# Elements successful, because, for a given number of nodes, they provide the most locations at which some component of the gradient of the solution can be discontinuous (the element edges). This argument is hardly rigorous, but it is, nevertheless, true that first-order elements tend to be preferred for such cases. The incompatible mode elements can represent discontinuities particularly well. They are also able to represent strain localization such as occurs in shear bands. One should realize, however, that better defined shear localization increases the strain magnitude and, hence, tends to increase the number of increments and iterations required for the analysis. All of the solid elements in ABAQUS, except the infinite elements, are written to include finite-strain effects. When these elements are used with elastomeric material definition (the \*HYPERELASTIC material option), the constitutive behavior is calculated directly from the deformation gradient matrix, F. When the elements are used for geometrically nonlinear analysis with any other material definition (at finite strain this means the material has some inelastic behavior, since all of the elasticity definitions in ABAQUS except the hyperelasticity models assume that the elastic strains are small), the strains are calculated as the integral of the rate of deformation, $$ \mathbf {D} = \mathrm{sym} \left(\frac {\partial \mathbf {v}}{\partial \mathbf {x}}\right), $$ with the effects of material rotation with respect to the coordinate system taken into consideration. In all cases the solid elements report stress as the "true" (Cauchy) stress. In all cases except when the \*ORIENTATION option is used with an element, stress and strain components are given as physical components referred to the global spatial directions. When the \*ORIENTATION option is used with a solid element, the strain and stress components are given in the local system defined in the \*ORIENTATION option: this system rotates with the average material rotation calculated at each material point. # 3.2.2 Solid element formulation All the solid elements in ABAQUS allow for finite strain and rotation in large-displacement analysis. For kinematically linear analysis the strain is defined as $$ \varepsilon = \mathrm{sym} \left(\frac {\partial \mathbf {u}}{\partial \mathbf {X}}\right), $$ where u is the total displacement and X is the spatial position of the point under consideration in the original configuration. As discussed in Chapter 1, "Introduction and Basic Equations," this measure of strain is useful only if the strains and rotations are small (all components of the strain and rotation matrices are negligible compared to unity). For cases where the strains and/or rotations are no longer small, two ways of measuring strain are used in the solid elements in ABAQUS. When the \*HYPERELASTIC or \*HYPERFOAM material definition is used with an element, ABAQUS internally uses the stretch values calculated directly from the deformation gradient matrix, F, to compute the material behavior. With any other material behavior it is assumed that any elastic strains are small compared to unity, so the appropriate reference # Elements configuration for the elasticity is only infinitesimally different from the current configuration, and the appropriate stress measure is, therefore, the Cauchy ("true") stress. (More precisely, the appropriate stress measure should be the Kirchhoff stress defined