Graham Baker

An anisotropic thermomechanical damage model for concrete at transient elevated temperatures

The behaviour of concrete at elevated temperatures is important for an assessment of integrity (strength and durability) of structures exposed to a high-temperature environment, in applications such as fire exposure, smelting plants and nuclear installations. In modelling terms, a coupled thermomechanical analysis represents a generalization of the computational mechanics of fracture and damage. Here, we develop a fully coupled anisotropic thermomechanical damage model for concrete under high stress and transient temperature, with emphasis on the adherence of the model to the laws of thermodynamics. Specific analytical results are given, deduced from thermodynamics, of a novel interpretation on specific heat, evolution of entropy and the identification of the complete anisotropic, thermomechanical damage surface. The model is also shown to be stable in a computational sense, and to satisfy the laws of thermodynamics.

Thermodynamics in solid mechanics: a commentary

This commentary on thermodynamics in solid mechanics aims to provide an overview of the main concepts of thermodynamic processes as they apply to, and may be exploited for, studies in nonlinear solid mechanics. We give a descriptive commentary on the (physical) interpretation of these concepts, and relate these where appropriate to behaviour of solids under thermo-mechanical conditions. The motivation is firstly that students of solid mechanics have often had less exposure to thermodynamics than those in other branches of science and engineering, yet there is great value in analytical formulations of material behaviour derived from the principles of thermodynamics. It also sets the contributions in this Theme Issue in context. Along with the deliberately descriptive treatment of thermodynamics, we do outline the main mathematical statements that define the subject, knowing that full details are provided by the authors in their corresponding contributions to this issue. The commentary ends on a lighter note. In order to aid understanding and to stimulate discussion of thermodynamics in solid mechanics, we have invented a number of very basic and completely fictitious materials. These have strange and extreme behaviours that describe certain thermodynamics concepts, such as entropy, in isolation from the complexities of real material behaviour.

An energy model for bifurcation analysis of a double-notched concrete panel: continuum model

In this paper, the uni-axial tension of double-notched concrete specimens is analyzed by using a continuum plasticity model. As in a companion paper, the concept of minimization of the second-order energy is used as the criterion for judging a bifurcation. The energy computation is formulated in standard continuum plasticity. The analysis confirms that the unsymmetrical crack propagation (i.e. strains localizing on one side of the specimen) may occur either before or after the peak load. Influences on bifurcation of three factors, the notch depth to panel width, the local constitutive law, and the ratio of the panel width to panel length are investigated. A larger ratio of the notch depth to panel width, a steeper softening constitutive law, and a smaller ratio of the panel width to panel length, leads to an earlier bifurcation. These conclusions are consistent with those obtained from a simplified model.

Material softening and structural instability

This note discusses the relationship between two kinds of instability problems: material failure and structural instability. Material failure is governed by the second-order work at the material point concerned, whereas structural instability is governed by the second-order work of the whole structure. Structural instability is not only related to material instability but also to the structural topology, boundary conditions, and the mathematical model used. Material failure only indicates that the structure cannot support some forms of loading further. If the mathematical modelling does not reflect these forms of the loading, the structure may be stable but with material failure. The important conclusion is that at a structural level, we should examine global not local stability. As an example, the stability of localized and non-localized solutions is evaluated with the aid of the second-order work expressions. A theoretical explanation is presented to the interesting phenomenon in softening solids that increasing the finite element space will reveal more unstable solutions and will “turn” those that were previously found “stable” into unstable solutions.

Enhanced Approach to Consistency in Gradient-Dependent Plasticity

This note proposes a modification of gradient-dependent plasticity to improve convergence. In gradient-dependent plasticity, the consistency condition, which results in a differential equation with respect to the plastic multiplier, is solved simultaneously with the equilibrium equation. In each iteration, the consistency condition is not really satisfied and the stress is generally not on the yield surface; this results in poor convergence. A modification is proposed, in which gradient-dependent plasticity is recast into the classical plasticity framework and a strict stress mapping strategy is established. Instead of solving a differential equation simultaneously with the equilibrium equation, the plastic multiplier is solved by minimizing a functional separately. The consistency condition can be satisfied and the stress is mapped back to the yield surface.

