Stress-equilibrium equations

Even though in classical lamination theory by virtue of the Kirchhoff hypothesis we assume the stresses and are zero, we can still obtain these stresses approximately by integration of the stress equilibrium equations... 

The distribution of x z isotropic beam of rectangular cross section comes from integration of the the stress-equilibrium equation... 

Continuing this process for the other two axes and using the symmetry relationships gives the three stress equilibrium equations, i.e.. 

In the absence of body forces, the stress equilibrium equations (Eq. (2.49)) become... 

The final term in Eq. (5.23) vanishes as a result of the continuity equation (Eq. (5.20)). If one generalizes this approach for the other stress equilibrium equations, one obtains... 

The perturbed stress field that arises as a result of the nonuniform composition can be found by straightforward application of the stress equilibrium equations and Hooke s law of linear response for an isotropic elastic material, subject to the constraints that the perturbation alters neither the mean extensional strain in any direction in the a z—plane nor the zero net force per wavelength in the y—direction. The mean normal stress implied by these constraints is... 

On substituting these relations into Eq. (14), the stress equilibrium equations can be replaced by a pair of second-order differential equations, the Lame equations ... 

The stress components are related by the stress-equilibrium equations ... 

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

In integrated photoelasticity it is impossible to achieve a complete reconstruction of stresses in samples by only illuminating a system of parallel planes and using equilibrium equations of the elasticity theory. Theory of the fictitious temperature field allows one to formulate a boundary-value problem which permits to determine all components of the stress tensor field in some cases. If the stress gradient in the axial direction is smooth enough, then perturbation method can be used for the solution of the inverse problem. As an example, distribution of stresses in a bow tie type fiber preforms is shown in Fig. 2 [2]. 

We shall consider an equilibrium problem with a constitutive law corresponding to a creep, in particular, the strain and integrated stress tensor components (IT ), ay(lT ) will depend on = (lT, w ), where (lT, w ) are connected with (IT, w) by (3.1). In this case, the equilibrium equations will be nonlocal with respect to t. 

Verify that the lamina mechanical and thermal stresses in Equations (4.118) and (4.119) for a three-layered cross-ply laminate with M =. 2 satisfy the larninate equilibrium conditions. 

Upon substitution of the displacement field. Equation (4.163), in the stress-displacement relations and subsequently in the stress-equilibrium differential equations. Equation (4.164), the displacement-equilibrium equations are, for each layer,... 

Pagano s exact solution for the stresses and displacements is too complex to present here. The corresponding classical lamination theory result stems from the equilibrium equations, Equations (5.6) to (5.8), which simplify to... 

For the cylindrical coordinates of the fiber push-out model shown in Fig. 4.36 where the external (compressive) stress is conveniently regarded as positive, the basic governing equations and the equilibrium equations are essentially the same as the fiber pull-out test. The only exceptions are the equilibrium condition of Eq. (4.15) and the relation between the IFSS and the resultant interfacial radial stress given by Eq. (4.29), which are now replaced by ... 

The temperature, fiber tension, stresses, and strains vary only in the radial directions. An elasticity solution is employed to calculate the six components of the stresses and strains. The solution procedure follows the established techniques of elasticity solutions. A displacement field is assumed that satisfies the equilibrium equations and the compatibility conditions. The latter requires that at each interface the displacements and the normal stresses in adjacent... 

The tensor of stress, Oy, has the meaning of the force in direction j on an infinitesimal area with normal in the direction i and is again a symmetric tensor with 6 independent components. In classical elasticity only the force resultant at any point is considered, the couple that must also exist is assumed to be negligible by comparison. However, in polar field theories of elasticity, couple stresses are considered and additional equations of equilibirum required. Classically however only the equation of stress equilibrium... 

The fundamental equations treated in structural analyses are the mechanical equilibrium, strain-displacement relation, and stress-strain relation. The equilibrium equations in an elementary volume can be expressed ... 

Thus Equation (10.33) is solved as the new equilibrium equation. To calculate the thermal expansion behavior of the model, the thermal expansion coefficient is necessary as the calculating parameter. If the temperature of the model is even, the initial temperature and the final temperature are used as just a calculating parameter. If a temperature distribution exists in the model, the temperature distribution data is dispensable for the stress calculation. The temperature is firstly calculated by a CFD and the calculated data is used as the boundary condition in the stress calculation. If the thermal expansion coefficient is temperature dependent, the temperature dependence must be considered in the calculation. Here the temperature data at the nodes is transferred from STAR-CD to the ABAQUS. 

