김경종
279 lines
24 KiB
Markdown
279 lines
24 KiB
Markdown
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# Uniaxial tests
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The uniaxial deformation mode is characterized in terms of the principal stretches, $\lambda _ { i } ,$ a s
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$$
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\lambda_ {1} = \lambda_ {U}, \quad \lambda_ {2} = \lambda_ {3} = 1 / \sqrt {\lambda_ {U}},
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$$
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where $\lambda _ { U }$ is the stretch in the loading direction. The nominal strain is defined by $\varepsilon _ { U } = \lambda _ { U } - 1$
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To derive the uniaxial nominal stress $T _ { U }$ , we invoke the principle of virtual work:
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$$
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\delta U = T _ {U} \delta \lambda_ {U},
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$$
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so that
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$$
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T _ {U} = \frac {\partial U}{\partial \lambda_ {U}} = 2 (1 - \lambda_ {U} ^ {- 3}) \left(\lambda_ {U} \frac {\partial U}{\partial \overline {{I}} _ {1}} + \frac {\partial U}{\partial \overline {{I}} _ {2}}\right).
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$$
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The uniaxial tension test is the most common of all the tests and is usually performed by pulling a “dog-bone” specimen. The uniaxial compression test is performed by loading a compression button between lubricated surfaces. The loading surfaces are lubricated to minimize any barreling effect in the button that would cause deviations from a homogeneous uniaxial compression stress-strain state.
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Input File Usage: \*UNIAXIAL TEST DATA
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; Input source: Test data and Test Data→Uniaxial Test Data
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# Equibiaxial tests
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The equibiaxial deformation mode is characterized in terms of the principal stretches, $\lambda _ { i }$ , as
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$$
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\lambda_ {1} = \lambda_ {2} = \lambda_ {B}, \quad \lambda_ {3} = 1 / \lambda_ {B} ^ {2},
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$$
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where $\lambda _ { B }$ is the stretch in the two perpendicular loading directions. The nominal strain is defined by $\varepsilon _ { B } = \lambda _ { B } - 1$
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To develop the expression for the equibiaxial nominal stress, $T _ { B }$ , we again use the principle of virtual work (assuming that the stress perpendicular to the loading direction is zero),
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$$
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\delta U = 2 T _ {B} \delta \lambda_ {B},
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$$
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so that
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$$
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T _ {B} = \frac {1}{2} \frac {\partial U}{\partial \lambda_ {B}} = 2 \left(\lambda_ {B} - \lambda_ {B} ^ {- 5}\right) \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \lambda_ {B} ^ {2} \frac {\partial U}{\partial \overline {{I}} _ {2}}\right).
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$$
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In practice, the equibiaxial compression test is rarely performed because of experimental setup difficulties. In addition, this deformation mode is equivalent to a uniaxial tension test, which is straightforward to conduct.
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A more common test is the equibiaxial tension test, in which a stress state with two equal tensile stresses and zero shear stress is created. This state is usually achieved by stretching a square sheet in a biaxial testing machine. It can also be obtained by inflating a circular membrane into a spheroidal shape (like blowing up a balloon). The stress field in the middle of the membrane then closely approximates equibiaxial tension, provided that the thickness of the membrane is very much smaller than the radius of curvature at this point. However, the strain distribution will not be quite uniform, and local strain measurements will be required. Once the strain and radius of curvature are known, the nominal stress can be derived from the inflation pressure.
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Input File Usage: \*BIAXIAL TEST DATA
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; Input source: Test data and Test Data→Biaxial Test Data
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# Planar tests
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The planar deformation mode is characterized in terms of the principal stretches, $\lambda _ { i } ,$ , as
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$$
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\lambda_ {1} = \lambda_ {S}, \lambda_ {2} = 1, \lambda_ {3} = 1 / \lambda_ {S},
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$$
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where $\lambda _ { S }$ is the stretch in the loading direction. Then, the nominal strain in the loading direction is $\varepsilon _ { S } = \lambda _ { S } - 1$
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This test is also called a “pure shear” test since, in terms of logarithmic strains,
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$$
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\varepsilon_ {1} = \ln \lambda_ {1} = - \ln \lambda_ {3} = - \varepsilon_ {3}, \quad \varepsilon_ {2} = \ln \lambda_ {2} = 0,
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$$
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which corresponds to a state of pure shear at an angle of $4 5 ^ { \circ }$ to the loading direction.