Incompatible 4-node element for gradient-dependent plasticity

In gradient-dependent plasticity theory, the yield strength depends on the Laplacian of an equivalent plastic strain measure (hardening parameter), and the consistency condition results in a differential equation with respect to the plastic multiplier. The plastic multiplier is then discretized on the mesh, in addition to the usual discretization of the displacements, and the consistency condition is solved simultaneously with the equilibrium equations. The notorious disadvantage is that the plastic multiplier requires a Hermitian interpolation which has four degrees of freedom at each node. However, in this article, an incompatible (trigonometric) interpolation is proposed for the plastic multiplier. This incompatible interpolation uses only the function values of each node, but both the function and first-order derivatives are continuous across element boundaries. It greatly reduces the degrees of freedom for a problem, and is shown through numerical examples on localization to give good results.

Analysis of Crack Spacing in Reinforced Concrete by a Lattice Model

A new numerical method is proposed for prediction of crack spacing in reinforced concrete. It is assumed that the deformation pattern of crack spacing consumes the least energy among all kinematically admissible deformations, and the energy minimization approach is applied to predict crack spacing. To simplify the problem, a lattice model is used, in which the cracking process is represented by softening of the concrete bar elements. The crack spacing due to tension and bending is investigated. The results confirm that crack spacing can be predicted by energy minimization. To account for the influence of the bond slip between the concrete and the reinforcing steel, the traditional bond-link element is employed. The influence of the bond slip on the cracking pattern is studied through numerical examples. The results show that without considering bond-slip, the damage near the reinforcement is distributed rather than localized, whereas considering bond-slip, the damage near the reinforcement is localized. The influence of bond slip on crack spacing is significant.

One-Dimensional Nonlinear Model for Prediction of Crack Spacing in Concrete Pavements

This paper proposes a one-dimensional non-linear model to predict the minimum and maximum crack spacings due to shrinkage in concrete pavements. The proposed model consists of two cohesive cracks and an elastic bar restrained by distributed elastic springs. The cohesive crack is characterized by an exponential softening constitutive relation. A set of non-linear equilibrium conditions are obtained. By varying the length of the elastic bar of the proposed model, the tensile forces acting on the cohesive cracks and the energy profiles are investigated. It is demonstrated that the cracking pattern varies with the length of the elastic bar (i.e. the spacing between the two possible cracks), from which the minimum and maximum crack spacings are obtained. Numerical analyses are made of a model pavement and the results indicate that it is the energy minimization principle that governs the cracking pattern. The proposed model provides physical insight into the mechanism of crack spacing in concrete pavements.

Energy Profile And Bifurcation Analysis in Softening Plasticity

Bifurcations of solutions and energy profile in softening plasticity are discussed in this paper. The localized and non-localized solutions are obtained for a simple softening bar; the second-order derivatives of the incremental energy are evaluated. The second-order derivatives along the fundamental path demonstrate a discontinuity at the bifurcation point; the eigen-analysis of the tangential stiffness matrix fails to identify the post-bifurcation paths. The energy variation near the bifurcation point is investigated; the relationship between the stationary points of the energy profile and post-bifurcation solutions is established. Beyond the bifurcation point, the single stable loading path splits into several post-bifurcation paths and the incremental energy exhibits several competing minima. Among the multiple post-bifurcation equilibrium states, the localized solutions correspond to the minimum points of the energy profile, while the non-localized solutions correspond to the saddles and local maximum points. To determine the real post-bifurcation path, the concept of minimization of the second-order energy is used as the criterion for the bifurcation analysis involved in softening plasticity. As an application, a lattice model of a beam is analyzed and damage localization is obtained.