with respect to the elastic reference configuration but the assumption that this reference configuration and the current configuration are only infinitesimally different makes the Kirchhoff and Cauchy stress measures almost the same: the differences are of the order of the elastic strains compared to unity). The conjugate strain rate to Cauchy stress is the rate of deformation: $$ \mathbf {D} = \mathrm{sym} \left(\frac {\partial \mathbf {v}}{\partial \mathbf {x}}\right), $$ where D is the rate of deformation, v is the velocity at a point, and x are the current spatial coordinates of the point. The strain is, therefore, defined as the integral of the rate of deformation. This integration is nontrivial, particularly in the general case where the principal axes of strain rotate during the deformation. In ABAQUS the total strain is constructed by integrating the strain rate approximately over the increment by the central difference algorithm; and, when the strain components are referred to a fixed coordinate basis, the strain at the start of the increment must also be rotated to account for the rigid body rotation that occurs in the increment. This is also done approximately, using the Hughes-Winget (1980) method. This integration algorithm defines the integration of a tensor associated with the material behavior as $$ \mathbf {a} _ {t + \Delta t} = \Delta \mathbf {R} \cdot \mathbf {a} _ {t} \cdot \Delta \mathbf {R} ^ {T} + \Delta \check {\mathbf {a}} (\Delta \mathbf {D}), $$ where a is the tensor; $\Delta \check { \mathbf { a } }$ is the increment in the tensor associated with the material's constitutive behavior, and, therefore, dependent on the strain increment, $\Delta \mathbf { D }$ , defined by the central difference formula as $$ \Delta \mathbf {D} = \mathrm{sym} \left(\frac {\partial \Delta \mathbf {u}}{\partial \mathbf {x} _ {t + \Delta t / 2}}\right), $$ where $x _ { t + \Delta t / 2 } = ( 1 / 2 ) ( \mathbf { x } _ { t } + \mathbf { x } _ { t + \Delta t } )$ ; and $\Delta \mathbf { R }$ is the increment in rotation, defined by Hughes and Winget as $$ \Delta \mathbf {R} = \left(\mathbf {I} - \frac {1}{2} \Delta \pmb {\omega}\right) ^ {- 1} \cdot \left(\mathbf {I} + \frac {1}{2} \Delta \pmb {\omega}\right), $$ where $\Delta \omega$ is the central difference integration of the rate of spin: $$ \Delta \pmb {\omega} = \mathrm{asym} \left(\frac {\partial \Delta \mathbf {u}}{\partial \mathbf {x} _ {t + \Delta t / 2}}\right). $$ A somewhat different algorithm to calculate ¢R is used for the Green-Naghdi rate in ABAQUS/Explicit. # Elements For example, the stress is integrated by this method as $$ \pmb {\sigma} _ {t + \Delta t} = \Delta \mathbf {R} \cdot \pmb {\sigma} _ {t} \cdot \Delta \mathbf {R} ^ {T} + \Delta \check {\pmb {\sigma}} (\Delta \mathbf {D}), $$ where $\Delta \check { \pmb { \sigma } } ( \Delta \mathbf { D } )$ is the stress increment caused by the straining of the material during this time increment and ¾ is the Kirchhoff (¼ Cauchy) stress. The subscripts t and $t + \Delta t$ refer to the beginning and the end of the increment, respectively. As shown in \`\`Procedures: overview and basic equations, '' Section 2.1.1, the contribution of the internal work terms to the Jacobian of the Newton method that is often used in ABAQUS/Standard is Equation 3.2.2-1 $$ \int_ {V ^ {o}} \left(d \pmb {\sigma}: \delta \mathbf {D} + \pmb {\sigma}: d \delta \mathbf {D}\right) d V, $$ where d¾ and ¾ are evaluated