Cellular and Porous Materials in Structures and Processes

Most of foam materials crush in compression, while in tension fail by propagating of single cracks. Rigid polymer foams have a linear-elastic behavior intension up to the brittle fracture, so they can be treated using Linear Elastic Fracture Mechanics. This chapter presents the Linear Elastic Fracture Mechanics parameters and criteria with application for foam materials. Methods for experimental determination for tear strength and fracture toughness are presented. The obtained experimental results are discussed and compared with other published data. Main micromechanical models for predicting fracture toughness of cellular materials are also discussed.

Cellular materials are characterized by a low apparent, density and a discrete micro-structure which is on a distinctly lower length-scale than the one of components made from them. Consequently, the effective behavior of cellular materials is rooted in the mechanical behavior of the struts and/or cell walls on the length scale of individual cells. These lecture notes describe methods of modeling cellular materials by the finite element method. Topics include the setup of micro-mechanical models for open and closed-cell foams as well as sintered hollow sphere foam, the transition between the different mechanical length scales, and the optimization of the density distribution in components ma.de from functionally graded foam.

This chapter gives in the first part a summary of some important elements in continuum mechanics, i.e. the decomposition of the stress tensor in its spherical and the deviatoric part and the use of stress invariants to describe the physical content of the stress tensor. In the next part, the elastic behaviour of isotropic materials based on generalised Hooke’s law is summarised and a notation appropriate for computer implementation is introduced. The constitutive description is then extended to plastic material behaviour and the description based on a yield condition, flow rule and hardening law is introduced. The concept of invariants is consistently applied and explained for the characterisation of yield conditions. A classical simple cubic cell model based on beams (Gibson/Ashby model) is investigated in the next chapter in order to highlight the assumptions and the derivation of the macroscopic material properties (elastic constants and yield stress). In the following, a strategy to determine the influence of the hydrostatic stress on the yield behaviour is proposed and conceptionally realised by a state of plane strain and a state of uniaxial strain. In addition, alternative ways to determine the complete set of elastic constants are shown. The last part covers the implementation of yield conditions into finite element codes. The understanding of the predictor-corrector concept is required to provide new constitutive equations in commercial computational codes.

Thin-walled structures made of foams, composites, sandwiches or functionally graded materials have many applications in aircraft, automotive, and other industries since they combine functionality, high specific stiffness with light weight. The analysis of such structures can be performed on the base of the three-dimensional theory, but often theories with a lower dimension are applied.

The classical plate theory elaborated by Kirchhoff can be applied to thin plates made of classical materials like steel. But this approach is connected with various limitations (e.g., constant material properties in the thickness direction). In addition, mathematical inconsistencies (for example, the order of the governing equation does not correspond to the number of boundary conditions) exist. Many suggestions for improvements of the classical plate theory are made. The engineering direction of improvements was ruled by applications (e.g., the use of laminates or sandwiches as the plate material), and new hypotheses are introduced. In addition, some mathematical approaches like power series expansions or asymptotic integration techniques are applied.

A conceptional different direction is the direct approach in the plate theory which is applied to plates made of the above mentioned advanced materials. Within this theory static and the vibration problems are solved assuming linear-elastic and linear-viscoelastic behavior. The material properties are changing in the thickness direction only. It is shown that the results based on the direct approach differ significantly from the classical estimates which can be explained by the non-classical estimation of the transverse shear stiffness. In the last part a general linear six-parametric theory is presented. The kinematics of the plate is described by using the vector of translation and the vector of rotation as the independent variables. The relations between the equilibrium conditions of a three-dimensional micropolar plate-like body and the two-dimensional equilibrium equations of the deformable surface are established.

The present chapter mainly deals with several non-standard issues in the theory of plasticity for traditional metals as well as porous and powder metals. The flow formulation is adopted throughout the chapter. All the material models considered are rigid plastic, i.e. the elastic portion of the strain rate tensor is neglected. Assuming a rigid/perfectly plastic material model, it is shown that the velocity fields adjacent to surfaces of maximum friction must be describable by non-differentiable functions where the equivalent strain rate approaches infinity. This result is extended to the double-shearing model. Qualitative behavior of solutions based on various models of pressure-dependent plasticity is considered by means of problems permitting closed-form solutions. In particular, such features of the solutions as non-uniqueness, non-existence, singularity and rigid zones are emphasized. One possible application of the aforementioned singular solutions to describing intensive plastic deformation in a narrow layer near friction surfaces is shortly discussed.

Cellular materials are used as impact energy absorbers due to their large densification strain at the plateau stress during the plastic compression. For a cellular rod struck by a rigid object, the critical impact velocity is determined. If the impact velocity is higher than the critical impact velocity, the elastic wave will be followed by plastic shock waves. Plastic shock waves and shock arrest are investigated analytically for longitudinal impacts. Shock behaviors are characterized for material design purpose and will be used for impact protection.