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2695x

000483

Dipl.-Ing. (FH) Robert Vogl

2023-04-27

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Results by Solid

You can display the results for solids graphically via the navigator category Solids. The numerical solid results can be found in the table category Results by Solid.

Results by Solid in Table

Info

The table and graphic show the results available on the boundary surfaces of the solid. To check the results inside the solid, activate the On FE mesh nodes option under the lower Values on Surfaces category. You can then read the values in the solid via a clipping plane (see the chapter Clipping Planes).

Deformations

The image Results by solid in table shows the table with the deformations of the boundary surfaces. The displacements and rotations are displayed at the surface grid points (see the chapter Surfaces ).

Tip

For small surfaces, the default grid size of 0.5 m can result in only a few grid points. In this case, adjust the number or the distance of the grid points to the surface size.

The deformations mean:

\|u\|	Absolute value of total displacement
u_X	Displacement in direction of global X-axis
u_Y	Displacement in direction of global Y-axis
u_Z	Displacement in direction of global Z-axis
φ_X	Rotation about the global X-axis
φ_Y	Rotation about the global Y-axis
φ_Z	Rotation about the global Z-axis

Stresses

Selecting Solid Stresses in Navigator

Stresses of Solids in Table

Specify in the navigator which stresses you want to display on the boundary surfaces of the solids. The table lists the stresses of these surfaces according to the specifications set in the Results Table Manager .

The solid stresses are divided into the following categories:

Basic stresses
Principal stresses
Equivalent stresses
Stress invariants

Basic Stresses

Solid stresses cannot be described with simple equations like surface stresses. The basic stresses σ_x, σ_y, and σ_z including the shear stresses τ_yz, τ_xz, and τ_xy are determined directly by the calculation kernel.

If a cube with the lengths of edges d_x, d_y, and d_z is cut out of a body with multi-axial loading, the stresses in each cube surface can be decomposed into normal and shear stresses. If we ignore the space force and also the stress differences on parallel surfaces, the stress condition in the local coordinate system of the cube can be described by nine stress components.

Solid Element with Stress Components

The matrix of the stress tensor is:

S = [\begin{matrix} σ_{x} & τ_{xy} & τ_{xz} \\ τ_{yx} & σ_{y} & τ_{yz} \\ τ_{zx} & τ_{zy} & σ_{z} \end{matrix}]

Matrix of Stress Tensor

Principal Stresses

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

\det (S - σ E) = 0

E	3x3 Einheitsmatrix

Principal Stresses

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

τ_{\max} = \frac{1}{2} (σ_{1} - σ_{3})

Maximum Shear Stress

Tip

With the navigator entry σ₁₂₃, you can display the trajectories of the principal stresses graphically.

Equivalent Stresses

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

σ_{v, Mises} = \sqrt{\frac{1}{2} [(σ_{1} - σ_{2})^{2} + (σ_{1} - σ_{3})^{2} + (σ_{2} - σ_{3})^{2}]}

Equivalent Stress σ_v,Mises from Principal Stresses

σ_{v, Mises} = \sqrt{σ_{x}^{2} + σ_{y}^{2} + σ_{z}^{2} - σ_{x} σ_{y} - σ_{x} σ_{z} - σ_{y} σ_{z} + 3 (τ_{xy}^{2} + τ_{xz}^{2} + τ_{yz}^{2})}

Equivalent Stress σ_v,Mises from Basic Stresses

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

σ_{v, Tresca} = max (|σ_{1} - σ_{2}|; |σ_{2} - σ_{3}|; |σ_{3} - σ_{1}|)

Equivalent Stress σ_v,Tresca

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

σ_{v, Rankine} = max (|σ_{1}|; |σ_{2}|; |σ_{3}|)

Equivalent Stress σ_v,Rankine

To determine the equivalent stress σ_v according to Bach , the principal stress differences are examined considering Poisson's ratio ν to determine the maximum value.

σ_{v, Bach} = max [|σ_{1} - ν (σ_{2} + σ_{3})|; |σ_{2} - ν (σ_{3} + σ_{1})|; |σ_{3} - ν (σ_{1} + σ_{2})|]

Equivalent Stress σ_{v, Bach}

Stress Invariants

Stress invariants allow a coordinate-independent and thus objective description of the material's stress condition. As scalar quantities, they remain unchanged under any rotations of the coordinate system and capture the physically relevant properties of these conditions independently of the chosen tensor representation. Their particular importance lies in the fact that many mechanical phenomena – especially plastic yielding, damage, and failure – do not depend on individual stress components, but on invariant measures. Thus, stress invariants form the basis of numerous established yield and failure criteria, such as the von Mises, Tresca, or Drucker-Prager theories.

The mean stress p is linked to the first stress invariant I₁ and describes the hydrostatic stress. It results from the arithmetic mean of the three principal stresses and maps the distance of the stress point from the coordinate origin on the space diagonal.

p = \frac{σ_{1} + σ_{2} + σ_{3}}{3} = \frac{I_{1}}{3}

σ₁ , σ₂ , σ₃	Principal stresses
I₁	First stress invariant

Mean Stress p

It characterizes the mean normal stress condition and is significantly responsible for volume changes. Physically, p corresponds to a uniform compressive or tensile condition, which causes no shape modification, only compression or dilation. In many materials, especially in soil and rock mechanics and in pressure-sensitive materials, p significantly influences the strength and deformation behavior.

