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Elements
\delta W = F _ {\alpha} \delta u _ {\alpha} + M _ {\alpha} \delta \psi_ {\alpha}.
We assume that the behavior of the joint is defined by
F _ {\alpha} = F _ {\alpha} (u _ {\alpha}, \dot {u} _ {\alpha}) \quad \mathrm{and} \quad M _ {\alpha} = M _ {\alpha} (\psi_ {\alpha}, \dot {\psi} _ {\alpha}) \quad \mathrm{(nosumon} \alpha).
The contribution to the operator matrix for the Newton solution is
d \delta W = \sum_ {\alpha} \delta u _ {\alpha} \left(\frac {\partial F _ {\alpha}}{\partial u _ {\alpha}} + \frac {\partial F _ {\alpha}}{\partial \dot {u} _ {\alpha}} \frac {\partial \dot {u}}{\partial u}\right) d u _ {\alpha} + \delta \psi_ {\alpha} \left(\frac {\partial M _ {\alpha}}{\partial \psi_ {\alpha}} + \frac {\partial M _ {\alpha}}{\partial \dot {\psi} _ {\alpha}} \frac {\partial \dot {u}}{\partial u}\right) d \psi_ {\alpha} + F _ {\alpha} d \delta u _ {\alpha} + M _ {\alpha} d \delta \psi_ {\alpha},
where \partial \dot { u } / \partial u is defined by the dynamic time integration operator.
3.9.8 Rotary inertia element
The MASS and ROTARYI elements allow the inertia of a rigid body to be introduced at a node. In this section the formulation used with these elements is defined.
It is assumed that the node at which the mass and rotary inertia are introduced is the center of mass of the body. We refer to the node as the rigid body reference node, C. Let the local principal axes of inertia of the body be \mathbf { e } _ { \alpha } , \alpha = 1 , 2 , 3 . Let r be the vector between C and some point in the rigid body with current coordinates x, so that
\mathbf {r} = \mathbf {x} - \mathbf {x} _ {C} = x _ {\alpha} \mathbf {e} _ {\alpha}, \mathrm{say},
where x _ { \alpha } are local coordinates in the rigid body. The mass of the rigid body is the integral of the mass density \rho ( x _ { \alpha } ) over the body
m = \int_ {V} \rho d V.
Since C is assumed to be the center of mass of the body,
\int_ {V} \rho x _ {\alpha} d V = 0.
Since the \mathbf { e } _ { \alpha } are the principal axes of the body,
\int_ {V} \rho x _ {\alpha} x _ {\beta} d V = 0 \quad \mathrm{for} \quad \alpha \neq \beta .
Let I _ { 1 1 } , I _ { 2 2 } , and I _ { 3 3 } be the second moments of inertia of the body about its principal axes \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } . , and \mathbf { e } _ { 3 } , then
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I _ {1 1} = \int_ {V} \rho \left(\left(x _ {2}\right) ^ {2} + \left(x _ {3}\right) ^ {2}\right) d V
I _ {2 2} = \int_ {V} \rho \bigl ((x _ {3}) ^ {2} + (x _ {1}) ^ {2} \bigr) d V
I _ {3 3} = \int_ {V} \rho \big ((x _ {1}) ^ {2} + (x _ {2}) ^ {2} \big) d V.
The rotary inertia tensor is written
\mathbf {I} = \sum_ {\alpha = 1} ^ {3} I _ {\alpha \alpha} \mathbf {e} _ {\alpha} \mathbf {e} _ {\alpha}.
For a rigid body the velocity of any point in the body is given by
\dot {\mathbf {u}} = \dot {\mathbf {u}} _ {C} + \boldsymbol {\omega} \times \mathbf {r},
where \omega = \dot { \phi } is the angular velocity of the body. Taking a second time derivative, the acceleration is
\ddot {\mathbf {u}} = \ddot {\mathbf {u}} _ {C} + \dot {\boldsymbol {\omega}} \times \mathbf {r} + \boldsymbol {\omega} \times (\boldsymbol {\omega} \times \mathbf {r}).
