Sheared woven fibers

Laminate Composites supports shearing of Simcenter 3D orthotropic materials as well as Laminate Composites woven ply materials. The anisotropic properties are obtained by considering a combination of virtual unidirectional plies aligned with the warp and weft orientations.

When the ply constituent properties are known, the software uses a laminate analogy [5] to calculate the equivalent properties of a woven ply. The software calculates the equivalent properties of the laminate using the properties of each ply and their orientation. For more information, see Woven fibers.

In the absence of the ply’s fiber and matrix materials properties, the software reverse engineers the virtual ply properties (denoted by numerical indices) using the woven fiber nominal properties (denoted by alphabetical indices).

Geometric Properties

The material is assumed to maintain the same volume through the draping process. This results in a thickness increase as the material is being sheared.

Nominal (Unsheared) Material	Sheared Material

t = \frac{t_0}{cos\beta}

where,

• t0 is the nominal material thickness.

• t is the material thickness after shearing.

• β is the shear angle.

The shear angle is the difference between the original yarn angle and the distorted yarn angle. The yarn angle is the angle between the warp and weft fibers that changes after shearing.

When you inflate plies that exhibit shearing, the software extrudes the sheared elements using the material thickness after shearing, t. The software manages the thickness transitions between the elements of adjacent zones.

Elastic properties

The following graphic represents a 2D model of a virtual ply with the warp and weft fibers oriented at 0° and 90°, respectively.

Warp virtual ply	Weft virtual ply

The virtual ply elastic properties are:

[Q_{warp}]=\begin{bmatrix}\frac{{{E}{1}}}{1-{{\nu}{12}},{{\nu}{21}}} & \frac{{{E}{2}},{{\nu}{12}}}{1-{{\nu}{12}},{{\nu}{21}}} & 0\\frac{{{E}{1}},{{\nu}{21}}}{1-{{\nu}{12}},{{\nu}{21}}} & \frac{{{E}{2}}}{1-{{\nu}{12}},{{\nu}{21}}} & 0\0 & 0 & {{G}_{12}}\end{bmatrix}

[Q_{weft}]=\begin{bmatrix}\frac{{{E}{2}}}{1-{{\nu}{12}},{{\nu}{21}}} & \frac{{{E}{1}},{{\nu}{21}}}{1-{{\nu}{12}},{{\nu}{21}}} & 0\\frac{{{E}{2}},{{\nu}{12}}}{1-{{\nu}{12}},{{\nu}{21}}} & \frac{{{E}{1}}}{1-{{\nu}{12}},{{\nu}{21}}} & 0\0 & 0 & {{G}_{12}}\end{bmatrix}

where,

{{\nu}{21}}=\frac{{{E}{2}},{{\nu}{12}}}{{{E}{1}}}

The overall ply elastic property relates to the nominal ply properties as:

[Q]=k,{{[Q_{\mathit{warp}}]}}+\left( 1-k\right) ,{{[Q_{\mathit{weft}}]}}=\begin{bmatrix}\frac{1}{{{E}{x}}} & -\frac{{{\nu}{\mathit{xy}}}}{{{E}{x}}} & 0\-\frac{{{\nu}{\mathit{xy}}}}{{{E}{x}}} & \frac{1}{{{E}{y}}} & 0\0 & 0 & \frac{1}{{{G}_{\mathit{xy}}}}\end{bmatrix}^{-1}

where k is the relative importance of the warp virtual ply to the woven fabric properties.

This leads to the following set of equations:

{{E}{x}}=\frac{{{E}{1}},\left( {{E}{2}^{2}},{{k}^{2}}-2{{E}{1}},{{E}{2}},{{k}^{2}}+{{E}{1}^{2}},{{k}^{2}}-{{E}{2}^{2}}k+2{{E}{1}},{{E}{2}}k-{{E}{1}^{2}}k+{{E}{2}^{2}},{{\nu}{12}^{2}}-{{E}{1}},{{E}{2}}\right) }{\left( {{E}{2}},{{\nu}{12}^{2}}-{{E}{1}}\right) ,\left( {{E}{2}}k-{{E}{1}}k+{{E}{1}}\right) }

