김경종
344 lines
25 KiB
Markdown
344 lines
25 KiB
Markdown
<!-- source-page: 111 -->
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where U is the strain energy per unit of reference volume; N is a material parameter; $C _ { i 0 }$ and $D _ { i }$ are temperature-dependent material parameters; $\overline { { I } } _ { 1 }$ is the first deviatoric strain invariant defined as
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$$
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\overline {{I}} _ {1} = \overline {{\lambda}} _ {1} ^ {2} + \overline {{\lambda}} _ {2} ^ {2} + \overline {{\lambda}} _ {3} ^ {2},
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$$
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where the deviatoric stretches ${ \overline { { \lambda } } } _ { i } = J ^ { - { \frac { 1 } { 3 } } } \lambda _ { i } ;$ J is the total volume ratio; $J ^ { e \ell }$ is the elastic volume ratio as defined below in “Thermal expansion”; and $\lambda _ { i }$ are the principal stretches. The initial shear modulus and bulk modulus are given by
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$$
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\mu_ {0} = 2 C _ {1 0}, \qquad K _ {0} = \frac {2}{D _ {1}}.
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$$
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# Van der Waals form
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The form of the Van der Waals strain energy potential is
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$$
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U = \mu \biggl \{- (\lambda_ {m} ^ {2} - 3) \biggl [ \ln (1 - \eta) + \eta \biggr ] - \frac {2}{3} a \biggl (\frac {\tilde {I} - 3}{2} \biggr) ^ {\frac {3}{2}} \biggr \} + \frac {1}{D} \biggl (\frac {J _ {e \ell} ^ {2} - 1}{2} - \ln J _ {e \ell} \biggr),
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$$
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where
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$$
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\tilde {I} = (1 - \beta) \overline {{I}} _ {1} + \beta \overline {{I}} _ {2} \quad \mathrm{and} \quad \eta = \sqrt {\frac {\tilde {I} - 3}{\lambda_ {m} ^ {2} - 3}}.
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$$
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Here, U is the strain energy per unit of reference volume; $\mu$ is the initial shear modulus; $\lambda _ { m }$ is the locking stretch; a is the global interaction parameter; $\beta$ is an invariant mixture parameter; and D governs the compressibility. These parameters can be temperature-dependent. $\overline { { I } } _ { 1 }$ and $\overline { { I } } _ { 2 }$ are the first and second deviatoric strain invariants defined as
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$$
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\overline {{I}} _ {1} = \overline {{\lambda}} _ {1} ^ {2} + \overline {{\lambda}} _ {2} ^ {2} + \overline {{\lambda}} _ {3} ^ {2} \qquad \mathrm{and} \qquad \overline {{I}} _ {2} = \overline {{\lambda}} _ {1} ^ {(- 2)} + \overline {{\lambda}} _ {2} ^ {(- 2)} + \overline {{\lambda}} _ {3} ^ {(- 2)},
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$$
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where the deviatoric stretches ${ \overline { { \lambda } } } _ { i } = J ^ { - { \frac { 1 } { 3 } } } \lambda _ { i } ;$ J is the total volume ratio; $J ^ { e \ell }$ is the elastic volume ratio as defined below in “Thermal expansion”; and $\lambda _ { i }$ are the principal stretches. The initial shear modulus and bulk modulus are given by
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$$
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\mu_ {0} = \mu , \qquad K _ {0} = \frac {2}{D}.