The local or strong form of the equilibrium equations represents the balance of linear momentum and balance of angular momentum; these two equilibrium equations are
m \ddot {\mathbf {u}} _ {C} = \bar {\mathbf {f}},
\mathbf {I} \cdot \dot {\boldsymbol {\omega}} + \boldsymbol {\omega} \times \mathbf {I} \cdot \boldsymbol {\omega} = \bar {\mathbf {m}}.
The variational or weak form of equilibrium is
\delta W _ {A} + \delta W _ {e x t} = 0.
The internal or d'Alembert force contribution is
\delta W _ {A} = - \int_ {V} \rho \delta \mathbf {u} \cdot \ddot {\mathbf {u}} d V = - m \ddot {\mathbf {u}} _ {C} \cdot \delta \mathbf {u} _ {C} - (\mathbf {I} \cdot \dot {\pmb {\omega}} + \pmb {\omega} \times \mathbf {I} \cdot \pmb {\omega}) \cdot \delta \pmb {\theta},
where \delta { \bf u } = \delta { \bf u } _ { C } + \delta \pmb { \theta } \times { \bf r } is the variation of the position of a point in the body. Here \delta \mathbf { u } _ { C } is the variation of the position of the rigid body reference node, and \delta \pmb { \theta } is the variation of the rotation of the rigid body reference node. The external loading contribution is
\delta W _ {e x t} = \bar {\mathbf {f}} \cdot \delta \mathbf {u} _ {C} + \bar {\mathbf {m}} \cdot \delta \pmb {\theta}.
Introducing component expressions relative to the principal axes of inertia, the rotational contribution to the weak form becomes
Elements
\begin{array}{l} \delta W _ {A} = - m \delta \mathbf {u} _ {C} \cdot \ddot {\mathbf {u}} _ {C} - \delta \theta^ {1} \left(I _ {1 1} \dot {\omega} ^ {1} + \left(I _ {3 3} - I _ {2 2}\right) \omega^ {2} \omega^ {3}\right) \\ - \delta \theta^ {2} \left(I _ {2 2} \dot {\omega} ^ {2} + \left(I _ {1 1} - I _ {3 3}\right) \omega^ {3} \omega^ {1}\right) \\ - \delta \theta^ {3} \left(I _ {3 3} \dot {\omega} ^ {3} + \left(I _ {2 2} - I _ {1 1}\right) \omega^ {1} \omega^ {2}\right). \\ \end{array}
When the inertia of a rigid body is used with implicit time integration, the Jacobian contribution of \delta W _ { A } is required: this is
- d \delta W _ {A} = m \delta \mathbf {u} _ {C} \cdot d \ddot {\mathbf {u}} _ {C} + \delta \pmb {\theta} \cdot \left[ \mathbf {I} \cdot d \dot {\pmb {\omega}} + d \pmb {\omega} \times \mathbf {I} \cdot \pmb {\omega} + \pmb {\omega} \times \mathbf {I} \cdot d \pmb {\omega} + d \pmb {\theta} \times \mathbf {I} \cdot \dot {\pmb {\omega}} + \mathbf {I} \cdot (\dot {\pmb {\omega}} \times d \pmb {\theta}) \right]
\left. + (\boldsymbol {\omega} \cdot \mathbf {I} \cdot \boldsymbol {\omega}) d \boldsymbol {\theta} - (\boldsymbol {\omega} \cdot d \boldsymbol {\theta}) \mathbf {I} \cdot \boldsymbol {\omega} + \boldsymbol {\omega} \times \mathbf {I} \cdot (\boldsymbol {\omega} \times d \boldsymbol {\theta}) \right].
3.9.9 Distributing coupling elements
The distributing coupling elements in ABAQUS/Standard provide a means to connect a reference node to a group of coupling nodes in a way that distributes loads according to weight factors that are prescribed individually at each coupling node. The element distributes forces and moments at the reference node as a coupling node-force distribution only. This section defines this load distribution relationship and the resultant element development.
