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The engineering notation matrix in this case is shown by Eq. The fill method axisymmetric_rz is used for materials which are isotropic with respect to an axis of symmetry, such as a material composed of fibers which are parallel to the axis of symmetry ( Slaughter, 2012). (7) (7) where is the permutation tensor and is the summation index. The elasticity tensor is built using the symmetries shown in Eq. The fill method antisymmetric_isotropic is used for an antisymmetric isotropic material in a shear case. (6) Please see the documentation page for ComputeIsotropicElasticityTensor for details and examples of the input file syntax for linear elastic isotropic elasticity tensors. These two fill methods use the symmetries shown in Eq. The two constant isotropic symmetry fill methods symmetric_isotropic and symmetric_isotropic_E_nu are used in the dedicated isotropic elasticity tensor ComputeIsotropicElasticityTensor. For cases where axes of orthotropy do not coincide with global axes, Euler angles can be provided to perform a general rotation of the elasticity tensor at the beginning of the simulation. (modules/tensor_mechanics/test/tests/finite_strain_elastic_anisotropy/3d_bar_orthotropic_full_rotation.i)įor the orthotropic fill method, the material parameters need to be referred to the global frame and introduced in the following order. Note that, in this case, the method symmetric9 is used to enter an isotropic elasticity tensor. (2), the parameter C_ijkl expects the elasticity components in the order C_ijkl = '1111 1122 1133 2222 2233 3333 2323 3131ġ212' for the symmetric9 fill method option. In the Einstein index notation shown in Eq. (modules/tensor_mechanics/test/tests/finite_strain_elastic/finite_strain_elastic_new_test.i) That is, the elasticity tensor is computed via moduli of elasticity, Poisson's ratios, and shear stiffnesses, see Eq.
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The user can also select the fill method orthotropic, which generates an elasticity tensor based directly on material parameters. (2), for an orthotropic material is given in Eq. The enginering elasticity tensor notation, Eq. The fill method symmetric9 is appropriate for materials with three orthotropic planes of symmetry ( Malvern, 1969), and is often used for simulations of anistropic materials such as cubic crystals. Which shows the expected order of the elasticity tensor components in the input argument string. (modules/combined/test/tests/linear_elasticity/tensor.i) (3) are used to determine the independent components ( Slaughter, 2012). The fill method symmetric21 is used to create the elasticity tensor for a linear hyperelastic material with 21 independent components: the symmetries shown in Eq. There are several different material symmetry options that a user can apply to build the elasticity tensor for a mechanics simulation that are discussed below.
#Cohesion and coupling in software engineering ppt full
Nonetheless, the full Rank-4 tensor with all 81 components is created by ComputeElasticityTensor.
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Rotations to the elasticity tensor are applied as (1) See below for examples of tensor rotations.įor a general stiffness tensor with 21 independent components, the elasticity tensor within the tensor mechanics module can be represented with the notation shown in Eq. Alternatively, the 3x3 rotation matrix can be defined directly in the input file. The initial rotation is performed if the user provides arguments to the three Euler angle parameters the Bunge Euler angles provided in this class are used to perform passive (from the sample to the crystal) rotations using the extrinsic convention to build the rotation matrix. ComputeElasticityTensor also rotates the elasticity tensor during the initial time step only this class does not rotate the elasticity tensor during the simulation.
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The material ComputeElasticityTensor builds the elasticity (stiffness) tensor with various user-selected material symmetry options.
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