4. Calculation of the maximum shear plane#
Here we use the definition of the shear plane introduced in paragraph [§2.4]. Practically, for us point \(M\) of the continuous medium will be a Gauss point.
4.1. Expression of shear stresses in plane \(\Delta\)#
For reasons of symmetry we vary the unit normal \(n\) along a half-sphere using the angles \(\gamma\) and \(\varphi\), cf. [Figure 4.1-a].
In coordinate system \((O,x,y,z)\), the unit normal vector \(n\) is defined by:
\({n}_{x}\mathrm{=}\text{sin}\gamma \text{cos}\varphi {n}_{y}\mathrm{=}\text{sin}\gamma \text{sin}\varphi {n}_{z}\mathrm{=}\text{cos}\gamma\). eq 4.1-1
We are introducing a new coordinate system \((O,u,v,n)\) where \(n\) is perpendicular to the shear plane \(\Delta\) and where \(u\) and \(v\) are in this plane, cf. [Figure 4.1-a]. In coordinate system \((O,x,y,z)\) the unit vectors \(u\) and \(v\) are respectively defined by:
\({u}_{x}=-\text{sin}\varphi {u}_{y}=\text{cos}\varphi {u}_{z}=0\), eq 4.1-2
\({v}_{x}\mathrm{=}\mathrm{-}\text{cos}\gamma \text{cos}\varphi {v}_{y}\mathrm{=}\mathrm{-}\text{cos}\gamma \text{sin}\varphi {v}_{z}\mathrm{=}\text{sin}\gamma\). Eq 4.1-3
Figure 4.1-a: Finding the normal \(n\) to a plane by the angles \(\gamma\) and \(\varphi\)
In plane \(\Delta\), the components \({\tau }_{u}\) and \({\tau }_{v}\) of the vector \(\tau\) representing the shear stress are obtained by the relationships:
\({\tau }_{u}=u\cdot \tau ={u}_{x}{\tau }_{x}+{u}_{y}{\tau }_{y}+{u}_{z}{\tau }_{z}\), eq 4.1-4
\({\tau }_{v}=v\cdot \tau ={v}_{x}{\tau }_{x}+{v}_{y}{\tau }_{y}+{v}_{z}{\tau }_{z}\). eq 4.1-5
On the [Figure 4.1-b], we represented the shear stresses in the \(\Delta\) plane as well as the \((O,u,v,n)\) coordinate system.
Figure 4.1-b: Representation of the shear stress vector \(\tau\) in the plane \(\Delta\)
Now our problem is to determine for each Gauss point or each node of a mesh the plane of normal \(n\) such that \(\mid \tau \mid\) is maximum. To do this we vary the unit normal \(n\).
4.2. Exploring shear planes#
The method we present here comes from the reference [bib4]. Its principle is as follows. As indicated in paragraph [§4.1], for reasons of symmetry we vary the unit normal \(n\) according to a hemisphere using the angles \(\gamma\) and \(\varphi\), cf. [Figure 4.1-a]. The immediate question is what should be the step of variation between angles \(\gamma\) and \(\varphi\). In fact, it is necessary to find an optimum between the fineness of exploration and a reasonable calculation time insofar as it is necessary to perform this operation at each Gauss point of the mesh. The author of the reference [bib4] proposes to divide the surface of the hemisphere into facets of equal surfaces at the center of which the unit normal \(n\) is positioned, cf. [Figure 4.2-a]. In practice, the surfaces are not strictly equal but of the same order of magnitude.
The change step value of \(\gamma\), \(\Delta \gamma\) is equal to 10 degrees. The angle \(\varphi\) varies according to a step \(\Delta \varphi\) which is a function of the angle \(\gamma\). Plus \(\gamma\) is faint or close to 180 degrees and more \(\Delta \varphi\) must be taller to maintain a roughly constant facet area. It is in the vicinity of \(\gamma =\text{90}°\) that \(\Delta \varphi\) is the smallest. The [Tableau 4.2-1] summarizes the division that was selected.
With this method, the number of facets, and therefore the number of normal vectors to be explored, is equal to 209 for a half-sphere.
Figure 4.2-a: Dividing the surface of the hemisphere into facets
\(\gamma °\) |
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\(\Delta \Phi °\) |
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Number of facets |
1 |
3 |
3 |
3 |
6 |
6 |
9 |
12 |
14 |
16 |
\(\gamma °\) |
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\(\Delta \Phi °\) |
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Number of facets |
17 |
18 |
18 |
18 |
18 |
17 |
16 |
14 |
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\(\gamma °\) |
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\(\Delta \Phi °\) |
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Number of facets |
12 |
9 |
9 |
6 |
6 |
3 |
1 |
Table 4.2-1: Number of facets as a function of \(\gamma\) and \(\Delta \Phi\)
In order to determine the normal vector \(n\) which will give the maximum shear plane with good precision, the author recommends using four additional successive refinements. The first involves exploring eight new facets around the initial normal vector, as illustrated in [Figure 4.2-b].
Figure 4.2-b: Representation of the eight additional facets around \(n\)
In this case \(\Delta \gamma\) is equal to two degrees and for \(\gamma \mathrm{\in }\mathrm{]}0°,\text{180}°\mathrm{[}\), \(\mathrm{\Delta \varphi }=\mathrm{\Delta \gamma }/\text{sin}\gamma\). For the last three refinements, \(\Delta \gamma\) equals 1, 0.5, and 0.25 degrees, respectively.
Special case. In the case where facet \({F}_{m}\) is perpendicular to \(y\), we consider the six facets all around it located at \(\gamma \mathrm{=}5°\) and defined respectively by \(\varphi =0°\), \(\Phi \mathrm{=}\text{60}°\), \(\Phi \mathrm{=}\text{120}°\), \(\Phi \mathrm{=}\text{180}°\), \(\Phi \mathrm{=}\text{240}°\) and \(\Phi \mathrm{=}\text{300}°\), cf. [Figure 4.2-c].
Figure 4.2-c: Location of explored facets when \({F}_{m}\) is normal to \(y\)
For each Gauss point or each node we explore the 209 normal vectors \(n\). Each normal vector is associated with a shear history expressed by a certain number of points located in the shear plane \(\Delta\) with axes \(u\) and \(v\). Now it is a question of finding the circle circumscribed to the points belonging to the shear plane in order to derive/to establish the half-shear amplitude.