The reference node has displacement ( \mathbf { u } ^ { R } ) and rotation ( \phi ^ { R } ) degrees of freedom. The coupling nodes have only displacement ( \boldsymbol { \mathbf { u } } ^ { n } ) degrees of freedom active in this element. Each coupling node has a weight factor w ^ { n } assigned, which determines the proportion of load carried by the element that is transmitted through the coupling node. Weight factors are dimensionless, and their magnitude is significant only in a relative sense. Hereafter, normalized weights are used:
\hat {w} ^ {n} = \frac {w ^ {n}}{\sum w ^ {n}}.
Load distribution
Let \mathbf { F } ^ { R } and \mathbf { M } ^ { R } be the load and moment applied to the reference node. The statically admissible force distribution \mathbf { F } ^ { n } among the coupling nodes satisfies
Equation 3.9.9-1
\sum_ {n} \mathbf {F} ^ {n} = \mathbf {F} ^ {R}
\sum_ {n} \mathbf {x} ^ {n} \times \mathbf {F} ^ {n} = \mathbf {M} ^ {R} + \mathbf {x} ^ {R} \times \mathbf {F} ^ {R},
where \mathbf { x } ^ { R } and \mathbf { x } ^ { n } are the positions of the reference and coupling nodes, respectively. For an arbitrary number of coupling nodes there is no unique solution to Equation 3.9.9-1.
The force distribution adopted in ABAQUS has the property that the linearized motion of the reference node is compatible with the coupling node group motion in an average sense. This compatibility can be
Elements
described by considering the momentum of a moving coupling node group in a case where weight factors are considered as masses. In this case the reference node motion is identical to that of a point on a rigid body occupying the position of the reference node, where the center of mass of the rigid body is the center of mass of the coupling nodes and the rigid body moves with the same linear and angular momentum as the coupling node group. Since the element mass is distributed this way, the dynamic behavior of the element also has this property.
\mathbf {F} ^ {n} \stackrel {{\text { def }}} {{=}} \hat {w} ^ {n} \left(\mathbf {F} ^ {R} + \left(\mathbf {T} ^ {- 1} \cdot \hat {\mathbf {M}} ^ {R}\right) \times \mathbf {r} ^ {n}\right),
where
\mathbf {\hat {M}} ^ {R} = \mathbf {M} ^ {R} + \mathbf {r} ^ {R} \times \mathbf {F} ^ {R},
\mathbf {r} ^ {n} = \mathbf {x} ^ {n} - \bar {\mathbf {x}},
\bar {\mathbf {x}} = \frac {\sum_ {n} w ^ {n} \mathbf {x} ^ {n}}{\sum_ {n} w ^ {n}} = \sum_ {n} \hat {w} ^ {n} \mathbf {x} ^ {n},
and the coupling node arrangement inertia tensor is
\mathbf {T} = \sum_ {n} \hat {w} ^ {n} \left[ \left(\mathbf {r} ^ {n} \cdot \mathbf {r} ^ {n}\right) \mathbf {I} - \left(\mathbf {r} ^ {n} \mathbf {r} ^ {n}\right) \right],
where I is the second-order identity tensor. This force distribution is recognized to be equivalent to the classic bolt-pattern force distribution when the weight factors are interpreted as bolt cross-section areas.
Constraint expression
The load distribution results in the following linearized constraint on node motions:
\delta \mathbf {x} ^ {R} = \sum_ {n} \hat {w} ^ {n} \delta \mathbf {x} ^ {n} + \left(\mathbf {T} ^ {- 1} \cdot \sum_ {n} \hat {w} ^ {n} \left(\mathbf {r} ^ {n} \times \delta \mathbf {x} ^ {n}\right)\right) \times \mathbf {r} ^ {R}
\delta \boldsymbol {\omega} ^ {R} = \mathbf {T} ^ {- 1} \cdot \sum_ {n} \hat {w} ^ {n} \left(\mathbf {r} ^ {n} \times \delta \mathbf {x} ^ {n}\right),
where \mathbf { r } ^ { R } = \mathbf { x } ^ { R } - \bar { \mathbf { x } }
Finite motion
Finite displacement and rotation terms take the form of a constraint on the motion of the reference node as a function of the coupling-node finite incremental motions. A measure of the finite rotation of the coupling node arrangement is developed first and is based on the mid-increment position of the coupling nodes, defined as
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\check {\mathbf {r}} ^ {n} \stackrel {\mathrm{def}} {=} \mathbf {r} _ {0} ^ {n} + \frac {1}{2} \Delta \mathbf {x} ^ {n},
and the mid-increment inertia tensor is
\mathbf {J} = \sum_ {n} \hat {w} ^ {n} \left[ (\check {\mathbf {r}} ^ {n} \cdot \check {\mathbf {r}} ^ {n}) \mathbf {I} - (\check {\mathbf {r}} ^ {n} \check {\mathbf {r}} ^ {n}) \right].