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The principle of virtual work gives
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$$
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\delta U = T _ {S} \delta \lambda_ {S},
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$$
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where $T _ { S }$ is the nominal planar stress, so that
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$$
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T _ {S} = \frac {\partial U}{\partial \lambda_ {S}} = 2 (\lambda_ {S} - \lambda_ {S} ^ {- 3}) \left(\frac {\partial U}{\partial \overline {{I}} _ {1}} + \frac {\partial U}{\partial \overline {{I}} _ {2}}\right).
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$$
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For the (general) polynomial and Ogden models and for the coefficient $\beta$ in the Van der Waals model this equation alone will not determine the constants uniquely. The planar test data must be augmented by uniaxial test data and/or biaxial test data to determine the material parameters.
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Planar tests are usually done with a thin, short, and wide rectangular strip of material fixed on its wide edges to rigid loading clamps that are moved apart. If the separation direction is the 1-direction and the thickness direction is the 3-direction, the comparatively long size of the specimen in the 2-direction
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and the rigid clamps allow us to use the approximation $\lambda _ { 2 } = 1$ ; that is, there is no deformation in the wide direction of the specimen. This deformation mode could also be called planar compression if the 3-direction is considered to be the primary direction. All forms of incompressible plane strain behavior are characterized by this deformation mode. Consequently, if plane strain analysis is performed, planar test data represent the relevant form of straining of the material.
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Input File Usage: \*PLANAR TEST DATA
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; Input source: Test data and Test Data→Planar Test Data
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# Volumetric tests
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The following discussion describes procedures for obtaining $D _ { i }$ values (or $D ,$ for the Arruda-Boyce and Van der Waals models) corresponding to the actual material behavior. With these values you can compare the material’s initial bulk modulus, $K _ { 0 } = 2 / D _ { 1 }$ , to its initial shear modulus $( \mu _ { 0 } = 2 ( C _ { 1 0 } + C _ { 0 1 } )$ for the polynomial model, $\begin{array} { r } { \mu _ { 0 } = \sum _ { i = 1 } ^ { N } \mu _ { i } } \end{array}$ for Ogden’s model) and then judge whether $D _ { i }$ values that will provide results are sufficiently realistic. For Abaqus/Explicit caution should be used; $K _ { 0 } / \mu _ { 0 }$ should be less than 100. Otherwise, noisy solutions will be obtained and time increments will be excessively small (see “Compressibility in Abaqus/Explicit” above). The $D _ { i }$ and D can be calculated from data obtained in pure volumetric compression of a specimen (volumetric tension tests are much more difficult to perform). In a pure volumetric test $\lambda _ { 1 } = \lambda _ { 2 } = \lambda _ { 3 } = \lambda _ { V } ;$ therefore, $\overline { { I } } _ { 1 } = \overline { { I } } _ { 2 } = 3$ and $J = { \lambda _ { V } } ^ { 3 } = V / V _ { 0 }$ (the volume ratio). Using the polynomial form of the strain energy potential, the total pressure stress on the specimen is obtained as
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$$
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p = - \left(\frac {\sigma_ {1} + \sigma_ {2} + \sigma_ {3}}{3}\right) = - \sum_ {i = 1} ^ {N} 2 i \frac {1}{D _ {i}} (\lambda_ {V} ^ {3} - 1) ^ {2 i - 1}.
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$$
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This equation can be used to determine the $D _ { i }$ . If we are using a second-order polynomial series for $U ,$ w e have $N = 2$ , and so two $D _ { i }$ are needed. Therefore, a minimum of two points on the pressure-volume ratio curve are required to give two equations for the $D _ { i }$ . For the Ogden and reduced polynomial potentials $D _ { i }$ can be determined for up to $N = 6$ . A linear least-squares fit is performed when more than N data points are provided.
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An approximate way of conducting a volumetric test consists of using a cylindrical rubber specimen that fits snugly inside a rigid container and whose top surface is compressed by a rigid piston. Although both volumetric and deviatoric deformation are present, the deviatoric stresses will be several orders of magnitude smaller than the hydrostatic stresses (because the bulk modulus is much higher than the shear modulus) and can be neglected. The compressive stress imposed by the rigid piston is effectively the pressure, and the volumetric strain in the rubber cylinder is computed from the piston displacement.
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Nonzero values of $D _ { i }$ affect the uniaxial, equibiaxial, and planar stress results. However, since the material is assumed to be only slightly compressible, the techniques described for obtaining the deviatoric coefficients should give sufficiently accurate values even though they assume that the material is fully incompressible.
