Material behavior

The idea of using mathematical modeling for describing materials behavior under loads is well known. Some physical phenomena, which can be observed in materials during testing, have time dependent quantitative characteristics. It gives a possibility to consider them as time series and use well developed models for their analysis [1, 2]. Usually applied... 

Calculations of mutual locations of poles and zeros for these TF models allow to trace dynamics of moving of the parameters (poles and zeros) under increasing loads. Their location regarding to the unit circle could be used for prediction of stability of the system (material behavior) or the process stationary state (absence of AE burst ) [7]. 

R. J. Bowers and J. C. Lippold, Introduction to Materials Behavior, NEMJET, The Ohio State University, Columbus, Ohio, 1996, p. 13. 

K. M. Kratsch, J. C. Schutzler, and D. A. Pitman, Carbon—Carbon 3D Orthogonal Material Behavior Ad AA Paper No. IThA, AlA A-ASME-S AE 13th Stmctural Dynamics and Materials Conference, 1972), American Institute of Aeronautics and Astronautics, New York, 1972. 

Eracture mechanics concepts can also be appHed to fatigue crack growth under a constant static load, but in this case the material behavior is nonlinear and time-dependent (29,30). Slow, stable crack growth data can be presented in terms of the crack growth rate per unit of time against the appHed R or J, if the nonlinearity is not too great. Eor extensive nonlinearity a viscoelastic analysis can become very complex (11) and a number of schemes based on the time rate of change of/have been proposed (31,32). 

M. Roland, Uiscoelasticity—Material Behavior and Measurement Technique, at the CL Meeting of the 147th Rubber Division, Philadelphia, Pa., May 2—5, 1995, American Chemical Society, Washington, D.C. 

The challenge is to understand how the laws of mechanics, physics, and chemistry, and how materials behaviors, control the processes. 

In this chapter, we will review the effects of shock-wave deform.ation on material response after the completion of the shock cycle. The techniques and design parameters necessary to implement successful shock-recovery experiments in metallic and brittle solids will be discussed. The influence of shock parameters, including peak pressure and pulse duration, loading-rate effects, and the Bauschinger effect (in some shock-loaded materials) on postshock structure/property material behavior will be detailed. 

Wallace [15], [16] gives details on effects of nonlinear material behavior and compression-induced anisotropy in initially isotropic materials for weak shocks, and Johnson et ai. [17] give results for infinitesimal compression of initially anisotropic single crystals, but the forms of the equations are the same as for (7.10)-(7.11). From these results it is easy to see where the micromechanical effects of rate-dependent plastic flow are included in the analysis the micromechanics (through the mesoscale variables and n) is contained in the term y, as given by (7.1). 

The lone remaining aspect of this topic that requires additional discussion is the fact that the mechanical threshold stress evolution is path-dependent. The fact that (df/dy)o in (7.41) is a function of y means that computations of material behavior must follow the actual high-rate deformational path to obtain the material strength f. This becomes a practical problem only in dealing with shock-wave compression. 

Underlying all continuum and mesoscale descriptions of shock-wave compression of solids is the microscale. Physical processes on the microscale control observed dynamic material behavior in subtle ways sometimes in ways that do not fit nicely with simple preconceived macroscale ideas. The repeated cycle of experiment and theory slowly reveals the micromechanical nature of the shock-compression process. 

Numerical simulations offer several potential advantages over experimental methods for studying dynamic material behavior. For example, simulations allow nonintrusive investigation of material response at interior points of the sample. No gauges, wires, or other instrumentation are required to extract the information on the state of the material. The response at any of the discrete points in a numerical simulation can be monitored throughout the calculation simply by recording the material state at each time step of the calculation. Arbitrarily fine resolution in space and time is possible, limited only by the availability of computer memory and time. 

The shock-compression pulse carries a solid into a state of homogeneous, isotropic compression whose properties can be described in terms of perfect-crystal lattices in thermodynamic equilibrium. Influences of anisotropic stress on solid materials behaviors can be treated as a perturbation to the isotropic equilibrium state. ... 

In the low-signal limit in which nonlinearities in material behavior are negligible and u/U I the analysis given above can easily be extended to stress pulses of arbitrary form, with the result [65G01]... 

Lithium niobate is strongly ferroelectric, yet the material behavior under elastic shock loading is apparently fully described by nonlinear piezoelec-... 

Beeause of its emphasis on eonsistent thermodynamies, the csq eode does not permit the use of a P-a model for the erush-up behavior of the powder. Thus, it was neeessary to draw upon the experience in the one-dimensional simulation to select appropriate shock-compression materials behaviors. The... 

There is a view developing concerning the accomplishments of shock-compression science that the initial questions posed by the pioneers in the field have been answered to a significant degree. Indeed, the progress in technology and description of the process is impressive by any standard. Impressive instrumentation has been developed. Continuum models of materials behavior have been elaborated. Techniques for numerical simulation have been developed in depth. 

Micromechanics is the study of composite material behavior wherein the interaction of the constituent materials is examined on a microscopic scale to determine their effect on the properties of the composite material. 

Fiber-reinforced composite materials such as boron-epoxy and graphite-epoxy are usually treated as linear elastic materials because the essentially linear elastic fibers provide the majority of the strength and stiffness. Refinement of that approximation requires consideration of some form of plasticity, viscoelasticity, or both (viscoplasticity). Very little work has been done to implement those models or idealizations of composite material behavior in structural applications. 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

An appropriate division of the efforts just mentioned is helped by defining two areas of composite material behavior, micromechanics and macromechanics ... 

The elasticity approaches depend to a great extent on the specific geometry of the composite material as well as on the characteristics of the fibers and the matrix. The fibers can be hollow or solid, but are usually circular in cross section, although rectangular-cross-section fibers are not uncommon. In addition, fibeie rejjsuallyjsotropic, but can have more complex material behavior, e.g., graphite fibers are transversely isotropic. 

Dow and Rosen s results are plotted in another form, composite material strain at buckling versus fiber-volume fraction, in Figure 3-62. These results are Equation (3.137) for two values of the ratio of fiber Young s moduius to matrix shear modulus (Ef/Gm) at a matrix Poisson s ratio of. 25. As in the previous form of Dow and Rosen s results, the shear mode governs the composite material behavior for a wide range of fiber-volume fractions. Moreover, note that a factor of 2 change in the ratio Ef/G causes a factor of 2 change in the maximum composite material compressive strain. Thus, the importance of the matrix shear modulus reduction due to inelastic deformation is quite evident. 

The next problem area of micromechanics is initially very attractive in some respects. We look to the fundamental definition of a composite material made up in this case of, say, a fiber and a matrix and attempt to actually design that material. Let us change the proportions of fibers and matrix so that we get the kind of material behavior characteristics we want. That objective is admirable, but achieving that objective in all cases is not entirely realistic. 

Let s address the issue of nonlinear material behavior, i.e., nonlinear stress-strain behavior. Where does this nonlinear material behavior come from Generally, any of the matrix-dominated properties will exhibit some degree of material nonlinearity because a matrix material is generally a plastic material, such as a resin or even a metal in a metal-matrix composite. For example, in a boron-aluminum composite material, recognize that the aluminum matrix is a metal with an inherently nonlinear stress-strain curve. Thus, the matrix-dominated properties, 3 and Gj2i generally have some level of nonlinear stress-strain curve. 

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