at the end of the increment. Using the integration definition above, it can be shown that $$ d \pmb {\sigma} _ {t + \Delta t} = d \Delta \mathbf {R} \cdot \Delta \mathbf {R} ^ {T} \cdot (\pmb {\sigma} _ {t + \Delta t} - \mathbf {C}: \Delta \mathbf {D}) + (\pmb {\sigma} _ {t + \Delta t} - \mathbf {C}: \Delta \mathbf {D}) \cdot \Delta \mathbf {R} \cdot d \Delta \mathbf {R} ^ {T} + \mathbf {C}: d \Delta \mathbf {D}, $$ where C is the Jacobian matrix of the constitutive model: $$ \mathbf {C} = \frac {\partial d \check {\pmb {\sigma}}}{\partial d \Delta \mathbf {D}}. $$ However, rather than computing the tangent matrix for the Newton method on this basis, we approximate this by using $$ d \pmb {\sigma} _ {t + \Delta t} = d \pmb {\Omega} \cdot \pmb {\sigma} _ {t + \Delta t} + \pmb {\sigma} _ {t + \Delta t} \cdot d \pmb {\Omega} ^ {T} + \mathbf {C}: d \mathbf {D}, $$ which yields the Jacobian $$ \int_ {V} \left(\delta \mathbf {D}: \mathbf {C}: d \mathbf {D} - \frac {1}{2} \pmb {\sigma}: \delta \left(2 \mathbf {D} \cdot \mathbf {D} - \frac {\partial \mathbf {v}}{\partial \mathbf {x}} ^ {T} \cdot \frac {\partial \mathbf {v}}{\partial \mathbf {x}}\right)\right) d V. $$ This Jacobian is the tangent stiffness of the rate form of the problem. Experience with practical cases suggests that this approximation provides an acceptable rate of convergence in the Newton iterations in most applications with real materials. The strain and rotation measures described above are approximations. Probably the most limiting aspect of these approximations is the definition of the rotation increment $\Delta \mathbf { R }$ . While this measure does give a representation of the rotation of the material at a point in some average sense (both in ABAQUS/Standard and ABAQUS/Explicit), it is clear that each of the individual material fibers at a point has a different rotation (unless the material point undergoes rigid body motion only or, as an approximate extension, if the strains at the point are small). This suggests that the formulation described above will not be suitable for applications where the strains and rotations are large and where the material exhibits some form of anisotropic behavior. A common example of such cases is the induction of anisotropy through straining, as in "kinematic hardening" plasticity models. The integration methods described above are not suitable for such material models at large strains (for practical purposes with typical material parameters this means that the solutions will be quite wrong when the strains are greater than 20%-30%). Therefore, the use of the kinematic hardening model in ABAQUS at such strain levels is not recommended. There is extensive literature on this subject; see Agah-Tehrani et al. (1986), for example. # 3.2.3 Hybrid incompressible solid element formulation Many problems involve the prediction of the response of almost incompressible materials. This is especially true at large strains, since most solid materials show relatively incompressible behavior under large deformations. In this section we describe the augmented virtual work basis provided in ABAQUS/Standard for such cases. The method is described in the context of incompressible elasticity theory, since that is where it is most likely to be used. When the material response is incompressible, the solution to a problem cannot be obtained in terms of the displacement history only, since a purely hydrostatic pressure can be added