Modeling High Temperature Materials Behavior for Structural Analysis

Bars are structural members that support uni-axial tensile or compressive loadings. Chapter 1 presents elementary examples for bars and bar systems and gives an introduction to inelastic stress analysis. Basic features of inelastic structural responses including creep, relaxation, stress redistribution and others are discussed. In Sect. 1.1 governing equations a for two-bar system are introduced and initial value problems for one-dimensional inelastic stress analysis are formulated. Section 1.2 presents elementary solutions for stresses in linear thermo-elastic bars subjected to non-uniform heating. Closed-form solutions for a two-bar system under assumption of linear viscous material behavior are presented in Sect. 1.3. Various force controlled and displacement-controlled loading profiles are discussed. In addition to analytical solutions, examples for numerical time-step methods are introduced. They include one-step explicit and implicit time integration methods. Results are compared with closed form solutions to conclude on numerical accuracy and stability. Section 1.3 gives an overview of uni-axial constitutive models describing idealized non-linear inelastic behavior. Hardening, softening and damage processes are neglected to make the analysis transparent. Solutions are presented for different loading paths illustrating stress-range dependent creep, creep recovery, relaxation and tensile behaviors. Finally time-step methods are discussed to show basic features of numerical analysis for non-linear inelasticity problems.

The objective of Chap. 2 is to introduce the governing mechanical equations to describe inelastic behavior in three-dimensional solids and to discuss numerical solution procedures. The set of equations includes material independent equations, constitutive and evolution equations, as well as the initial and boundary conditions. The formulated initial-boundary value problem (IBVP) can be solved by numerical methods. Explicit and implicit time integration methods were introduced in Chap. 1 for bars. In Chap. 2 they are generalized to analyze three-dimensional solids. Applying time-step procedures, linearized boundary value problems should be solved within time and/or iteration steps. The attention will be given to the variational formulations and the use of direct variational methods.

Beams are structural members that are designed to support lateral forces and bending moments. Beams can be also subjected to combined bending, torsion as well as axial tensile or compressive loads. In the case of linear elasticity the laterally loaded beams, rods subjected to torque as well as axially loaded rods can be analyzed separately and the superposition principle can be applied to establish the resultant stress and deformation states. For nonlinear material behavior such a superposition is not possible and combined loadings should be considered. Furthermore, inelastic material response may be different for tensile and compressive loadings leading to a shift of the neutral plane under pure bending. Beams are also important in testing of materials. Three or four point bending tests are frequently used to analyze inelastic behavior experimentally. Examples are presented in Chuang (1986); Scholz et al (2008); Xu et al (2007) for homogeneous beams, in Weps et al (2013) for laminated beams and in Nordmann et al (2018) for beams with coatings. Beams are discussed in monographs and textbooks on creep mechanics (Boyle and Spence, 1983; Hult, 1966; Kachanov, 1986; Kraus, 1980; Malinin, 1975, 1981; Odqvist, 1974; Penny and Mariott, 1995; Skrzypek, 1993), where the Bernoulli-Euler beam theory and elementary constitutive equations, such as the Norton-Bailey constitutive law for steady-state creep are applied.

Chapter 3 presents examples of inelastic structural analysis for beams. In Sect 3.1 the classical Bernoulli-Euler beam theory is introduced. Governing equations and variational formulations for inelastic analysis are introduced. Closed-form solutions and approximate analytical solutions are derived for beams from materials that exhibit power law creep and stress regime dependent creep. Numerical solutions by the Ritz and finite element methods are discussed in detail. Creep and creep-damage constitutivemodels are applied to illustrate basic features of stress redistribution and damage evolution in beams. Furthermore, several benchmark problems for beams are introduced. The reference solutions for these problems obtained by the Ritz method are applied to verify user-defined creep-damage material subroutines and the general purpose finite element codes.

For many materials inelastic behavior depends on the kind of stress state. Examples for stress state effects including different creep rates under tension, compression and torsion are discussed in Sect. 3.2. For such kind of material behavior, the classical beam theory may lead to errors in computed deformations and stresses. Section 3.3 presents a refined beam theory which includes the effect of transverse shear deformation (Timoshenko-type theory). Based on several examples, classical and refined theories are compared as they describe creep-damage processes in beams.

Many structural members can be analyzed applying simplifying assumptions of plane stress or plane strain state. Examples include plates under in-plane loading, thick pipes under internal pressure, rotating discs, etc. Although many problems of this type can be solved in a closed analytical form assuming linear-elastic material behavior (Altenbach et al, 2016; Lurie, 2005; Timoshenko and Goodier, 1951), only few solutions for elementary examples exist, where plasticity and/or creep are taken into account (Boyle and Spence, 1983; Malinin, 1975, 1981; Odqvist, 1974; Skrzypek, 1993). Chapter 4 presents examples of inelastic structural analysis for plane stress and plane stress problems. In Sect. 4.1 basic assumptions are discussed and governing equations are introduced. Elementary structures including a pressurised thick cylinder, Sect. 4.2, a rotating disc, Sect. 4.3 and a plate with a circular hole, Sect. 4.4, are introduced to illustrate basic features of inelastic behavior under plane multi-axial stress and strain states. Classical results assuming the power law type creep as well as solutions with stress regime dependent inelastic behavior are presented.

Thin and moderately thick shell structures are designed as structural components in many engineering applications because of light weight and high load-carrying capacity. In many cases they are subjected to high temperature environment and mechanical loadings, such that inelastic material behavior must be taken into account. Examples of high-temperature shell components include pressure vessels, boiler tubes, steam transfer lines, thin coatings, etc. A steam transfer line under long-term operation considering creep-damage material behavior is discussed in Naumenko and Altenbach (2016, Chap. 1). Chapter 5 presents examples of inelastic structural analysis of plates and shells. Section 5.1 gives an overview of modeling approaches including various theories of plates and shells as well as various constitutive models of inelastic material behavior. Governing equations of the first order shear deformation theory of plates are presented in Sect. 5.2. An emphasis is placed on the direct formulation of inelastic constitutive laws. Section 5.3 illustrates examples of steady-state creep analysis of circular plates. Advanced constitutive models with internal state variables, such as the damage parameter require the use of advanced plate theories to consider edge effects. Section 5.4 illustrates an example of a rectangular plate with different types of boundary conditions. The results based on the plate theory are compared with the results according to the three-dimensional theory. Section 5.5 presents governing equations and the solution procedure for the creep behavior of a thin-walled pipe subjected to the internal pressure and the bending moment.