must be specified: c0, c1, tS0, w_h, ncoeff, a)  
 shmode shrinkage mode; no shrinkage; average shrinkage (the following parameters must be specified: ks, vs, hum and additionally alpha1 alpha2 for and kt EpsSinf q5 for ; point shrinkage (needed: es0, r, rprime, at), w_h ncoeff a; point shrinkage based on MPS theory (needed: parameter kSh or value of kSh can be approximately determined if following parameters are given: inithum, finalhum, alpha1 and alpha2)  
 begoftimeofinterest optional parameter; lower boundary of time interval with good approximation of the compliance function [day]; default 0.1 day  
 endoftimeofinterest optional parameter; upper boundary of time interval with good approximation of the compliance function [day]  
 timefactor scaling factor transforming the simulation time units into days  
 relMatAge relative material age [day] 

Now consider the same conditions for “B3solidmat”.
In all the examples below,
the input record with the material description can start by
B3solidmat 1 d 2420. n 0.2 talpha 12.e6 begtimeofinterest 1.e2
endtimeofinterest 3.e4 timefactor 86400. relMatAge 28.
Parameters begoftimeofinterest 1.e2 and endoftimeofinterest 3.e4 mean that the computed response (e.g., deflection) should be accurate in the range from 0.01 day to 30,000 days after load application. Parameter timefactor 86400. means that the time unit used in the finite element analysis is 1 second (because 1 day = 86,400 seconds). Note that the values of begtimeofinterest, endtimeofinterest and relMatAge are always specified in days, independently of the actual time units in the analysis. Parameter EmoduliMode is not specified, which means that the moduli of the Kelvin chain will be determined using the default method, based on the continuous retardation spectrum.
Additional parameters depend on the specific type of analysis:
Description  Microprestresssolidification theory material model for concrete creep 
Record Format  mps d(rn) # n(rn) # talpha(rn) # referencetemperature(rn) # mode(in) # [ CoupledAnalysisType(in) #] [ begoftimeofinterest(rn) #] [ endoftimeofinterest(rn) #] timefactor(rn) # relMatAge(rn) # lambda0(rn) # fc(rn) # cc(rn) # w/c(rn) # a/c(rn) # stiffnessfactor(rn) # q1(rn) # q2(rn) # q3(rn) # q4(rn) # ksh(rn) # [ mus(rn) #] k3(rn) # [ alphaE(rn) #] [ alphaR(rn) #] [ alphaS(rn) #] [ QEtoR(rn) #] [ QRtoR(rn) #] [ QStoR(rn) #] kTm(rn) # [ kTc(rn) #] [ p(rn) #] [ p_tilde(rn) #] [ alpha_as(rn) #] [ eps_cas0(rn) #] [ b4_eps_au_infty(rn) #] [ b4_tau_au(rn) #] [ b4_alpha(rn) #] [ b4_r_t(rn) #] [ b4_cem_type(rn) #] [ temperInCelsius ] 
Parameters   num material model number 
 d material density  
 n Poisson ratio  
 talpha coefficient of thermal expansion  
 referencetemperature reference temperature only to thermal expansion of material  
 mode optional parameter; if (default), parameters are predicted from composition of the concrete mixture (parameters fc, cc, w/c, a/c and stiffnessfactor need to be specified). Otherwise values of parameters are expected.  
 CoupledAnalysisType basic creep; (default) drying creep, shrinkage, temperature transient creep and creep at elevated temperature; drying creep, shrinkage; temperature transient creep and creep at elevated temperature; for choice # 1, 2, 3 the problem must be run as a staggered problem with preceding analysis of humidity and/or temperature distribution; Following parameters must be specified: mus or k3 (according to exponent p), kTm (compulsory for choice #3 otherwise optional)  
 lambda0 scaling factor equal to 1.0 day in time units of analysis (e.g. 86400 if the analysis runs in seconds)  
 begoftimeofinterest lower boundary of time interval with good approximation of the compliance function; default value = 0.01  
 endoftimeofinterest upper boundary of time interval with good approximation of the compliance function; default value = 10000.  
 timefactor scaling factor, for mps material must be equal to 1.0  
 relMatAge relative material age = age at time when the material is cast in the structure 

Finally consider the same conditions for “MPS material”.
In all the examples below,
the input record with the material description can start by
mps 1 d 2420. n 0.2 talpha 12.e6 referencetemperature 296.
Additional parameters depend on the specific type of analysis:
Final recommendations:
Borek Patzak