Matrix stress

In this array, the stress components in the first row act on a plane perpendicular to the X axis, the stress components in the second row act on a plane perpendicular to the y axis, and the stress components in the third row act on a plane perpendicular to the z axis. The stress matrix of Eq. (5.6) is symmetric, that is, the complementary shear stress components are equivalent—for example, % = Xyx, and so on. 

Although the two approaches are very similar, the value of A Tc in Boccaccini s model does not depend on the interfacial shear strength t, as a result of the model chosen for the value of matrix cracking stress. Blissett et al. (1997) suggested that their method was valid for the UD material providing that some key parameters (interfacial shear stress, matrix fracture energy) were determined independently. 

Chiquet, M. Regulation of extracellular gene expression by mechanical stress. Matrix Biol. 1999 18 417 126. 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require cjxy=(jyx, stress components in the symmetric stress matrix are only six. Similarly there are only six independent strain components. Therefore there can only be six stress and six strain components for an elastic body which has unequal elastic responses in x, y and z directions as in a completely anisotropic solid. The representation of elastic properties become simple by following the well known Einstein convention. The subscript xx, yy, zz, yz, zx and xy are respectively represented by 1, 2, 3, 4, 5 and 6. Therefore Hooke s law may now be written in a generalized form as. 

From the above equations it is possible to calculate the size of the largest drop that exists in a fluid undergoing distortion at any shear rate. In these equations, the governing parameters for droplet breakup are the viscosity ratio p (viscosity of the dispersed phase to that of the matrix) the type of flow (elongational, shear, combined, etc.) the capillary number Cfl, which is the ratio between the deforming stress (matrix viscosity x shear rate) imposed by the flow on the droplet and the interfacial forces a/R, where ais the interfacial... 

One major diflerence between the two mechanisms of deformation is illustrated in Fig. 5.7, which shows a failure envelope for PMMA under biaxial loading. The pure-shear line, defined by er,j = —<722> marks the boundary between hydrostatic compression and hydrostatic tension. Below this line, crazing and other hole-forming processes do not take place because the pressure component of the stress matrix tends to reduce rather than to increase volume above the line, crazing is the principal mechanism of failure. 

As mentioned in Sections 1 and 3, a rheological constitutive equation relates the components of the three-dimensional extra-stress (matrix) to the components of the strain (matrix) or the rate of strain (matrix) in any given flow field. Extensional viscosity data of the kind shown in Figure 5 should therefore be explainable on the basis of a proper constitutive equation. Unfortunately, there is no simple constitutive equation that accurately predicts the behavior of a polymer melt in all the commonly encountered flow situations. Some equations do a remarkably good job... 

The above equation is generalized to three dimensions by replacing the stress component T by the stress matrix and the strain component 7 by the strain matrix. This procedure works well as long as one confines oneself to small strains. For large strains, the time derivative of the stress requires special treatment to ensure that the principle of material objectivity(78) is not violated. This principle requires that the response of a material not depend on the position or motion of the observer. It turns out that one can construct several different time derivatives all of which satisfy this requirement and also reduce to the ordinary time derivative for infinitesimal strains. By experience over many years, it has been found (see Chapter 3 of Reference 79) that the Oldroyd contravariant derivative also called the codeformational derivative or the upper convected derivative, gives the most realistic results. This derivative can be written in Cartesian coordinates as(79)... 

The six independent components of the symmetric stress matrix are recorded here for later reference. The stress field arising from the edge component of the Burgers vector is conveniently represented in terms of the Airy stress function... 

Using equilibrium, it is easy to show that the stress matrix is symmetric or,... 

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