The mid-increment "spin" is then
\Delta \pmb {\theta} ^ {R} = \mathbf {J} ^ {- 1} \cdot \left(\sum_ {n} \hat {w} ^ {n} \check {\mathbf {r}} ^ {n} \times \Delta \mathbf {x} ^ {n}\right).
The finite incremental rotation tensor \Delta \mathbf { R } is deduced from the above expression according to the Hughes and Winget (1980) formula,
\Delta \mathbf {R} = \left(\mathbf {I} - \frac {1}{2} \widehat {\Delta \pmb {\theta}} ^ {R}\right) ^ {- 1} \cdot \left(\mathbf {I} + \frac {1}{2} \widehat {\Delta \pmb {\theta}} ^ {R}\right).
This orthogonal tensor yields an incremental finite-rotation vector \Delta \stackrel { \prime } { \phi } ^ { R } . From this rotation description comes the constraint expressions for finite displacement and rotation:
\Delta \mathbf {x} ^ {R} = \sum_ {n} \hat {w} ^ {n} \Delta \mathbf {x} ^ {n} + \Delta \mathbf {R} \cdot \mathbf {r} _ {0} ^ {R} - \mathbf {r} _ {0} ^ {R},
\Delta \phi^ {R} = \Delta \acute {\phi} ^ {R}.
The compatibility tolerance applied to these expressions can be controlled using the *CONTROLS, PARAMETERS=CONSTRAINTS option.
Virtual work contribution
The virtual work expression for the attached structure is augmented with the contribution of the constraint
Equation 3.9.9-2
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\begin{array}{l} \delta \Pi^ {*} = \delta \Pi + \pmb {\lambda} _ {f} \cdot \left[ \delta \mathbf {x} ^ {R} - \sum_ {n} \hat {w} ^ {n} \delta \mathbf {x} ^ {n} - \delta \mathbf {R} ^ {R} \cdot \mathbf {r} _ {0} ^ {R} \right] + \\ \pmb {\lambda} _ {m} \cdot \left[ \delta \pmb {\omega} ^ {R} - \delta \pmb {\acute {\phi}} ^ {R} \right] + \\ \delta \pmb {\lambda} _ {f} \cdot \left[ \Delta \mathbf {x} ^ {R} - \sum_ {n} \hat {w} ^ {n} \Delta \mathbf {x} ^ {n} - \Delta \mathbf {R} \cdot \mathbf {r} _ {0} ^ {R} + \mathbf {r} _ {0} ^ {R} \right] + \\ \delta \pmb {\lambda} _ {m} \cdot \left[ \Delta \pmb {\phi} ^ {R} - \Delta \pmb {\acute {\phi}} ^ {R} \right], \\ \end{array}
where \delta \Pi ^ { * } is the augmented virtual work expression, ±¦ is the virtual work expression for the attached structure, and \lambda _ { f } and \lambda _ { m } are the respective Lagrange multiplier variables for force and moment.