without changing the displacements. The nearly incompressible case (that is, when the bulk modulus is much larger than the shear modulus or Poisson's ratio, º, is greater than 0.4999999) exhibits behavior approaching this limit, in that a very small change in displacement produces extremely large changes in pressure, so that a purely displacement-based solution is too sensitive to be useful numerically (for example, round-off on the computer may cause the method to fail). We remove this singular behavior in the system by treating the pressure stress as an independently interpolated basic solution variable, coupled to the displacement solution through the constitutive theory and the compatibility condition, with this coupling implemented by a Lagrange multiplier. This independent interpolation of pressure stress is the basis of these "hybrid" elements. More precisely, they are "mixed formulation" elements, using a mixture of displacement and stress variables with an augmented variational principle to approximate the equilibrium equations and compatibility conditions. The hybrid elements also remedy the problem of volume strain "locking," which can occur at much lower values of º (i.e., º = 0.49). Volume strain locking occurs if the finite element mesh cannot properly represent incompressible deformations. Volume strain locking can be avoided in regular displacement elements by fully or selectively reduced integration, as described in \`\`Solid isoparametric quadrilaterals and hexahedra, '' Section 3.2.4. We begin by writing the internal virtual work: Equation 3.2.3-1 $$ \delta W = \int_ {V} \pmb {\sigma}: \delta \pmb {\varepsilon} d V, $$ where ±" is the virtual strain: $$ \delta \pmb {\varepsilon} = \mathrm{sym} \left(\frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}}\right), $$ # Elements where ±u is the virtual displacement field; ¾ is the true (Cauchy) stress; V is the current volume; and ±W is the virtual work as defined by this equation. See \`\`Equilibrium and virtual work,'' Section 1.5.1, for a detailed discussion of the virtual work concept. In a displacement-based formulation the Cauchy stress, ¾, is obtained with the constitutive equations from the deformation, usually in rate form: Equation 3.2.3-2 $$ d \pmb {\sigma} = \mathbf {C}: d \pmb {\varepsilon} + d \pmb {\Omega} \cdot \pmb {\sigma} - \pmb {\sigma} \cdot d \pmb {\Omega}, $$ where C is the "material stiffness matrix" and d− is the rate of rotation (spin) of the material. We modify the Cauchy stress by introducing an independent hydrostatic pressure field $\hat { p }$ as follows: Equation 3.2.3-3 $$ \overline {{\pmb {\sigma}}} = \pmb {\sigma} + (1 - \rho) \mathbf {I} (p - \hat {p}), $$ where $$ p = - \frac {1}{3} \mathrm{trace} (\pmb {\sigma}) $$ is the hydrostatic pressure stress and $\rho$ is a small number. $\operatorname { I f } \rho$ was set equal to zero, the hydrostatic component in $\overline { { \pmb { \sigma } } }$ would be identical to the independent pressure field $\hat { p } ,$ corresponding to a pure "mixed" formulation. The small nonzero value $( 1 0 ^ { - 9 } )$ is chosen to avoid equation solver difficulties. This relation is used in incremental form: Equation 3.2.3-4 $$ \overline {{\pmb {\sigma}}} = \overline {{\pmb {\sigma}}} ^ {0} + \Delta \pmb {\sigma} + (1 - \rho) \mathbf {I} (\Delta p - \Delta \hat {p}), $$ where $\overline { { \pmb { \sigma } } } ^ { 0 }$ is the modified Cauchy stress at the start of the