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$$
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# Yeoh form
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The form of the Yeoh strain energy potential is
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$$
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\begin{array}{l} U = C _ {1 0} (\overline {{{I}}} _ {1} - 3) + C _ {2 0} (\overline {{{I}}} _ {1} - 3) ^ {2} + C _ {3 0} (\overline {{{I}}} _ {1} - 3) ^ {3} \\ + \frac {1}{D _ {1}} (J ^ {e \ell} - 1) ^ {2} + \frac {1}{D _ {2}} (J ^ {e \ell} - 1) ^ {4} + \frac {1}{D _ {3}} (J ^ {e \ell} - 1) ^ {6}, \\ \end{array}
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$$
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<!-- source-page: 112 -->
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where U is the strain energy per unit of reference volume; $C _ { i 0 }$ and $D _ { i }$ are temperature-dependent material parameters; $\overline { { I } } _ { 1 }$ is the first deviatoric strain invariant defined as
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$$
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\overline {{I}} _ {1} = \overline {{\lambda}} _ {1} ^ {2} + \overline {{\lambda}} _ {2} ^ {2} + \overline {{\lambda}} _ {3} ^ {2},
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$$
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where the deviatoric stretches ${ \overline { { \lambda } } } _ { i } = J ^ { - \frac { 1 } { 3 } } \lambda _ { i } ;$ ; J is the total volume ratio; $J ^ { e \ell }$ is the elastic volume ratio as defined below in “Thermal expansion”; and $\lambda _ { i }$ are the principal stretches. The initial shear modulus and bulk modulus are given by
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$$
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\mu_ {0} = 2 C _ {1 0}, \qquad K _ {0} = \frac {2}{D _ {1}}.
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$$
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# Thermal expansion
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Only isotropic thermal expansion is permitted with the hyperelastic material model.
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The elastic volume ratio, $J ^ { e \ell }$ , relates the total volume ratio, J, and the thermal volume ratio, $J ^ { t h }$ :
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$$
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J ^ {e \ell} = \frac {J}{J ^ {t h}}.
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$$
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$J ^ { t h }$ is given by
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$$
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J ^ {t h} = (1 + \varepsilon^ {t h}) ^ {3},
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$$
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where $\varepsilon ^ { t h }$ is the linear thermal expansion strain that is obtained from the temperature and the isotropic thermal expansion coefficient (“Thermal expansion,” Section 26.1.2).
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# Defining the hyperelastic material response
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The mechanical response of a material is defined by choosing a strain energy potential to fit the particular material. The strain energy potential forms in Abaqus are written as separable functions of a deviatoric component and a volumetric component; i.e., $U = U _ { d e v } ( \overline { { I } } _ { 1 } , \overline { { I } } _ { 2 } ) + U _ { v o l } ( J _ { e \ell } )$ . Alternatively, in Abaqus/Standard you can define the strain energy potential with user subroutine UHYPER, in which case the strain energy potential need not be separable.
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Generally for the hyperelastic material models available in Abaqus, you can either directly specify material coefficients or provide experimental test data and have Abaqus automatically determine appropriate values of the coefficients. An exception is the Marlow form: in this case the deviatoric part of the strain energy potential must be defined with test data. The different methods for defining the strain energy potential are described in detail below.
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The properties of rubberlike materials can vary significantly from one batch to another; therefore, if data are used from several experiments, all of the experiments should be performed on specimens taken from the same batch of material, regardless of whether you or Abaqus compute the coefficients.
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# Viscoelastic and hysteretic materials
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The elastic response of viscoelastic materials (“Time domain viscoelasticity,” Section 22.7.1, and “Parallel rheological framework,” Section 22.8.2) and hysteretic materials (“Hysteresis in elastomers,” Section 22.8.1) can be specified by defining either the instantaneous response or the long-term response of such materials. To define the instantaneous response, the experiments outlined in the “Experimental tests” section that follows have to be performed within time spans much shorter than the characteristic relaxation times of these materials.
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Input File Usage: \*HYPERELASTIC, MODULI=INSTANTANEOUS
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; any Strain energy potential except Unknown: Moduli time scale (for viscoelasticity): Instantaneous
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If, on the other hand, the long-term elastic response is used, data from experiments have to be collected after time spans much longer than the characteristic relaxation times of these materials. Longterm elastic response is the default elastic material behavior.
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Input File Usage: \*HYPERELASTIC, MODULI=LONG TERM
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; any Strain energy potential except Unknown: Moduli time scale (for viscoelasticity): Long-term
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# Accounting for compressibility
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Compressibility can be defined by specifying nonzero values for $D _ { i }$ (except for the Marlow model), by setting the Poisson’s ratio to a value less than 0.5, or by providing test data that characterize the compressibility. The test data method is described later in this section. If you specify the Poisson’s ratio for hyperelasticity other than the Marlow model, Abaqus computes the initial bulk modulus from the initial shear modulus
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$$
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D _ {1} = \frac {2}{K _ {0}} = \frac {3 (1 - 2 \nu)}{\mu_ {0} (1 + \nu)}.
