Principal stress invariants

Isotropic hyperelastic materials For this model, the strain energy density function is written in terms of the principal stress invariants I, h, h). Equation 1 becomes... 

Recall from the discussion in Section 2.5.4 that the stress tensor, like the strain-rate tensor, has certain invariants. For any known stress tensor, these invariant relationships can be used to determine the principal stresses. 

Thermal conductivity Mass density Isochoric specific heat Tensile yield stress Mean stress First stress invariant Principal stresses... 

Then, to have all measures in the same units, we take (sumn) and (summ). To emphasize the difference between the invariants and the principal stresses, we use the symbols S, S, and Thus... 

For numerical examples, principal stresses are related to invariants thus ... 

Appendix 8C describes a procedure for specifying the stress state at a point by means of three invariants rather than three principal stresses the invariants are totally free of any link to directions. The point is also made that the Laplacian operator has the same attribute any first derivative with respect to position, d/dx, is a vector, and a single second derivative d /8x is also linked to the direction x, but the Laplacian has no links to direction... 

This model is based on the mean features of the Mohr-Coulomb model and is expressed with stress invariants [Maleki (1999)] instead of principal stresses. Until plasticity is reached, a linear elastic behaviour is assumed. It is fully described by the drained elastic bulk and shear moduli. The yield surface of the perfectly plastic model is given by equation 7. Function 7i(0) is chosen so that the shape of the criterion in the principal stress space is close to the Lade criterion. 

The value of E depends upon the values of the elonents in the stress and strain tensors. Under plane stress conditions, one of die principal stresses is zero and E is equal to Young s modulus, E. However, under plane strain conditions, the strain in one of the principal axes is zero and E = E/(l — v ) where v is Poisson s ratio. For most polymers 0.3 < v < 0.5 and the values of both Gic and Kic invariably are much greater when measured in plane stress. For the purposes both of toughness comparisons and component design, die plane strain values of Gic and Kic are preferred because th are the minimum values fm any given material. In order to achieve plane strain conditions, the following criteria need to be satisfied ... 

Jl, J2 and J3 are invariant coefficients of stress and, therefore, independent of the choice of coordinate system. liquation (1.23) is a third-order equation for stress, a, with three roots, ttmt. Iso termed principal stresses . Such solutions may be inserted into Eq. (1.22c) by utilizing the relation ... 

The boundary of the crazed region coincided to a good approximation with contour plots showing lines of constant major principal stress 0, as shown in Figure 12.14(b) where the contour numbers are per unit of applied stress. At low applied stresses it is not possible to discriminate between the contours of constant cTi and contours showing constant values of the first stress invariant I = Oi+ a. However, the consensus of the results is in accord with a craze-stress criterion based on the former rather than on the latter and, as we have seen, the direction of the crazes is consistent with the former. 

In isotropic materials, the yield surface must not depend on the orientation of the load. Thus, the function / describing the yield surface can only contain those parts of the deviatoric stress tensor that do not change during coordinate transformations. This is already ensured if the principal stresses are used because the hydrostatic stress (Thyd is also coordinate invariant. 

Apart from the principal stresses, a stress tensor has another set of invariants, called the principal invariants Ji, J2, and J3 (see appendix A. 7). They are defined as... 

Voorhees analysis assumes that the creep-rupture life of a vessel under complex stressing is controlled by an equivalent stress, J, termed the shear-stress invariant. This average stress is also known as the octahedral shear stress, the effective stress, the intensity of stress, and the quadratic invariant. The theory for the biaxial-stress condition was developed by Von Mises (205), and this theory was further developed to apply to the triaxial-stress condition independently by Hencky (206, 207, 208) and by Huber (209). A derivation of the relationship between the equivalent stress, /, and the three principal stresses, /i, /2, and/s where /i > /2 > /s was given by Eichinger (210). The relationship between these stresses is ... 

Voorhees analyzed the experimental data obtained on notched-bar samples tested at elevated temperatures (211, 214) and on pressure vessels tested at elevated temperatures and high pressures (204, 211), He concluded that the equivalent stress, / (shear-stress invariant) was more useful in correlating these experimental data than the maximum principal stress or the maximum shear stress. Voorhees also reviewed the lirork of other investigators in this lietd and condudefl that these studies alim indicated the u ful-ness of the equivalent stress (211). 

Since in a simple tension test, the lateral surfaces of the specimen are supposed to be unloaded, the principal stresses corresponding to the directions 2 and 3 vanish. For deformation (59) the strain invariants become ... 

The simplest theories of plasticity exclude time as a variable and ignore any feature of the behaviour, which takes place below the yield point. In other words, we assume a rigid-plastic material whose stress-strain relationship in tension is shown in Figure 12.9. For stresses below the yield stress there is no deformation. Yield can be produced by a wide range of stress states, not just simple tension. In general, it must therefore be assumed that the yield condition depends on a function of the three-dimensional stress field. In a Cartesian axis set, this is defined by the six components of stress, an, a22, <733, ayi, < 22 and a 31. However, the numerical values of these components depend on the orientation of the axis set, and it is crucial that the yield criterion be independent of the observer s chosen viewpoint the yield criterion must be invariant with respect to changes in the axis set. It is often convenient to make use of the principal stresses. If the material itself is such that its tendency to yield is independent of the direction of the stresses - that is if it is isotropic -then the yield criterion is a function of the principal stresses only... 

This general expression first accounts for the principal of causality by stating that the state of stress at a time t is dependent on the strains in the past only. Secondly, by using the time dependent Finger tensor B, one extracts from the fiow fields only those properties which produce stress and eliminates motions like translations or rotations of the whole body which leave the stress invariant. Equation (7.128) thus provides us with a suitable and sound basis for further considerations. 

There is only one commonly used invariant of a vector its magnitude. However there are three possible invariant scalar functions of a tensor. For the stress tensor we can give these three invariants physical meaning through the principal stresses. 

Determine the invariants and the magnitudes and directions of the principal stresses for the stress tensor given in Example 1.2.2. Check the values for the invariants using the principal stress magnitudes. 

At the microlevel, initial damage in bonded joints is defined by whichever of the two strain invariants reaches its critical value first. At the ends of the overlap in a badly designed bonded joint or test coupon, the dilatational Ji failure mechanism would always occur first. There are no significant peel stresses in the interior of the joint, so the distortional invariant will be exceeded first. The visual consequences of this failure may seem to be one of dilatation because the first indication is the formation of a series of hackles formed at roughly 45° from the adherends being bonded together, as shown in Fig. 44.3. (Exactly the same model applies for in-plane-shear failures in the matrix between parallel fibers in composite laminates.) These are, in reality, failures by distortion, which result in tensile fractures perpendicular to the highest principal stress. 

Maier [10] represented first the plastic strain to fracture against the ratio of the average of the three principal stresses and of the Misesian yield stress (see Equation 8). The importance of Ae Maier s stress parameter is that it is the ratio of two principal components of the stress state, i.e. it is the quotient of the hydrostatic sphere-tensor and the flow stress, which is connected to the second invariant of the deviator-tensor (see Equation 15). 

If the invariants are known for some arbitrary strain-rate state, then it is clear that the three equations above form a system of equations from which the principal strain rates can be uniquely determined. This analysis is explained more fully in Appendix A. Using the principal axes greatly facilitates subsequent analysis, wherein quantitative relationships are established between the strain-rate and stress tensors. 

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