increment. We use the modified Cauchy stress in the virtual work expression and augment the expression with the Lagrange multiplier enforced constraint $\Delta p - \Delta \hat { p } = 0$ : Equation 3.2.3-5 $$ \delta \overline {{W}} = \int_ {V} [ \overline {{\pmb {\sigma}}}: \delta \pmb {\varepsilon} + J ^ {- 1} \delta \lambda (\Delta p - \Delta \hat {p}) ] d V, $$ with J the volume change ratio (Jacobian) and ±¸ a Lagrange multiplier whose interpolation must still be determined. $\Delta \hat { p }$ will be interpolated over each element so that the constraint is satisfied in an integrated (average) sense. Since $\Delta p$ is the value of the equivalent pressure stress increment computed from the kinematic solution, Equation 3.2.3-4 does not make sense if the material is fully incompressible because then $\Delta p$ cannot be computed. For the purpose of development we regard the bulk modulus as finite, and we will be able to show that the final formulation approaches a usable limit as we allow the bulk modulus to approach infinity. # Elements For the formulation of the tangent stiffness (the Jacobian), we need to define the rate of change of $\delta \overline { { W } }$ . Therefore, we rewrite the virtual work equation in terms of the reference volume $V ^ { 0 }$ : Equation 3.2.3-6 $$ \delta \overline {{W}} = \int_ {V ^ {0}} \left[ J \overline {{\pmb {\sigma}}}: \delta \pmb {\varepsilon} + \delta \lambda (\Delta p - \Delta \hat {p}) \right] d V ^ {0}. $$ The rate of change $d \delta \overline { { W } }$ is then readily obtained as Equation 3.2.3-7 $$ d \delta \overline {{{W}}} = \int_ {V ^ {0}} \left[ d J \overline {{{\pmb {\sigma}}}}: \delta \pmb {\varepsilon} + J d \overline {{{\pmb {\sigma}}}}: \delta \pmb {\varepsilon} + J \overline {{{\pmb {\sigma}}}}: d \delta \pmb {\varepsilon} + \delta \lambda (d p - d \hat {p}) + d \delta \lambda (\Delta p - \Delta \hat {p}) \right] d V ^ {0}. $$ We rewrite this expression in terms of the current volume: Equation 3.2.3-8 $$ d \delta \overline {{{W}}} = \int_ {V} [ d \overline {{{\pmb {\sigma}}}}: \delta \pmb {\varepsilon} + d \pmb {\varepsilon}: \mathbf {I} \overline {{{\pmb {\sigma}}}}: \delta \pmb {\varepsilon} + \overline {{{\pmb {\sigma}}}}: d \delta \pmb {\varepsilon} + J ^ {- 1} \delta \lambda (d p - d \hat {p}) + J ^ {- 1} d \delta \lambda (\Delta p - \Delta \hat {p}) ] d V, $$ where we used the identity $J ^ { - 1 } d J = \mathbf { I } : d { \boldsymbol { \varepsilon } }$ . The rate of the modified stress follows from Equation 3.2.3-4 and the constitutive equations: Equation 3.2.3-9 $$ d \overline {{\boldsymbol {\sigma}}} = \mathbf {C}: d \boldsymbol {\varepsilon} + (1 - \rho) \mathbf {I} (d p - d \hat {p}) + d \boldsymbol {\Omega} \cdot \overline {{\boldsymbol {\sigma}}} - \overline {{\boldsymbol {\sigma}}} \cdot d \boldsymbol {\Omega}, $$ where $$ d p = - \frac {1}{3} \mathrm{trace} (d \pmb {\sigma}) = - \frac {1}{3} \mathbf {I}: \mathbf {C}: d \pmb {\varepsilon}; $$ and we used the fact that $d \Omega \cdot { \overline { { \sigma } } } - { \overline { { \sigma } } } \cdot d \Omega = d \Omega \cdot \sigma - \sigma \cdot d \Omega$ , since $\overline { { \pmb { \sigma } } }$ and $\pmb { \sigma }$ differ only in the hydrostatic part. Substituting these expressions into the expression for the rate of virtual work yields Equation 3.2.3-10 $$ d \delta \overline {{{W}}} = \int_ {V} \Big \{\delta \pmb {\varepsilon}: \mathbf {C}: d \pmb {\varepsilon} + \delta \pmb {\varepsilon}: \overline {{{\pmb {\sigma}}}} \mathbf {I}: d \pmb {\varepsilon} - \frac {1}{3} (1 - \rho) \delta \pmb {\varepsilon}: \mathbf {I} \mathbf {I}: \mathbf {C}: d \pmb {\varepsilon} - \overline {{{\pmb {\sigma}}}} \mathbf {I} \mathbf {I}: d \pmb {\varepsilon} \Big \}. $$ $$ (1 - \rho) \delta \pmb {\varepsilon}: \mathbf {I} d \hat {p} - \frac {1}{3} J ^ {- 1} \delta \lambda \mathbf {I}: \mathbf {C}: d \pmb {\varepsilon} - J ^ {- 1} \delta \lambda d \hat {p} + $$ $$ J ^ {- 1} d \delta \lambda (\Delta p - \Delta \hat {p}) + \overline {{{\pmb {\sigma}}}}: d \delta \pmb {\varepsilon} + \delta \pmb {\varepsilon}: (d \pmb {\Omega} \cdot \overline {{{\pmb {\sigma}}}} - \overline {{{\pmb {\sigma}}}} \cdot d \pmb {\Omega}) \Big \} d V. $$ It remains to choose ±¸. To get a symmetric expression for the rate of virtual work, we choose Equation 3.2.3-11 $$ \delta \lambda = (1 - \rho) J \left(\frac {1}{3 K} \mathbf {I}: \mathbf {C}: \delta \pmb {\varepsilon} - \mathbf {I}: \delta \pmb {\varepsilon} + \frac {1}{K} \delta \hat {p}\right), $$ where Equation 3.2.3-12 $$ K = \frac {1}{9} \mathbf {I}: \mathbf {C}: \mathbf {I} $$ is the (instantaneous) bulk modulus. This is a suitable choice for $\delta \lambda ,$ because the (independent) term proportional to $\delta \hat { p }$ ensures that the modified incremental pressure field, $\Delta \hat { p } ,$ is properly constrained to the incremental pressure, $\Delta p .$ . If we assume that the volumetric moduli I : C and K change slowly with strain and ignore changes in volume, we can write for the second variation d±¸: $$ d \delta \lambda = (1 - \rho) J \left(\frac {1}{3 K} \mathbf {I}: \mathbf {C} - \mathbf {I}\right): d \delta \pmb {\varepsilon}. $$ Equation 3.2.3-13 Hence, we find for the virtual work expression: $$ \delta \overline {{W}} = \int_ {V} \left[ \tilde {\pmb {\sigma}}: \delta \pmb {\varepsilon} + (1 - \rho) (\Delta p - \Delta \hat {p}) \frac {1}{K} \delta \hat {p} \right] d V, $$ Equation 3.2.3-14 where $$ \tilde {\pmb {\sigma}} = \overline {{\pmb {\sigma}}} + (1 - \rho) (\Delta p - \Delta \hat {p}) \left(\frac {1}{3 K} \mathbf {I}: \mathbf {C} - \mathbf {I}\right). $$ Equation 3.2.3-15 For the rate of change of virtual work we find $$ d \delta \overline {{W}} = \int_ {V} \left\{\delta \pmb {\varepsilon}: \mathbf {C}: d \pmb {\varepsilon} - \frac {1}{9 K} (1 - \rho) \delta \pmb {\varepsilon}: \mathbf {C} ^ {T}: \mathbf {I I}: \mathbf {C}: d \pmb {\varepsilon} + \delta \pmb {\varepsilon}: \overline {{\pmb {\sigma}}} \mathbf {I}: d \pmb {\varepsilon} - \right. $$ $$ \frac {1}{3 K} (1 - \rho) \left(\delta \hat {p} \mathbf {I}: \mathbf {C}: d \pmb {\varepsilon} + \delta \pmb {\varepsilon}: \mathbf {C} ^ {T}: \mathbf {I} d \hat {p}\right) - \frac {1}{K} (1 - \rho) \delta \hat {p} d \hat {p} + $$ $$ \left. \tilde {\boldsymbol {\sigma}}: d \delta \boldsymbol {\varepsilon} + \overline {{\boldsymbol {\sigma}}}: \left(\delta \boldsymbol {\varepsilon} \cdot d \boldsymbol {\Omega} - d \boldsymbol {\Omega} \cdot \delta \boldsymbol {\varepsilon}\right) \right\} d V. $$ Equation 3.2.3-16 The initial stress term can be approximated by $$ \tilde {\boldsymbol {\sigma}}: (d \delta \boldsymbol {\varepsilon} + \delta \boldsymbol {\varepsilon} \cdot d \boldsymbol {\Omega} - d \boldsymbol {\Omega} \cdot \delta \boldsymbol {\varepsilon}) d V, $$ which can be written as $$ \tilde {\boldsymbol {\sigma}}: \left[ \left(\frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}}\right) ^ {T} \cdot \frac {\partial d \mathbf {u}}{\partial \mathbf {x}} - 2 \delta \boldsymbol {\varepsilon} \cdot d \boldsymbol {\varepsilon} \right], $$ so that the final expression for the rate of virtual work becomes Equation 3.2.3-17 $$ \begin{array}{l} d \delta \overline {{{W}}} = \int_ {V} \left\{\delta \pmb {\varepsilon}: \mathbf {C}: d \pmb {\varepsilon} - \frac {1}{9 K} (1 - \rho) \delta \pmb {\varepsilon}: \mathbf {C} ^ {T}: \mathbf {I I}: \mathbf {C}: d \pmb {\varepsilon} + \delta \pmb {\varepsilon}: \overline {{{\pmb {\sigma}}}} \mathbf {I}: d \pmb {\varepsilon} - \right. \\ \frac {1}{3 K} (1 - \rho) \left(\delta \hat {p} \mathbf {I}: \mathbf {C}: d \pmb {\varepsilon} + \delta \pmb {\varepsilon}: \mathbf {C} ^ {T}: \mathbf {I} d \hat {p}\right) - \frac {1}{K} (1 - \rho) \delta \hat {p} d \hat {p} + \\ \tilde {\pmb {\sigma}}: \left[ \left(\frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}}\right) ^ {T} \cdot \frac {\partial d \mathbf {u}}{\partial \mathbf {x}} - 2 \delta \pmb {\varepsilon} \cdot d \pmb {\varepsilon} \right] \Biggr \} d V. \\ \end{array} $$ The asymmetric term ±" : $\overline { { \pmb { \sigma } } } \mathbf { I }$ : d" is only significant if large volume changes occur. Hence, the term is ignored except for material models with volumetric plasticity, such as the (capped) Drucker-Prager model and the Cam-clay model. For these models the constitutive matrix C is usually asymmetric anyway so that the addition of this nonsymmetric term does not affect the cost of the analysis. It was assumed in the expression for $d \delta \lambda$ that the (volumetric) moduli change only slowly with strain. This is not the case for material models with volumetric plasticity, in which these moduli can change abruptly. This may lead to slow convergence or even convergence failures. Failures usually occur only in higher-order elements, since in lower-order elements $\Delta p - \Delta \hat { p }$ approaches zero at every point and the error in d±¸ has no impact. # 3.2.4 Solid isoparametric quadrilaterals and hexahedra The library of solid elements in ABAQUS contains first- and second-order isoparametric elements. The first-order elements are the 4-node quadrilateral for plane and axisymmetric analysis and the 8-node brick for three-dimensional cases. The library of second-order isoparametric elements includes "serendipity" elements: the 8-node quadrilateral and the 20-node brick, and a "full Lagrange" element, the 27-node (variable number of nodes) brick. The term "serendipity" refers to the interpolation, which is based on corner and midside nodes only. In contrast, the full Lagrange interpolation uses product forms of the one-dimensional Lagrange polynomials to provide the two- or three-dimensional interpolation functions. All these isoparametric elements are available with full or reduced integration. Gauss integration is almost always used with second-order isoparametric elements because it is efficient and the Gauss points corresponding to reduced integration are the Barlow points ( Barlow, 1976) at which the strains are most accurately predicted if the elements are well-shaped. The three-dimensional brick elements can also be used for the analysis of laminated composite solids. Several layers of different material, in different orientations, can be specified in each solid element. The material layers or lamina can be stacked in any of the three isoparametric coordinates, parallel to opposite faces of the