理论

The 有限元方法 is basically concerned with the determination of 场变量. The most important ones are the 应力和应变场. As basic measure of strain in CalculiX the 拉格朗日应变张量 E is used for 弹性介质, the 偏弹左柯西-格林张量 for incremental plasticity, the logarithmic (or Hencky) strain [10] for some other plasticity models as 变形塑性 and 约翰逊-库克硬化 and linear strains where appropriate, i.e. for 小变形 combined with 小转动. The Lagrangian strain satisfies ([27]):

$$E_{KL}=(U_{K,L}+U_{L,K}+U_{M,K} U_{M,L})/2,\;\;\;K,L,M=1,2,3$$ (2)

where $U_K$ are the 位移分量 in the 材料参考系 and repeated indices imply summation over the appropriate range. In a 线性分析, this reduces to the familiar form:

$$\tilde{E}_{KL}=(U_{K,L}+U_{L,K})/2,\;\;\;K,L=1,2,3.$$ (3)

The 偏弹左柯西-格林张量 is defined by ([90]):

$$\bar{b}^e_{kl}=J^{e-2/3}x^e_{k,K}x^e_{l,K}$$ (4)

where $J^e$ is the 弹性雅可比行列式 and $x^e_{k,K}$ is the elastic deformation gradient. Finally, the 对数应变 satisfies:

$$e_{\text{ln}}:= \sum_{i} \ln \lambda_i \boldsymbol{n}_i \otimes \boldsymbol{n}_i,$$ (5)

where $\lambda_i^2$ are the 特征值 of the 柯西-格林张量 $\boldsymbol{C}:= \boldsymbol{F}^T \cdot \boldsymbol{F}$ and $\boldsymbol{n}_i$ are obtained from the 特征向量 $\boldsymbol{N}_i$ of $\boldsymbol{C}$ through

$$\boldsymbol{n}_i = \boldsymbol{R} \cdot \boldsymbol{N}_i,$$ (6)

where $\boldsymbol{R}$ is the rotation tensor obtained from the well known decomposition of the deformation gradient $\boldsymbol{F}$ into a product of an orthogonal matrix $\boldsymbol{R}$ and a symmetric matrix $\boldsymbol{U}$ in the form $\boldsymbol{F}= \boldsymbol{R} \cdot \boldsymbol{U}$. The above formulas apply to Cartesian coordinate systems.

The stress measure consistent with the Lagrangian strain is the second Piola-Kirchhoff stress $\boldsymbol{S}$. This stress, which is internally used in CalculiX for all applications (the so-called total Lagrangian approach, see [10]), can be transformed into the first Piola-Kirchhoff stress $\boldsymbol{P}$ (the so-called engineering stress, a non-symmetric tensor) and into the Cauchy stress $\boldsymbol{\sigma }$ (true stress). All CalculiX input (e.g. distributed loading) and output is in terms of true stress. The stress measures are related by:

$$\boldsymbol{P} = J \boldsymbol{F}^{-1} \cdot \boldsymbol{\sigma }$$ (7)

and

$$\boldsymbol{S} = J \boldsymbol{F}^{-1} \cdot \boldsymbol{\sigma } \cdot \boldsymbol{F}^{-T},$$ (8)

where $J=\det(\boldsymbol{F})$.

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 $\overline{\alpha} \Delta T \boldsymbol{I}$, where $\overline{ \alpha}$ is the (secant) 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:

$$\tilde{E}_{KL}^{\text{mech}} = \tilde{E}_{KL} - \overline{\alpha} \Delta T \delta_{KL}.$$ (9)

In a non线性分析 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 $\boldsymbol{F}_{\text{th}}=(1+\overline{\alpha} \Delta T) \boldsymbol{I}$ and consequently:

$$\boldsymbol{F}_{\text{th}}^{-1} \approx (1-\overline{\alpha} \Delta T) \boldsymbol{I}.$$ (11)

Therefore one obtains:

$$(F_{\text{mech}})_{kK} \approx F_{kK}(1-\overline{\alpha} \Delta T) = (1+u_{k,K})(1-\overline{\alpha} \Delta T)$$

$$\approx 1+u_{k,K}-\overline{\alpha} \Delta T.$$ (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)

Since the stretches $\lambda$ are the 特征值 of the deformation gradient, subtracting $\overline{\alpha} \Delta T \boldsymbol{I}$ from $\boldsymbol{F}$ amounts to subtracting $\overline{\alpha} \Delta T$ from $\lambda = L/L_0 = 1 + \Delta L/L_0$. Therefore, the thermal strain get the meaning of a length change divided by an initial length. Infinitesimally one obtains:

$$\frac{dL}{L_0} = \alpha d T,$$ (14)

leading to

$$\frac{L-L_0}{L_0} = \int_{T_0}^{T} \alpha (\xi) d \xi$$

$$=: \overline{\alpha }(T) (T-T_0)$$

$$= \overline{\alpha } (T) \Delta T,$$ (15)

from which

$$L = L_0 ( 1 + \overline{\alpha }(T) \Delta T.$$ (16)

Here, $T_0$ is the temperature for which the specimen length is $L_0$, $\alpha$ is the instantaneous or tangent expansion coefficient and $\overline{ \alpha}$ is the secant expansion coefficient. We observe that the extension is linear in the temperature. This is also the way in which the expansion coefficients are usually measured, i.e. with respect to the initial specimen length.

Notice that the same approach is taken in CalculiX for the calculation of the corotational 对数应变 needed for an Abaqus User Material Routine. First, the thermal strain is subtracted from the total stretch to obtain the mechanical stretch and then the corational logarithmic mechanical strain is built:

$$\hat{E}_{\text{ln,mech}} = \sum_{i} \ln \lambda_{\text{mech},i} \boldsymbol{N}_i \otimes \boldsymbol{N}_i,$$ (17)

In Abaqus, however, it is obtained according to:

$$\hat{E}_{\text{ln,mech}} = \sum_{i} (\ln \lambda_i - \overline{\alpha} \Delta T) \boldsymbol{N}_i \otimes \boldsymbol{N}_i,$$ (18)

which implies

$$\frac{dL}{L} = \alpha d T,$$ (19)

leading to

$$\ln \frac{L}{L_0} = \int_{T_0}^{T} \alpha (\xi) d \xi$$

$$= \overline{\alpha } (T) \Delta T,$$ (20)

or

$$L = L_0 \exp[\overline{\alpha }(T) \Delta T].$$ (21)

This establishes an exponential expansion relationship. This is fine, as long as the thermal strain was also measured according to Equation(19), i.e. with reference to the actual length. The latter approach, used by a lot of Finite Element codes requires linear expansion coefficients for linear calculations (using linear strain) and exponential ones for nonlinear geometric calculations (using 对数应变). The approach in CalculiX avoids this.