By symmetry, the two principal directions of stress (and strain) are in the meridian direction, Tin, and the circumferential direction 7133. The third principal stress is zero. Show that if body and acceleration forces are neglected, the following equilibrium equations are obtained for thin membranes ... 

The analysis of flaw reorientation in the drawn material is rudimentary. However, it is not obvious that greater detail such as a general formulation using the equations of stress equilibrium and compatibility (several partial differential equations are called for) would significantly enhance the model. This question will be resolved experimentally. 

If the displacements on the surface of the body are given (second boundary problem), the stress-displacement relationships are obtained first, and their substitution into the equilibrium equations permits us to eliminate the stress variables and thus to obtain the three equilibrium equations in terms of the displacements (see Navier equations in... 

The substitution of the components of the stress tensor, given in the preceding section, into the equilibrium equations [Eq. (4.14) with = 0] leads to... 

What this equation tells us is that a particular state of stress is nothing more than a linear combination (albeit perhaps a tedious one) of the entirety of components of the strain tensor. The tensor Cijn is known as the elastic modulus tensor or stiffness and for a linear elastic material provides nearly a complete description of the material properties related to deformation under mechanical loads. Eqn (2.52) is our first example of a constitutive equation and, as claimed earlier, provides an explicit statement of material response that allows for the emergence of material specificity in the equations of continuum dynamics as embodied in eqn (2.32). In particular, if we substitute the constitutive statement of eqn (2.52) into eqn (2.32) for the equilibrium case in which there are no accelerations, the resulting equilibrium equations for a linear elastic medium are given by... 

Given this definition of the Airy stress function, show that the equilibrium equations are satisfied. 

This expression insures that our integrations are organized such that the integrands are well behaved in the regions of interest. If we use the fact that the stress fields satisfy the equilibrium equations (i.e. Oijj = 0) in conjunction with the divergence theorem, this expression may be rewritten as... 

Our fundamental assertion concerning the geometry of pile-ups is that they reflect the equilibrium spacing of the various dislocations which are participants in such a pile-up. From a discrete viewpoint, what one imagines is an equilibrium between whatever applied stress is present and the mutual interactions of the dislocations. In simple terms, using the geometry depicted schematically in fig. 11.12, we argue that each dislocation satisfies an equilibrium equation of the form... 

By way of contrast, the resisting force offered by the obstacle at the moment of breakaway corresponding to the Orowan process is dictated by the equilibrium equation written above as eqn (11.52). In particular, since (/> = 0 in this case, we have Fmax = 2r, and therefore the critical stress is... 

As an application of the ideas on dislocation pile-ups described in section 11.4.2, consider a pile-up of three dislocations in which the leading dislocation i , fixed at (0,0). The other dislocations have coordinates (xi, 0) and (X2, 0) which are to be determined by applying the equilibrium equations presented as eqn (11.30). Assume that the externally applied stress is constant and is denoted by r. 

The finite element method with the sintering stress taken into account has been formulated [4,6]. In this case, the equilibrium equation is given by dojj dSjj ... 

The minus sign in this equation is a matter of convention t(n) is considered positive when it acts inward on a surface whereas n is the outwardly directed normal, andp is taken as always positive. The fact that the magnitude of the pressure (or surface force) is independent of n is self-evident from its molecular origin but also can be proven on purely continuum mechanical grounds, because otherwise the principle of stress equilibrium, (2 25), cannot be satisfied for an arbitrary material volume element in the fluid. The form for the stress tensor T in a stationary fluid follows immediately from (2 59) and the general relationship (2-29) between the stress vector and the stress tensor ... 

Balance of momentum for the medium as a whole is reduced to the equation of stress equilibrium together with a mechanical constitutive model to relate stresses with strains. Strains are defined in terms of displacements. 

The layout of the CERTI concept being considered in a hard geological medium, a simple thermo-elastic law has been chosen to describe the behavior of the rock. Together with the mechanical equilibrium equation, it leads to a relation between the displacement vector u and the temperature T besides, deformations and stresses tensors can be derived from the classical following equations ... 

The stress expression in Equation (3) is used in the following quasi-static equilibrium equation assuming the acceleration term to be negligible. 

As the effects of (rock matrix) swelling are mainly on effective stress, they can be taken into account by keeping track of changes in rock water potential or effective stresses, and the strain induced (swelling and hydrational stress model. Equations 14 to 16). The hydrational stress can be considered as additional internal stress in the equilibrium equations. 

According to the revised Terzaghi s effective stress principle, the equilibrium equation of the reservoir matrix can be written... 

---