Simulating squeeze flows in multiaxial laminates: towards fully 3D mixed formulations

Abstract

Thermoplastic composites are widely considered in structural parts. In this paper attention is paid to squeeze flow of continuous fiber laminates. In the case of unidirectional prepregs, the ply constitutive equation is modeled as a transversally isotropic fluid, that must satisfy both the fiber inextensibility as well as the fluid incompressibility. When laminate is squeezed the flow kinematics exhibits a complex dependency along the laminate thickness requiring a detailed velocity description through the thickness. In a former work the solution making use of an in-plane-out-of-plane separated representation within the PGD – Poper Generalized Decomposition – framework was successfully accomplished when both kinematic constraints (inextensibility and incompressibility) were introduced using a penalty formulation for circumventing the LBB constraints. However, such a formulation makes difficult the calculation on fiber tractions and compression forces, the last required in rheological characterizations. In this paper the former penalty formulation is substituted by a mixed formulation that makes use of two Lagrange multipliers, while addressing the LBB stability conditions within the separated representation framework, questions never until now addressed.

Appendix: A: Separated representation of the Stokes weak form

The efficient computer implementation of the separated representation constructor discussed in “Separated representation constructor” section needs a separated form of the flow problem expressed in a weak form (7). For that purpose we first consider the second term D ^∗:D, that takes into account expression (5) as follows

$$\begin{array}{@{}rcl@{}} 4 \mathbf{D}^{\ast} : \mathbf{D} &=& \nabla \mathbf{v}^{\ast} : \nabla \mathbf{v} + \nabla \mathbf{v}^{\ast} : (\nabla \mathbf{v})^{T} \\&&+ (\nabla \mathbf{v}^{\ast})^{T} : \nabla \mathbf{v} + (\nabla \mathbf{v}^{\ast})^{T} : (\nabla \mathbf{v})^{T}. \end{array} $$

(54)

The simplest choice of the test function v ^∗(x,z) is

$$ \mathbf{v}^{\ast}(\mathbf{x}, z) = \mathbf{P}^{\ast} (\mathbf{x}) \circ \mathbf{T}(z) + \mathbf{P}(\mathbf{x}) \circ \mathbf T^{\ast} (z), $$

(55)

from which the velocity gradient is:

$$ \nabla \mathbf{v}^{\ast} = \mathbb{P}^{\ast} \circ \mathbb{T} + \mathbb{P} \circ \mathbb{T}^{\ast}. $$

(56)

The choice of $\mathbb {P}$ and $\mathbb {T}$ in Eq. 56 is discussed in “Separated representation constructor” section.

Developing the first term in Eq. 54 (the other terms follow the same rationale) taking into account Eq. 3 results

$$ \nabla \mathbf{v}^{\ast} : \nabla \mathbf{v} \approx \left(\mathbb{P}^{\ast} \circ \mathbb{T} + \mathbb{P} \circ \mathbb{T}^{\ast} \right) : \left(\sum \limits_{j=1}^{N} \mathbb{P}_{j} \circ \mathbb{T}_{j} \right). $$

(57)

It is easy to note that if matrices $\mathbb {M}(\mathbf {x})$ and $\mathbb {N}(\mathbf {x})$ depend on the in-plane coordinates x, and matrices $\mathbb {U}(z)$ and $\mathbb {V}(z)$ depend on the out-of-plane coordinate z, we have

$$ \left(\mathbb{M} \circ \mathbb{U} \right) : \left(\mathbb{N} \circ \mathbb{V} \right) = \left(\mathbb{M} \circ \mathbb{N} \right) : \left(\mathbb{U} \circ \mathbb{V} \right). $$

(58)

Using this equality, Eq. 57 can be written as

$$\begin{array}{@{}rcl@{}} \nabla \mathbf{v}^{\ast} : \nabla \mathbf{v} &\approx& \sum \limits_{j=1}^{N} \left\{ \left(\mathbb{P}^{\ast} \circ \mathbb{P}_{j} \right) : \left(\mathbb{T} \circ \mathbb{T}_{j} \right)\right.\\ &&\left.+ \left(\mathbb{P} \circ \mathbb{P}_{j} \right) : \left(\mathbb{T}^{\ast} \circ \mathbb{T}_{j} \right) \right\}, \end{array} $$

(59)

and the other terms involved in Eq. 54 as

$$\begin{array}{@{}rcl@{}} \nabla \mathbf{v}^{\ast} : \left(\nabla \mathbf{v} \right)^{T} &\approx& \sum \limits_{j=1}^{N} \left\{ \left(\mathbb{P}^{\ast} \circ \mathbb{P}_{j}^{T} \right) : \left(\mathbb{T} \circ \mathbb{T}_{j}^{T} \right) \right.\\&&\left.+ \left(\mathbb{P} \circ \mathbb{P}_{j}^{T} \right) : \left(\mathbb{T}^{\ast} \circ \mathbb{T}_{j}^{T} \right) \right\} , \end{array} $$