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$$
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For the Marlow model the specified Poisson’s ratio represents a constant value, which determines the volumetric response throughout the deformation process. If $D _ { 1 }$ is equal to zero, all of the $D _ { i }$ must be equal to zero. In such a case the material is assumed to be fully incompressible in Abaqus/Standard, while Abaqus/Explicit will assume compressible behavior with $K _ { 0 } / \mu _ { 0 } = 2 0$ (Poisson’s ratio of 0.475).
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Input File Usage: \*HYPERELASTIC, POISSON=
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; any Strain energy potential except Unknown or User-defined: Input source: Test data: Poisson's ratio:
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# Specifying material coefficients directly
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The parameters of the hyperelastic strain energy potentials can be given directly as functions of temperature for all forms of the strain energy potential except the Marlow form.
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Input File Usage: Use one of the following options:
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*HYPERELASTIC, ARRUDA-BOYCE
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*HYPERELASTIC, MOONEY-RIVLIN
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*HYPERELASTIC, NEO HOOKE
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*HYPERELASTIC, OGDEN, N=n (n ≤ 6)
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*HYPERELASTIC, POLYNOMIAL, N=n (n ≤ 6)
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*HYPERELASTIC, REDUCED POLYNOMIAL, N=n (n ≤ 6)
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*HYPERELASTIC, VAN DER WAALS
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*HYPERELASTIC, YEOH
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; Input source: Coefficients and Strain energy potential: Arruda-Boyce, Mooney-Rivlin, Neo Hooke, Ogden, Polynomial, Reduced Polynomial, Van der Waals, or Yeoh
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# Using test data to calibrate material coefficients
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The material coefficients of the hyperelastic models can be calibrated by Abaqus from experimental stress-strain data. In the case of the Marlow model, the test data directly characterize the strain energy potential (there are no material coefficients for this model); the Marlow model is described in detail below. The value of N and experimental stress-strain data can be specified for up to four simple tests: uniaxial, equibiaxial, planar, and, if the material is compressible, a volumetric compression test. Abaqus will then compute the material parameters. The material constants are determined through a least-squares-fit procedure, which minimizes the relative error in stress. For the n nominal-stress–nominal-strain data pairs, the relative error measure E is minimized, where
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$$
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E = \sum_ {i = 1} ^ {n} \left(1 - T _ {i} ^ {\mathrm{th}} / T _ {i} ^ {\mathrm{test}}\right) ^ {2}.
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$$
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$T _ { i } ^ { \mathrm { t e s t } }$ is a stress value from the test data, and $T _ { i } ^ { \mathrm { t h } }$ comes from one of the nominal stress expressions derived below (see “Experimental tests”). Abaqus minimizes the relative error rather than an absolute error measure since this provides a better fit at lower strains. This method is available for all strain energy potentials and any order of N except for the polynomial form, where a maximum of $N \ = \ 2$ is allowed. The polynomial models are linear in terms of the constants $C _ { i j } ;$ therefore, a linear leastsquares procedure can be used. The Arruda-Boyce, Ogden, and Van der Waals potentials are nonlinear in some of their coefficients, thus necessitating the use of a nonlinear least-squares procedure. “Fitting of hyperelastic and hyperfoam constants,” Section 4.6.2 of the Abaqus Theory Guide, contains a detailed derivation of the related equations.
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It is generally best to obtain data from several experiments involving different kinds of deformation over the range of strains of interest in the actual application and to use all of these data to determine the parameters. This is particularly true for the phenomenological models; i.e., the Ogden and the polynomial models. It has been observed that to achieve good accuracy and stability, it is necessary to fit these models using test data from more than one deformation state. In some cases, especially at large strains, removing the dependence on the second invariant may alleviate this limitation. The Arruda-Boyce, neo-Hookean, and Van der Waals models with $\beta = 0$ offer a physical interpretation and provide a better prediction of general deformation modes when the parameters are based on only one test. An extensive discussion of this topic can be found in “Hyperelastic material behavior,” Section 4.6.1 of the Abaqus Theory Guide.
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This method does not allow the hyperelastic properties to be temperature dependent. However, if temperature-dependent test data are available, several curve fits can be conducted by performing a data check analysis on a simple input file. The temperature-dependent coefficients determined by Abaqus can then be entered directly in the actual analysis run.
