Lines Matching refs:sigma

16 The effect of these linearizations is sketched in the diagram below, where $\bm \sigma$ and $\bm \e…
23 …@V{\text{geometric}}V{\begin{smallmatrix}\bm E \to \bm \epsilon \\ \bm S \to \bm \sigma \end{small…
24 …@V{\begin{smallmatrix}\bm E \to \bm \epsilon \\ \bm S \to \bm \sigma \end{smallmatrix}}V{\text{geo…
25   {\underbrace{\bm \sigma(\bm \epsilon)}_\text{Small Strain Hyperelastic}}
27   {\underbrace{\bm \sigma = \mathsf C \bm \epsilon}_\text{Linear Elastic}}
47 \nabla \cdot \bm{\sigma} + \bm{g} = \bm{0}
50 where $\bm{\sigma}$ and $\bm{g}$ are stress and forcing functions, respectively.
54 \int_{\Omega}{ \nabla \bm{v} \tcolon \bm{\sigma}} \, dV
55 - \int_{\partial \Omega}{\bm{v} \cdot \left(\bm{\sigma} \cdot \hat{\bm{n}}\right)} \, dS
60 where $\bm{\sigma} \cdot \hat{\bm{n}}|_{\partial \Omega}$ is replaced by an applied force/traction …
65 In their most general form, constitutive models define $\bm \sigma$ in terms of state variables.
73 This constitutive model $\bm \sigma(\bm \epsilon)$ is a linear tensor-valued function of a tensor-v…
78 Q \bm \sigma(\bm \epsilon) Q^T = \bm \sigma(Q \bm \epsilon Q^T),
81 which means that we can change our reference frame before or after computing $\bm \sigma$, and get …
82 Constitutive relations in which $\bm \sigma$ is uniquely determined by $\bm \epsilon$ while satisfy…
86 \bm \sigma(\bm \epsilon) = \frac{\partial \Phi}{\partial \bm \epsilon}.
108 \bm\sigma = \lambda (\operatorname{trace} \bm\epsilon) \bm I_3 + 2 \mu \bm\epsilon,
120 \bm{\sigma} = \mathsf{C} \!:\! \bm{\epsilon}.
123 For notational convenience, we express the symmetric second order tensors $\bm \sigma$ and $\bm \ep…
153 \bm{\sigma} = \lambda \log(1 + \operatorname{trace} \bm\epsilon) \bm{I}_3 + 2\mu \bm{\epsilon}
164 \diff \bm{\sigma} = \dfrac{\partial \bm{\sigma}}{\partial \bm{\epsilon}} \tcolon \diff \bm{\epsilon}
182 \diff \bm{\sigma}  = \bar{\lambda} \cdot \operatorname{trace} \diff \bm{\epsilon} \cdot \bm{I}_3 + …
266 …eq}`linear-stress-strain`, which  expresses the relationship between $\bm\sigma$ and $\bm\epsilon$.