(60)

$$\begin{array}{@{}rcl@{}} \left(\nabla \mathbf{v} \right)^{\ast T} : \nabla \mathbf{v} &\approx& \sum \limits_{j=1}^{N} \left\{ \left(\mathbb{P}^{\ast T} \circ \mathbb{P}_{j} \right) : \left(\mathbb{T}^{T} \circ \mathbb{T}_{j} \right) \right.\\&&\left.+ \left(\mathbb{P}^{T} \circ \mathbb{P}_{j} \right) : \left(\mathbb{T}^{\ast T} \circ \mathbb{T}_{j} \right) \right\}, \end{array} $$

(61)

and

$$\begin{array}{@{}rcl@{}} \left(\nabla \mathbf{v} \right)^{\ast T} : \left(\nabla \mathbf{v} \right)^{T} &\approx& \sum \limits_{j=1}^{N} \left\{ \left(\mathbb{P}^{\ast T} \circ \mathbb{P}_{j}^{T} \right) : \left(\mathbb{T}^{T} \circ \mathbb{T}_{j}^{T} \right) \right.\\&&\left.+ \left(\mathbb{P}^{T} \circ \mathbb{P}_{j}^{T} \right) : \left(\mathbb{T}^{\ast T} \circ \mathbb{T}_{j}^{T} \right) \right\}. \end{array} $$

(62)

Thus, we finally obtain

$$\begin{array}{@{}rcl@{}} 4\mathbf{D}^{\ast} : \mathbf{D} \!&\approx&\! \sum \limits_{j=1}^{N} \left\{ \left(\mathbb{P}^{\ast} \circ \mathbb{P}_{j} \right) \!:\! \left(\mathbb{T} \circ \mathbb{T}_{j} \right) \,+\, \left(\mathbb{P} \circ \mathbb{P}_{j} \right) \!:\! \left(\mathbb{T}^{\ast} \circ \mathbb{T}_{j} \right) \right\} \\&&+ \sum \limits_{j=1}^{N} \left\{ \left(\mathbb{P}^{\ast} \circ \mathbb{P}_{j}^{T} \right) : \left(\mathbb{T} \circ \mathbb{T}_{j}^{T} \right)\right.\\ &&\hspace*{2.5pc}\left.+ \left(\mathbb{P} \circ \mathbb{P}_{j}^{T} \right) : \left(\mathbb{T}^{\ast} \circ \mathbb{T}_{j}^{T} \right) \right\} \\&&+ \sum \limits_{j=1}^{N} \left\{ \left(\mathbb{P}^{\ast T} \circ \mathbb{P}_{j} \right) : \left(\mathbb{T}^{T} \circ \mathbb{T}_{j} \right)\right.\\ &&\hspace*{2.5pc}\left.+ \left(\mathbb{P}^{T} \circ \mathbb{P}_{j} \right) : \left(\mathbb{T}^{\ast T} \circ \mathbb{T}_{j} \right) \right\} \\&&+ \sum \limits_{j=1}^{N} \left\{ \left(\mathbb{P}^{\ast T} \circ \mathbb{P}_{j}^{T} \right) : \left(\mathbb{T}^{T} \circ \mathbb{T}_{j}^{T} \right)\right.\\ &&\hspace*{2.5pc}\left.+ \left(\mathbb{P}^{T} \circ \mathbb{P}_{j}^{T} \right) : \left(\mathbb{T}^{\ast T} \circ \mathbb{T}_{j}^{T} \right) \right\} \\&=& \sum \limits_{j=1}^{N} \sum \limits_{k=1}^{4} \left\{ \mathbb{A}_{jk}^{\ast}(\mathbf{x}) : \mathbb{B}_{jk}(z) + \mathbb{A}_{jk}(\mathbf{x}) : \mathbb{B}_{jk}^{\ast}(z) \right\},\\ \end{array} $$

(63)

where ∀j, j=1,⋯,N

$$ \mathbb{A}_{jk}^{\ast} = \left\{ \begin{array}{ll} \mathbb{P}^{\ast} \circ \mathbb{P}_{j} ,& \mathrm{if } \ k=1\\ \mathbb{P}^{\ast} \circ \mathbb{P}_{j}^{T}, & \mathrm{if } \ k=2 \\ \mathbb{P}^{\ast T} \circ \mathbb{P}_{j}, & \mathrm{if }\ k=3\\ \mathbb{P}^{\ast T} \circ \mathbb{P}_{j}^{T}, & \mathrm{if }\ k=4 \end{array} \right. , $$