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Optionally, the parameter $\beta$ in the Van der Waals model can be set to a fixed value while the other parameters are found using a least-squares curve fit.
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As many data points as required can be entered from each test. It is recommended that data from all four tests (on samples taken from the same piece of material) be included and that the data points cover the range of nominal strains expected to arise in the actual loading. For the (general) polynomial and Ogden models and for the coefficient $\beta$ in the Van der Waals model, the planar test data must be accompanied by the uniaxial test data, the biaxial test data, or both of these types of test data; otherwise, the solution to the least-squares fit will not be unique.
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The strain data should be given as nominal strain values (change in length per unit of original length). For the uniaxial, equibiaxial, and planar tests stress data are given as nominal stress values (force per unit of original cross-sectional area). These tests allow for entering both compression and tension data. Compressive stresses and strains are entered as negative values.
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If compressibility is to be specified, the $D _ { i }$ or $D$ can be computed from volumetric compression test data. Alternatively, compressibility can be defined by specifying a Poisson’s ratio, in which case Abaqus computes the bulk modulus from the initial shear modulus. If no such data are given, Abaqus/Standard assumes that D or all of the $D _ { i }$ are zero, whereas Abaqus/Explicit assumes compressibility corresponding to a Poisson’s ratio of 0.475 (see “Compressibility in Abaqus/Explicit” above). For these compression tests the stress data are given as pressure values.
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Input File Usage: Use one of the following options to select the strain energy potential:
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*HYPERELASTIC, TEST DATA INPUT, ARRUDA-BOYCE
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*HYPERELASTIC, TEST DATA INPUT, MOONEY-RIVLIN
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*HYPERELASTIC, TEST DATA INPUT, NEO HOOKE
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*HYPERELASTIC, TEST DATA INPUT, OGDEN, N=n (n ≤ 6)
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*HYPERELASTIC, TEST DATA INPUT, POLYNOMIAL, N=n (n ≤ 2)
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*HYPERELASTIC, TEST DATA INPUT, REDUCED POLYNOMIAL, N=n (n ≤ 6)
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*HYPERELASTIC, TEST DATA INPUT, VAN DER WAALS
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*HYPERELASTIC, TEST DATA INPUT, VAN DER WAALS,
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<!-- source-page: 116 -->
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BETA= ( )
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\*HYPERELASTIC, TEST DATA INPUT, YEOH
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In addition, use at least one and up to four of the following options to give the test data (see “Experimental tests” below):
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\*UNIAXIAL TEST DATA
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\*BIAXIAL TEST DATA
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\*PLANAR TEST DATA
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\*VOLUMETRIC TEST DATA
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# Abaqus/CAE Usage:
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Property module: material editor: Mechanical→Elasticity→Hyperelastic:
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Material type: Isotropic; Input source: Test data and Strain energy potential: Arruda-Boyce, Mooney-Rivlin, Neo Hooke, Ogden, Polynomial, Reduced Polynomial, Van der Waals (Beta: Fitted value or Specify), or Yeoh
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In addition, use at least one and up to four of the following options to give the test data (see “Experimental tests” below):
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Test Data→Uniaxial Test Data Test Data→Biaxial Test Data Test Data→Planar Test Data Test Data→Volumetric Test Data
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Alternatively, you can select Strain energy potential: Unknown to define the material temporarily without specifying a particular strain energy potential. Then select Material→Evaluate to have Abaqus/CAE evaluate the material to determine the optimal strain energy potential.
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# Specifying the Marlow model
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The Marlow model assumes that the strain energy potential is independent of the second deviatoric invariant $\overline { { I } } _ { 2 }$ . This model is defined by providing test data that define the deviatoric behavior, and, optionally, the volumetric behavior if compressibility must be taken into account. Abaqus will construct a strain energy potential that reproduces the test data exactly, as shown in Figure 22.5.1–1. The interpolation and extrapolation of stress-strain data with the Marlow model is approximately linear for small and large strains. For intermediate strains in the range 0.1 to 1.0 a noticeable degree of nonlinearity may be observed in the interpolation/extrapolation with the Marlow model; for example, some nonlinearity is apparent between the 4th and 5th data points in Figure 22.5.1–1. To minimize undesirable nonlinearity, make sure that enough data points are specified in the intermediate strain range.