The deviatoric stress q is linked to the second invariant of the stress deviator J₂. It is determined as follows:

q = \sqrt{{σ_{1}}^{2} + {σ_{2}}^{2} + {σ_{3}}^{2} - σ_{1} σ_{2} - σ_{2} σ_{3} - σ_{3} σ_{1}}

= \frac{1}{\sqrt{2}} \sqrt{{(σ_{1} - σ_{2})}^{2} + {(σ_{2} - σ_{3})}^{2} + {(σ_{3} - σ_{1})}^{2}} = \sqrt{{I_{1}}^{2} - 3 I_{2}} = \sqrt{3 J_{2}}

I₁	First stress invariant
I₂	Second stress invariant
J₂	Second deviatoric stress invariant

Deviatoric Stress q

It describes the part of the stress condition responsible for shape modifications (shear distortions) without changing the volume. The deviatoric part particularly drives plastic yielding and failure in ductile materials. The von Mises yield criterion is based directly on J₂ or q and clarifies that plastic deformation is primarily controlled by deviatoric stresses.

The Lode angle θ indicates the position of the stress point in the deviator plane. The deviator plane is divided into six sectors, so that −30° ≤ θ ≤ 30° applies. The angle is determined as follows:

θ = \frac{1}{3} \sin^{- 1} (- \frac{3 \sqrt{3}}{2} \frac{J_{3}}{{J_{2}}^{3 / 2}})

J₂	Second deviatoric stress invariant: 1/6 [(σ₁ – σ₂)² + (σ₂ – σ₃)^2 + (σ₃ – σ₂)²]
J₃	Third deviatoric stress invariant: 1/27 (2σ₁ – σ₂ – σ₃) (2σ₂ – σ₃ – σ₁) (2σ₃ – σ₁ – σ₂)

Lode Angle θ

A pure shear stress results for θ = 0, while for θ = 30° the stress condition σ₁ > σ₂ = σ₃ occurs, which corresponds to a triaxial compression test. From θ = −30°, the stress condition of a triaxial tensile test results with σ₁ < σ₂ = σ₃.

Strains

Selecting Solid Strains in Navigator

Strains on Solids in Table

Specify in the navigator which strains you want to display on the boundary surfaces of the solids. The table lists the expansions of these surfaces according to the specifications set in the Results Table Manager .

The solid strains are divided into the following categories:

Basic total strains
Principal total strains
Equivalent total strains
Strain invariants

Basic Total Strains

The basic total strains including the shear distortions are determined directly by the calculation kernel. For the spatial strain condition, the general definition of the tensor is:

ε = [\begin{matrix} ε_{x x} & ε_{x y} & ε_{x z} \\ ε_{y x} & ε_{y y} & ε_{y z} \\ ε_{z x} & ε_{z y} & ε_{z z} \end{matrix}]

Tensor for Strain State

The elements of the tensor are defined as follows:

ε_{i j} = \frac{1}{2} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})

Tensor Elements

Principal Total Strains

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

Tip

With the navigator entry ε₁₂₃, you can display the trajectories of the principal strains graphically.

Equivalent Total Strains

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

ε_{v, Mises} = \frac{1}{1 + ν} \sqrt{{ε_{x}}^{2} + {ε_{y}}^{2} + {ε_{z}}^{2} - ε_{x} ε_{y} - ε_{y} ε_{z} - ε_{z} ε_{x} + \frac{3}{4} ({γ_{x y}}^{2} + {γ_{y z}}^{2} + {γ_{x z}}^{2})}

Comparative Total Strain ε_v,Mises

ε_{v, Tresca} = \max (|R_{1} - R_{2}|; |R_{2} - R_{3}|; |R_{3} - R_{1}|)

R	Matrix (see below)

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

ε_{v, Rankine} = \max (|R_{1}|; |R_{2}|; |R_{3}|)

R	Matrix (siehe unten)

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

ε_{v, Bach} = \max (|R_{1} - ν (R_{2} + R_{3})|; |R_{2} - ν (R_{3} + R_{1})|; |R_{3} - ν (R_{1} + R_{2})|)

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 + ν} [\begin{matrix} \frac{(1 - ν) ε_{x} + ν (ε_{y} + ε_{z})}{1 - 2 ν} & \frac{γ_{x y}}{2} & \frac{γ_{x z}}{2} \\ \frac{γ_{x y}}{2} & \frac{(1 - ν) ε_{y} + ν (ε_{x} + ε_{z})}{1 - 2 ν} & \frac{γ_{y z}}{2} \\ \frac{γ_{x z}}{2} & \frac{γ_{y z}}{2} & \frac{(1 - ν) ε_{z} + ν (ε_{x} + ε_{z})}{1 - 2 ν} \end{matrix}]

Matrix R

Strain Invariants

Strain invariants are characteristic values of the strain tensor that remain independent of the orientation of the coordinate system. They enable a clear separation between volume change and shape modification of a material. The distinction is central for the analysis of material behavior, strength criteria, and plasticity models.

The volumetric strain invariant ε_v corresponds to the isotropic part of the total strains. It is determined from the principal strains:

ε_{v} = ε_{11} + ε_{22} + ε_{33}

Volumetric Strain Invariant, ε_v

The deviatoric strains ε_q or also shear strains γ_s describe the pure shape modification without volume change. They are determined as follows:

ε_{q} = \frac{1}{3} \sqrt{2 [{(ε_{11} - ε_{22})}^{2} + {(ε_{22} - ε_{33})}^{2} + {(ε_{33} - ε_{11})}^{2}] + 3 (2 {ε_{12}}^{2} + 2 {ε_{23}}^{2} + 2 {ε_{31}}^{2})}

Shear Strain ε_q

Parent Chapter

Results