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Stress softening
The response of typical anisotropic hyperelastic materials, such as reinforced rubbers and biological tissues, under cyclic loading and unloading usually displays stress softening effects during the first few cycles. After a few cycles the response of the material tends to stabilize and the material is said to be preconditioned. Stress softening effects, often referred to in the elastomers literature as Mullins effect, can be accounted for by using the anisotropic hyperelastic model in combination with the pseudo-elasticity model for Mullins effect in Abaqus (see “Mullins effect,” Section 22.6.1). The stress softening effects provided by this model are isotropic.
Elements
The anisotropic hyperelastic material model can be used with solid (continuum) elements, finite-strain shells (except S4), continuum shells, and membranes. When used in combination with elements with plane stress formulations, Abaqus assumes fully incompressible behavior and ignores any amount of compressibility specified for the material.
Pure displacement formulation versus hybrid formulation in Abaqus/Standard
For continuum elements in Abaqus/Standard anisotropic hyperelasticity can be used with the pure displacement formulation elements or with the “hybrid” (mixed formulation) elements. Pure displacement formulation elements must be used with compressible materials, and “hybrid” (mixed formulation) elements must be used with incompressible materials.
In general, an analysis using a single hybrid element will be only slightly more computationally expensive than an analysis using a regular displacement-based element. However, when the wavefront is optimized, the Lagrange multipliers may not be ordered independently of the regular degrees of freedom associated with the element. Thus, the wavefront of a very large mesh of second-order hybrid tetrahedra may be noticeably larger than that of an equivalent mesh using regular second-order tetrahedra. This may lead to significantly higher CPU costs, disk space, and memory requirements.
Incompatible mode elements in Abaqus/Standard
Incompatible mode elements should be used with caution in applications involving large strains. Convergence may be slow, and in hyperelastic applications inaccuracies may accumulate. Erroneous stresses may sometimes appear in incompatible mode anisotropic hyperelastic elements that are unloaded after having been subjected to a complex deformation history.
Procedures
Anisotropic hyperelasticity must always be used with geometrically nonlinear analyses (“General and linear perturbation procedures,” Section 6.1.3).
Output
In addition to the standard output identifiers available in Abaqus (“Abaqus/Standard output variable identifiers,” Section 4.2.1, and “Abaqus/Explicit output variable identifiers,” Section 4.2.2), local material directions will be output whenever element field output is requested to the output database. The local directions are output as field variables (LOCALDIR1, LOCALDIR2, LOCALDIR3) representing the direction cosines; these variables can be visualized as vector plots in the Visualization module of Abaqus/CAE (Abaqus/Viewer).
Output of local material directions is suppressed if no element field output is requested or if you specify not to have element material directions written to the output database (see “Specifying the directions for element output in Abaqus/Standard and Abaqus/Explicit” in “Output to the output database,” Section 4.1.3).
Additional references
Gasser, T. C., R. W. Ogden, and G. A. Holzapfel, “Hyperelastic Modelling of Arterial Layers with Distributed Collagen Fibre Orientations,” Journal of the Royal Society Interface, vol. 3, pp. 15–35, 2006.
• Holzapfel, G. A., T. C. Gasser, and R. W. Ogden, “A New Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models,” Journal of Elasticity, vol. 61, pp. 1–48, 2000.
• Spencer, A. J. M., “Constitutive Theory for Strongly Anisotropic Solids,” A. J. M. Spencer (ed.), Continuum Theory of the Mechanics of Fibre-Reinforced Composites, CISM Courses and Lectures No. 282, International Centre for Mechanical Sciences, Springer-Verlag, Wien, pp. 1–32, 1984.
22.6 Stress softening in elastomers
• “Mullins effect,” Section 22.6.1
• “Energy dissipation in elastomeric foams,” Section 22.6.2
22.6.1 MULLINS EFFECT
Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
References
• “Material library: overview,” Section 21.1.1
• “Combining material behaviors,” Section 21.1.3
• “Elastic behavior: overview,” Section 22.1.1
• “Hyperelastic behavior of rubberlike materials,” Section 22.5.1
• “Anisotropic hyperelastic behavior,” Section 22.5.3
• “Permanent set in rubberlike materials,” Section 23.7.1
• “Energy dissipation in elastomeric foams,” Section 22.6.2
• *HYPERELASTIC
• *MULLINS EFFECT
• *PLASTIC
• *UNIAXIAL TEST DATA
• *BIAXIAL TEST DATA
• *PLANAR TEST DATA
• “Mullins effect” in “Defining damage,” Section 12.9.3 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
Overview
The Mullins effect model:
• is intended for modeling stress softening of filled rubber elastomers under quasi-static cyclic loading, a phenomenon referred to in the literature as Mullins effect;
• provides an extension to the well-known isotropic hyperelastic models;
• is based on the theory of incompressible isotropic elasticity modified by the addition of a single variable, referred to as the damage variable;
• assumes that only the deviatoric part of the material response is associated with damage;
• is intended for modeling material response in situations where different parts of the model undergo different levels of damage resulting in a different material response;
• is applied to the long-term modulus when combined with viscoelasticity; and
• cannot be used with hysteresis.
