Elastic-plastic bending

J.G. Williams, H. Hadavinia, B. Cotterell, Anisotropic elastic and elastic-plastic bending solutions for edge constrained beams. Int. J. Solids Stmct. 42, 4927-4946 (2005)... 

When Afo > Af then elastic-plastic bending occurs which may be analysed using plastic bending theory, and it is convenient to describe the analysis via the non-work hardening case [28-30,44] in which any bending moment M > M may be written as ... 

Figure 20. Elastic-Plastic Solution for Bending of Blast Loaded Beams. (Reprinted with permission from ref. 15. Copyright 1983 Elsevier Science.)...

One of the major areas of RubCon application is in structures that operate in aggressive environments where crack resistance of the material is important. Our experiments have shown that steel fibers show the greatest opportunity to increase crack resistance and allow production of a material with high elastic-plastic properties. For this reason, we have undertaken research on the influence of fiber reinforcement ratio and aspect ratio on RubCon strength at compression, tension, and bend. 

FIGURE 41.8 Representative stress-strain curve for a cellular solid. The plateau region for compression in the case of elastomeric foam (a rubbery polymer) represents elastic buckling for an elastic-plastic foam (such as metallic foam), it represents plastic yield, and for an elastic-brittle foam (such as ceramic) it represents crushing. On the tension side, point A represents the transition between cell wall bending and cell wall alignment In elastomeric foam, the alignment occurs elastically, in elastic plastic foam it occurs plastically, and an elastic-brittle foam fractures at A. 

Especially at three-point bending impact loading the elastic modulus (dynamic modulus) is proportional to the slope of the linear part of the load - deflection diagram of the unnotched samples assuming that the material behaviour can be approximated to be linear-elastic-plastic in that case (for more information see [13Gre] and [OlGre]). 

Different materials can be used, such as nylon, polyester (TS), and epoxy, but TS polyurethane (PUR) is predominantly used. Almost no other plastic has the range of properties of PUR—a modulus of elasticity in bending of 200 to 1,400 MPa and heat resistance from 90°C to over 200°C (the higher values are for chopped glass fiber reinforced RIM, or RRIM see Chapter... 

Fundamental Properties of Polymers, Metals and Ceramics (e.g., strength in compression, tension and bending elasticity/plasticity failure mechanisms phase diagrams transition temperatures surface roughness hydrophobicity) Mechanical Properties of Biological Tissues (e.g., elastic viscoelastic, hysteresis, creep, stress relaxation)... 

Figure 14.19. (a) Elastic bending causing detachment of fitm. (b) Plastic bending. 

Hart-Smith (references 5.25, 5.26, 5.30 and 5.31) has conducted extensive studies of bonded joints using the elastic—plastic model for the adhesive. He has covered the analysis of lap, strap, scarf and step-lap joints. He has modified the load eccentricity induced peel stress approach by using a modified bending stiffness. He has studied the effects of non-uniform adhesive thickness, adhesive non-uniform moisture absorbtion and defects in the bondline. He has also included thermal stresses in his models. 

As mentioned above, some ceramics may be made plastic even at room temperature. As such, producing them in nano-size form may induce low-temperature plasticity. Figure 4.4 is an illustration of in situ TEM observation of the Plastic deformation features of a SiC nanowire [henceforth NW]. The in situ SiC NW elastic, elastic-plastic and Plastic Deformations were conducted using an ultra-HRTEM by bending SiC NWs via the mechanical force produced by the TEM specimen-supporting grid under the irradiation of an electron beam. Figure 4.4... 

Fig. 4.5 SiCNW morphologies of the elastic bend (a, c) and the final plastic bend of (b) and (d). The location indicated by the white arrow in (c) is the triggering point of plastic defcmnation. e is an enlarged TEM toight field image. The black arrows in (e) show the top and bottom deformation-induced amcnphous zones. Re ninted with permission from Han et al. [5]. With kind permission of the American Chemical Society...

ANSYS 5.3 (Swanson Analysis, 1994) nonlinear elastic-plastic large deformation finite-element (FE) models were made of the bellows and assembly illustrated in Fig. 42.8. The bellows and 0.71-m OD stub pipes were fabricated from 316L SS the 0.508-m OD dogleg pipe bridge was made from four sections of 4.8-mm-thick 304L SS pipe with two 80° miter bends (Foley, 1974a). 

The evolution of the Kg factor has been based on simple analyses and limited test data. Krempl studied the low cycle fatigue behavior of notched cylinders and plates of three pipe materials.Tagart formulated design rules for the Nuclear Piping Code B31.7 based partly on these tests, which led to Eq. (11.3b). Langer performed elastic-plastic analyses of a beam in bending and a tapered bar in tension which led to Eq. (11.3c). 

Figure C.2 Elastic-plastic and fully plastic bending of a beam.

The RIM process predominantly uses TS polyurethane (PUR) plastics. Others include nylon, TS polyester, and epoxy. PUR offers a large range of product performance properties. As an example PUR has a modulus of elasticity in bending of 200 to 1400 MPa (29,000 to 203,000 psi) and heat resistance in the range of 90 to 200°C (122 to 392°F). The higher values are obtained when glass-fiber reinforces the PUR (also with nylon, etc.). The reinforced RIM process is called RRIM or... 

Rice was the first to recognize the potential use of J-integral in elastic-plastic fracture mechanics, after Eshelby was apparently the first to derive the integral [1 3]. The J-integral of a critical crack, Jc, is actually equivalent to Rc defined in Eq. (2) combined with the schematic procedure in Fig. 2. For the case of a plate containing a deep notch and subjected to pure bending, Rice found that [1 3]... 

Fracture then occurs in an elastic-plastic rather than a purely elastic strain field. For example, in the double cantilever beam test (Figure 12.4) the maximum bending stress is CT = 7>GcE where b is the width of the beam. Hence, the critical width for the transition from yielding to brittle failure is... 

For the unbonded strip which is used in the calibration tests, then Gc = 0 and for elastic bending dr = 0 . However, for the case of plastic bending then, again Gc = 0, but now G will increase as D is increased until it becomes constant at a value determined by r, say Gj. Thus, the calibration curve becomes ... 

The elastic-plastic solution, presented in Fig.5, indicates (1) the flatness of the (average) stress distribution near the top and the bottom of the beam, (2) the bending moment as a result of the bending stresses is smaller because the moment arms of the applied forces at the free end become smaller due to large deformation, and (3) the spread of stresses within an element becomes pronounced especially when the magnitude of the average stress becomes large. 

Many techniques have been developed to measure the Young s modulus and the stress of the mesoscopic systems [12, 13]. Besides the traditional Vickers microhardness test, techniques mostly used for nanostructures are tensile test using an atomic force microscope (AFM) cantilever, a nanotensile tester, a transmission electron microscopy (TEM)-based tensile tester, an AFM nanoindenter, an AFM three-point bending tester, an AFM wire free-end displacement tester, an AFM elastic-plastic indentation tester, and a nanoindentation tester. Surface acoustic waves (SAWs), ultrasonic waves, atomic force acoustic microscopy (AFAM), and electric field-induced oscillations in AFM and in TEM are also used. Comparatively, the methods of SAWs, ultrasonic waves, field-induced oscillations, and an AFAM could minimize the artifacts because of their nondestructive nature though these techniques collect statistic information from responses of all the chemical bonds involved [14]. 

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