شبیه سازی رفتار اندازه اثر از طریق تجزیه و تحلیل حساسیت

Sensitivity analysis techniques are applied to the simulation of size effect behaviour. The scale factor is included in the discretised equilibrium equations. A gradient-enhanced damage model is used. The sensitivity of the equilibrium path with respect to the loading factor is then obtained through the direct differentiation method. Particular attention is paid to the proper differentiation of constitutive internal variables. The predictive possibilities of the algorithm are illustrated by means of an example.

Advanced methods of design sensitivity analysis are not only needed in optimisation and reliability problems, but are also a valuable complement to any generic finite element computation. In this context sensitivity is understood as the derivative of any measure of the structural response with respect to a design parameter. Size-effect phenomena are essentially a matter of structural performance vs. a scale factor [1] and provide therefore a preferential environment for the application of design sensitivity algorithms. Several techniques exist for evaluation of the sensitivity to design parameters [10]. Among these, only the direct differentiation method (DDM) seems to be adequate for solids with material non-linearities. The DDM is essentially based on the application of the implicit function theorem to the non-linear algebraic equations which result from the finite element discretisation. When advanced material models are used the dependence of internal parameters on the design parameters must also be considered. This is of special importance when local loading/unloading conditions depend on global variables, as in gradient-enhanced models. This paper provides insight in how the size effect behaviour can be studied by means of sensitivity algorithms. In particular, the DDM is applied to a gradient-enhanced damage model for concrete. First, the scale factor is incorporated into the discrete equilibrium equations. Then, the sensitivity of the peak load to the scale factor is formulated. The exact computation of the peak load is also commented on. The possibilities of the proposed method are illustrated by means of an example.

The sensitivity of a structure to the scale factor can be obtained from a single computation. For this purpose the scale factor is included in the finite element formulation and differentiation is carried out taking into account the characteristics of the material model. The procedure is meaningful only if the discretisation error is sufficiently small, otherwise this error is explicitly reflected in the computed sensitivity.