The state of stress

These three forces are then resolved into their nine components in the 1,2 and 3 directions as follows  

The first subscript refers to the direction of the normal to the plane on which the stress acts, and the second subscript to the direction of the stress. In the absence of body torques, the total torque acting on the cube must also be zero, and this implies three equalities  

Therefore, the components of stress are defined by six independent quantities CTii, 022 and CT33, the normal stresses, and o 2, 023 and 0-31, the shear stresses. 

These form the six independent components of the stress tensor L or a,j  

The state of stress at a point in a body is determined when we can specify the normal components and the shear components of stress acting on a plane drawn in any direction through the point. If we know these six components of stress at a given point, the stresses acting on any plane through this point can be calculated. (See Reference [1], Section 67 and Reference [2], Section 47.) 

however, customary when considering yield behaviour to envisage that hydrostatic pressure causes an increase in the yield stress. For this reason the hydrostatic pressurep in Chapters 10 and 11 is defined as/ = — Oxx + Oyy + o z). 

An Introduction to the Mechanical Properties of Solid Polymers L M. Ward and J. Sweeney o 2004 John Wiley Sons, Ltd ISBN 0471 49625 1 (HB) 0471 49626 X (PB) 

The components of stress are therefore defined by six independent quantities the normal stresses o, Oyy and o z, together with the shear stresses oyz and Ozx- It is usual to write these components as the elements of a matrix, which is called the stress tensor Oy (for an explanation of tensors see Appendix 1) 

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The eddy current method allows to evalute the state of stress in ferromagnetic material. The given method is used for determining own stress as well as that formed in effect of outside load. With regard to physical principles of own stress analysis, the dependence between the magnetic permeability and the distance between atomic surfaces is utilized. 

The state of stress in a cylinder subjected to an internal pressure has been shown to be equivalent to a simple shear stress, T, which varies across the wall thickness in accordance with equation 5 together with a superimposed uniform (triaxial) tensile stress (6). 

The laminate stress-analysis elements are affected by the state of the material and, in turn, determine the state of stress. For example, the laminate stiffnesses are usually a function of temperature and can be a function of moisture, too. The laminae hygrothermomechanical properties, thicknesses, and orientations are important in determining the directional characteristics of laminate strength. The stacking sequence... 

Finally, both the state of the material and the state of stress affect the laminate strength evaluation. That is, the actual temperature and moisture conditions influence the laminae strengths. Taken together with the laminae stresses, the laminae strengths and the laminate loads lead to an evaluation of the laminate capabilities. 

The objective of this chapter is to address introductory sketches of some fundamental behavior issues that affect the performance of composite materials and structures. The basic questions are, given the mechanics of the problem (primarily the state of stress) and the materials basis of the problem (essentially the state of the material) (1) what are the stiffnesses, (2) what are the strengths, and (3) what is the life of the composite material or structure as influenced by the behavioral or environmental issues in Figure 6-1 ... 

The state of stress at a point in a structural member under a complex system of loading is described by the magnitude and direction of the principal stresses. The principal stresses are the maximum values of the normal stresses at the point which act on planes on which the shear stress is zero. In a two-dimensional stress system, Figure 13.2, the principal stresses at any point are related to the normal stresses in the x and y directions ax and ay and the shear stress rxy at the point by the following equation ... 

It is also evident that this phenomenological approach to transport processes leads to the conclusion that fluids should behave in the fashion that we have called Newtonian, which does not account for the occurrence of non-Newtonian behavior, which is quite common. This is because the phenomenological laws inherently assume that the molecular transport coefficients depend only upon the thermodyamic state of the material (i.e., temperature, pressure, and density) but not upon its dynamic state, i.e., the state of stress or deformation. This assumption is not valid for fluids of complex structure, e.g., non-Newtonian fluids, as we shall illustrate in subsequent chapters. 

In Section 1.9 it is explained that the state of stress can be described by nine terms. In the above example, the wall shear stress is a particular value of one stress component, that denoted by rrx. In this notation, the second subscript denotes the direction in which the stress component acts, here,... 