{{E}{y}}=-\frac{{{E}{1}},\left( {{E}{2}^{2}},{{k}^{2}}-2{{E}{1}},{{E}{2}},{{k}^{2}}+{{E}{1}^{2}},{{k}^{2}}-{{E}{2}^{2}}k+2{{E}{1}},{{E}{2}}k-{{E}{1}^{2}}k+{{E}{2}^{2}},{{\nu}{12}^{2}}-{{E}{1}},{{E}{2}}\right) }{\left( {{E}{2}},{{\nu}{12}^{2}}-{{E}{1}}\right) ,\left( {{E}{2}}k-{{E}{1}}k-{{E}{2}}\right) }

{{G}{\mathit{xy}}}={{G}{12}}

{{\nu}{\mathit{xy}}}=\frac{{{E}{2}},{{\nu}{12}}}{{{E}{2}}k-{{E}{1}}k+{{E}{1}}}

Besides the trivial G12, there are four unknowns: k, E1, E2, and ν12 with three remaining equations. The solution is not unique, and we must assume one property. There is a theoretical maximum on the value of E1 in relation to the ply properties:

{{E}{1}}<\frac{\left( 2{{E}{x}},{{\nu}{\mathit{xy}}}-{{E}{x}}\right) ,{{E}{y}}-{{E}{x}^{2}}}{{{\nu}{\mathit{xy}}^{2}},{{E}{y}}-{{E}_{x}}}

The ply fibrous influence coefficient, KE, is introduced to express E1 as a fraction of this theoretical maximum.

{{E}{1}}=\frac{{{K}{E}},\left( \left( 2{{E}{x}},{{\nu}{\mathit{xy}}}-{{E}{x}}\right) ,{{E}{y}}-{{E}{x}^{2}}\right) }{{{\nu}{\mathit{xy}}^{2}},{{E}{y}}-{{E}{x}}}

KE should preferably be determined experimentally. However, it is possible to estimate KE using the invariant-based theory developed in [6]. The invariant of the stiffness matrix for the material is:

I=Q_{11}+Q_{22}+2Q_{66}=\frac{\left( 2{{G}{\mathit{xy}}},{{\nu}{\mathit{xy}}^{2}}-{{E}{x}}\right) ,{{E}{y}}-2{{E}{x}},{{G}{\mathit{xy}}}-{{E}{x}^{2}}}{{{\nu}{\mathit{xy}}^{2}},{{E}{y}}-{{E}{x}}}

Rewriting E1 as the product of the invariant and a specific constant leads to:

E_1=K_{Tsai_Melo}\frac{\left( 2{{G}{\mathit{xy}}},{{\nu}{\mathit{xy}}^{2}}-{{E}{x}}\right) ,{{E}{y}}-2{{E}{x}},{{G}{\mathit{xy}}}-{{E}{x}^{2}}}{{{\nu}{\mathit{xy}}^{2}},{{E}{y}}-{{E}{x}}}

It is therefore possible to relate KE to the Tsai-Melo Constant with:

K_E=K_{Tsai_Melo}\frac{\left( 2{{G}{\mathit{xy}}},{{\nu}{\mathit{xy}}^{2}}-{{E}{x}}\right) ,{{E}{y}}-2{{E}{x}},{{G}{\mathit{xy}}}-{{E}{x}^{2}}}{\left( 2{{E}{x}},{{\nu}{\mathit{xy}}}-{{E}{x}}\right) ,{{E}{y}}-{{E}{x}^{2}}}

Tsai and Melo’s experiments show that, in average, carbon/epoxy materials have KTsai_Melo =0.88. This implies that a KE value close to, but less than 1 works well for carbon/epoxy fabrics. The default KE value implemented in the software is 0.988. Ultimately, this value must be determined experimentally.