Abaqus provides a similar capability that can be applied to elastomeric foams (see “Energy dissipation in elastomeric foams,” Section 22.6.2).
Material behavior
The real behavior of filled rubber elastomers under cyclic loading conditions is quite complex. Certain idealizations have been made for modeling purposes. In essence, these idealizations result in two main components to the material behavior: the first component describes the response of a material point (from an undeformed state) under monotonic straining, and the second component is associated with damage and describes the unloading-reloading behavior. The idealized response and the two components are described in the following sections.
Idealized material behavior
When an elastomeric test specimen is subjected to simple tension from its virgin state, unloaded, and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved during the initial loading. This stress softening phenomenon is known as the Mullins effect and reflects damage incurred during previous loading. This type of material response is depicted qualitatively in Figure 22.6.1–1.
line
| stretch | stress |
|---|---|
| a | 0 |
| b | b |
| b' | b' |
| c | c |
| c' | c' |
| d | d |
Figure 22.6.1–1 Idealized response of the Mullins effect model.
This figure and the accompanying description is based on work by Ogden and Roxburgh (1999), which forms the basis of the model implemented in Abaqus. Consider the primary loading path a b { b ^ { ' } } of a previously unstressed material, with loading to an arbitrary point b ^ { ' } . On unloading from b ^ { ' } , the path is followed. When the material is loaded again, the softened path is retraced as a B b ^ { ' } . If further loading is then applied, the path b ^ { ' } c is followed, where b ^ { ' } c is a continuation of the primary loading path (which is the path that would be followed if there was no unloading). If loading is now stopped at c ^ { ' } , the path c ^ { ' } C a is followed on unloading and then retraced back to on reloading. If no further
loading beyond c ^ { ' } is applied, the curve a C c ^ { ' } represents the subsequent material response, which is then elastic. For loading beyond c ^ { ' } , the primary path is again followed and the pattern described is repeated.
This is an ideal representation of Mullins effect since in practice there is some permanent set upon unloading and/or viscoelastic effects such as hysteresis. Points such as b ^ { ' } and c ^ { ' } may not exist in reality in the sense that unloading from the primary curve followed by reloading to the maximum strain level attained earlier usually results in a stress that is somewhat lower than the stress corresponding to the primary curve. In addition, the cyclic response for some filled elastomers shows evidence of progressive damage during unloading from and subsequent reloading to a certain maximum strain level. Such progressive damage usually occurs during the first few cycles, and the material behavior soon stabilizes to a loading/unloading cycle that is followed beyond the first few cycles. More details regarding the actual behavior and how test data that display such behavior can be used to calibrate the Abaqus model for Mullins effect are discussed later and in “Analysis of a solid disc with Mullins effect and permanent set,” Section 3.1.7 of the Abaqus Example Problems Guide.
The loading path will henceforth be referred to as the “primary hyperelastic behavior.” The primary hyperelastic behavior is defined by using a hyperelastic material model.
Stress softening is interpreted as being due to damage at the microscopic level. As the material is loaded, the damage occurs by the severing of bonds between filler particles and the rubber molecular chains. Different chain links break at different deformation levels, thereby leading to continuous damage with macroscopic deformation. An equivalent interpretation is that the energy required to cause the damage is not recoverable.
