Stress coordinate transformation

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

The objectivity of the spatial stress rate relation (5.154) may be investigated by applying the coordinate transformation (A.50) representing a rotation and translation of the coordinate frame. The spatial strain and its convected rate are indifferent by (A.58) and (A.62). So are the stress and its Truesdell rate. It is readily verified from (5.151), (5.152), and the fact that K has been assumed to be invariant, that k and its Truesdell rate are also indifferent. Using these facts together with (A.53) in (5.154) with c and b given by (5.155)... 

The dynamical history of stress-relaxation in a star-linear blend begins life in just the same way as a star-star blend,because when t r gp the linear chain relaxation is dominated by pathlength fluctuation and behaves as a two-arm star with M =Mii /2. So very early Rouse fluctuation (Eq. 25) crosses over to activated fluctuation in self-consistent potentials. These are calculated via the coordinate transformation used in the star-star case above. For example, the effective potential for the star component in this regime is... 

The stress tensor thus allows us to completely describe the state of stress in a continuum in terms of quantities that depend on position and time only, not on the orientation of the surface on which the stress acts. More precisely, the stress tensor should be referred to as a tensor of second order or tensor of second rank because its components transform as squares of the coordinates. We shall, however, simply use the term tensor, since tensors of order higher than second generally are not dealt with in fluid mechanics. We note in passing that a vector is a tensor of first order, its components transforming like the coordinates themselves, and a scalar is a tensor of zeroth order, a scalar being invariant under coordinate transformation. 

By changing the coordinate system e e2 into (Fig. 4.13), according to equation (4.6), the stress tensor transforms as... 

If we change the coordinate system, the components of the stress tensor a (its matrix representation) change, but it still describes the same state of stress. The transformation rules are detailed in appendix A.5. 

The stresses (cij) and ffi j ) describe the same state of stress, Therefore, a coordinate transformation must transform (ffij) to ([Pg.48]

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. 

According to equation (3.25), the hydrostatic stress is (Thyd = di/3. This yields the invariance of the hydrostatic stress with respect to coordinate transformations, which was already used above. 

Other less well-known types of nonlinearities include interaction and intermode . In the former, stress-strain response for a fundamental load component (e.g. shear) in a multi-axial stress state is not equivalent to the stress-strain response in simple one component load test (e.g. simple shear). For example. Fig. 10.3 shows that the stress-strain curve under pure shear loading of a composite specimen varies considerably from the shear stress-strain curve obtained from an off-axis specimen. In this type of test, a unidirectional laminate is tested in uniaxial tension where the fiber axis runs 15° to the tensile loading axis. A 90° strain gage rosette is applied to the specimen oriented to the fiber direction and normal to the fiber direction and thus obtain the strain components in the fiber coordinate system. Using simple coordinate transformations, the shear response of the unidirectional composite can be found (Daniel, 1993, Hyer, 1998). At small strains in the linear range, the shear response from the two tests coincide. 

At this point, we recall from elementary mechanics of materials the transformation equations for expressing stresses in an x-y coordinate system in temis of stresses in a 1-2 coordinate system. 

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

As an alternative to the foregoing procedure, we can express the strains in terms of the stresses in body coordinates by either (1) inversion of the stress-strain relations in Equation (2.84) or (2) transformation of the strain-stress relations in principal material coordinates from Equation (2.61),... 

Compare the transformed orthotropic compliances in Equation (2.88) with the anisotropic compliances in terms of engineering constants in Equation (2.91). Obviously an apparenf shear-extension coupling coefficient results when an orthotropic lamina is stressed in non-principal material coordinates. Redesignate the coordinates 1 and 2 in Equation (2.90) as X and y because, by definition, an anisotropic material has no principal material directions. Then, substitute the redesignated Sy from Equation (2.91) in Equation (2.88) along with the orthotropic compliances in Equation (2.62). Finally, the apparent engineering constants for an orthotropic iamina that is stressed in non-principal x-y coordinates are... 

The foregoing example is but one of the difficulties encountered in analysis of orthotropic materials with different properties in tension and compression. The example is included to illustrate how basic information in principal material coordinates can be transformed to other useful coordinate directions, depending on the stress field under consideration. Such transformations are simply indications that the basic information. 

For each of the failure criteria, we will generate biaxial stresses by off-axis loading of a unidirectionally reinforced lamina. That is, the uniaxial off-axis stress at 0 to the fibers is transformed into biaxial stresses in the principal material coordinates as shown in Figure 2-35. From the stress-transformation equations in Figure 2-35, a uniaxial loading obviously cannot produce a state of mixed tension and compression in principal material coordinates. Thus, some other loading state must be applied to test any failure criterion against a condition of mixed tension and compression. 

In applications of the maximum stress criterion, the stresses in the body under consideration must be transformed to stresses in the principal material coordinates. For example, Tsai [2-21] considered a unidirec-tionally reinforced composite lamina subjected to uniaxial load at angle 6 to the fibers as shown in Figure 2-35. The biaxial stresses in the principal material coordinates are obtained by transformation of the uniaxial stress, a, as... 

