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4.4.6 Models for crushable foams
The constitutive model described here is available in ABAQUS for the analysis of crushable foams typically used in energy absorption structures. The model is based on the critical state theory, which is widely accepted as a framework for describing porous materials such as soils and rocks ( Schofield et al., 1968, and Parry, 1972). In the case of foams, the ability of the material to deform volumetrically (in compression) is enhanced by cell wall buckling processes as described by Gibson et al. (1982), Gibson and Ashby (1982), and Maiti et al. (1984). It is assumed that the resulting deformation is not recoverable instantaneously and can, thus, be idealized as being plastic for short duration events. In tension, on the other hand, cell walls break readily and as a result the tensile bearing capacity of foam is considerably smaller than its compressive strength.
The model uses a yield surface with an elliptical dependence of deviatoric stress on pressure stress. In the ¦-plane yielding is assumed to depend on the third invariant of deviatoric stress (this part of the model is identical to the critical state and granular material models). The evolution of the yield surface is controlled by the inelastic volume strain experienced by the material: compactive inelastic strains produce hardening while dilatant inelastic volume strains lead to softening. Nonassociated flow is assumed and is based on simple observations described later. The mechanical behavior of foam is also known to be sensitive to the rate of straining. This effect can be introduced by a piecewise linear law or by the overstress power law model.
The strain rate decomposition
The volume change is decomposed as
Equation 4.4.6-1
J = J ^ {e l} \cdot J ^ {p l},
where J is the ratio of current volume to original volume, J ^ { e l } is the elastic (recoverable) part of the ratio of current to original volume of the foam volume, and J ^ { p l } is the plastic--nonrecoverable--part of the ratio of current to original volume of the foam volume.
Volumetric strains are defined as
\varepsilon_ {\mathrm{vol}} = \ln J,
\varepsilon_ {\mathrm{vol}} ^ {e l} = \ln J ^ {e l},
\varepsilon_ {\mathrm{vol}} ^ {p l} = \ln J ^ {p l}.
These definitions and Equation 4.4.6-1 result in the usual additive strain rate decomposition for volumetric strains:
Equation 4.4.6-2
d \varepsilon_ {\mathrm{vol}} = d \varepsilon_ {\mathrm{vol}} ^ {e l} + d \varepsilon_ {\mathrm{vol}} ^ {p l}.
The model also assumes the deviatoric strain rates decompose additively, so that the total strain rates decompose as
d \pmb {\varepsilon} = d \pmb {\varepsilon} ^ {e l} + d \pmb {\varepsilon} ^ {p l}.
Elastic behavior
The elastic behavior can be modeled as linear or by using the porous elasticity model, typically with a nonzero tensile strength, as described in ``Porous elasticity,'' Section 4.4.1.
Plastic behavior
The yield surface used with this model is defined in terms of the equivalent pressure stress,
p = - \frac {1}{3} \mathrm{trace} \pmb {\sigma} = - \frac {1}{3} \pmb {\sigma}: \mathbf {I};
the Mises equivalent stress,
q = \sqrt {\frac {3}{2} (\mathbf {S} : \mathbf {S})};
and the third invariant of deviatoric stress,
r = (\frac {9}{2} \mathbf {S} \cdot \mathbf {S}: \mathbf {S}) ^ {\frac {1}{3}}.
We also define a deviatoric stress measure,
t = \frac {q}{2} \left[ 1 + \frac {1}{K} - \left(1 - \frac {1}{K}\right) \left(\frac {r}{q}\right) ^ {3} \right],
where K = K ( \theta , f _ { i } ) is a material parameter that may be a function of temperature, \theta , and other predefined field variables, f _ { i } , i = 1 , 2 . . . . This measure of deviatoric stress is used because it allows matching of different stress values in tension and compression in the deviatoric plane, thereby providing flexibility in fitting experimental results. In uniaxial tension ( r / q ) ^ { 3 } = 1 . 0 ; therefore, t = q / K . In uniaxial compression ( r / q ) ^ { 3 } = - 1 . 0 ; therefore, t = q . When K { = } 1 . 0 _ { ; } , the dependence on the third deviatoric stress invariant is removed, and the Mises circle is recovered in the deviatoric plane: t = q . . Figure 4.4.6-2 shows the dependence of t on K. To ensure convexity of the yield surface 0 . 7 7 8 \leq K \leq 1 . 0 .
