Internal Stress Calculation

Global Formulation

The time integration of stresses has been stated earlier (Stress Rates) as:(1)

σ_{i j} (t + Δ t) = σ_{i j} (t) + {˙ σ}_{i j} Δ t

$σ_{i j} (t + Δ t) = σ_{i j} (t) + {\dot{σ}}_{i j} Δ t$

The stress rate is comprised of two components:(2)

{˙ σ}_{i j} = {˙ σ}_{i j}^{v} + {˙ σ}_{i j}^{r}

${\dot{σ}}_{i j} = {\dot{σ}}_{i j}^{v} + {\dot{σ}}_{i j}^{r}$

Where,

${˙ σ}_{i j}^{r}$ ${\dot{σ}}_{i j}^{r}$: Stress rate due to the rigid body rotational velocity
${˙ σ}_{i j}^{v}$ ${\dot{σ}}_{i j}^{v}$: Jaumann objective stress tensor derivative

The correction for stress rotation from time

t

$t$ to time

t + Δ t

$t + Δ t$ is given by ¹:(3)

{˙ σ}_{i j}^{r} = σ_{i k} Ω_{k j} + σ_{j k} Ω_{k i}

${\dot{σ}}_{i j}^{r} = σ_{i k} Ω_{k j} + σ_{j k} Ω_{k i}$

Where, $Ω$ $Ω$ is the rigid body rotational velocity tensor (Kinematic Description, Equation 14).

The Jaumann objective stress tensor derivative ${˙ σ}_{i j}^{v}$ ${\dot{σ}}_{i j}^{v}$ is the corrected true stress rate tensor without rotational effects. The constitutive law is directly applied to the Jaumann stress rate tensor.

Deviatoric stresses and pressure (Stresses in Solids) are computed separately and related by:(4)

σ_{i j} = s_{i j} - p δ_{i j}

$σ_{i j} = s_{i j} - p δ_{i j}$

Where,

$s_{i j}$ $s_{i j}$: Deviatoric stress tensor
$p$ $p$: Pressure or mean stress - defined as positive in compression
$δ_{i j}$ $δ_{i j}$: Substitution tensor or unit matrix

Co-rotational Formulation

A co-rotational formulation for bricks is a formulation where rigid body rotations are directly computed from the element's node positions. Objective stress and strain tensors are computed in the local (co-rotational) frame. Internal forces are computed in the local frame and then rotated to the global system.

So, when co-rotational formulation is used, Deviatoric Stress Calculation, Equation 2

{˙ σ}_{i j} = {˙ σ}_{i j}^{v} + {˙ σ}_{i j}^{r}

${\dot{σ}}_{i j} = {\dot{σ}}_{i j}^{v} + {\dot{σ}}_{i j}^{r}$ reduces to:(5)

{˙ σ}_{i j} = {˙ σ}_{i j}^{v}

${\dot{σ}}_{i j} = {\dot{σ}}_{i j}^{v}$

Where,

${˙ σ}_{i j}^{v}$ ${\dot{σ}}_{i j}^{v}$: Jaumann objective stress tensor derivative expressed in the co-rotational frame

Figure 1 orthogonalization, when one of the r, s, t directions is orthogonal to the two other directions.

When large rotations occur, this formulation is more accurate than the global formulation, for which the stress rotation due to rigid body rotational velocity is computed in an incremental way.

Co-rotational formulation avoids this kind of problem. Consider this test:

The increment of the rigid body rotation vector during time step

Δ t

$Δ t$ is:(6)

Δ Ω = Δ t / 2 \cdot ⎡ ⎢ ⎢ ⎣ \begin{matrix} (\partial v_{x} / \partial y - \partial v_{y} / \partial x) = 0 (\partial v_{x} / \partial z - \partial v_{z} / \partial x) = \partial v_{x} / \partial z (\partial v_{y} / \partial x - \partial v_{x} / \partial y) = 0 \end{matrix}

$Δ Ω = Δ t / 2 \cdot [\begin{array}{l} (\partial v_{x} / \partial y - \partial v_{y} / \partial x) = 0 \\ (\partial v_{x} / \partial z - \partial v_{z} / \partial x) = \partial v_{x} / \partial z \\ (\partial v_{y} / \partial x - \partial v_{x} / \partial y) = 0 \end{array}$

So, $Δ Ω_{y} = α Δ T / 2$ $Δ Ω_{y} = α Δ T / 2$

Where, $α = v / h$ $α = v / h$ equals the imposed velocity on the top of the brick divided by the height of the brick (constant value).

