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2816x

000483

Dipl.-Ing. (FH) Robert Vogl

2023-04-27

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Results by Solid

You can graphically display the results for volumes via the Solids navigator category. You can find the numerical solid results in the Results by Solid table category.

Results by Solid in Table

Info

The table and the graphic show the results present on the boundary surfaces of the solid. To check the results inside the solid, activate the On FE mesh nodes option in the Values on Surfaces category below. You can then read the values in the solid via a clipping plane (see 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 in the surface grid points (see Chapter Surfaces ).

Tip

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

The deformations mean:

\|u\|	Absolute value of the total displacement
u_X	Displacement in the direction of the global X-axis
u_Y	Displacement in the direction of the global Y-axis
u_Z	Displacement in the direction of the 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

In the navigator, specify which stresses are to be displayed on the boundary surfaces of the solids. The table lists the stresses of these surfaces according to the specifications defined in the Results Table Manager .

The solid stresses are divided into the following categories:

Base stresses
Principal stresses
Equivalent stresses
Stress invariants

Base Stresses

Solid stresses cannot be described with simple equations like surface stresses. The base 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 core.

If a cube with edge lengths d_x, d_y, and d_z is cut out of a body subjected to multiaxial stress, the stresses in each cube face can be decomposed into normal and shear stresses. Neglecting the body force and also the stress differences on parallel faces, the stress state can be described in the local coordinate system of the cube by nine stress components.

Solid Element with Stress Components

The matrix of the stress tensor is:

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

Stress Tensor Matrix

Principal Stresses

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

\det (σ - σ E) = 0

E	3x3 Einheitsmatrix

Principal stresses

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

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

Maximum Shear Stress

Tip

You can graphically display the trajectories of the principal stresses with the σ₁₂₃ navigator entry.

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 base 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.

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 in order 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 enable a coordinate-independent and therefore objective description of the stress state of a material. As scalar quantities, they remain unchanged under arbitrary rotations of the coordinate system and capture the physically relevant properties of these states independent of the chosen tensor representation. Their particular significance lies in the fact that many mechanical phenomena – especially plastic flow, failure, and fracture – 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 is obtained from the arithmetic mean of the three principal stresses and represents 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 state and is primarily responsible for volume changes. Physically, p corresponds to a uniform compressive or tensile state that does not cause any change in shape, but solely compression or dilation. In many materials, especially in soil and rock mechanics as well as 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 portion of the stress state that is responsible for changes in shape (shear strains) without changing the volume. The deviatoric portion particularly drives plastic flow and failure in ductile materials. The von Mises yield criterion is based directly on J₂ or q and illustrates that plastic deformation is primarily controlled by deviatoric stresses.

The Lode angle θ indicates the position of the stress point in the deviatoric plane. The deviatoric 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 loading results for θ = 0, while for θ = 30° the stress state σ₁ > σ₂ = σ₃ occurs, which corresponds to a triaxial compression test. From θ = −30° results the stress state of a triaxial tension test with σ₁ < σ₂ = σ₃.

Strains

Selecting Solid Strains in Navigator

Strains on Solids in Table

In the navigator, specify which strains are to be displayed on the boundary surfaces of the solids. The table lists the strains of these surfaces according to the specifications defined in the Results Table Manager .

The solid strains are divided into the following categories:

Base total strains
Principal total strains
Equivalent total strains
Strain invariants

Base Total Strains

The base total strains, including the shear strains, are determined directly by the calculation core. For the spatial strain state, the general definition of the tensor is:

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

Strain State Tensor

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 base strains.

Tip

You can graphically display the trajectories of the principal strains with the ε₁₂₃ navigator entry.

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})}

γ_xy	2ϵ_xy
γ_yz	2ϵ_yz
γ_xz	2ϵ_xz

Equivalent 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 the eigenvalue differences according to Matrix R)

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

R	Matrix (siehe unten)

Comparison total strain ε_v,Rankine (maximum of the 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 the eigenvalue differences according to Matrix R considering 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_{y})}{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 change 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 portion of the total strains.

ε_{v} = ε_{x} + ε_{y} + ε_{z}

Volumetric Strain Invariant, ε_v

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

ε_{q} = \frac{1}{3} \sqrt{2 [{(ε_{y} - ε_{z})}^{2} + {(ε_{x} - ε_{z})}^{2} + {(ε_{x} - ε_{y})}^{2}] + 3 (γ_{y z}^{2} + γ_{x z}^{2} + γ_{x y}^{2})}

Shear Strain ε_q

Parent Chapter

Results