Initial stress stiffness terms
The initial stress stiffness terms are derived from a suitable approximation of the exact virtual work expression shown in Equation 3.9.9-2. This approximation is based on an assumption of infinitesimal incremental motions, \Delta \mathbf { r } ^ { n } and \Delta { \phi } ^ { R } , that implies
\delta \pmb {\phi} ^ {R} = \delta \pmb {\omega} ^ {R} \quad \mathrm{and}
\delta \acute {\boldsymbol {\phi}} ^ {R} = \mathbf {T} ^ {- 1} \cdot \sum_ {n} \hat {w} ^ {n} \left(\mathbf {r} ^ {n} \times \delta \mathbf {x} ^ {n}\right).
An approximate virtual work expression is obtained:
\widetilde {\delta \Pi} ^ {*} = \delta \Pi + \pmb {\lambda} _ {f} \cdot \left[ \delta \mathbf {x} ^ {R} - \sum_ {n} \hat {w} ^ {n} \delta \mathbf {x} ^ {n} - \delta \pmb {\omega} ^ {R} \times \mathbf {r} ^ {R} \right] +
\pmb {\lambda} _ {m} \cdot \left[ \delta \pmb {\omega} ^ {R} - \mathbf {T} ^ {- 1} \cdot \sum_ {n} \hat {w} ^ {n} \left(\mathbf {r} ^ {n} \times \delta \mathbf {x} ^ {n}\right) \right] +
\delta \pmb {\lambda} _ {f} \cdot \left[ \Delta \mathbf {x} ^ {R} - \sum_ {n} \hat {w} ^ {n} \Delta \mathbf {x} ^ {n} - \Delta \mathbf {R} \cdot \mathbf {r} _ {0} ^ {R} + \mathbf {r} _ {0} ^ {R} \right] +
\delta \pmb {\lambda} _ {m} \cdot \left[ \Delta \pmb {\phi} ^ {R} - \Delta \acute {\pmb {\phi}} ^ {R} \right].
This expression yields the following initial stress stiffness terms:
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\pmb {\lambda} _ {f} \cdot \left[ \left(\delta \pmb {\omega} ^ {R} \cdot \mathrm{d} \pmb {\omega} ^ {R}\right) \mathbf {r} ^ {R} - \frac {1}{2} \left(\delta \pmb {\omega} ^ {R} \cdot \mathbf {r} ^ {R}\right) \mathrm{d} \pmb {\omega} ^ {R} - \frac {1}{2} \left(\mathrm{d} \pmb {\omega} ^ {R} \cdot \mathbf {r} ^ {R}\right) \delta \pmb {\omega} ^ {R} \right] +
\boldsymbol {\lambda} _ {m} \cdot \mathbf {T} ^ {- 1} \cdot \left[ \delta \boldsymbol {\omega} ^ {R} \sum_ {n} \hat {w} ^ {n} \mathbf {r} ^ {n} \cdot \mathrm{d} \mathbf {r} ^ {n} - \frac {1}{2} \sum_ {n} \hat {w} ^ {n} \mathrm{d} \mathbf {r} ^ {n} \mathbf {r} ^ {n} \cdot \delta \boldsymbol {\omega} ^ {R} - \frac {1}{2} \sum_ {n} \hat {w} ^ {n} \mathbf {r} ^ {n} \mathrm{d} r ^ {n} \cdot \delta \boldsymbol {\omega} ^ {R} + \right.
\left. \mathrm{d} \pmb {\omega} ^ {R} \sum_ {n} \hat {w} ^ {n} \mathbf {r} ^ {n} \cdot \delta \mathbf {r} ^ {n} - \frac {1}{2} \sum_ {n} \hat {w} ^ {n} \delta \mathbf {r} ^ {n} \mathbf {r} ^ {n} \cdot \mathrm{d} \pmb {\omega} ^ {R} - \frac {1}{2} \sum_ {n} \hat {w} ^ {n} \mathbf {r} ^ {n} \delta r ^ {n} \cdot \mathrm{d} \pmb {\omega} ^ {R} \right].
Mass
The distributing coupling elements also distribute masses to each coupling node according to the weight distribution. A prescribed element mass of M is distributed to the cloud nodes according to
m ^ {n} = \hat {w} ^ {n} M,
where m ^ { n } is the cloud node mass. Masses are distributed only to the cloud nodes; no mass is associated with the reference node.