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Input File Usage: \*VOLUMETRIC TEST DATA
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; Input source: Test data and Test Data→Volumetric Test Data
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# Equivalent experimental tests
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The superposition of a tensile or compressive hydrostatic stress on a loaded, fully incompressible elastic body results in different stresses but does not change the deformation. Thus, Figure 22.5.1–3 shows that some apparently different loading conditions are actually equivalent in their deformations and, therefore, are equivalent tests:
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• Uniaxial tension Equibiaxial compression
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• Uniaxial compression Equibiaxial tension
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• Planar tension Planar compression
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On the other hand, the tensile and compressive cases of the uniaxial and equibiaxial modes are independent from each other: uniaxial tension and uniaxial compression provide independent data.
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# Smoothing the test data
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Experimental test data often contain noise in the sense that the test variable is both slowly varying and also corrupted by random noise. This noise can affect the quality of the strain energy potential that Abaqus derives. This noise is particularly a problem with the Marlow form, where a strain energy potential that exactly describes the test data that are used to calibrate the model is computed. It is less of a concern with the other forms, since smooth functions are fitted through the test data.
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Abaqus provides a smoothing technique to remove the noise from the test data based on the Savitzky-Golay method. The idea is to replace each data point by a local average of its surrounding data points, so that the level of noise can be reduced without biasing the dominant trend of the test data. In the implementation a cubic polynomial is fitted through each data point i and n data points to the immediate left and right of that point. A least-squares method is used to fit the polynomial through these points. The value of data point i is then replaced by the value of the polynomial at the same position. Each polynomial is used to adjust one data point except near the ends of the curve, where a polynomial is used to adjust multiple points, because the first and last few points cannot be the center of the fitting set of data points. This process is applied repeatedly to all data points until two consecutive passes through the data produce nearly the same results.
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By default, the test data are not smoothed. If smoothing is specified, the default value is n=3. Alternatively, you can specify the number of data points to the left and right of a data point in the moving window within which a least-squares polynomial is fit.
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# Input File Usage:
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For the Marlow form, use one of the first three options and, optionally, the fourth option; for the other potential forms, use one and up to four of the following options:
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\*UNIAXIAL TEST DATA, SMOOTH=n ( )
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\*BIAXIAL TEST DATA, SMOOTH=n ( )
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The stresses, $\sigma _ { \mathrm { i } } ,$ shown here are true (Cauchy) stresses and not nominal stresses.
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Figure 22.5.1–3 Equivalent deformation modes through superposition of hydrostatic stress.
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\*PLANAR TEST DATA, SMOOTH=n ( )
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\*VOLUMETRIC TEST DATA, SMOOTH=n $\left( n \geq 2 \right)$
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Property module: material editor: Mechanical→Elasticity→Hyperelastic: Material type: Isotropic; Input source: Test data and Test Material type: Isotropic; Input source: Test data and Test Data→Uniaxial Test Data, Biaxial Test Data, Planar Test Data→Uniaxial Test Data, Biaxial Test Data, Planar Test Data, or Volumetric Test Data Data, or Volumetric Test Data
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In each of the Test Data Editor dialog boxes, toggle on Apply smoothing, and select a value for n $\left( n \geq 2 \right)$ .
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Once the strain energy potential is determined, the behavior of the hyperelastic model in Abaqus is established. However, the quality of this behavior must be assessed: the prediction of material behavior under different deformation modes must be compared against the experimental data. You must judge whether the strain energy potentials determined by Abaqus are acceptable, based on the correlation between the Abaqus predictions and the experimental data. You can evaluate the hyperelastic behavior automatically in Abaqus/CAE. Alternatively, single-element test cases can be used to derive the nominal stress–nominal strain response of the material model.
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See “Fitting of rubber test data,” Section 3.1.4 of the Abaqus Benchmarks Guide, which illustrates the entire process of fitting hyperelastic constants to a set of test data.
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# Hyperelastic material stability
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An important consideration in judging the quality of the fit to experimental data is the concept of material or Drucker stability. Abaqus checks the Drucker stability of the material for the first three deformation modes described above.
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The Drucker stability condition for an incompressible material requires that the change in the stress, , following from any infinitesimal change in the logarithmic strain, , satisfies the inequality
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$$
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d \pmb {\sigma}: d \pmb {\varepsilon} > 0.
