This paper discusses the issues involved in the development of combined finite/discrete element methods; both from a fundamental theoretical viewpoint and some related algorithmic considerations essential for the efficient numerical solution of large scale industrial problems. The finite element representation of the solid region is combined with progressive fracturing, which leads to the formation of discrete elements, which may be composed of one or more deformable finite elements. The applicability of the approach is demonstrated by the solution of a range of examples relevant to various industrial sections.

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SUMMARYFormulation of the scale transition equations coupling the microscopic and macroscopic variables in the second-order computational homogenization of heterogeneous materials and the enforcement of generalized boundary conditions for the representative volume element (RVE) are considered. The proposed formulation builds on current approaches by allowing any type of RVE boundary conditions (e.g. displacement, traction, periodic) and arbitrary shapes of RVE to be applied in a unified manner. The formulation offers a useful geometric interpretation for the assumptions associated with the microstructural displacement fluctuation field within the RVE, which is here extended to second-order computational homogenization. A unified approach to the enforcement of the boundary conditions has been undertaken using multiple constraint projection matrices. The results of an illustrative shear layer model problem indicate that the displacement and traction RVE boundary conditions provide the upper and lower bounds of the response determined via second-order computational homogenization, and the solution associated with the periodic RVE boundary conditions lies between them.

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Negative pore pressures existing in semi-saturated conditions provide a substantial ‘cohesion’ of the soil. This cohesion is of importance in the dynamic response of embankments and dams. The paper extends the formulation presented in part I to problems of semi-saturated behaviour with the assumption of free air ingress. An approximate reconstruction of the failure of the lower San Fernando dam during the 1971 earthquake is presented.

The fracture energy of high-performance concrete (HPC) with a compressive strength of 67·1 MPa was studied by conducting three-point bending tests on eighty notched beams of 500×100×100 mm at high temperatures up to 450°C (hot) and in cooled-down states (cold). The temperatures in the furnace and inside the concrete and the weight loss of concrete were continuously monitored. If the exposure time was long enough, 16 h in this study, both thermal and hygric equilibriums could be reached. The fracture energy sustained a decrease–increase tendency with the increasing heating temperature for the hot concrete, and an increase–decrease tendency for the cold concrete. The modulus of rupture generally decreased with the heating temperature for the hot concrete but sustained an increase–decrease tendency for the cold concrete. There was a sudden drop at 105°C for the hot concrete owing to high vapour pressure inside the concrete. The residual compressive and tensile strengths both decreased with the increasing heating temperature, with sudden drops between 105°C and 150°C owing to residual stresses caused by high vapour pressures inside the concrete during heating. The tensile strength decreased more rapidly than the compressive strength for the same heating scenario. The residual Young's modulus of concrete monotonically decreased with the increasing heating temperature and this could be expressed using a linear relationship. The fracture energy and other material properties of concrete could be closely related to the ultimate weight loss.

The effects of elevated temperatures (Tm), related to the exposure period (th) and the curing age (ta), on the residual fracture properties of normal-strength concrete (NSC) and high-strength concrete (HSC) were investigated by conducting three-point bending tests on 87 notched preheated beams. Most beams were exposed to temperatures between 100°C and 600°C for 12 h at 14 days, while some NSC beams were heated either for various exposure periods up to 168 h at 14 days or for 12 h at 7, 28 and 90 days. The weight loss (ω) was also monitored. The measured residual properties included the energy parameter (fracture energy GF), a number of strength parameters (compressive strength fcu, tensile strength f′t and modulus of rupture fr), stiffness parameters (Young's modulus Ec and Poisson's ratio νc) and the brittleness parameter (the characteristic length lch). ω increased with Tm and th but decreased with ta. There existed a transition point for ω at 200°C which could mark a distinction between physical and chemical processes. GF increased with Tm and ω up to 300°C and then decreased. GF also increased with th at lower temperatures but decreased at higher temperatures, and increased with ta as well; fcu, f′t and fr did not change very much with Tm up to 200°C and decreased thereafter. A longer th had an intensifying effect on all strengths at lower temperatures but a damaging effect at higher temperatures. All strength parameters increased with ta. Ec and νc decreased continuously with Tm, th and ω. Ec increased whereas νc decreased with ta. The concrete became less brittle with increasing Tm, th and ω or with decreasing ta. In this study, all fracture parameters tended to become stable at 90 days. Finally, a linear relationship between GF and fcu existed not only for room temperature but also for higher temperatures.

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