4. Mechanical Constitutive Theories
4.1 Overview
4.1.1 Mechanical constitutive models
A wide variety of materials is encountered in stress analysis problems, and for any one of these materials a range of constitutive models is available to describe the material's behavior. For example, a component made from a standard structural steel can be modeled as an isotropic, linear elastic, material with no temperature dependence. This simple model would probably suffice for routine design, so long as the component is not in any critical situation. However, if the component might be subjected to a severe overload, it is important to determine how it might deform under that load and if it has sufficient ductility to withstand the overload without catastrophic failure. The first of these questions might be answered by modeling the material as a rate-independent elastic, perfectly plastic material, or--if the ultimate stress in a tension test of a specimen of the material is very much above the initial yield stress--isotropic work hardening might be included in the plasticity model. A nonlinear analysis (with or without consideration of geometric nonlinearity, depending on whether the analyst judges that the structure might buckle or undergo large geometry changes during the event) is then done to determine the response. But the severe overload might be applied suddenly, thus causing rapid straining of the material. In such circumstances the inelastic response of metals usually exhibits rate dependence: the flow stress increases as the strain rate increases. A "viscoplastic" (rate-dependent) material model might, therefore, be required. (Arguing that it is conservative to ignore this effect because it is a strengthening effect is not necessarily acceptable--the strengthening of one part of a structure might cause load to be shed to another part, which proves to be weaker in the event.) So far the analyst can manage with relatively simple (but nonlinear) constitutive models. But if the failure is associated with localization--tearing of a sheet of material or plastic buckling--a more sophisticated material model might be required because such localizations depend on details of the constitutive behavior that are usually ignored because of their complexity (see, for example, Needleman, 1977). Or if the concern is not gross overload, but gradual failure of the component because of creep at high temperature or because of low cycle fatigue, or perhaps a combination of these effects, then the response of the material during several cycles of loading, in each of which a small amount of inelastic deformation might occur, must be predicted: a circumstance in which we need to model much more of the detail of the material's response.
So far the discussion has considered a conventional structural material. We can broadly classify the materials of interest as those that exhibit almost purely elastic response, possibly with some energy dissipation during rapid loading by viscoelastic response (the elastomers, such as rubber or solid propellant); materials that yield and exhibit considerable ductility beyond yield (such as mild steel and other commonly used metals, ice at low strain rates, and clay); materials that flow by rearrangement of particles that interact generally through some dominantly frictional mechanism (such as sand); and brittle materials (rocks, concrete, ceramics). The constitutive library provided in ABAQUS contains a range of linear and nonlinear material models for all of these categories of materials. In general the library has been developed to provide those models that are most usually required for practical applications. There are several distinct models in the library; and for the more commonly encountered
materials (metals, in particular), several ways of modeling the material are provided, each suitable to a particular type of analysis application. But the library is far from comprehensive: the range of physical material behavior is far too broad for this ever to be possible. The analyst must review the material definitions provided in ABAQUS in the context of his particular application. If there is no model in the library that is useful for a particular case, ABAQUS/Standard contains a user subroutine--UMAT--and, similarly, ABAQUS/Explicit contains a user subroutine--VUMAT. In these routines the user can code a material model (or call other routines that perform that task). This "user subroutine" capability is a powerful resource for the sophisticated analyst who is able to cope with the demands of programming a complex material model.
Theoretical aspects of the material models that are provided in ABAQUS are described in this chapter, which is intended as a definition of the details of the material models that are provided: it also provides useful guidance to analysts who might have to code their own models in UMAT or VUMAT.
From a numerical viewpoint the implementation of a constitutive model involves the integration of the state of the material at an integration point over a time increment during a nonlinear analysis. (The implementation of constitutive models in ABAQUS assumes that the material behavior is entirely defined by local effects, so each spatial integration point can be treated independently.) Since ABAQUS/Standard is most commonly used with implicit time integration, the implementation must also provide an accurate "material stiffness matrix" for use in forming the Jacobian of the nonlinear equilibrium equations; this is not necessary for ABAQUS/Explicit.