As With the shear strength, the maximum shear strain is unaffected by the sign of the shear stress. The strains in principal material coordinates, 1- yi2 be found from the strains in body coordinates by transformation before the criterion can be applied. 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

In conclusion, classical lamination theory enables us to calculate forces and moments if we know the strains and curvatures of the middle surface (or vice versa). Then, we can calculate the laminae stresses in laminate coordinates. Next, we can transform the laminae stresses from laminate coordinates to lamina principal material directions. Finally, we would expect to apply a failure criterion to each lamina in its own principal material directions. This process seems straightfonward in principle, but the force-strain-curvature and moment-strain-curvature relations in Equations (4.22) and (4.23) are difficult to completely understand. Thus, we attempt some simplifications in the next section in order to enhance our understanding of classical lamination theory. 

Vectors are commonly used for description of many physical quantities such as force, displacement, velocity, etc. However, vectors alone are not sufficient to represent all physical quantities of interest. For example, stress, strain, and the stress-strain iaws cannot be represented by vectors, but can be represented with tensors. Tensors are an especially useful generalization of vectors. The key feature of tensors is that they transform, on rotation of coordinates, in special manners. Tsai [A-1] gives a complete treatment of the tensor theory useful in composite materials analysis. What follows are the essential fundamentals. 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

The next level of complexity in the treatment is to orient the apphed stress at an angle, 6, to the lamina fiber axis, as illustrated in Figure 5.119. A transformation matrix, [T], must be introduced to relate the principal stresses, o, 02, and tu, to the stresses in the new x-y coordinate system, a, ay, and t j, and the inverse transformation matrix, [T]- is used to convert the corresponding strains. The entire development will not be presented here. The results of this analysis are that the tensile moduli of the composite along the x and y axes, E, and Ey, which are parallel and transverse to the applied load, respectively, as well as the shear modulus, Gxy, can be related to the lamina tensile modulus along the fiber axis, 1, the transverse tensile modulus, E2, the lamina shear modulus, Gu, Poisson s ratio, vn, and the angle of lamina orientation relative to the applied load, 6, as follows ... 

Developing the stress-strain-rate relationships is greatly facilitated in the principal coordinate directions. Since isotropy requires that the constitutive relationships be independent of coordinate orientation, the principal-direction relationships can be transformed to any other coordinate directions. At every point in a flow field the strain-rate and stress state... 

The principal coordinates provide an extraordinarily useful conceptual framework within which to develop the fundamental relationships between stress and strain rate. For practical application, however, it is essential that a common coordinate system be used for all points in the flow. The coordinate system is usually chosen to align as closely as possible with the natural boundaries of a particular problem. Thus it is essential that the stress-strain-rate relationships can be translated from the principal-coordinate setting (which, in general, is oriented differently at all points in the flow) to the particular coordinate system or control-volume orientation of interest. Accomplishing this objective requires developing a general transformation for the rotation between the principal axes and any other set of axes. 

If the principal stresses had had shear components, which by definition they don t, then, in general, those shear components would have contributed to the stress vector on the rotated z plane. The a vector completely defines the stress state on the rotated z face. However, our objective is to determine the stress-state vector on the z face that aligns with the rotated coordinate system (z,r,G) x--, x-r, and x-e. The a vector itself has no particular value in its own right. Therefore one more transformation from cs to r is required ... 

In addition to representing a vector in a rotated coordinate system, the matrix of direction cosines can also be used to transform a tensor (e.g., the stress tensor) into a rotated coordinate system as... 

Referring to Fig. A.2, assume that the principal coordinates align with z, r, and O. The unit vectors (direction cosines) just determined correspond with the row of the transformation matrix N. Thus, if the principal stress tensor is... 

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... 

We have mentioned above the tendency of atoms to preserve their coordination in solid state processes. This suggests that the diffusionless transformation tries to preserve close-packed planes and close-packed directions in both the parent and the martensite structure. For the example of the Bain-transformation this then means that 111) -> 011). (J = martensite) and <111> -. Obviously, the main question in this context is how to conduct the transformation (= advancement of the p/P boundary) and ensure that on a macroscopic scale the growth (habit) plane is undistorted (invariant). In addition, once nucleation has occurred, the observed high transformation velocity (nearly sound velocity) has to be explained. Isothermal martensitic transformations may well need a long time before significant volume fractions of P are transformed into / . This does not contradict the high interface velocity, but merely stresses the sluggish nucleation kinetics. The interface velocity is essentially temperature-independent since no thermal activation is necessary. 

The previous section used the constant strain three-noded element to solve Poisson s equation with steady-state as well as transient terms. The same problems, as well as any field problems such as stress-strain and the flow momentum balance, can be formulated using isoparametric elements. With this type of element, the same (as the name suggests) shape functions used to represent the field variables are used to interpolate between the nodal coordinates and to transform from the xy coordinate system to a local element coordinate system. The first step is to discretize the domain presented in Fig. 9.12 using the isoparametric quadrilateral elements as shown in Fig. 9.15. 

Therefore the scaling transformation of the quantum-mechanical force field is an empirical way to account for the electronic correlation effects. As far as the conditions listed above are not always satisfied (e.g. in the presence of delocalized 7r-electron wavefunctions) the real transformation is not exactly homogeneous but rather of Puley s type, involving n different scale constants. The need of inhomogeneous Puley s scaling also arises due to the fact that the quantum-mechanical calculations are never performed in the perfect Hartree-Fock level. The realistic calculations employ incomplete basis sets and often are based on different calculation schemes, e.g. semiempirical hamiltonians or methods which account for the electronic correlations like Cl and density-functional techniques. In this context we want to stress that the set of scale factors for the molecule under consideration is specific for a given set of internal coordinates and a given quantum-mechanical method. 

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