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The deviatoric behavior is defined by specifying uniaxial, biaxial, or planar test data. Generally, you can specify either the data from tension tests or the data from compression tests because the tests are equivalent (see “Equivalent experimental tests). However, for beams, trusses, and rebars, the data from tension and compression tests can be specified together. Volumetric behavior is defined by using one of the following three methods:
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<!-- source-page: 117 -->
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<details>
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<summary>line</summary>
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| Nominal Strain | Nominal Stress (x10⁶) |
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| -------------- | --------------------- |
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| 0.0 | 0.0 |
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| 0.2 | 0.15 |
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| 0.4 | 0.25 |
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| 0.6 | 0.45 |
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| 0.8 | 0.55 |
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| 1.0 | 0.60 |
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| 1.2 | 0.65 |
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| 1.4 | 0.70 |
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| 1.6 | 0.75 |
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| 1.8 | 0.80 |
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| 2.0 | 0.85 |
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| 2.2 | 0.90 |
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| 2.4 | 0.95 |
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| 2.5 | 1.0 |
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</details>
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Figure 22.5.1–1 The results of the Marlow model with test data.
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• Specify nominal lateral strains, in addition to nominal stresses and nominal strains, as part of the uniaxial, biaxial, or planar test data.
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• Specify Poisson’s ratio for the hyperelastic material.
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• Specify volumetric test data directly. Both hydrostatic tension and hydrostatic compression data can be specified. If only hydrostatic compression data are available, as is usually the case, Abaqus will assume that the hydrostatic pressure is an antisymmetric function of the nominal volumetric strain, $\epsilon _ { v o l } = J _ { v o l } - 1$ .
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If you do not define volumetric behavior, Abaqus/Standard assumes fully incompressible behavior, while Abaqus/Explicit assumes compressibility corresponding to a Poisson’s ratio of 0.475.
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Material test data in which the stress does not vary smoothly with increasing strain may lead to convergence difficulty during the simulation. It is highly recommended that smooth test data be used to define the Marlow form. Abaqus provides a smoothing algorithm, which is described in detail later in this section.
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The test data for the Marlow model can also be given as a function of temperature and field variables. You must specify the number of user-defined field variable dependencies required.
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Uniaxial, biaxial, and planar test data must be given in ascending order of the nominal strains; volumetric test data must be given in descending order of the volume ratio.
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<!-- source-page: 118 -->
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Input File Usage: To define the Marlow test data as a function of temperature and/or field variables, use the following option:
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\*HYPERELASTIC, MARLOW
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with one of the following first three options and, optionally, the fourth option:
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\*UNIAXIAL TEST DATA, DEPENDENCIES=n
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\*BIAXIAL TEST DATA, DEPENDENCIES=n
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\*PLANAR TEST DATA, DEPENDENCIES=n
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\*VOLUMETRIC TEST DATA, DEPENDENCIES=n
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic:
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Material type: Isotropic; Input source: Test data and
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Strain energy potential: Marlow
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In addition, select one of the following first three options and, optionally, the fourth option to give the test data (see “Experimental tests” below):
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Test Data→Uniaxial Test Data
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Test Data→Biaxial Test Data
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Test Data→Planar Test Data
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Test Data→Volumetric Test Data
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In each of the Test Data Editor dialog boxes, you can toggle on Use temperature-dependent data to define the test data as a function of temperature and/or select the Number of field variables to define the test data as a function of field variables.
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Alternatively, you can select Material→Evaluate to have Abaqus/CAE evaluate the material. If you included temperature dependencies, field variable dependencies, or lateral nominal strain in the test data—which can only be defined in the Marlow hyperelastic definition—Marlow will be the only strain energy potential available for evaluation.
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# User subroutine specification in Abaqus/Standard
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An alternative method provided in Abaqus/Standard for defining the hyperelastic material parameters allows the strain energy potential to be defined in user subroutine UHYPER. Either compressible or incompressible behavior can be specified. Optionally, you can specify the number of property values needed as data in the user subroutine. The derivatives of the strain energy potential with respect to the strain invariants must be provided directly through user subroutine UHYPER. If needed, you can specify the number of solution-dependent variables (see “User subroutines: overview,” Section 18.1.1).
