2.3. Derivatives of the energy-density The load terms in (6) from body and surface forces may be treated exactly as in linear finite element analysis [22,23]. Combining this with (20) or (23) gives a numerically integrated variational principle to be minimized over Sh. We will denote this principle by the same (u,) and understand that throughout the following discussion the minimum is to be taken over S". The equations of equilibrium of the finite element model are then where fier is the ith component of the distributed nodal force from body and surface forces acting on node r of element e [22,23]. Thus fi is just the usual finite element load vector, and (24-26) express the fact that the assembly of the equilibrium equations of the nonlinear model obeys the same rule as assembly of load vectors in linear analysis [22,23]. The equations of equilibrium, F = 0, will be solved by a combination of incremental loading with the Newton-Raphson method [22]. More details of these procedures will be described in section 4. Both incremental loading and the Newton-Raphson method require calculation of the "tangent stiffness matrix" [22], T Since K, is the matrix of second derivatives with respect to u¦ of π, which is a continuously differentiable function of the nodal-values, K, is a symmetric "hessian" matrix. In other words, (27-29) imply that K, assembles by the same rule by which stiffness matrices assemble in linear analysis [22,23]. Beyond that, Kr will have the same banded structure which a stiffness matrix in Ꭲ linear analysis would have, using the same mesh [22,23]. This follows because the band-structure is determined by Kr will have (2w+1)d bands, where d = 3 for the 3-D model and d=2 for plane-stress. Clearly w is determined by the form of A and not the form of the energy expression. Boundary conditions are also imposed on in the same manner as in linear analysis [22,23]. a. Separation of tensorial factors In what follows, let U, represent generically either a compressible element energy density, U(I,II,III.), or one of the terms of an incompressible element energy density, W, or G.. From the previous section we may conclude that We point out again that there is no summation on e; in fact, since in what follows we will always be concerned with elemental expressions, we will drop the index e. At this point the strain-energy derivatives are seen to be the sum of terms which are separated into factors, the second of which, მდ. მი, მდ. Ст Спе (32) is determined from the elemental isoparametrics and local shape functions, and is no different in character from the corresponding factors in linear isoparametric problems [22,23]. What distinguishes the finite elasticity case is the complexity of the first factors. It is these derivatives of the energy density with respect to displacement-gradients to which we now turn our attention. Actually, we may further restrict our attention to the derivatives of strain-invariants, since au au au + similar terms for II and III etc., depend on the form of the energy density for a specific material and are not usually difficult to evaluate. So the problem is to compute the derivatives of the invariants. b. Derivatives of the strain-invariants Derivatives of the strain invariants, I, Iijke, II, . . ., etc., are defined in a fashion similar to U1, and Uijke· The expression for I follows immediately from equation (5), as does III, by remembering that the derivative of a determinant with respect to the matrix itself is the matrix of cofactors. Next we observe that the derivative of the matrix of cofactors with respect to the matrix itself can be written where is the Cartesian permutation symbol. Using the chain-rule and collecting terms by application of symmetries gives an expression for II. So we have διδκ (35) jkl = Calculating I is trivial. Using the chain-rule and the identity upp dikdje - dit dkj leads readily to an expression for Iljke. To find IIIke, use can be made of the inverse of Ju, (J) = |Jxe. Then III becomes Although the terms with the ex may look forbidding, they may be computed using no floating point operations other than "change sign," and the same calculations can be used for II, and II. Alternative expressions (of comparable complexity) for II, and IIjke can be derived without using the identity: II = (1/2X12 – G1G1). 3. Two- and three-dimensional problems In this section we describe the assumptions which are made in reducing a three-dimensional problem in finite elasticity to a two-dimensional problem. We describe how this reduction is incorporatd in the framework of the results of section 2. We also will describe the similarity and differences between the element shape-functions and the isoparametric transformations in the two- and three-dimensional cases. It will be easier to describe the implementation of the isoparametric transformations in two-dimensions. The threedimensional implementation can be viewed as a straight-forward generalization of the 2-D case. Finally, as will be explained in more detail in section 5, linear elastic problems serve an important purpose in computer code verification. They are also interesting in their own right. We therefore conclude this section with a discussion of two- and three-dimensional linear elasticity problems. The linear two-dimensional model incorporates compressible, nearly incompressible and incompressible materials in states of plane stress or plane strain. Penalties are used to enforce plane-strain incompressibility. 3.1. Three-dimensional problems The results of section 2 were derived in full three-dimensional form. They may be applied directly to the coding of the element matrix, element load-vector and assembly subroutines [22,23]. Details of how the mesh is described and how the isoparametric transformations are generated will be described later in this section. But the two-dimensional, plane-stress model requires additional development. It may be thought of as a three-dimensional model in which specific simplifying assumptions are imposed on the stress and strain tensors. To see how this is accomplished, it is useful to look first at those tensors in the fully three-dimensional case and observe an example of a particular material model. a. The general incompressible formulation In the 3-D case J, G1, T1, and all the tensors of the previous section are of dimension three in each rank. The Lagrange stress tensor can be written in terms of (35) For incompressible, 3-D problems, U is assumed to have the form (21), so (39) becomes Recall that z is a large penalty parameter forcing G(III-1) to be small in the finite element solution. Therefore the pressure is given by so that (40) gives an incompressible constitutive equation in the z-∞ limit. For stress calculations from the finite element model, it is more physically meaningful to compute the Euler stress tensor, referred to the strained body: = Finally it is desired to have a stress-free undeformed state, but for Judy (47) gives (48) (49) (50) All constituents of (50) are easy to compute given J. Symmetry implies only six of the components of ou need be calculated. This is a penalty function version of the Mooney-Rivlin constitutive equation [9,10,19, and 21]; c, and c2 are material constants. This equation is intended to model exactly incompressible materials, and in fact, it has some continuum mechanical inadequacies if significant compressive deformation is allowed [10]. Therefore it should only be employed with large values of z. The exactly incompressible, plane-stress version of the constitutive model has been found to give reasonable agreement with experimental results in some problems [21]. How well the 3-D penalty version succeeds will be discussed below. The symmetric stress tensor for Mooney-Rivlin material is given by It should be noted that this differs slightly from the H used elsewhere [9,19,21], because c2Cd, is considered a pressure term. This means that H defined in (52) has the value c2-c1 in the undeformed state. [In [19], since the c2CS term is included with the deviatoric terms, the pressure in the undeformed state is -c1-2c2.] A quantity of more interest than H, then, is the pressure above the undeformed state: z(III-1)III + cz(C-3) 3.2. Finite plane stress a. The general incompressible formulation The assumptions in the plane-stress model are that (53) 1. The body is very thin, with uniform thickness, 2t, in the x3-coordinate. How thin the body must be will be dealt with further below. 2. The applied forces are in the x'-x2 plane; the upper and lower surfaces in the x1-x2 plane are free surfaces. 3. 031 = 032 = 033 = 0 throughout the thickness, i.e. for t≤ x3 ≤ t. |