Quasi-static deformation

Shock loading in most metals and alloys produces greater hardening than quasi-static deformation to the same total strain, particularly if the metal undergoes a polymorphic phase transition, such as is observed in pure iron [1]-[10]. Figure 6.1 compares the stress-strain response of an annealed... 

Figure 6.1. Stress-strain behavior of shock-loaded copper compared to the annealed starting condition illustrating an enhanced flow stress following shock-wave deformation compared to quasi-static deformation (based on an equivalent strain basis).

Many authors have worked on drop deformation and breakup, beginning with Taylor. In 1934, he published an experimental work [138] in which a unique drop was submitted to a quasi-static deformation. Taylor provided the first experimental evidence that a drop submitted to a quasi-static flow deforms and bursts under well-defined conditions. The drop bursts if the capillary number Ca, defined as the ratio of the shear stress a over the half Laplace pressure (excess of pressure in a drop of radius R. Pl = where yint is the interfacial tension) ... 

Quasi-static deformation 63 —, stress-strain curve 78... 

Componsation of (Quasi)Static Doformations. This topic aims to stiffen machine tool structures and thus protect them from thermal and (quasi) static deformations caused by other effects. To this end, researchers must examine sensor-actuator combinations integrated into supporting and coupling components and intelligently modify the feed motion performed by the machine axes. 

Figure 8.7 Schematic of a cyclic shear apparatus used to obtain homogeneous shear. This geometry also allows quasi-static deformation making it possible to obtain position and velocity information on the particles inside tbe bulk using particle index-matching techniques.

Here ij denotes a plastic deformation velocity. Adding the relations (1.10), (1.11), we obtain the quasi-static elastoplastic model... 

In this study, the appearance and evolution sequence of planar slip bands, in addition to a dislocation cell structure with increasing e,, is identical to that observed in quasi-static studies of the effects of stress path changes on dislocation substructure development [27]. The substructure evolution in copper deformed quasi-statically is known to be influenced by changes in stress path [27]. Deforming a sample in tension at 90° orthogonal to the... 

P.S. Follansbee and G.T. Gray III, The Response of Single Crystal and Polycrystal Nickel to Quasi-Static and Shock Deformation, in Advances in Plasticity 1989 (edited by A.S. Khan and M. Tokuda), Pergamon Press, Oxford, 1989, pp. 385-388. 

Dynamic loading in the present context is taken to include deformation rates above those achieved on the standard laboratorytesting machine (commonly designated as static or quasi-static). These slower tests may encounter minimal time-dependent effects, such as creep and stress-relaxation, and therefore are in a sense dynamic. Thus the terms static and dynamic can be overlapping. 

The quasi-static acquisition and interpretation of the force-distance curves is straightforward for elastic materials. The information is generally incomplete and less reproducible for polymers which demonstrate viscoelastic contact, plastic deformation, and ploughing type friction. Moreover, they exhibit a wide spectrum of relaxation times from 105 to 109 Hz [121]. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

The basic theories of elastic deformations associated with various contact forces under static contact conditions have been introduced in the last section. Assuming that an impact process of two solids can be regarded as quasi-static, the theories given in 2.3 are used directly to link the dynamic deformations of the colliding solids with the impact forces. In this section, the collisions of elastic spheres are described. 

Collisions between particles with smooth surfaces may be reasonably approximated as elastic impact of frictionless spheres. Assume that the deformation process during a collision is quasi-static so that the Hertzian contact theory can be applied to establish the relations among impact velocities, material properties, impact duration, elastic deformation, and impact force. 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

According to these considerations, we assume that for quasi-static, cyclic deformations of filler reinforced rubbers up to large strain the total free energy density consists of two contributions ... 

The success of the developed model in predicting uniaxial and equi-biaxi-al stress strain curves correctly emphasizes the role of filler networking in deriving a constitutive material law of reinforced rubbers that covers the deformation behavior up to large strains. Since different deformation modes can be described with a single set of material parameters, the model appears well suited for being implemented into a finite element (FE) code for simulations of three-dimensional, complex deformations of elastomer materials in the quasi-static Emit. 

It is demonstrated that the quasi-static stress-strain cycles of carbon black as well as silica filled rubbers can be well described in the scope of the theoretic model of stress softening and filler-induced hysteresis up to large strain. The obtained microscopic material parameter appear reasonable, providing information on the mean size and distribution width of filler clusters, the tensile strength of filler-filler bonds, and the polymer network chain density. In particular it is shown that the model fulfils a plausibility criterion important for FE applications. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. 

The quadrupolar quasi-static coupling in NF3 — 7.068 MHz— is very close to the value obtained from microwave measurements — 7.07 MHz — this feature may be interpreted as a clue that intermolecular effects are negligible in this compound unless, by a fortuitous coincidence, thes intermolecular effects are cancelled by some other mechanism, such as a molecular deformation in the solid (the coupling is very sensitive to slight variations of the pyramidal angle, as discussed below). 

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