Figure 4.4.6-1 Typical yield surfaces in the deviatoric plane for the foam model.
radar
| Curve | K |
|---|---|
| a | 1.0 |
| b | 0.8 |
With this expression for the deviatoric stress measure, the yield surface is defined as
Equation 4.4.6-3
F = f - f _ {0} = \left[ \left(\frac {p _ {t} - p _ {c}}{2} + p\right) ^ {2} + \left(\frac {t}{M}\right) ^ {2} \right] ^ {\frac {1}{2}} - \frac {p _ {c} + p _ {t}}{2} = 0,
where p _ { t } = p _ { t } ( \theta , f _ { i } ) is the strength of the material in hydrostatic tension, p _ { c } ( \varepsilon _ { v o l } ^ { p l } , \theta , f _ { i } ) is the yield stress in hydrostatic compression, and M ( \theta , f _ { i } ) is the slope of the critical state line in the p { - } t plane. This yield surface is depicted in Figure 4.4.6-2.
Figure 4.4.6-2 Yield surfaces in t-p plane for the foam model.
text_image
t Uniaxial compression M original surface softened surface hardened surface -p_t σ_o 3 p_o p_o + p_t 2 p_o p_c p
M is computed from the yield stress in uniaxial compression test as
M = \sigma_ {0} / \sqrt {p _ {t} p _ {c} | _ {0} - \frac {1}{3} \sigma_ {0} (p _ {t} - p _ {c} | _ {0}) - \frac {1}{9} \sigma_ {0} ^ {2}},
where \sigma _ { 0 } ( \theta , f _ { i } ) is the initial yield stress in uniaxial compression (given as a positive value) and p _ { c } | _ { 0 } is the initial value of p _ { c } .
The yield criterion of Equation 4.4.6-3 defines an elliptic yield surface in the p { - } t plane. The yield locus intersects the p-axis at points - p _ { t } and p _ { c } . We assume that p _ { t } remains fixed throughout any plastic deformation process. By contrast, the compressive strength p _ { c } evolves as a result of compaction or dilation of the material (Figure 4.4.6-3).
Figure 4.4.6-3 Typical hardening/softening rule for the foam model.
text_image
softening hardening t -p_t p_c p_c l_o p p_c l_o -t -p_t p_c l_o p_c p -ε^pl_vol l_o -ε^pl_vol -p_t
This can be modeled by an exponential law since for many materials it is observed experimentally that during plastic deformation,
Equation 4.4.6-4
d e = - \lambda d \left(\ln \frac {p + p _ {t}}{p _ {t}}\right),
where \lambda = \lambda ( \theta , f _ { i } ) is a material parameter (see Figure 4.4.6-4).
Figure 4.4.6-4 Pure hydrostatic compression behavior for the crushable foam model (exponential hardening).
voids ratio,e
text_image
plastic slope \frac{de}{d \left( \ln \frac{p + p_t}{p_t} \right)} = -\lambda elastic slope \frac{de}{d \left( \ln \frac{p + p_t}{p_t} \right)} = -\kappa
In P+P) P： (p= pressure stress)
Using the previous definitions of strain rate decomposition and porous elasticity, we can write the volumetric response of the material as the following exponential hardening/softening law
Equation 4.4.6-5
p _ {c} = - p _ {t} + (p _ {c} | _ {0} + p _ {t}) \exp \left[ (1 + e _ {0}) \frac {1 - J ^ {p l}}{\lambda - \kappa J ^ {p l}} \right],
where p _ { c } | _ { 0 } is the initial value of p _ { c } and J ^ { p l } is the volumetric plastic strain, which controls hardening and softening.
The evolution of the yield surface can alternatively be defined by giving a table of values of the yield surface size on the hydrostatic stress axis, p _ { c } + p _ { t } . , as a function of the value of volumetric compacting plastic strain, - \varepsilon _ { \mathrm { v o l } } ^ { p l } (Figure 4.4.6-5).
Figure 4.4.6-5 Typical piecewise linear foam hardening.
line
| ε_vol | p_c + p_t | p_c | p_t |
|---|---|---|---|
| + | |||
| 0 | |||
| + | |||
| + | |||
| - | |||
| + |
These entries must be given in increasing magnitude of - \varepsilon _ { \mathrm { v o l } } ^ { p l } . Since the material may soften, an arbitrary origin must be used for - \varepsilon _ { \mathrm { v o l } } ^ { p l } so that the values of p _ { c } + p _ { t } cover the entire range of equivalent pressure stress values to which the material may be subjected.
