Plastic strain defined

The mechanical concepts of stress are outlined in Fig. 1, with the axes reversed from that employed by mechanical engineers. The three salient features of a stress-strain response curve are shown in Fig. la. Initial increases in stress cause small strains but beyond a threshold, the yield stress, increasing stress causes ever increasing strains until the ultimate stress, at which point fracture occurs. The concept of the yield stress is more clearly realised when material is subjected to a stress and then relaxed to zero stress (Fig. Ih). In this case a strain is developed but is reversed perfectly - elastically - to zero strain at zero stress. In contrast, when the applied stress exceeds the yield stress (Fig. Ic) and the stress relaxes to zero, the strain does not return to zero. The material has irreversibly -plastically - extended. The extent of this plastic strain defines the residual strain. 

Forming Limit Analysis. The ductihty of sheet and strip can be predicted from an analysis that produces a forming limit diagram (ELD), which defines critical plastic strains at fracture over a range of forming conditions. The ELD encompasses the simpler, but limited measures of ductihty represented by the percentage elongation from tensile tests and the minimum bend radius from bend tests. 

For low-cycle fatigue of un-cracked components where (imax or iCT inl am above o-y, Basquin s Law no longer holds, as Fig. 15.2 shows. But a linear plot is obtained if the plastic strain range defined in Fig. 15.3, is plotted, on logarithmic scales, against the cycles to failure, Nf (Fig. 15.4). TTiis result is known as the Coffin-Manson Law ... 

It follows that s = In Aq/A for large plastic strains. The quantity A0/A is a measure of the reduction in area due to deformation. One can define a quantity the true reduction in area (f), in a manner analogous to the definition of true strain ... 

It is possible to make models which give microscopic meaning to the Weibull parameter 8 of the statistical description of fibril breakdown given in Eq. (31), or the related fibril stability — e. The necessary inputs are values for Pg CSi) which is defined as the P at a craze stress corresponding to a plastic strain e = 1. Modi-... 

Linearity. The strain at which the modulus is no longer proportional to the applied stress is always quite small for plastic fats. Defining the linear region as that over which the change in modulus is at... 

The total strain is defined as the sum of elastic-plastic strain from mechanical action, strain due to thermal effects and strain due to suction. Then, the constitutive relationship in a rate form for the coupled THM process in the unsaturated soils is obtained. 

The state of stress just before plastic strain begins to appear is known as the proportional limit, or elastic limit, and is defined by the stress level and the corresponding value of elastic strain. The proportional limit is expressed in pounds per square inch. For load intensities beyond the proportional limit, the deformation consists of both elastic and plastic strains. 

We will discuss here the anisotropic yield behaviour of oriented polymers but there is a need for a few preliminary remarks regarding the topic of yield in general. In describing the deformation of many crystalline materials, especially metals and ceramics, it is often convenient to introduce the idealisation of an elastic-plastic transition . The term elastic is used to describe the components of the strain which are proportional to the applied stresses, and which are completely recovered on removal of the stresses. Plastic strains are observed only for stresses greater than or equal to the yield stress and are not recovered on removal of the stress. The yield stress defines the elastic-plastic transition. 

It has been possible, for metals and ceramic materials, to demonstrate by direct observation the existence of lattice defects called dislocations, using the techniques of transmission electron microscopy. These studies have shown that it is often adequate to assume that dislocation motion is responsible for the observed plastic, or permanent, deformation, and that this motion is negligible at stresses below the yield stress. Although very refined microstrain measurements and internal friction experiments have failed to define a stress range in which dislocation motion is completely absent, there is still a clear distinction for these materials between elastic and plastic strain, both on a macroscopic level, in terms of permanency of deformation, and on a microscopic level in terms of large scale dislocation motion. 

A common example is to identify r with the cumulated plastic deviatoric strain defined... 

Here, tmr H) is the yield stress caused by the apphed magnetic field H, j is the shear rate and r/p is the field-independent plastic viscosity defined as the slope of the measured shear stress against the shear strain rate. 

The significance of the argument at this stage relates to the failure of plastics in the ductile state. Orowan [3] first pointed out that for ductile materials the ultimate stress is entirely determined by the stress-strain curve, i.e. by the plastic behaviour of the material, without any reference to its fracture properties provided that fracture does not occur before the load maximum corresponding to da/dl = a/A is reached. Yield stress is thus an important property in many plastics, and defines the practical limit of behaviour much more than the ultimate fracture, unless the plastic fails by brittle fracture. 

It is important to appreciate that plasticity is different in kind from elasticity, where there is a unique relationship between stress and strain defined by a modulus or stiffness constant. Once we achieve the combination of stresses required to produce yield in an idealized rigid plastic material, deformation can proceed without altering stresses and is determined by the movements of the external constraints, e.g. the displacement of the jaws of the tensometer in a tensile test. This means that there is no unique relationship between the stresses and the total plastic deformation. Instead, the relationships that do exist relate the stresses and the incremental plastic deformation, as was first recognized by St Venant, who proposed that for an isotropic material the principal axes of the strain increment are parallel to the principal axes of stress. 

Rp is the yield strength. Here, we did not specify the plastic strain used to define the yield strength. In metals, iipo.2 is the most common choice (see section 3.2). If the principal stresses are sorted by their size, a Roman subscript is used if they are unsorted, the subscript uses Arabic numbers, see section 2.2.1. 

To take hardening into account, we need to find a quantity that can describe the deformation history of the material. This quantity has to increase during plastic deformation, regardless of the deformation orientation, for, in general, any plastic deformation causes hardening. A frequently used quantity is the so-called equivalent plastic strain To define this strain, we need... 

Since the equivalent plastic strain rate defined this way is positive for all plastic strain rates, the equivalent plastic strain increases for any plastic deformation, regardless of the deformation orientation. 

A simple isotropic hardening law can be written down for the case of linear hardening, defined by the flow stress increasing linearly with the plastic strain. Its rate formulation is [65]. 

According to section 3.2, the yield strength iJpo.2 is defined as the stress corresponding to a plastic strain of 0.2%. For this, some amount of dislocation movement is necessary. 

Acceleration Factors. One of the challenges that must be addressed when attempting to employ Eq. 59.9 directly to assess fatigue life is the determination of the plastic strain and the value of the constant yl. It is crucial to remember that the goal of this analysis is to develop an acceleration transform (or acceleration factor) that can be employed to use known fatigue-life data obtained under controlled laboratory conditions to estimate the number of cycles to failure under field conditions. The acceleration factor AF is defined as the ratio of the number of cycles to fail in the field Nfr to the number of cycles to fail in the lab NfL (see Eq. 59.10). 

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