Evaluate Cauchy strain tensor on a surface region.
See CauchyStrainTerm.
Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.
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\int_{\Gamma} \ull{e}(\ul{w})
\mbox{vector for } K \from \Ical_h: \int_{T_K} \ull{e}(\ul{w}) / \int_{T_K} 1
\ull{e}(\ul{w})|_{qp}
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ev_cauchy_strain_s | (parameter) |
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Evaluate Cauchy strain tensor.
It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [11, 22, 12]. The last three (non-diagonal) components are doubled so that it is energetically conjugate to the Cauchy stress tensor with the same storage.
Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.
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\int_{\Omega} \ull{e}(\ul{w})
\mbox{vector for } K \from \Ical_h: \int_{T_K} \ull{e}(\ul{w}) / \int_{T_K} 1
\ull{e}(\ul{w})|_{qp}
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ev_cauchy_strain | (parameter) |
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Evaluate fading memory Cauchy stress tensor.
It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [11, 22, 12].
Assumes an exponential approximation of the convolution kernel resulting in much higher efficiency.
Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.
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\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}
\mbox{vector for } K \from \Ical_h: \int_{T_K} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} / \int_{T_K} 1
\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}|_{qp}
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ev_cauchy_stress_eth | (ts, material_0, material_1, parameter) |
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Evaluate fading memory Cauchy stress tensor.
It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [11, 22, 12].
Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.
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\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}
\mbox{vector for } K \from \Ical_h: \int_{T_K} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} / \int_{T_K} 1
\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}|_{qp}
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ev_cauchy_stress_th | (ts, material, parameter) |
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Evaluate Cauchy stress tensor.
It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [11, 22, 12].
Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.
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\int_{\Omega} D_{ijkl} e_{kl}(\ul{w})
\mbox{vector for } K \from \Ical_h: \int_{T_K} D_{ijkl} e_{kl}(\ul{w}) / \int_{T_K} 1
D_{ijkl} e_{kl}(\ul{w})|_{qp}
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ev_cauchy_stress | (material, parameter) |
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This term has the same definition as dw_lin_elastic_th, but assumes an exponential approximation of the convolution kernel resulting in much higher efficiency. Can use derivatives.
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\int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v})
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dw_lin_elastic_eth | (ts, material_0, material_1, virtual, state) |
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Isotropic linear elasticity term.
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\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) \mbox{ with } D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) + \lambda \ \delta_{ij} \delta_{kl}
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dw_lin_elastic_iso | (material_1, material_2, virtual, state) |
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Fading memory linear elastic (viscous) term. Can use derivatives.
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\int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v})
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dw_lin_elastic_th | (ts, material, virtual, state) |
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General linear elasticity term, with D_{ijkl} given in the usual matrix form exploiting symmetry: in 3D it is 6\times6 with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it is 3\times3 with the indices ordered as [11, 22, 12]. Can be evaluated. Can use derivatives.
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\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
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dw_lin_elastic | (material, virtual, state) |
(material, parameter_1, parameter_2) |
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Linear prestress term, with the prestress \sigma_{ij} given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [11, 22, 12]. Can be evaluated.
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\int_{\Omega} \sigma_{ij} e_{ij}(\ul{v})
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dw_lin_prestress | (material, virtual) |
(material, parameter) |
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Linear (pre)strain fiber term with the unit direction vector \ul{d}.
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\int_{\Omega} D_{ijkl} e_{ij}(\ul{v}) \left(d_k d_l\right)
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dw_lin_strain_fib | (material_1, material_2, virtual) |
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Sensitivity analysis of the linear elastic term.
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\int_{\Omega} \hat{D}_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
\hat{D}_{ijkl} = D_{ijkl}(\nabla \cdot \ul{\Vcal}) - D_{ijkq}{\partial \Vcal_l \over \partial x_q} - D_{iqkl}{\partial \Vcal_j \over \partial x_q}
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d_sd_lin_elastic | (material, parameter_w, parameter_u, parameter_mesh_velocity) |
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