The State of Stress in Laminar Shear Flow and its Relation to Flow... 

The state of stress in a flowing liquid is assumed to be describable in the same way as in a solid, viz. by means of a stress-ellipsoid. As is well-known, the axes of this ellipsoid coincide with directions perpendicular to special material planes on which no shear stresses act. From this characterization it follows that e.g. the direction perpendicular to the shearing planes cannot coincide with one of the axes of the stress-ellipsoid. A laboratory coordinate system is chosen, as shown in Fig. 1.1. The x- (or 1-) direction is chosen parallel with the stream lines, the y- (or 2-) direction perpendicular to the shearing planes. The third direction (z- or 3-direction) completes a right-handed Cartesian coordinate system. Only this third (or neutral) direction coincides with one of the principal axes of stress, as in a plane perpendicular to this axis no shear stress is applied. Although the other two principal axes do not coincide with the x- and y-directions, they must lie in the same plane which is sometimes called the plane of flow, or the 1—2 plane. As a consequence, the transformation of tensor components from the principal axes to the axes of the laboratory system becomes a simple two-dimensional one. When the first principal axis is... 

The forces and stresses applied to a body may be resolved in three vectors, one normal to an arbitrarily selected element of area and two tangential. For the yz plane, the stress vectors are a, and on, a, respectively. Six analogous stresses exist for tile other orthogonal orientations, giving a total of nine quantities, of which three exist as commutative pairs (arl = crSr). The state of stress, therefore, is defined by three tensiic or normal components (pxx,oyy. o--f) and three shear or tangential components (crIy,cr.I ,CTy,), The shear components are most readily applicable to the determination of jj and G,... 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

The following conclusions can be drawn a) the AE occurrence rate descriptor is useful for determination of the periods at which the state of stress becomes destructive for the material b) three possible groups of AE occurrence rates may arise during drying first, during the heating of the material second, when the tensional stresses at the surface reach their maximum third, when the tensional stresses in the core reach their maximum after the stress reverse c) the descriptor of total energy reflects accurately the stress state in the material and indicates whether the material suffers the destruction or not. 

The Mohr-Coulomb failure criterion can be recognized as an upper bound for the stress combination on any plane in the material. Consider points A, B, and C in Fig. 8.4. Point A represents a state of stresses on a plane along which failure will not occur. On the other hand, failure will occur along a plane if the state of stresses on that plane plots a point on the failure envelope, like point B. The state of stresses represented by point C cannot exist since it lies above the failure envelope. Since the Mohr-Coulomb failure envelope characterizes the state of stresses under which the material starts to slide, it is usually referred to as the yield locus, YL. 

Figure 8.8. Mohr circle for the state of stresses near the hopper wall.

The average force per unit area is Af,-/AS. This quantity attains a limiting nonzero value as AS approaches zero at point P (Cauchy s stress principle). This limiting quantity is called the stress vector, or traction vector T. But T depends on the orientation of the area element, that is, the direction of the surface defined by normal n. Thus it would appear that there are an infinite number of unrelated ways of expressing the state of stress at point P. 

It turns out, however, that the state of stress at P can be completely specified by giving the stress vector components in any three mutually perpendicular planes passing through the point. That is, only nine components, three for each vector, are needed to define the stress at point P. Each component can be described by two indices ij, the first denoting the orientation of the surface and the second, the direction of the force. Figure 2.3 gives these components for three Cartesian planes. The nine stress vector components form a second-order Cartesian tensor, the stress tensor8 n. ... 

The state of strain in a body is fully described by a second-rank tensor, a strain tensor , and the state of stress by a stress tensor, again of second rank. Therefore the relationships between the stress and strain tensors, i.e. the Young modulus or the compliance, are fourth-rank tensors. The relationship between the electric field and electric displacement, i.e. the permittivity, is a second-rank tensor. In general, a vector (formally regarded as a first-rank tensor) has three components, a second-rank tensor has nine components, a third-rank tensor has 27 components and a fourth-rank tensor has 81 components. 

