Inelastic strain

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

The inelastic strain as an internal state variable is obtained from a linear evolution equation formulated with respect to the intermediate configuration ... 

Figure 3 presents the time evolution of tangential stresses at the surface and the core of the cylinder according to the viscoelastic and elastic models. The stress reverse can be explained as follows when the body dries, the drier surface attempts to shrink but is restrained by the wet core. The surface is stressed in tension and the core in compression and inelastic strain occurs. Later, under a surface with reduced shrinkage, the core dries and attempts to shrink causing the stress state to reverse [4],... 

The initial intent of this review is to address the mechanisms of stress redistribution upon monotonic and cyclic loading, as well as the mechanics needed to characterize the notch sensitivity.5 13 This assessment is conducted primarily for composites with 2-D reinforcements. The basic phenomena that give rise to inelastic strains are matrix cracks and fiber failures subject to interfaces that debond and slide (Fig. 1.1).14-16 These phenomena identify the essential constituent properties, which have the typical values indicated in Table 1.1. 

When d a) has been established in this manner, stress-strain curves can be simulated for 1-D materials. Based on this approach, simulations have been used to conduct sensitivity studies of the effects of constituent properties on the inelastic strain. Examples (Fig. 1.27) indicate the spectrum of possibilities for CMCs. 

Relatively complete matrix cracking and inelastic strain measurements have been made on two unidirectional CMCs 5,18,48,89 SiC/CAS, as well as SiC/SiC (produced by CVI). The stress-strain curves for these two materials (Fig. 1.28) indicate a contrast in inelastic strain capability, which demand... 

Using this simplified approach, simulations of stress-strain curves have been conducted.89,97 These curves have been compared with experimental measurements for several 2-D CMCs. One result is summarized in Fig. 1.35. It is apparent that the simulations lead to somewhat larger flow strengths than the experiments, especially at small inelastic strains. To address this discrepancy, further modeling is in progress, which attempts to couple the behavior of the tunnel cracks with the matrix cracks in the 0° plies. 

Before concentrating on specific factors and mechanisms that influence crazing, it is instructive to take account of some general principles on how inelastic strain and toughness result from crazing. A similar but more restricted discussion has been given earlier by Brown... 

Here, e. and Epj are homogeneous inelastic strains defined in the respective phase domains L and are dimensionless constants satisfying + /ps Lj =1(1 the identity tensor of rank 4) M, and Mj are constants having the same dimension as the elastic modulus and satisfying +/pszM2 = 0 N, and Nj are also dimensionless constants relating to... 

Thus, the center deflection of the beam is a product of the maximum stress at the outer elements of the beam, and at the center a geometrical term (L / 6h)) divided by Young s modulus E(i), which is now time-dependent because of the viscoelastic relaxations in the beam, and decreases with time under stress as additional inelastic strains build up. However, the stresses in the viscoelastic beam continue to remain unaltered since they depend only on the applied forces and moments, which remain constant. 

J. C. McNulty, F. W. Zok, G. M. Genin, and A. G. Evans, Notch-sensitivity of fiber-reinforced ceramic-matrix composites effects of inelastic straining and volume dependent strength, J.Am. Ceram. Soc., 82 [5] 1217-1228 (1999). 

The material constant in Equation (5.3) is called the amorphous pre-exponential inelastic strain rate and fP is the effective equivalent inelastic deformation rate of a glassy polymer subjected to the effective equivalent shear stress, t, which is defined at the absolute temperature, 0, as follows ... 

Equation (5.3) represents the magnitude of the amorphous inelastic strain rate, while the direction of the amorphous inelastic flow rate,, is governed by the deviatoric driving stress, s y. The following flow rule is proposed for the inelastic deformation in the amorphous phase [88] ... 

The return mapping techniques in inelastic solutions are a natural consequence of splitting the total strain into elastic and inelastic strains. Let tensor uy, an incremental field to describe the deformation, and its gradient, Vt/,y, show the deformation rate. The solution is implemented by the following steps. Step 1 introduces a loading condition such as F = (/, 4- V ,t)Fj." where ly is the unity second-rank tensor and the superscripts n and n 4-1 represent, respectively, the previous and current load steps. In step 2 the material is elastically stretched... 

Averaged experimental creep compliance data are represented by dots in Figure 12.12 as a function of log time. The tests show that even after 14 months the creep continued and deformations kept increasing. After tests of 10000 h the compliance of the materials examined exceeds the instantaneous elastic compliance by a factor varying from 3.8 (for pure PP) to 3.2 (for the 80% PP + 20% PLC blend). This shows that the inelastic strain of these materials manifests itself substantially methods of accelerated testing and prediction have to be subjected to very detailed scrutiny. 

Is assumed, where is the elastic strain and denotes for the time being all forms of irreversible (i.e. inelastic) strains. In order to satisfy the CD inequality (11), a common practice is to assume that ip so that Ip scalar... 

Houtman, J.L., Inelastic strains from thermal shock. Machine Design, 46,190-194,1974. 

It is evident from the discussion presented in Section 7.4 that observation of substrate curvature, interpreted as mean stress in a film bonded to an elastic substrate, during temperature cycling provide a convenient framework with which the inelastic strain relaxation characteristics of thin film materials can be inferred. Indeed, substrate curvature measurement made in the course of temperature cycling is among the most common experimental methods... 

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