Stresses second-order

Because the strain energy function for rubber is valid at large strains, and yields stress-strain relations which are nonlinear in character, the stresses depend on the square and higher powers of strain, rather than the simple proportionality expected at small strains. A striking example of this feature of large elastic deformations is afforded by the normal stresses tn,(22,133 that are necessary 

FICURE1.15 Stresses required to maintain a simple shear deformation of amount y. The normal stress t i is set equal to zero (from Gent, 1958). 

FICURE1.16 Stresses required to maintain a simple shear deformation of amount y. The normal 

They are represented schematically in Figs. 15 and 16 for two different choices of the arbitrary hydrostatic pressure p, chosen so as to give the appropriate reference (zero) stress. In Fig. 15, for example, the normal stress 61 in the shear direction is set equal to zero this condition would arise near the front and rear surfaces of a sheared block. In Fig. 16, the normal stress 33 is set equal to zero this condition would arise near the side surfaces of a sheared block. In each case a compressive stress 22 is found to be necessary to maintain the simple shear deformation. In its absence the block would tend to increase in thickness on shearing. 

When the imposed deformation consists of an inhomogeneous shear, as in torsion, the normal forces generated (corresponding to the stresses 22 in Figs. 15 and 16) vary from point to point over the cross-section (Fig. 17). The exact way in which they are distributed depends on the particular form of strain energy function obeyed by the rubber, i.e., on the values of Wi and W2 which obtain under the imposed deformation state [31]. 

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Second-order stress is difficult to observe and much less extensively studied. The causes of internal stress are still a matter for investigation. There are broad generalisations, e.g. frozen-in excess surface energy and a combination of edge dislocations of similar orientation , and more detailed mechanisms advanced to explain specific examples. 

The element ryl of the second-order stress tensor t is the force-per-unit area acting in the x direction on an elemental fluid surface perpendicular to the y direction. The stress tensor is symmetric, and so ryx = tiv. 

Here ma is the bulk solid-fluid interaction force, T.s the partial Cauchy stress in the solid, p/ the hydrostatic pressure in the perfect fluid, IIS the second-order stress in the solid, ha the density of partial body forces, ta the partial surface tractions, ts the traction corresponding to the second-order stress tensor in the solid and dvs/dn the directional derivative of v.s. along the outward unit normal n to the boundary cXl of C. 

When the shear waves propagate through the elastic layer, or the elastic plate and reach the steady state, the type of the wave, SH wave for example, and it s dispersion relation are determined by the boundary conditions at the plate surfaces [7]. We have assumed that the sound waves modulate the stress fields at the tip of the crack, and then solved the wave equations with the boundary conditions at the surfaces of the crack and the plate. If the analysis is extended to derive the higher order fields and the dispersion relation of the wave is then obtained, such a wave do exist in the steady state. In this case we could confirm the existence of such "new wave" associated with the crack. Much algebra is required to obtain the higher order fields, however, it is not difficult to see the structure of the fields with the boundary condition at the plate surfaces. We find the boundary conditions at the plate surfaces for the second order stress fields are satisfied by the factor, cos /5 z, in the similar manner to Eq. 

Any physical qnantity along with its spatial and geometric dimensions can also be characterized by the physical dimensionality of the components, which is the same for all components. The components of a second-order stress tensor have the dimensions of N/m = Pa = 10 dyn/cm = 10 kg-force/cm. At the same time, a second-order strain tensor s components are dimensionless (of zeroth-order physical dimensionality). This is true for both 2D and 3D models. 

Normally, the extra stress in the equation of motion is substituted in terms of velocity gradients and hence this equation includes second order derivatives of... 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

It may first be noted that the referential symmetric Piola-Kirchhoff stress tensor S and the spatial Cauchy stress tensor s are related by (A.39). Again with the back stress in mind, it will be assumed in this section that the set of internal state variables is comprised of a single second-order tensor whose referential and spatial forms are related by a similar equation, i.e., by... 

If the stress is at the primary time step loeation and the veloeities are at the middle of the time step, then the resulting finite-difference equation is second-order accurate in space and time for uniform time steps and elements. If all quantities are at the primary time step, then a more complicated predictor-corrector procedure must be used to achieve second-order accuracy. A typical predictor-corrector scheme predicts the stresses at the middle of the time step and uses them to calculate the divergence of the stress tensor. 

In this chapter the regimes of mechanical response nonlinear elastic compression stress tensors the Hugoniot elastic limit elastic-plastic deformation hydrodynamic flow phase transformation release waves other mechanical aspects of shock propagation first-order and second-order behaviors. 

Curran [61C01] has pointed out that under certain unusual conditions the second-order phase transition might cause a cusp in the stress-volume relation resulting in a multiple wave structure, as is the case for a first-order transition. His shock-wave compression measurements on Invar (36-wt% Ni-Fe) showed large compressibilities in the low stress region but no distinct transition. 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

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