Due to first order approximation, the increment of stress

σ_{x x}

$σ_{x x}$ due to the rigid body motion is:(7)

Δ σ_{x x}^{r} = Δ Ω_{y} (τ_{x z} + τ_{z x}) = 2 Δ Ω_{y} τ_{x z} = α Δ T τ_{x z}

$Δ σ_{x x}^{r} = Δ Ω_{y} (τ_{x z} + τ_{z x}) = 2 Δ Ω_{y} τ_{x z} = α Δ T τ_{x z}$

Increment of stress

σ_{z z}

$σ_{z z}$ due to the rigid body motion:(8)

Δ σ_{z z}^{r} = - Δ Ω_{y} (τ_{x z} + τ_{z x}) = - 2 Δ Ω_{y} τ_{x z} = - α Δ T τ_{x z}

$Δ σ_{z z}^{r} = - Δ Ω_{y} (τ_{x z} + τ_{z x}) = - 2 Δ Ω_{y} τ_{x z} = - α Δ T τ_{x z}$

Increment of shear stress

τ_{x z}

$τ_{x z}$ due to the rigid body motion:(9)

Δ τ_{x z}^{r} = Δ Ω_{y} (σ_{z z} - σ_{x x}) = 2 Δ Ω_{y} σ_{z z} = α Δ T σ_{z z}

$Δ τ_{x z}^{r} = Δ Ω_{y} (σ_{z z} - σ_{x x}) = 2 Δ Ω_{y} σ_{z z} = α Δ T σ_{z z}$

Increment of shear strain:(10)

Δ γ_{x z} = Δ T (\partial v_{x} / \partial z + \partial v_{z} / \partial x) = α Δ T

$Δ γ_{x z} = Δ T (\partial v_{x} / \partial z + \partial v_{z} / \partial x) = α Δ T$

Increment of stress

σ_{z z}

$σ_{z z}$ due to strain:(11)

Δ σ_{z z}^{v} = 0

$Δ σ_{z z}^{v} = 0$

and increment of shear stress due to strain is:(12)

Δ τ_{x z}^{v} = G Δ γ_{x z} = G α Δ T

$Δ τ_{x z}^{v} = G Δ γ_{x z} = G α Δ T$

Where, $G$ $G$ is the shear modulus (material is linear elastic).

From Equation 8 to Equation 12, you have:(13)

[\begin{matrix} Δ τ_{x z} = α Δ T σ_{z z} + G α Δ T Δ σ_{z z} = - α Δ τ_{x z} \end{matrix}]

$[\begin{array}{l} Δ τ_{x z} = α Δ T σ_{z z} + G α Δ T \\ Δ σ_{z z} = - α Δ τ_{x z} \end{array}]$

System Equation 13 leads to:(14)

Δ τ_{x z} / Δ T^{2} = - α^{2} τ_{x z}

$Δ τ_{x z} / Δ T^{2} = - α^{2} τ_{x z}$

So, shear stress is sinusoidal and is not strictly increasing.

So, it is recommended to use co-rotational formulation, especially for visco-elastic materials such as foams, even if this formulation is more time consuming than the global one.

Co-rotational Formulation and Orthotropic Material

When orthotropic material and global formulation are used, the fiber is attached to the first direction of the isoparametric frame and the fiber rotates a different way depending on the element numbering.

On the other hand, when the co-rotational formulation is used, the orthotropic frame keeps the same orientation with respect to the local (co-rotating) frame, and is therefore also co-rotating.

Wilkins M., “Calculation of elastic plastic flow” LLNL, University of California UCRL-7322, 1981.