Strain-displacement equations

These equations are identical to those previously obtained. The strain-displacement equations can also be transformed in a similar way. 

In fact, the solution for a plane stress problem can be determined from the solution of the corresponding plane strain problem and vice versa. Note that in contrast to the plane strain case, the remaining stresses in the plane stress are not required to be independent of z. In fact, the three-dimensionality of plane stress is closely linked to the fact that the conditions fulfilled by the stresses no longer lead to a single nontrivial compatibility equation. In other words, if the remaining stresses ctyy, and <5xy are functions of only X and y, the strain-displacement equations cannot in general be satisfied. 

Problems in the cylindrical coordinate system are also frequently encotmtered, and with similar assumptions regarding second-order terms, the corresponding strain-displacement equations... 

Essentially, the foregoing can be considered as the theory of infinitesimal strain and to be valid for problems involving the elasticity of metals. It is apparent that the use of these relations for finite strain would lead to considerable error whether or not the strain was uniform. The difficulty can be overcome by realizing that since the foregoing equations are valid for infinitesimal total strains, then they must also be valid for infinitesimal increments of finite strains. The value of the total finite strain can then be determined fi-om an integration of these increments. The symbol E is normally used to denote finite linear strain, the strain-displacement equations henceforth being valid with 8e replacing e ... 

Eor this reason and the confusion that can arise in defining the momentum and strain-displacement equations, one must use care in solving problems with the contracted notation. The value of p in terms of i,j varies notoriously between authors, but the IEEE standard [4] is the most commonly accepted version here p= (i +f) (1 - sgn(i -j) I) -b sgn(j -j) (9 - i -j) for i,j = 1,2,3 here the vertical bars represent an absolute value while sgn(-) is the signum, zero when x is zero and the sign of x otherwise. 

The first approach is based upon direct solution involving the displacements. In the most basic sense, a strategy can be found to solve the 15 coupled differential equations directly. However, other approaches are more expedient. The most classical approach is to develop the Navier equations by putting the strain-displacement equations (Elq. 9.26) into the constitutive equations (Eq. 9.27) to obtain the stresses, a, in terms of the displacements, Uj. The result is then inserted into the equilibrium equations (Eq, 9.25), yielding three, coupled, second order partial differential equations on the three displacements, Uj. These three equations can then be solved for the displacements. Upon solution the stresses and strains can be found by substitution of the displacements in to the appropriate expressions. 

Since one of the boundary conditions in Eki. 9.46 is now a displacement boundary condition, we also require the expressions for the displacements in terms of the constants A and C. These are found first by using the stress-strain-displacement relations and then by integrating the strain components to determine the displacements. The stress-strain-displacement equations for the condition of plane strain in cylindrical coordinates are,... 

We can now use the strain-displacement equations (Chapter 2) to derive the normal engineering strains en, 622 and 633 ... 

By small strains , we mean that only linear terms in the strain-displacement equations are required (Section 3.1.5). 

Carpenter and Barsoum [14] formulated a specific finite element to simulate various closed form solutions to the stress and strain fields within a single lap Joint. It was shown that the theoretical singularities within such a Joint could be removed through use of incomplete strain-displacement equations. Beer [15] gave the formulation of a simplified finite element chiefly concerned with the correct representation of the mechanical properties of an adhesive within a stmctural model rather than the prediction of detailed stresses within the adhesive. 

Equations (1.11.30) are supplemented by stress dynamical equations and strain-displacement equations which take the same form as in classical elasticity. Likewise, boundary conditions may take any of the forms appropriate to the classical (dynamic) theory of elasticity in particular the type of boundary condition specified at a point on the boundary of a body may change with time and further, the boundary of the body may change with time, through the process of ablation, for example. 

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

Then, integration of the strain-displacement relation. Equation (2.2), with respect to z (with w assumed to be independent of z) yields... 

The moment relations are obtained from integration 6f the stress-strain relations, Equation (6.27), after the strain-displacerrfent relations. Equation (6.22), and the displacement relations. Equation (6.31), are substituted ... 

The notion of fluid strains and stresses and how they relate to the velocity field is one of the fundamental underpinnings of the fluid equations of motion—the Navier-Stokes equations. While there is some overlap with solid mechanics, the fact that fluids deform continuously under even the smallest stress also leads to some fundamental differences. Unlike solid mechanics, where strain (displacement per unit length) is a fundamental concept, strain itself makes little practical sense in fluid mechanics. This is because fluids can strain indefinitely under the smallest of stresses—they do not come to a finite-strain equilibrium under the influence of a particular stress. However, there can be an equilibrium relationship between stress and strain rate. Therefore, in fluid mechanics, it is appropriate to use the concept of strain rate rather than strain. It is the relationship between stress and strain rate that serves as the backbone principle in viscous fluid mechanics. 

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 ... 

Or, normalizing the displacement with respect to the original length of the bond we can write this equation in terms of the strain, Af (Equation 13-17) ... 

The necessary conditions to be fulfilled are the equilibrium conditions, the strain-displacement relationships (kinematic equations), and the stress-strain relationships (constitutive equations). As in linear elasticity theory (12), these conditions form a system of 15 equations that permit us to obtain 15 unknowns three displacements, six strain components, and six stress components. 

Equation (3.12) is the governing partial differential equation for two-dimensional elasticity. Any function that satisfies this fourth-order partial differential equation will satisfy all of the eight equations of elasticity namely, the equilibrium equations, Hooke s law, and the strain-displacement relations. 

In general, we measure the homogeneous strain of a solid by the relative displacement of two points and P2 separated by the vector r, keeping the coordinate system invariant (Fig. 4.9). The strain displaces the point Piix ) to the point P Xi + < j) and the point P2( i + ) to P 2 Xi H- H- u-). The vector r H- u gives the relative position of the two points of the strained solid. By analogy with equation (4.34) and (4.35), the strain tensor eexpresses the displacement u per unit... 

It is noted that the above strain can also be expressed by the displacement gradient. For instance, the Green-Lagrange strain in Equation (4.20), after operation in terms of the tensor index, can be rewritten as... 

The VFM is based on the fundamental equations of solid mechanics the equilibrium equation, through the principle of virtual work (PVW), the constitutive equations and the strain-displacement relationships. For an arbitrary solid in equilibrium, the PVW can be written as... 

A state of plane stress exists with respect to x-y plane when Ox, T yz and r z are zero, and a, Oy and x y are functions of x and y only. The equilibrium equations are given by (A.17) for the x and y directions. In this case, however, cXx is zero, but is nonzero. The strain displacement relations are the same as Eq. (A.17) with the additional relationship... 

The dashed line (c) in Fig. 2 corresponds to a typical stress (force) versus strain (displacement) curve for a material which obeys Hooke s Law (elastic) and curve (b) represents a viscoelastic behavior. It can be observed from the figure that the curve from the thermoplastic material is almost linear for a force (stress) below 300 N. In the linear region of the curve, the material behaves like an elastic material and obeys Hooke s Law. The curve in Fig. 2 can be represented by the following equation ... 

Thus, we have arrived at an alternative form of equilibrium equations where differentiability requirements are relaxed for the stress. Therefore, the stresses can be related through the constitutive and strain-displacement relationships to the primary variable, displacements, in the discretized equations. 

The number of equations in (8.9) is equal to the number of nodes times the number of degrees of freedom (dof) at a node. We note that the number of elements and nodes can be of the order of millions to achieve an accurate solution for tire problems. All material and geometric nonlinearities in (8.9) are in the first term where the stress o,j is a nonlinear function of the velocities through the constitutive Eq. (8.2) and the strain displacement Eq. (8.3). 

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