Results by Solid

You can display the results for solids graphically using the Solids navigator category. You will find the numerical results of solids in the Results by Solid table category.

Results by Solid in Table

Deformations

The image Results by Solid in Table shows the table with deformations of the boundary surfaces. Displacements and rotations are displayed in the surface grid points (see Chapter Surfaces ).

The deformations have the following meanings:

Stress

Selecting Solid Stresses in Navigator

Stresses of Solids in Table

In the navigator, define the stresses to be displayed on the boundary surfaces of solids. The table lists the stresses of these surfaces according to the specifications set in the Result Table Manager .

The solid stresses are divided into the following categories:

• Basic stresses
• Principal stresses
• Equivalent stresses

Unlike surface stresses, solid stresses cannot be described by simple equations. The basic stresses σ_x, σ_y, and σ_z as well as the shear stresses τ_yz, τ_xz, and τ_xy are determined directly by the analysis core.

If a cube with the edge lengths d_x, d_y, and d_z is cut from a 3D object with multiaxial loading, the stresses in each cubic surface can be split into normal and shear stresses. If neither the spatial force nor stress differences on parallel surfaces are considered, the stress condition in the cube's local coordinate system can be described by nine stress components.

Solid Element with Stress Components

The matrix of the stress tensor is the following:

The principal stresses σ₁, σ₂, and σ₃ result from the eigenvalues of the tensor as follows:

The maximum shear stress" τ_max is determined according to Mohr's circle:

$${\tau }_{\max }=\frac{1}{2}\left({\sigma }_1-{\sigma }_3\right)$$

Maximum Shear Stress

Tip

The σ₁₂₃ navigator entry allows you to graphically display the trajectories of the principal stresses.

The equivalent stresses σ_v according to von Mises can be determined by two equivalent formulas.

$${\mathrm{\sigma }}_{\mathrm{v},\mathrm{Mises}}=\sqrt{\frac{1}{2}\left[({\mathrm{\sigma }}_1-{\mathrm{\sigma }}_2{)}^2+({\mathrm{\sigma }}_1-{\mathrm{\sigma }}_3{)}^2+({\mathrm{\sigma }}_2-{\mathrm{\sigma }}_3{)}^2\right]}$$

Equivalent Stress σ_v,Mises from Principal Stresses

$${\mathrm{\sigma }}_{\mathrm{v},\mathrm{Mises}}=\sqrt{{\mathrm{\sigma }}_{\mathrm{x}}^2+{\mathrm{\sigma }}_{\mathrm{y}}^2+{\mathrm{\sigma }}_{\mathrm{z}}^2-{\mathrm{\sigma }}_{\mathrm{x}}{\mathrm{\sigma }}_{\mathrm{y}}-{\mathrm{\sigma }}_{\mathrm{x}}{\mathrm{\sigma }}_{\mathrm{z}}-{\mathrm{\sigma }}_{\mathrm{y}}{\mathrm{\sigma }}_{\mathrm{z}}+3({\mathrm{\tau }}_{\mathrm{xy}}^2+{\mathrm{\tau }}_{\mathrm{xz}}^2+{\mathrm{\tau }}_{\mathrm{yz}}^2)}$$

Equivalent Stress σ_v,Mises from Basic Stresses

To determine the equivalent stress σ_v according to Tresca , the differences from the principal stresses are examined in order to determine the maximum value.

$${\mathrm{\sigma }}_{\mathrm{v},\mathrm{Tresca}}=\text{max}(\left|{\mathrm{\sigma }}_1-{\mathrm{\sigma }}_2\right|; \left|{\mathrm{\sigma }}_2-{\mathrm{\sigma }}_3\right|; \left|{\mathrm{\sigma }}_3-{\mathrm{\sigma }}_1\right|)$$

Equivalent Stress σ_v,Tresca

The equivalent stresses σ_v according to Rankine is determined from the greatest absolute values of the principal stresses.

$${\sigma }_{v,\mathrm{Rankine}}=\text{max}(\left|{\sigma }_1\right|; \left|{\sigma }_2\right|; \left|{\sigma }_3\right|)$$

Equivalent Stress σ_v,Rankine

To determine the equivalent stress σ_v according to Bach , the principal stress differences are examined, taking into account Poisson's ratio ν, in order to determine the maximum value.

$${\sigma }_{v,\mathrm{Bach}}=\text{max}[\left|{\sigma }_1-\nu ({\sigma }_2+{\sigma }_3)\right|; \left|{\sigma }_2-\nu ({\sigma }_3+{\sigma }_1)\right|; \left|{\sigma }_3-\nu ({\sigma }_1+{\sigma }_2)\right|]$$

Equivalent Stress σ_{v, Bach}

Strains

Selecting Solid Strains in Navigator

Strains on Solids in Table

In the navigator, define the strains to be displayed on the boundary surfaces of solids. The table lists the strains of these surfaces according to the specifications set in the Result Table Manager .