A few years ago the concept considered was introduced also in the low-temperature chemistry of the solid.31 Benderskii et al. have employed the idea of self-activation of a matrix due to the feedback between the chemical reaction and the state of stress in the frozen sample to explain the so called explosion during cooling observed by them in the photolyzed MCH + Cl2 system. The model proposed in refs. 31,48,49 is unfortunately not quite concrete, because it includes an abstract quantity called by the authors the excess free energy of internal stresses. No means of measuring this quantity or estimating its numerical values are proposed. Neither do the authors discuss the connection between this characteristic and the imperfections of a solid matrix. Moreover, they have to introduce into the model a heat-balance equation to specify the feedback, although they proceed from the nonthermal mechanisms of selfactivation of reactants at low temperatures. Nevertheless, the essence of their concept is clear and can be formulated phenomenologically as follows the... 

A dimple is a concave depression on the fracture surface resulting from microvoid growth in coalescence. As a result of the state of stress during fracture, the dimple may be elongated, oval or equiaxed. 

It is emphasized that defines the state of stress and should not be confused with the orientation angle p. For example, — 45° and p = 45° represents equal biaxial tension acting on a sheet whose ellipse is at 45° to one of the chosen external principal stress axes. This is not equivalent to — 0 and ft = 0, which represents uniaxial tension parallel to the elliptic major axis (which is equivalent to = 90° and p — 90°). Because of symmetry considerations, all possible combinations of elliptic... 

To represent completely the state of stress at each point in a solid requires use of a stress tensor, T. Each element of the stress tensor, Ty, represents the component of force per area acting on the face of an infinitesimal volume element. T allows the determination of the stress in any direction on any plane... 

Also shown on Fig. 1 is the important effect of the state of stress which may be characterized by the negative pressure CTq and the deviatoric stress Sq. These are given by... 

Consider two cases of spherical particles, one of KRO-1 morphology of a volume fraction of 0.23 of randomly wavy PB rods, and the other pure PB — both occupying a volume fraction of 0.22 in PS. We are interested in the craze initiation condition for these two particles at room temperature under a uniaxial tensile stresso. We consider the state of stress at a typical equatorial point A along the particle interface, on the PS side of the particle, as shown in Fig. 33. We determine first by standard methods the elastic properties of the particles and their thermal expansion coefficients, together with the elastic properties and thermal expansion coefficients of the composite matrix as a whole consisting of particles and the majority phase of PS. 

Constitutive equations relate stress tensor to various kinematic tensors and can be used to solve flow and other engineering problems. The development of constitutive equations to describe the state of stress continues to be an active area of research. 

Figure 1.18 The definition of the state-of-stress tensor in terms of force components acting on the faces of a cube. (From Larson 1988, with permission.)...

Figure 4.1 Schematic representation of the state of stress at a point of a material body.

One criterion of yield is the critical condition that has to be met by the applied stress tensor for yield to take place. In order to represent the state of stress of a body it is convenient to choose a suitable set of orthogonal coordinates such that the shear stresses are zero. In this case the stress state is described by three normal stresses [Pg.593]

The problem is specified as the determination of the state of stress and the deformation produced in viscoelastic half-space (z < 0) by a circular punch of radius a whose force is P (Fig. 16.4). As is well known (Ref. 15, p. 25), the displacement of a half-space caused by forces applied to its free surface with the condition of null deformation at infinite distance is given by... 

In the previous sections, the non-Newtonian viscosity rj) was used to characterize the rheology of the fluid. For a viscoelastic fluid, additional coefficients are required to determine the state of stress in any flow. For steady simple shear flow, the additional coefficients are given by... 

Because of the difference in form between Eqs. (2) and (3), the mechanisms of deformation and fracture change with the state of stress. For example, polystyrene yields by shear band formation under ccm ression, but crazes and frachues in a brittle matmer under tensile loading. Chants in failure nwchanian with state of stress are e cially important in particulate conqx tes, since the second phase can alter the local state of stress in the surrounding matrix. 

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