The rate-dependent version of the model is activated by using the *RATE DEPENDENT option in conjunction with the *FOAM option. This is intended for relatively high strain rate applications. One way of introducing strain rate effects is by using the overstress power law model
\frac {d \bar {\varepsilon} ^ {p l}}{d t} = D \left(\frac {\bar {f}}{f _ {0}} - 1\right) ^ {n} \quad \mathrm{for} \quad f \geq f _ {0},
where f _ { 0 } is the static yield stress; ¹f is the effective yield stress (at a nonzero strain rate) as defined in Equation 4.4.6-3; t is time; and D and n are material parameters that may be temperature dependent. The yield surface is then rewritten as
Equation 4.4.6-6
F = f - \bar {f} = f - f _ {0} \left[ 1 + \left(\frac {\Delta \bar {\varepsilon} ^ {p l}}{D \Delta T}\right) ^ {\frac {1}{n}} \right] = 0,
where \bar { \varepsilon } ^ { p l } is the equivalent plastic strain. Another way to introduce strain rate effects is to specify the yield stress ratios, \bar { f } / f _ { 0 } , directly as a function of the equivalent plastic strain rate, \dot { \bar { \varepsilon } } ^ { p l } .
Flow rule
Potential flow is assumed, so
Equation 4.4.6-7
d \varepsilon^ {p l} = d \bar {\varepsilon} ^ {p l} \frac {\partial h}{\partial \pmb {\sigma}},
where d \bar { \varepsilon } ^ { p l } is the incremental equivalent plastic strain and h is the flow potential, chosen in this model as
Equation 4.4.6-8
h = \sqrt {\frac {9}{2} p ^ {2} + q ^ {2}}.
A geometrical representation of this flow potential is shown in the q-p diagram of Figure 4.4.6-6.
Figure 4.4.6-6 Plastic potential surfaces in q-p plane for the foam model.
text_image
q dε^pl σ h p
Equation 4.4.6-8 gives a direction of flow that is identical to the stress direction for radial paths. This is motivated by simple laboratory experiments performed by Bilkhu (1987), which suggest that loading in any principal direction causes insignificant deformations in the other directions.
Calibration of material parameters for the crushable foam model
The calibration procedure of the material parameters for the exponential hardening version of the model is illustrated in ``Simple tests on a crushable foam specimen,'' Section 3.2.7 of the ABAQUS Benchmarks Manual. The calibration procedure for the piecewise linear hardening version of the model is outlined in ``Crushable foam plasticity model,'' Section 11.3.5 of the ABAQUS/Standard User's Manual and Section 10.3.3 of the ABAQUS/Explicit User's Manual.
4.5 Other inelastic models
4.5.1 An inelastic constitutive model for concrete
This section describes the model provided in ABAQUS/Standard for plain concrete. In ABAQUS/Explicit, plain concrete can be analyzed with the cracking model described in ``A cracking model for concrete and other brittle materials, '' Section 4.5.2. It is intended that reinforced concrete modeling be accomplished by combining standard elements, using this plain concrete model, with "rebar elements"--rods, defined singly or embedded in oriented surfaces, that use a one-dimensional strain theory and that may be used to model the reinforcing itself. These elements are superposed on the mesh of plain concrete elements and are used with standard metal plasticity models that describe the behavior of the rebar material. This modeling approach allows the concrete behavior to be considered independently of the rebar, so that this section discusses the plain concrete model only. Effects associated with the rebar/concrete interface, such as bond slip and dowel action, cannot be considered in this approach, except by modifying some aspects of the plain concrete behavior to mimic them, such as the use of "tension stiffening" to simulate load transfer across cracks through the rebar.
The theory described in this section is intended as a model of concrete behavior for relatively monotonic loadings under fairly low confining pressures (less than four to five times the largest compressive stress that can be carried by the concrete in uniaxial compression). Cracking is assumed to be the most important aspect of the behavior, and it dominates the modeling. Cracking is assumed to occur when the stresses reach a failure surface, which we call the "crack detection surface." Thissurface." This failure surface is taken to be a simple Coulomb line written in terms of the first and second stress invariants, p and q, that are defined below. The anisotropy introduced by cracking is assumed to beThe anisotropy important in the simulations for which the model is intended, so the model includes consideration of this anisotropy. The model is a smeared crack model, in the sense that it does not track individual "macro" cracks: rather, constitutive calculations are performed independently at each integration point of the finite element model, and the presence of cracks enters into these calculations by the way the cracks affect the stress and material stiffness associated with the integration point. Various objections have been raised against such smeared crack models. The principal concern is that this modeling approach inherently introduces mesh sensitivity in the solutions, in the sense that the finite element results do not converge to a unique result. For example, since cracking is associated with strain softening, mesh refinement will lead to narrower crack bands. Crisfield (1986) discusses this concern in detail and concludes that Hilleborg's (1976) approach, based on brittle fracture concepts, is adequate to deal with this issue for practical purposes. This aspect of the model is discussed below in the section on cracking. For simplicity of discussion in what follows, the term "crack" is used to mean a direction in which cracking has been detected at the single constitutive calculation point in question: the closest physical concept is that there exists a continuum of micro-cracks at the point, oriented as determined by the model.