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$$
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Using $d { \boldsymbol { \sigma } } = \mathbf { D } : d { \boldsymbol { \varepsilon } }$ , where is the tangent material stiffness, the inequality becomes
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$$
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d \boldsymbol {\varepsilon}: \mathbf {D}: d \boldsymbol {\varepsilon} > 0,
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$$
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thus requiring the tangential material stiffness to be positive-definite.
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For an isotropic elastic formulation the inequality can be represented in terms of the principal stresses and strains,
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$$
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d \sigma_ {1} d \varepsilon_ {1} + d \sigma_ {2} d \varepsilon_ {2} + d \sigma_ {3} d \varepsilon_ {3} > 0.
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$$
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As before, since the material is assumed to be incompressible, we can choose any value for the hydrostatic pressure without affecting the strains. A convenient choice for the stability calculation is $\sigma _ { 3 } = d \sigma _ { 3 } = 0$ , which allows us to ignore the third term in the above equation.
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The relation between the changes in stress and in strain can then be obtained in the form of the matrix
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$$
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\binom{d \sigma_ {1}}{d \sigma_ {2}} = \left( \begin{array}{c c} D _ {1 1} & D _ {1 2} \\ D _ {2 1} & D _ {2 2} \end{array} \right) \binom{d \varepsilon_ {1}}{d \varepsilon_ {2}},
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$$
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where $D _ { i j } = D _ { i j } ( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } )$ . For material stability must be positive-definite; thus, it is necessary that
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$$
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D _ {1 1} + D _ {2 2} > 0,
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$$
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$$
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D _ {1 1} D _ {2 2} - D _ {1 2} D _ {2 1} > 0.
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$$
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This stability check is performed for the polynomial models, the Ogden potential, the Van der Waals form, and the Marlow form. The Arruda-Boyce form is always stable for positive values of $( \mu , \lambda _ { m } )$ ; hence, it suffices to check the material coefficients to ensure stability.
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You should be careful when defining the $C _ { i j }$ or $( \mu _ { i } , \alpha _ { i } )$ for the polynomial models or the Ogden form: especially when $N > 1$ , the behavior at higher strains is strongly sensitive to the values of the $C _ { i j }$ or $( \mu _ { i } , \alpha _ { i } )$ , and unstable material behavior may result if these values are not defined correctly. When some of the coefficients are strongly negative, instability at higher strain levels is likely to occur.
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Abaqus performs a check on the stability of the material for six different forms of loading—uniaxial tension and compression, equibiaxial tension and compression, and planar tension and compression—for $0 . 1 \leq \lambda _ { 1 } \leq \ 1 0 . 0$ (nominal strain range of $- 0 . 9 \leq \epsilon _ { 1 } \leq 9 . 0 )$ at intervals $\Delta \lambda _ { 1 } = 0 . 0 1$ . If an instability is found, Abaqus issues a warning message and prints the lowest absolute value of $\epsilon _ { 1 }$ for which the instability is observed. Ideally, no instability occurs. If instabilities are observed at strain levels that are likely to occur in the analysis, it is strongly recommended that you either change the material model or carefully examine and revise the material input data. If user subroutine UHYPER is used to define the hyperelastic material, you are responsible for ensuring stability.
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# Improving the accuracy and stability of the test data fit
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Unfortunately, the initial fit of the models to experimental data may not come out as well as expected. This is particularly true for the most general models, such as the (general) polynomial model and the Ogden model. For some of the simpler models, stability is assured by following some simple rules.
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• For positive values of the initial shear modulus, $\mu ,$ and the locking stretch, $\lambda _ { m }$ , the Arruda-Boyce form is always stable.
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• For positive values of the coefficient $C _ { 1 0 }$ the neo-Hookean form is always stable.
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• Given positive values of the initial shear modulus, $\mu ,$ and the locking stretch, $\lambda _ { m }$ , the stability of the Van der Waals model depends on the global interaction parameter, a.
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• For the Yeoh model stability is assured if all $C _ { i 0 } > 0$ . Typically, however, $C _ { 2 0 }$ will be negative, since this helps capture the S-shape feature of the stress-strain curve. Thus, reducing the absolute value of $C _ { 2 0 }$ or magnifying the absolute value of $C _ { 1 0 }$ will help make the Yeoh model more stable.
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In all cases the following suggestions may improve the quality of the fit:
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• Both tension and compression data are allowed; compressive stresses and strains are entered as negative values. Use compression or tension data depending on the application: it is difficult to fit a single material model accurately to both tensile and compressive data.
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• Always use many more experimental data points than unknown coefficients.