The mechanical constitutive models that are provided in ABAQUS often consider elastic and inelastic response. The inelastic response is most commonly modeled with plasticity models. Several plasticity models are described in this chapter. Some of the constitutive models in ABAQUS also use damage mechanics concepts, the distinction being that in plasticity theory the elasticity is not affected by the inelastic deformation (the Young's modulus of a metal specimen is not changed by loading it beyond yield, until the specimen is very close to failure), while damage models include the degradation of the elasticity caused by severe loading (such as the loss of elastic stiffness suffered by a concrete specimen after it has been subjected to large uniaxial compressive loading).
In the inelastic response models that are provided in ABAQUS, the elastic and inelastic responses are distinguished by separating the deformation into recoverable (elastic) and nonrecoverable (inelastic) parts. This separation is based on the assumption that there is an additive relationship between strain rates:
Equation 4.1.1-1
\dot {\varepsilon} = \dot {\varepsilon} ^ {e l} + \dot {\varepsilon} ^ {p l},
where "_ is the total strain rate, \dot { \pmb { \varepsilon } } ^ { e l } is the rate of change of the elastic strain, and \dot { \varepsilon } ^ { p l } is the rate of change of inelastic strain.
A more general assumption is that the total deformation, F, is made up of inelastic deformation followed by purely elastic deformation (with the rigid body rotation added in at any stage in the process):
\mathbf {F} = \mathbf {F} ^ {e l} \cdot \mathbf {F} ^ {p l}.
In ``The additive strain rate decomposition, '' Section 1.4.4, the circumstances are discussed under which Equation 4.1.1-1 is a legitimate approximation to Equation 4.1.1-2. We conclude that, if
- the total strain rate measure used in Equation 4.1.1-1 is the rate of deformation:
\dot {\pmb {\varepsilon}} = \mathrm{sym} (\mathbf {L}) = \mathrm{sym} \left(\dot {\mathbf {F}} \cdot \mathbf {F} ^ {- 1}\right) = \mathrm{sym} \left(\frac {\partial \mathbf {v}}{\partial \mathbf {x}}\right),
where v is the velocity and x is the current spatial position of a material point; and
2. the elastic strains are small,
then the approximation is consistent. ABAQUS uses the rate of deformation as the strain rate measure in finite-strain problems for this reason. (The distinction between different strain measures matters only when the strains are not negligible compared to unity; that is, in finite-strain problems.) The elastic strains always remain small for many materials of practical interest; for example, the yield stress of a metal is typically three orders of magnitude smaller than its elastic modulus, implying elastic strains of order 1 0 ^ { - 3 } . However, some materials (polymers, for example) can undergo large elastic straining and also flow inelastically, in which case the additive strain rate decomposition is no longer a consistent approximation.
Various elastic response models are provided in ABAQUS. The simplest of these is linear elasticity:
\pmb {\sigma} = \mathbf {D} ^ {e l}: \pmb {\varepsilon} ^ {e l},
where \mathbf { D } ^ { e l } is a matrix that may depend on temperature but does not depend on the deformation (except when such dependency is introduced in damage models). This elasticity model is intended to be used for small-strain problems or to model the elasticity in an elastic-plastic model in which the elastic strains are always small. This type of behavior is defined in the *ELASTIC material option.
An extension of the elastic type of behavior is the *HYPOELASTIC option:
\dot {\pmb {\sigma}} = \mathbf {D} ^ {e l}: \dot {\pmb {\varepsilon}} ^ {e l},
where now \mathbf { D } ^ { e l } may depend on the deformation. In this case the elasticity may be nonlinear, but the implementation in ABAQUS still assumes that the elastic strains will always be small. In porous and granular media, the elastic properties are strongly dependent on the volume strain. This form of elastic behavior is modeled with the *POROUS ELASTIC model, which is described in ``Porous elasticity,'' Section 4.4.1.
The most general type of nonlinear elastic behavior is the *HYPERELASTIC option, in which we assume that there is a strain energy density potential--U --from which the stresses are defined (to within a hydrostatic stress value if the material is fully incompressible) by