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Input File Usage: Use one of the following two options:
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\*HYPERELASTIC, USER, TYPE=COMPRESSIBLE, PROPERTIES=n
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\*HYPERELASTIC, USER, TYPE=INCOMPRESSIBLE, PROPERTIES=n
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<!-- source-page: 119 -->
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# Abaqus/CAE Usage:
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Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; Input source: Coefficients and Strain energy potential: User-defined: optionally, toggle on Include compressibility and/or specify the Number of property values
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# Experimental tests
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For a homogeneous material, homogeneous deformation modes suffice to characterize the material constants. Abaqus accepts test data from the following deformation modes:
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• Uniaxial tension and compression
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• Equibiaxial tension and compression
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• Planar tension and compression (also known as pure shear)
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• Volumetric tension and compression
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These modes are illustrated schematically in Figure 22.5.1–2 and are described below. The most commonly performed experiments are uniaxial tension, uniaxial compression, and planar tension. Combine data from these three test types to get a good characterization of the hyperelastic material behavior.
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For the incompressible version of the material model, the stress-strain relationships for the different tests are developed using derivatives of the strain energy function with respect to the strain invariants. We define these relations in terms of the nominal stress (the force divided by the original, undeformed area) and the nominal, or engineering, strain defined below.
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The deformation gradient, expressed in the principal directions of stretch, is
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$$
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\mathbf {F} = \left[ \begin{array}{c c c} \lambda_ {1} & 0 & \\ 0 & \lambda_ {2} & 0 \\ 0 & 0 & \lambda_ {3} \end{array} \right],
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$$
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where $\lambda _ { 1 } , \lambda _ { 2 }$ , and $\lambda _ { 3 }$ are the principal stretches: the ratios of current length to length in the original configuration in the principal directions of a material fiber. The principal stretches, $\lambda _ { i }$ , are related to the principal nominal strains, $\epsilon _ { i } ,$ , by
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$$
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\lambda_ {i} = 1 + \epsilon_ {i}.
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$$
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Because we assume incompressibility and isothermal response, $J = \operatorname* { d e t } ( \mathbf { F } ) = 1$ and, hence, $\lambda _ { 1 } \lambda _ { 2 } \lambda _ { 3 } = 1$ . The deviatoric strain invariants in terms of the principal stretches are then
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$$
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\overline {{I}} _ {1} = \lambda_ {1} ^ {2} + \lambda_ {2} ^ {2} + \lambda_ {3} ^ {2},
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$$
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and
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$$
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\overline {{I}} _ {2} = \lambda_ {1} ^ {- 2} + \lambda_ {2} ^ {- 2} + \lambda_ {3} ^ {- 2}.
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$$
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<!-- source-page: 120 -->
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${ \sf T } _ { \cup } , \ { \sf \epsilon } _ { \cup }$
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$\lambda _ { 1 } = \lambda _ { \cup } = 1 + \epsilon _ { \cup } , \lambda _ { 2 } = \lambda _ { 3 } = 1 / \div \lambda _ { \cup }$
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${ \sf T } _ { \sf B } , ~ { \sf \epsilon } _ { \sf B }$
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$\lambda _ { 1 } = \lambda _ { 2 } = \lambda _ { \mathrm { B } } = 1 + \epsilon _ { \scriptscriptstyle \mathrm { B } } \ , \lambda _ { 3 } = 1 / \lambda _ { \scriptscriptstyle \mathrm { B } } ^ { 2 }$
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$\mathsf { T } _ { \mathsf { s } } , \mathsf { \Pi } \in _ { \mathsf { s } }$
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$\lambda _ { 1 } = \lambda _ { \mathrm { s } } = 1 + \epsilon _ { \mathrm { s } } , \ \lambda _ { 2 } = 1 , \ \lambda _ { 3 } = \ 1 / \lambda _ { \mathrm { s } }$
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p, VV 0
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λ1=λ2=λ3= λv , VV = $\lambda _ { 1 } = \lambda _ { 2 } = \lambda _ { 3 } = \lambda _ { v } , \frac { v } { V _ { 0 } } = \lambda _ { v } ^ { 3 }$
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Figure 22.5.1–2 Schematic illustrations of deformation modes.
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