master element (Figure 3.2.4-1). These elements use the same interpolation functions as the homogeneous elements, but the integration takes the variation of material properties in the stacking direction into account. Hybrid pressure-displacement versions of these elements are provided for use with incompressible and nearly incompressible constitutive models (see \`\`Hybrid incompressible solid element formulation,'' Section 3.2.3, and \`\`Hyperelastic material behavior,'' Section 4.6.1, for a detailed discussion of the formulations used). # Interpolation Isoparametric interpolation is defined in terms of the isoparametric element coordinates g, h, r shown in Figure 3.2.4-1. These are material coordinates, since ABAQUS is a Lagrangian code. They each span the range ¡1 to +1 in an element. The node numbering convention used in ABAQUS for isoparametric elements is also shown in Figure 3.2.4-1. Corner nodes are numbered first, followed by the midside nodes for second-order elements. The interpolation functions are as follows. First-order quadrilateral: $$ u = \frac {1}{4} (1 - g) (1 - h) u _ {1} + \frac {1}{4} (1 + g) (1 - h) u _ {2} + \frac {1}{4} (1 + g) (1 + h) u _ {3} + \frac {1}{4} (1 - g) (1 + h) u _ {4} $$ Second-order quadrilateral: $$ \begin{array}{l} u = - \frac {1}{4} (1 - g) (1 - h) (1 + g + h) u _ {1} - \frac {1}{4} (1 + g) (1 - h) (1 - g + h) u _ {2} \\ - \frac {1}{4} (1 + g) (1 + h) (1 - g - h) u _ {3} - \frac {1}{4} (1 - g) (1 + h) (1 + g - h) u _ {4} \\ + \frac {1}{2} (1 - g) (1 + g) (1 - h) u _ {5} + \frac {1}{2} (1 - h) (1 + h) (1 + g) u _ {6} \\ + \frac {1}{2} (1 - g) (1 + g) (1 + h) u _ {7} + \frac {1}{2} (1 - h) (1 + h) (1 - g) u _ {8} \\ \end{array} $$ First-order brick: $$ \begin{array}{l} u = \frac {1}{8} (1 - g) (1 - h) (1 - r) u _ {1} + \frac {1}{8} (1 + g) (1 - h) (1 - r) u _ {2} \\ + \frac {1}{8} (1 + g) (1 + h) (1 - r) u _ {3} + \frac {1}{8} (1 - g) (1 + h) (1 - r) u _ {4} \\ + \frac {1}{8} (1 - g) (1 - h) (1 + r) u _ {5} + \frac {1}{8} (1 + g) (1 - h) (1 + r) u _ {6} \\ + \frac {1}{8} (1 + g) (1 + h) (1 + r) u _ {7} + \frac {1}{8} (1 - g) (1 + h) (1 + r) u _ {8} \\ \end{array} $$ 20-node brick: # Elements $$ \begin{array}{l} u = - \frac {1}{8} (1 - g) (1 - h) (1 - r) (2 + g + h + r) u _ {1} - \frac {1}{8} (1 + g) (1 - h) (1 - r) (2 - g + h + r) u _ {2} \\ - \frac {1}{8} (1 + g) (1 + h) (1 - r) (2 - g - h + r) u _ {3} - \frac {1}{8} (1 - g) (1 + h) (1 - r) (2 + g - h + r) u _ {4} \\ - \frac {1}{8} (1 - g) (1 - h) (1 + r) (2 + g + h - r) u _ {5} - \frac {1}{8} (1 + g) (1 - h) (1 + r) (2 - g + h - r) u _ {6} \\ - \frac {1}{8} (1 + g) (1 + h) (1 + r) (2 - g - h - r) u _ {7} - \frac {1}{8} (1 - g) (1 + h) (1 + r) (2 + g - h - r) u _ {8} \\ + \frac {1}{4} (1 - g) (1 + g) (1 - h) (1 - r) u _ {9} + \frac {1}{4} (1 - h) (1 + h) (1 + g) (1 - r) u _ {1 0} \\ + \frac {1}{4} (1 - g) (1 + g) (1 + h) (1 - r) u _ {1 1} + \frac {1}{4} (1 - h) (1 + h) (1 - g) (1 - r) u _ {1 2} \\ + \frac {1}{4} (1 - g) (1 + g) (1 - h) (1 + r) u _ {1 3} + \frac {1}{4} (1 - h) (1 + h) (1 + g) (1 + r) u _ {1 4} \\ + \frac {1}{4} (1 - g) (1 + g) (1 + h) (1 + r) u _ {1 5} + \frac {1}{4} (1 - h) (1 + h) (1 - g) (1 + r) u _ {1 6} \\ + \frac {1}{4} (1 - r) (1 + r) (1 - g) (1 - h) u _ {1 7} + \frac {1}{4} (1 - r) (1 + r) (1 + g) (1 - h) u _ {1 8} \\ + \frac {1}{4} (1 - r) (1 + r) (1 + g) (1 + h) u _ {1 9} + \frac {1}{4} (1 - r) (1 + r) (1 - g) (1 + h) u _ {2 0} \\ \end{array} $$ Figure 3.2.4-1 Isoparametric master elements.