Extending to 3D, the virtual ply properties are expressed as:

[S_{warp}]=\begin{bmatrix}\frac{1}{{{E}{1}}} & -\frac{{{\nu}{12}}}{{{E}{1}}} & -\frac{{{\nu}{13}}}{{{E}{1}}} & 0 & 0 & 0\-\frac{{{\nu}{12}}}{{{E}{1}}} & \frac{1}{{{E}{2}}} & -\frac{{{\nu}{23}}}{{{E}{2}}} & 0 & 0 & 0\-\frac{{{\nu}{13}}}{{{E}{1}}} & -\frac{{{\nu}{23}}}{{{E}{2}}} & \frac{1}{{{E}{3}}} & 0 & 0 & 0\0 & 0 & 0 & \frac{1}{{{G}{12}}} & 0 & 0\0 & 0 & 0 & 0 & \frac{1}{{{G}{23}}} & 0\0 & 0 & 0 & 0 & 0 & \frac{1}{{{G}{13}}}\end{bmatrix}

[S_{weft}]=\begin{bmatrix}\frac{1}{{{E}{2}}} & -\frac{{{\nu}{12}}}{{{E}{1}}} & -\frac{{{\nu}{23}}}{{{E}{2}}} & 0 & 0 & 0\-\frac{{{\nu}{12}}}{{{E}{1}}} & \frac{1}{{{E}{1}}} & -\frac{{{\nu}{13}}}{{{E}{1}}} & 0 & 0 & 0\-\frac{{{\nu}{23}}}{{{E}{2}}} & -\frac{{{\nu}{13}}}{{{E}{1}}} & \frac{1}{{{E}{3}}} & 0 & 0 & 0\0 & 0 & 0 & \frac{1}{{{G}{12}}} & 0 & 0\0 & 0 & 0 & 0 & \frac{1}{{{G}{13}}} & 0\0 & 0 & 0 & 0 & 0 & \frac{1}{{{G}{23}}}\end{bmatrix}

[C_{warp}]=[S_{warp}]^{-1}

[C_{weft}]=[S_{weft}]^{-1}

Similarly, the nominal ply compliance and stiffness matrices are obtained with:

[S]=\begin{bmatrix}\frac{1}{{{E}{x}}} & -\frac{{{\nu}{\mathit{xy}}}}{{{E}{x}}} & -\frac{{{\nu}{\mathit{xz}}}}{{{E}{x}}} & 0 & 0 & 0\-\frac{{{\nu}{\mathit{xy}}}}{{{E}{x}}} & \frac{1}{{{E}{y}}} & -\frac{{{\nu}{\mathit{yz}}}}{{{E}{y}}} & 0 & 0 & 0\-\frac{{{\nu}{\mathit{xz}}}}{{{E}{x}}} & -\frac{{{\nu}{\mathit{yz}}}}{{{E}{y}}} & \frac{1}{{{E}{z}}} & 0 & 0 & 0\0 & 0 & 0 & \frac{1}{{{G}{\mathit{xy}}}} & 0 & 0\0 & 0 & 0 & 0 & \frac{1}{{{G}{\mathit{yz}}}} & 0\0 & 0 & 0 & 0 & 0 & \frac{1}{{{G}{\mathit{xz}}}}\end{bmatrix}

[C]=[S]^{-1}

From [7], we can relate the nominal properties of the ply to the properties of the virtual warp and weft:

\begin{aligned}&{{C}{\mathit{ij}}}=k,\left( {{C}{\mathit{warp_ {ij}}}}-\frac{{{C}{\mathit{warp{ i3}}}}{{C}{\mathit{warp {3j}}}}}{{{C}{{{\mathit{warp}}{33}}}}}+\frac{{{C}{\mathit{warp{ i3}}}}\left( \frac{k,{{C}{\mathit{warp {3j}}}}}{{{C}{{{\mathit{warp}}{33}}}}}+\frac{\left( 1-k\right) ,{{C}{\mathit{weft {3j}}}}}{{{C}{{{\mathit{weft}}{33}}}}}\right) }{{{C}{{{\mathit{warp}}{33}}}}\left( \frac{k}{{{C}{{{\mathit{warp}}{33}}}}}+\left( \frac{1-k}{{{C}{{{\mathit{weft}}{33}}}}}\right) \right) }\right) \&+k,\left( {{C}{\mathit{weft {ij}}}}-\frac{{{C}{\mathit{weft{ i3}}}}{{C}{\mathit{weft {3j}}}}}{{{C}{{{\mathit{weft}}{33}}}}}+\frac{{{C}{\mathit{weft{ i3}}}}\left( \frac{k,{{C}{\mathit{warp {3j}}}}}{{{C}{{{\mathit{warp}}{33}}}}}+\frac{\left( 1-k\right) ,{{C}{\mathit{weft {3j}}}}}{{{C}{{{\mathit{weft}}{33}}}}}\right) }{{{C}{{{\mathit{weft}}{33}}}}\left( \frac{k}{{{C}{{{\mathit{warp}}{33}}}}}+\left( \frac{1-k}{{{C}{{{\mathit{weft}}{33}}}}}\right) \right) }\right);(i,j=1,2,3,4) \end{aligned}