The solid strains are divided into the following categories:

• Basic total strains
• Principal total strains
• Equivalent total strains

The basic total strains including shear strains are determined directly by the computation kernel. The general definition of the tensor for the spatial strain state is as follows:

$$\epsilon =\left[\begin{array}{ccc}{\epsilon }_{xx}& {\epsilon }_{xy}& {\epsilon }_{xz}\\ {\epsilon }_{yx}& {\epsilon }_{yy}& {\epsilon }_{yz}\\ {\epsilon }_{zx}& {\epsilon }_{zy}& {\epsilon }_{zz}\end{array}\right]$$

Tensor for Strain State

The elements of the tensor are defined as follows:

$${\epsilon }_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})$$

Tensor Elements

The principal total strains ε₁, ε₂, and ε₃ are determined from the basic strains.

Tip

The ε₁₂₃ navigator entry allows you to graphically display the trajectories of the principal total strains.

The equivalent total strains ε_v are determined according to four different stress hypotheses as follows.

$${\epsilon }_{v,\mathrm{Mises}}=\frac{1}{1+\nu }\sqrt{{{\epsilon }_x}^2+{{\epsilon }_y}^2+{{\epsilon }_z}^2-{\epsilon }_x{\epsilon }_y-{\epsilon }_y{\epsilon }_z-{\epsilon }_z{\epsilon }_x+\frac{3}{4}\left({{\gamma }_{xy}}^2+{{\gamma }_{yz}}^2+{{\gamma }_{xz}}^2\right)}$$

Comparative Total Strain ε_v,Mises

$${\epsilon }_{v,\mathrm{Tresca}}=\max \left(\left|R_1-R_2\right|; \left|{\mathrm{R}}_2-{\mathrm{R}}_3\right|; \left|{\mathrm{R}}_3-{\mathrm{R}}_1\right|\right)$$

R	Matrix (see below)

Comparative Total Strain ε_v,Tresca (Maximum of Eigenvalue Differences According to Matrix R)

$${\epsilon }_{v,\mathrm{Rankine}}=\max \left(\left|R_1\right|; \left|{\mathrm{R}}_2\right|; \left|{\mathrm{R}}_3\right|\right)$$

R	Matrix (see below)

Comparative Total Strain ε_v,Rankine (Maximum of Eigenvalues According to Matrix R)

$${\epsilon }_{v,\mathrm{Bach}}=\max \left(\left|R_1-\nu \left(R_2+R_3\right)\right|; \left|{\mathrm{R}}_2-\nu \left(R_3+R_1\right)\right|; \left|{\mathrm{R}}_3-\nu \left(R_1+R_2\right)\right|\right)$$

R	Matrix (see below)

Comparative Total Strain ε_v,Bach (Maximum of Eigenvalue Differences According to Matrix R Taking into Account Poisson's Ratio ν)

$$R=\frac{1}{1+\nu }\left[\begin{array}{ccc}\frac{\left(1-\nu \right){\epsilon }_x+\nu \left({\epsilon }_y+{\epsilon }_z\right)}{1-2\nu }& \frac{{\gamma }_{xy}}{2}& \frac{{\gamma }_{xz}}{2}\\ \frac{{\gamma }_{xy}}{2}& \frac{\left(1-\nu \right){\epsilon }_y+\nu \left({\epsilon }_x+{\epsilon }_z\right)}{1-2\nu }& \frac{{\gamma }_{yz}}{2}\\ \frac{{\gamma }_{xz}}{2}& \frac{{\gamma }_{yz}}{2}& \frac{\left(1-\nu \right){\epsilon }_z+\nu \left({\epsilon }_x+{\epsilon }_z\right)}{1-2\nu }\end{array}\right]$$

Matrix R