A critical assessment of nonlocal strain softening methods in biaxial compression

1Dr Freya Caroline Summersgill 3 (Formerly Imperial College London) 4 Design Engineer 5 freya.summersgill@tonygee.com 6 Tony Gee &Partners LLP 7 Hardy House 8 140 High Street 9 EsherSurrey 10 KT10 9QJ 11 UK 12 13 2Dr Stavroula Kontoe (Corresponding author) 14 Senior Lecturer 15 stavroula.kontoe@imperial.ac.uk 16 Department of Civil & Environmental Engineering 17 Imperial College London 18 SW7 2AZ 19 UK 20 21


INTRODUCTION
One of the first formulations for a nonlocal strain model is presented in Equation 143 1. This was proposed by Eringen (1981)   (1) 7 where ε p is the accumulated plastic deviatoric strain, * denotes the nonlocal 149 parameter, xn is the point at which the calculation of the nonlocal strain, ε p* is 150 required, whereas xn' refers to all the surrounding locations, i.e. the location of 151 reference strains. Therefore, ε p (xn') equals the reference strain at the reference 152 location. The weighting function, ω(xn') is defined for all the reference locations, 153 but it is centred at the location xn.

154
The weighting function is the Gaussian or normal distribution which is shown in (2) 165 The weighting function is chosen such that it will not alter a uniform field of 166 strain.

167
The integral of the weighting function in the three dimensions x1, x2 and x3 is 168 referred to as the reference volume, Vω, as shown in Equation 3. This is used to 169 normalise the calculation of the nonlocal strain and for the Gaussian distribution This method, which will be referred to as the "original method" herein, has been 176 shown to have low mesh dependency when applied to strain softening analyses As it can be seen with reference to Equation 5, when α is greater than 1, the local 222 strain contribution is negative. This significantly reduces the value of nonlocal 223 strains calculated over the areas where local strain is high and increases the 224 nonlocal strain value calculated in the areas immediately adjacent to the local 225 strain distribution. If α = 1.0 the formula reverts to the original nonlocal 226 formulation. If α is less than 1.0, the local strain part of the equation contributes 227 by increasing the nonlocal strain to be greater than the local strain, which 228 contradicts the aims of the formulation. The alpha parameter, α, must therefore 229 always be greater than 1.0 (Vermeer & Brinkgreve, 1994). parameter on nonlocal strain is investigated and discussed in a following section.

318
A radius of influence can also be specified to limit the number of elements used 319 in the calculation of the nonlocal strain. The weighting function is an exponential 320 function based on distance, therefore the contribution of strain diminishes rapidly with distance, as can be seen in Figure 1. The exclusion of strains at a 322 distance greater than 4 times DL will alter the outcome of the nonlocal strain 323 calculation very little, but it will improve numerical efficiency. A specified radius 324 of influence of 0.4m was therefore considered as appropriate for these analyses.

325
The sensitivity of the analysis on the radius of influence is further investigated in 326 the last part of this study.  for the Over-nonlocal method again resulted in a sudden reduction in the 377 reaction load, identified as premature softening in the previous section, Figure   378 10(d). to evaluate the influence of these parameters.

415
The alpha parameter is only employed in the Over-nonlocal modified strain 416 softening method. The defined length, DL, is a required input for all the nonlocal 417 methods and influences the shape of the weighting function used to calculate 418 nonlocal strain, as shown in Figure 2 and as a consequence the softening rate.

419
The radius of influence, RI, is an optional parameter that improves the numerical As previously discussed, the parameter alpha controls the distribution of non- analyses with the finest mesh for each α value, which is shown in Figure 15.

439
There is a high value of strain concentrated in a smaller area similar in shape to 440 the local strain softening method results of Figure 9      Over-nonlocal α=1.5, 10x10 Over-nonlocal α=1.5, 20x20 Over-nonlocal α=1.5, 40x40