{{C}{\mathit{ij}}}={{C}{\mathit{ji}}}=0;(i=1,2,3,4;j=5,6)

{{C}{\mathit{ij}}}=\frac{\frac{k,{{C}{\mathit{warp_{ ij}}}}}{{{C}{{{\mathit{warp}}{55}}}}{{C}{{{\mathit{warp}}{66}}}}}+\frac{\left( 1-k\right) ,{{C}{\mathit{weft {ij}}}}}{{{C}{{{\mathit{weft}}{55}}}}{{C}{{{\mathit{weft}}{66}}}}}}{\frac{{{k}^{2}}}{{{C}{{{\mathit{warp}}{55}}}}{{C}{{{\mathit{warp}}{66}}}}}+\frac{k,\left( 1-k\right) }{{{C}{{{\mathit{warp}}{66}}}}{{C}{{{\mathit{weft}}{55}}}}}+\frac{\operatorname{k}\left( 1-k\right) }{{{C}{{{\mathit{warp}}{55}}}}{{C}{{{\mathit{weft}}{66}}}}}+\frac{{{\left( 1-k\right) }^{2}}}{{{C}{{{\mathit{weft}}{55}}}}{{C}{{{\mathit{weft}}{66}}}}}};(i,j=5,6)

Using the in-plane properties determined from the 2D approach, the remaining 3D virtual material properties can be obtained.

Applying a rotation to the weft virtual ply:

[C_{weft_sheared}]=[R_{\sigma}]^{-1}[C_{weft}][R_{\epsilon}]

where,

[R_{\sigma}]=\begin{bmatrix}{{\cos^{2}{\left( -\beta\right) }}} & {{\sin^{2}{\left( -\beta\right) }}} & 0 & 2\cos{\left( -\beta\right) },\sin{\left( -\beta\right) } & 0 & 0\{{\sin^{2}{\left( -\beta\right) }}} & {{\cos^{2}{\left( -\beta\right) }}} & 0 & -2 ,\cos{\left( -\beta\right) },\sin{\left( -\beta\right) } & 0 & 0\0 & 0 & 1 & 0 & 0 & 0\ -\cos{\left( -\beta\right) } ,\sin{\left( -\beta\right) } & \cos{\left( -\beta\right) },\sin{\left( -\beta\right) } & 0 & {{\cos^{2}{\left( -\beta\right) }}}-{{\sin^{2}{\left( -\beta\right) }}} & 0 & 0\0 & 0 & 0 & 0 & \cos{\left( -\beta\right) } & -\sin{\left( -\beta\right) }\0 & 0 & 0 & 0 & \sin{\left( -\beta\right) } & \cos{\left( -\beta\right) }\end{bmatrix}

[R_{\epsilon}]=\begin{bmatrix}{{\cos^{2}{\left( -\beta\right) }}} & {{\sin^{2}{\left( -\beta\right) }}} & 0 & \cos{\left( -\beta\right) },\sin{\left( -\beta\right) } & 0 & 0\{{\sin^{2}{\left( -\beta\right) }}} & {{\cos^{2}{\left( -\beta\right) }}} & 0 & - ,\cos{\left( -\beta\right) },\sin{\left( -\beta\right) } & 0 & 0\0 & 0 & 1 & 0 & 0 & 0\ -2\cos{\left( -\beta\right) } ,\sin{\left( -\beta\right) } & 2\cos{\left( -\beta\right) },\sin{\left( -\beta\right) } & 0 & {{\cos^{2}{\left( -\beta\right) }}}-{{\sin^{2}{\left( -\beta\right) }}} & 0 & 0\0 & 0 & 0 & 0 & \cos{\left( -\beta\right) } & -\sin{\left( -\beta\right) }\0 & 0 & 0 & 0 & \sin{\left( -\beta\right) } & \cos{\left( -\beta\right) }\end{bmatrix}