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• If $N \geq 3$ is used, experimental data should be available to at least 100% tensile strain or 50% compressive strain.
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• Perform different types of tests (e.g., compression and simple shear tests). Proper material behavior for a deformation mode requires test data to characterize that mode.
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• Check for warning messages about material instability or error messages about lack of convergence in fitting the test data. This check is especially important with new test data; a simple finite element model with the new test data can be run through the analysis input file processor to check the material stability.
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• Use the material evaluation capability in Abaqus/CAE to compare the response curves for different strain energy potentials to the experimental data. Alternatively, you can perform one-element simulations for simple deformation modes and compare the Abaqus results against the experimental data. The X–Y plotting options in the Visualization module of Abaqus/CAE can be used for this comparison.
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• Delete some data points at very low strains if large strains are anticipated. A disproportionate number of low strain points may unnecessarily bias the accuracy of the fit toward the low strain range and cause greater errors in the large strain range.
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• Delete some data points at the highest strains if small to moderate strains are expected. The high strain points may force the fitting to lose accuracy and/or stability in the low strain range.
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• Pick data points at evenly spaced strain intervals over the expected range of strains, which will result in similar accuracy throughout the entire strain range.
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• The higher the order of N, the more oscillations are likely to occur, leading to instabilities in the stress-strain curves. If the (general) polynomial model is used, lower the order of N from 2 to 1 (3 to 2 for Ogden), especially if the maximum strain level is low (say, less than 100% strain).
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• If multiple types of test data are used and the fit still comes out poorly, some of the test data probably contain experimental errors. New tests may be needed. One way of determining which test data are erroneous is to first calibrate the initial shear modulus $\mu _ { 0 } ^ { t e s t }$ of the material. Then fit each type of test data separately in Abaqus and compute the shear modulus, $\mu _ { 0 } ^ { f i t }$ , from the material constants using the relations
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$$
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\mu_ {0} ^ {f i t} = 2 (C _ {1 0} + C _ {0 1}) \quad \mathrm{(polynomialform)} \qquad \mathrm{or} \qquad \mu_ {0} ^ {f i t} = \sum_ {i = 1} ^ {N} \mu_ {i} \quad \mathrm{(Ogdenform)}.
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$$
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Alternatively, the initial Young’s modulus, $E _ { 0 } ^ { t e s t }$ , can be calibrated and compared with
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$$
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E _ {0} ^ {f i t} = 6 (C _ {1 0} + C _ {0 1}) \quad \text {(polynomial form)} \qquad \text {or} \qquad E _ {0} ^ {f i t} = 3 \sum_ {i = 1} ^ {N} \mu_ {i} \quad \text {(Ogden form)}.
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$$
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The values of $\mu _ { 0 } ^ { f i t }$ or $E _ { 0 } ^ { f i t }$ that are most different from $\mu _ { 0 } ^ { t e s t }$ or $E _ { 0 } ^ { t e s t }$ indicate the erroneous test data.
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# Elements
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The hyperelastic material model can be used with solid (continuum) elements, finite-strain shells (except S4), continuum shells, membranes, and one-dimensional elements (trusses and rebars). In Abaqus/Standard the hyperelastic material model can be also used with Timoshenko beams (B21, B22, B31, B31OS, B32, B32OS, PIPE21, PIPE22, PIPE31, PIPE32, and their “hybrid” equivalents). It cannot be used with Euler-Bernoulli beams (B23, B23H, B33, and B33H) and small-strain shells (STRI3, STRI65, S4R5, S8R, S8R5, S9R5).
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# Pure displacement formulation versus hybrid formulation in Abaqus/Standard
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For continuum elements in Abaqus/Standard hyperelasticity can be used with the pure displacement formulation elements or with the “hybrid” (mixed formulation) elements. Because elastomeric materials are usually almost incompressible, fully integrated pure displacement method elements are not recommended for use with this material, except for plane stress cases. If fully or selectively reduced-integration displacement method elements are used with the almost incompressible form of this material model, a penalty method is used to impose the incompressibility constraint in anything except plane stress analysis. The penalty method can sometimes lead to numerical difficulties; therefore, the fully or selectively reduced-integrated “hybrid” formulation elements are recommended for use with hyperelastic materials.
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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.
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# Incompatible mode elements in Abaqus/Standard
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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 hyperelastic elements that are unloaded after having been subjected to a complex deformation history.
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# Procedures
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Hyperelasticity must always be used with geometrically nonlinear analyses (“General and linear perturbation procedures,” Section 6.1.3).
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