E_{KL}=(U_{K,L}+U_{L,K}+U_{M,K} U_{M,L})/2,\;\;\;K,L,M=1,2,3 | (3) |
\tilde{E}_{KL}=(U_{K,L}+U_{L,K})/2,\;\;\;K,L=1,2,3. | (4) |
The Eulerian strain satisfies ([21]):
e_{kl}=(u_{k,l}+u_{l,k}-u_{m,k} u_{m,l})/2,\;\;\;k,l,m=1,2,3 | (5) |
Finally, the deviatoric elastic left Cauchy-Green tensor is defined by ([76]):
\bar{b}^e_{kl}=J^{e-2/3}x^e_{k,K}x^e_{l,K} | (6) |
where J^e is the elastic Jacobian and x^e_{k,K} is the elastic deformation gradient. The above formulas apply for Cartesian coordinate systems.
The stress measure consistent with the Lagrangian strain is the second Piola-Kirchhoff stress S. This stress, which is internally used in CalculiX for all applications (the so-called total Lagrangian approach, see [9]), can be transformed into the first Piola-Kirchhoff stress P (the so-called engineering stress, a non-symmetric tensor) and into the Cauchy stress t (true stress). All CalculiX input (e.g. distributed loading) and output is in terms of true stress. In a tensile test on a specimen with length L the three stress measures are related by:
t=P/(1-\epsilon)=S/{(1-\epsilon)^2} | (7) |
where \epsilon is the engineering strain defined by
\epsilon=dL/L. | (8) |
The treatment of the thermal strain depends on whether the analysis is geometrically linear or nonlinear. For isotropic material the thermal strain tensor amounts to \alpha \Delta T \boldsymbol{I}, where \alpha is the expansion coefficient, \Delta T is the temperature change since the initial state and \boldsymbol{I} is the second order identity tensor. For geometrically linear calculations the thermal strain is subtracted from the total strain to obtain the mechanical strain:
\tilde{E}_{KL}^{\text{mech}} = \tilde{E}_{KL} - \alpha \Delta T \delta_{KL}. | (9) |
In a nonlinear analysis the thermal strain is subtracted from the deformation gradient in order to obtain the mechanical deformation gradient. Indeed, assuming a multiplicative decomposition of the deformation gradient one can write:
d \boldsymbol{x} = \boldsymbol{F} \cdot d \boldsymbol{X} = \boldsymbol{F}_{\text{mech}} \cdot \boldsymbol{F}_{\text{th}} \cdot d \boldsymbol{X}, | (10) |
where the total deformation gradient \boldsymbol{F} is written as the product of the mechanical deformation gradient and the thermal deformation gradient. For isotropic materials the thermal deformation gradient can be written as \boldsymbol{F}_{\text{th}}=(1+\alpha \Delta T) \boldsymbol{I} and consequently:
\boldsymbol{F}_{\text{th}}^{-1} \approx (1-\alpha \Delta T) \boldsymbol{I}. | (11) |
Therefore one obtains:
\begin{eqnarray}(F_{\text{mech}})_{kK} &\approx & F_{kK}(1-\alpha \Delta T) = (1+u_{k,K})(1-\alpha \Delta T) \nonumber \\ &\approx & 1+u_{k,K}-\alpha \Delta T.\end{eqnarray} | \displaystyle \approx | \displaystyle F_{kK}(1-\alpha \Delta T) = (1+u_{k,K})(1-\alpha \Delta T) | |
\displaystyle \approx | \displaystyle 1+u_{k,K}-\alpha \Delta T. | (12) |
Based on the mechanical deformation gradient the mechanical Lagrange strain is calculated and subsequently used in the material laws:
2 \boldsymbol{E}_{\text{mech}} = \boldsymbol{F}_{\text{mech}}^T \cdot\boldsymbol{F}_{\text{mech}} - \boldsymbol{I}. | (13) |