The 3D sheared woven elastic properties are finally calculated as:

\begin{aligned}&{{C}{\mathit{ply_ sheared{ ij}}}}=k,\left( {{C}{\mathit{warp{ij}}}}-\frac{{{C}{\mathit{warp{ i3}}}}{{C}{\mathit{warp{3j}}}}}{{{C}{{{\mathit{warp}}{33}}}}}+\frac{{{C}{\mathit{warp{ i3}}}}\left( \frac{k,{{C}{\mathit{warp{ 3j}}}}}{{{C}{{{\mathit{warp}}{33}}}}}+\frac{\left({1-k}\right) ,{{C}{\mathit{weft_ sheared{ 3j}}}}}{{{C}{{{\mathit{weft_ sheared}}{33}}}}}\right) }{{{C}{{{\mathit{warp}}{33}}}}\left( \frac{k}{{{C}{{{\mathit{warp}}{33}}}}}+\left( \frac{1-k}{{{C}{{{\mathit{weft_ sheared}}{33}}}}}\right) \right) }\right) \&+k,\left( {{C}{\mathit{weft_ sheared{ ij}}}}-\frac{{{C}{\mathit{weft_ sheared{ i3}}}}{{C}{\mathit{weft_ sheared {3j}}}}}{{{C}{{{\mathit{weft_ sheared}}{33}}}}}+\frac{{{C}{\mathit{weft_ sheared{ i3}}}}\left( \frac{k,{{C}{\mathit{warp{ 3j}}}}}{{{C}{{{\mathit{warp}}{33}}}}}+\frac{\left( 1-k\right) ,{{C}{\mathit{weft_ sheared{ 3j}}}}}{{{C}{{{\mathit{weft_ sheared}}{33}}}}}\right) }{{{C}{{{\mathit{weft}}{33}}}}\left( \frac{k}{{{C}{{{\mathit{warp}}{33}}}}}+\left(\frac{1-k}{{{C}{{{\mathit{weft_ sheared}}{33}}}}}\right) \right) }\right) \ &;(i,j=1,2,3,4) \end{aligned}

{{C}{ply_sheared{\mathit{ij}}}}={{C}{ply_sheared{\mathit{ji}}}}=0;(i=1,2,3,4;j=5,6)

\begin{aligned}{C_{ply_sheared_{\mathit{ij}}}}&=\frac{\frac{k,{{C}{\mathit{warp{ ij}}}}}{{{C}{{{\mathit{warp}}{55}}}}{{C}{{{\mathit{warp}}{66}}}}}+\frac{\left( 1-k\right) ,{{C}{\mathit{weft_sheared {ij}}}}}{{{C}{{{\mathit{weft_sheared}}{55}}}}{{C}{{{\mathit{weft_sheared}}{66}}}}}}{\frac{{{k}^{2}}}{{{C}{{{\mathit{warp}}{55}}}}{{C}{{{\mathit{warp}}{66}}}}}+\frac{k,\left( 1-k\right) }{{{C}{{{\mathit{warp}}{66}}}}{{C}{{{\mathit{weft_sheared}}{55}}}}}+\frac{\operatorname{k}\left( 1-k\right) }{{{C}{{{\mathit{warp}}{55}}}}{{C}{{{\mathit{weft_sheared}}{66}}}}}+\frac{{{\left( 1-k\right) }^{2}}}{{{C}{{{\mathit{weft_sheared}}{55}}}}{{C}{{{\mathit{weft_sheared}}{66}}}}}}\&;(i,j=5,6)\end{aligned}

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