Primary hyperelastic behavior
Hyperelastic materials are described in terms of a “strain energy potential” function U ( \mathbf { F } ) that defines the strain energy stored in the material per unit reference volume (volume in the initial configuration). The quantity is the deformation gradient tensor. To account for Mullins effect, Ogden and Roxburgh propose a material description that is based on an energy function of the form { \cal U } ( { \bf F } , { \boldsymbol \eta } ) , where the additional scalar variable, \eta , represents damage in the material. The damage variable controls the material properties in the sense that it enables the material response to be governed by an energy function on unloading and subsequent submaximal reloading different from that on the primary (initial) loading path from a virgin state. Because of the above interpretation of \eta , , it is no longer appropriate to think of \boldsymbol { U } as the stored elastic energy potential. Part of the energy is stored as strain energy, while the rest is dissipated due to damage. The shaded area in Figure 22.6.1–1 represents the energy dissipated by damage as a result of deformation until the point c ^ { ' } , while the unshaded part represents the recoverable strain energy.
The following paragraphs provide a summary of the Mullins effect model in Abaqus. For further details, see “Mullins effect,” Section 4.7.1 of the Abaqus Theory Guide. In preparation for writing the constitutive equations for Mullins effect, it is useful to separate the deviatoric and the volumetric parts of the total strain energy density as
U = U _ {d e v} + U _ {v o l}.
In the above equation U , U _ { d e v ; } , and U _ { v o l } are the total, deviatoric, and volumetric parts of the strain energy density, respectively. All the hyperelasticity models in Abaqus use strain energy potential functions that are already separated into deviatoric and volumetric parts. For example, the polynomial models use a strain energy potential of the form
U = \sum_ {i + j = 1} ^ {N} C _ {i j} (\overline {{I}} _ {1} - 3) ^ {i} (\overline {{I}} _ {2} - 3) ^ {j} + \sum_ {i = 1} ^ {N} \frac {1}{D _ {i}} (J ^ {e \ell} - 1) ^ {2 i},
where the symbols have the usual interpretations. The first term on the right represents the deviatoric part of the elastic strain energy density function, and the second term represents the volumetric part.
Modified strain energy density function
The Mullins effect is accounted for by using an augmented energy function of the form
U (\overline {{\lambda}} _ {i}, \eta) = \eta \tilde {U} _ {d e v} (\overline {{\lambda}} _ {i}) + \phi (\eta) + \tilde {U} _ {v o l} (J ^ {e \ell}),
where \tilde { U } _ { d e v } ( \overline { { \lambda } } _ { i } ) is the deviatoric part of the strain energy density of the primary hyperelastic behavior, defined, for example, by the first term on the right-hand-side of the polynomial strain energy function given above; \tilde { U } _ { v o l } ( J ^ { e \ell } ) is the volumetric part of the strain energy density, defined, for example, by the second term on the right-hand-side of the polynomial strain energy function given above; \overline { { \lambda } } _ { i } ( i = 1 , 2 ) represent the deviatoric principal stretches; and J ^ { e \ell } represents the elastic volume ratio. The function \phi ( \eta ) is a continuous function of the damage variable and is referred to as the “damage function.” When the deformation state of the material is on a point on the curve that represents the primary hyperelastic behavior, \eta = 1 , \phi ( \eta ) = 0 , U ( \overline { { { \lambda } } } _ { i } , 1 ) = \tilde { U } _ { d e v } ( \overline { { { \lambda } } } _ { i } ) + \tilde { U } _ { v o l } ( J ^ { e \ell } ) , and the augmented energy function reduces to the strain energy density function of the primary hyperelastic behavior. The damage variable varies continuously during the course of the deformation and always satisfies 0 < \eta \leq 1 . The above form of the energy function is an extension of the form proposed by Ogden and Roxburgh to account for material compressibility.
Stress computation
With the above modification to the energy function, the stresses are given by
\pmb {\sigma} (\eta , \overline {{\lambda}} _ {i}, J ^ {e \ell}) = \eta \tilde {\mathbf {S}} (\overline {{\lambda}} _ {i}) - \tilde {p} (J ^ {e \ell}) \mathbf {I},
where \tilde { \bf S } is the deviatoric stress corresponding to the primary hyperelastic behavior at the current deviatoric deformation level \overline { { \lambda } } _ { i } and \tilde { p } is the hydrostatic pressure of the primary hyperelastic behavior at the current volumetric deformation level J ^ { e \ell } . Thus, the deviatoric stress as a result of Mullins effect is obtained by simply scaling the deviatoric stress of the primary hyperelastic behavior with the damage variable \eta . . The pressure stress is the same as that of the primary behavior. The model predicts loading/unloading along a single curve (that is different, in general, from the primary hyperelastic behavior) from any given strain level that passes through the origin of the stress-strain plot. It cannot
capture permanent strains upon removal of load. The model also predicts that a purely volumetric deformation will not have any damage or Mullins effect associated with it.