When the principal stress components are dominantly compressive, the response of the concrete is modeled by an elastic-plastic theory, using a simple form of yield surface written in terms of the first two stress invariants. Associated flow and isotropic hardening are used. This model significantlytwo stress invariants. Associated flow and isotropic hardening are used. This model significantly simplifies the actual behavior: the associated flow assumption generally overpredicts the inelastic volume strain; the simple yield surface used does not match all data very accurately (the third stress invariant would be needed to improve this aspect of the model); and, especially when the concrete is strained beyond the ultimate stress point, the assumption of constant elastic stiffness does not reproduce the observation that the unloading response is significantly weakened (the elastic response of the material appears to be damaged). In addition, when concrete is subjected to very high pressure stress, it exhibits inelastic response: no attempt has been made to build this behavior into the model. In spite of these limitations the model provides useful predictions for a variety of problems involving inelastic loading of concrete. The limitations are introduced for the sake of computational efficiency. In particular, assuming associated flow leads to enough symmetry in the Jacobian matrix of the constitutive model (the "material stiffness matrix") that the overall equilibrium equation solution usually does not require nonsymmetric equation solution for this reason. All of these limitations could be removed at some sacrifice in computational cost.
The cracking and compression responses of concrete that are incorporated in the model are illustrated by the uniaxial response of a specimen shown in Figure 4.5.1-1.
Figure 4.5.1-1 Uniaxial behavior of plain concrete.
flowchart
graph TD
A["Start of inelastic behavior"] --> B["Cracking failure"]
B --> C["Softening"]
C --> D["Failure point in compression (peak stress)"]
D --> E["Idealized (elastic) unload/reload response"]
E --> F["Unload/reload response"]
F --> G["Failure point in compression (peak stress)"]
style A fill:#f9f,stroke:#333
style B fill:#f9f,stroke:#333
style C fill:#f9f,stroke:#333
style D fill:#f9f,stroke:#333
style E fill:#f9f,stroke:#333
style F fill:#f9f,stroke:#333
style G fill:#f9f,stroke:#333
When concrete is loaded in compression, it initially exhibits elastic response. As the stress is increased, some nonrecoverable (inelastic) straining occurs, and the response of the material softens. An ultimate stress is reached, after which the material softens until it can no longer carry any stress. If the load is removed at some point after inelastic straining has occurred, the unloading response is softer than the initial elastic response: this effect is ignored in the model. When a uniaxial specimen is loaded into tension, it responds elastically until, at a stress that is typically 7-10% of the ultimate compressive stress, cracks form so quickly that--even on the stiffest testing machines available--it is very difficult to observe the actual behavior. For the purpose of developing the model we assume that the material loses strength through a softening mechanism, and that this is dominantly a damage effect, in the sense that open cracks can be represented by loss of elastic stiffness (as distinct from the nonrecoverable straining that is associated with classical plasticity effects, such as what we are using for the compressive behavior model). The model neglects any permanent strain associated with cracking; that is, we assume that the cracks can close completely when the stress across them becomes compressive.
In multiaxial stress states these observations can be generalized through the concept of surfaces of failure and of ultimate strength in stress space. These surfaces are defined below and are fitted to experimental data. Typical surfaces are shown in Figure 4.5.1-2 and Figure 4.5.1-3.
Figure 4.5.1-2 Concrete failure surfaces in plane stress.
text_image
"crack detection" surface uniaxial tension σ₂ uniaxial compression σ₁ biaxial tension "compression" surface biaxial compression
Figure 4.5.1-3 Concrete failure surfaces in the (p-q) plane.
line
| p/σ_c^u | q/σ_c^u (crack detection) | q/σ_c^u (compression) |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | 2 | 2 |
| 3 | 3 | 3 |
| 4 | 4 | 4 |
This model makes no attempt to include prediction of cyclic response or of the reduction in the elastic stiffness caused by inelastic straining because the model is intended for application to relatively monotonic loading cases. Nevertheless, it is likely that--even in such cases--the stress trajectories will not be entirely radial, and the model must predict the response in such cases in a reasonable way. An isotropically hardening "compressive" yield surface forms the basis of the model for the inelastic response when the principal stresses are dominantly compressive. In tension once cracking is defined