PSI - Issue 13

Josef Květoň et al. / Procedia Structural Integrity 13 (2018) 1367 – 1372 Josef Kveˇtonˇ / Structural Integrity Procedia 00 (2018) 000–000

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3

following system is obtained.

M u t +∆ t = F t +∆ t + M

˙ u t +

1 ¨ u t

K +

1 β ∆ t 2

1 β ∆ t

1 2 β −

1 β ∆ t 2

u t +

(2)

where on the left side, the part multiplied by unknown displacements u t +∆ t is the e ff ective sti ff ness and the right-hand side is the e ff ective loading vector. ∆ t is time step length and β and γ are parameters of Newmark method, chosen here as β = 0 . 3 and γ = 0 . 55.

2.3. Strain rate dependency of constitutive law

The discrete model is able to capture some part of strain rate dependency at macroscopic level by correctly ac counting for heterogeneity and inertia in the meso-structure. However, since used resolution does not capture all the possible cracking in the material undergoing fast damage and also viscous e ff ect of free water in the material are not explicitly addressed, some strain rate dependency needs to be incorporated phenomenologically. Calculation of s eq is then performed using Eq.3 that takes into account rate of crack opening, ˙ w , of contact element through increase function F ( ˙ w ) provided in Cusatis (2011). s eq = F ( ˙ w ) s 0 exp K 0 ( χ − F ( ˙ w ) s 0 / E 0 ) s 0 F ( ˙ w ) = 1 + c 1 arcsinh ˙ w c 0 (3) Here χ is history variable accounting for irreversibility of the damage. Variables K 0 and s 0 are dependent on straining direction, anyone interested in complete constitutive law description is referred to Cusatis and Cedolin (2007); Elia´sˇ (2016). Variables c 0 and c 1 are additional material parameters. Concrete exhibit spatially fluctuating properties. Some part of this randomness is included in the discrete model via direct representation of its heterogeneous structure. The remaining part can be conveniently incorporated with help of random field. Single random field h ( x ) is applied on tensile strength f t and tensile fracture energy G t according to Le et al. (2018): f t ( x ) = ¯ f t h ( x ) and G f ( x ) = ¯ G f ( h ( x )) 2 , where barred symbol denotes mean values. Since values of some other material parameters are derived from these two, random field a ff ects them as well. Distribution function of h ( x ) is considered normal (Gaussian) with grafted Weibull tail in the left part Bazˇant and Pang (2007). The mean value of h is 1. Autocorrelation function of h ( x ) is considered square exponential Elia´sˇ et al. (2015) with governing parameter called correlation length, here chosen as 80 mm. The random field is generated initially on regular grid via Karhunen– Loe`ve expansion and than projected onto the model by EOLE method Li and Der Kiureghian (1993). The autocor relation function allows decomposition of the problem into individual directions Voˇrechovsky´ (2008), which greatly simplifies the problem of searing for eigendecomposition of covariance matrix. 2.4. Random field

3. Numerical simulations

The numerical simulations of L-shaped specimen investigated experimentally in Ozˇbolt et al. (2015) are performed. The specimens has overall width and height 500 mm, square cut o ff producing shape of upside down letter L has side of length 250 mm. Thickness of specimen is 50 mm. Loading is applied vertically upwards under prescribed displacement rate (according to experimental study) and bottom part is fixed, as schematically shown in left upper part of Fig.1. Additionally, to prevent cracking in the area of bottom support, its rotation is allowed for rates greater than 0 . 5 m / s. For such rates, the peak load and main crack formation occurs much earlier than the bottom support starts to rotate.