Damage variable
The damage variable, , varies with the deformation according to
\eta = 1 - \frac {1}{r} \mathrm{erf} \left(\frac {U _ {d e v} ^ {m} - \tilde {U} _ {d e v}}{m + \beta U _ {d e v} ^ {m}}\right),
where U _ { d e v } ^ { m } is the maximum value of \tilde { U } _ { d e v } at a material point during its deformation history; r , \beta , and m are material parameters; and is the error function defined as
\operatorname{erf} (x) = \frac {2}{\sqrt {\pi}} \int_ {0} ^ {x} \exp (- w ^ {2}) d w.
When \tilde { U } _ { d e v } = U _ { d e v } ^ { m } , , corresponding to a point on the primary curve, \eta = 1 . 0 . On the other hand, attains its minimum value, \eta _ { m } , given by
\eta_ {m} = 1 - \frac {1}{r} \mathrm{erf} \left(\frac {U _ {d e v} ^ {m}}{m + \beta U _ {d e v} ^ {m}}\right),
upon removal of deformation, when \tilde { U } _ { d e v } ~ = ~ 0 . For all intermediate values of \tilde { U } _ { d e v } , ~ \eta varies monotonically between and \eta _ { m } . While the parameters r and \beta are dimensionless, the parameter m has the dimensions of energy. The equation for \eta reduces to that proposed by Ogden and Roxburgh when \beta ~ = ~ 0 . The material parameters may be specified directly or may be computed by Abaqus based on curve-fitting of unloading-reloading test data. These parameters are subject to the restrictions r > 1 , \beta \geq 0 , and m \geq 0 (the parameters \beta and m cannot both be zero). Alternatively, the damage variable can be defined through user subroutine UMULLINS in Abaqus/Standard and VUMULLINS in Abaqus/Explicit.
If the parameter \beta = 0 and the parameter m has a value that is small compared to U _ { d e v } ^ { m } , the slope of the stress-strain curve at the initiation of unloading from relatively large strain levels may become very high. As a result, the response may become discontinuous, as illustrated in Figure 22.6.1–2. This kind of behavior may lead to convergence problems in Abaqus/Standard. In Abaqus/Explicit the high stiffness will lead to very small stable time increments, thereby leading to a degradation in performance. This problem can be avoided by choosing a small value for \beta . . The choice \beta = 0 can be used to define the original Ogden-Roxburgh model. In Abaqus/Standard the default value of \beta is 0. In Abaqus/Explicit, however, the default value of \beta is 0.1. Thus, if you do not specify a value for \beta , , it is assumed to be 0 in Abaqus/Standard and 0.1 in Abaqus/Explicit.
The parameters r , \beta , , and m do not have direct physical interpretations in general. The parameter m controls whether damage occurs at low strain levels. If m = 0 , there is a significant amount of damage at low strain levels. On the other hand, a nonzero m leads to little or no damage at low strain levels. For further discussion regarding the implications of this model to the energy dissipation, see “Mullins
line
| stretch | stress |
|---|---|
| a | 0 |
| b | b |
| b' | b' |
| c | c |
| c' | c' |
| d | d |
Figure 22.6.1–2 Overly stiff response at the initiation of unloading.
effect,” Section 4.7.1 of the Abaqus Theory Guide. The qualitative effects of varying the parameters r and \beta individually, while holding the other parameters fixed, are shown in Figure 22.6.1–3.
text_image
stress increasing r stretch
line
| stretch | stress (σ̃) | stress (ηₘ(β₁)σ̃) | stress (ηₘ(β₂)σ̃) |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| increasing β | ~0.5 | ~0.3 | ~0.1 |
Figure 22.6.1–3 Qualitative dependence of damage on material properties.
The left figure shows the unloading-reloading curve from a certain maximum strain level for increasing values of \pmb { r } . It suggests that the parameter \pmb { r } controls the amount of damage, with decreasing damage for increasing \pmb { r } . This behavior follows from the fact that the larger the value of { \boldsymbol { r } } , the less the damage variable \eta can deviate from unity. The figure on the right shows the unloading-reloading curve from a certain maximum strain level for increasing values of { \bf \partial } \cdot \beta . The figure suggests that increasing \beta also leads to lower amounts of damage. It also shows that the unloading-reloading response approaches the asymptotic response given by \eta _ { m } \tilde { \sigma } , where \eta _ { m } is the minimum value of \eta , faster for lower values of \beta . .