(64)

$$ \mathbb{B}_{jk} = \left\{ \begin{array}{ll} \mathbb{T} \circ \mathbb{T}_{j} ,& \mathrm{if } \ k=1 \\ \mathbb{T} \circ \mathbb{T}_{j}^{T} , & \mathrm{if } \ k=2 \\ \mathbb{T}^{T} \circ \mathbb{T}_{j} , & \mathrm{if } \ k=3 \\ \mathbb{T}^{T} \circ \mathbb{T}_{j}^{T}, & \mathrm{if } \ k=4 \end{array} \right. , $$

(65)

$$ \mathbb{A}_{jk} = \left\{ \begin{array}{ll} \mathbb{P} \circ \mathbb{P}_{j} ,& \mathrm{if } \ k=1 \\ \mathbb{P} \circ \mathbb{P}_{j}^{T}, & \mathrm{if } \ k=2 \\ \mathbb{P}^{T} \circ \mathbb{P}_{j}, & \mathrm{if } \ k=3 \\ \mathbb{P}^{T} \circ \mathbb{P}_{j}^{T}, & \mathrm{if } \ k=4 \end{array} \right. , $$

(66)

and

$$ \mathbb{B}_{jk}^{\ast} = \left\{ \begin{array}{ll} \mathbb{T}^{\ast} \circ \mathbb{T}_{j} ,& \mathrm{if } \ k=1 \\ \mathbb{T}^{\ast} \circ \mathbb{T}_{j}^{T} , & \mathrm{if } \ k=2 \\ \mathbb{T}^{\ast T} \circ \mathbb{T}_{j} , & \mathrm{if } \ k=3 \\ \mathbb{T}^{\ast T} \circ \mathbb{T}_{j}^{T}, & \mathrm{if } \ k=4 \end{array} \right. , $$

(67)

On the other hand, the first term in Eq. 7 can be expressed as:

$$ \text{Tr}(\mathbf{D}^{\ast}) \cdot \text{Tr}(\mathbf{D}) = \text{Tr}(\nabla \mathbf{v}^{\ast}) \cdot \text{Tr}(\nabla \mathbf{v}), $$

(68)

with

$$ \text{Tr}(\nabla \mathbf{v}) \approx \left(\sum \limits_{i=1}^{N} \mathbb{P}_{i} \circ \mathbb{T}_{i} \right) : \mathbf I. $$

(69)

Thus, a generic term in Eq. 68 can be written as

$$ \left(\left(\mathbb{P}^{\ast} \circ \mathbb{T} + \mathbb{P} \circ \mathbb{T}^{\ast} \right) : \mathbf{I} \right) \cdot \left(\left(\mathbb{P}_{j} \circ \mathbb{T}_{j} \right) : \mathbf{I} \right). $$

(70)

Now, by defining $\mathcal {V}(\mathbb {J})$ the vector form of the diagonal matrix $\mathbb {J}$, Eq. 70 results

$$\begin{array}{@{}rcl@{}} &&\left(\left(\mathbb{P}^{\ast} \circ \mathbb{T} + \mathbb{P} \circ \mathbb{T}^{\ast} \right) : \mathbf{I} \right) \cdot \left(\left(\mathbb{P}_{j} \circ \mathbb{T}_{j} \right) : \mathbf{I} \right) \\ &=& \left(\mathcal{V}(\mathbb{P}^{\ast} \circ \mathbf{I}) \otimes \mathcal{V}(\mathbb{P}_{j} \circ \mathbf{I}) \right) : (\mathcal{V} (\mathbb{T} \circ \mathbf{I}) \otimes \mathcal{V}(\mathbb{T}_{j} \circ \mathbf{I})) \\ &&+ \left(\mathcal{V}(\mathbb{P} \circ \mathbf{I}) \otimes \mathcal{V}(\mathbb{P}_{j} \circ \mathbf{I}) \right) : (\mathcal{V} (\mathbb{T}^{\ast} \circ \mathbf{I}) \otimes \mathcal{V}(\mathbb{T}_{j} \circ \mathbf{I})),\\ \end{array} $$

(71)

that allows finally casting the first term in the weak form (7) as

$$ \text{Tr}(\mathbf{D}^{\ast}) \cdot \text{Tr}(\mathbf{D}) \approx \sum\limits_{j=1}^{N} \mathbb{F}_{j}^{\ast} (\mathbf{x}) : \mathbb{G}_{j} (z) + \mathbb{F}_{j}(\mathbf{x}) : \mathbb{G}_{j}^{\ast}(z) $$

(72)

with

$$ \mathbb{F}^{\ast}_{j} = \mathcal{V}(\mathbb{P}^{\ast} \circ \mathbf{I}) \otimes \mathcal{V}(\mathbb{P}_{j} \circ \mathbf{I}) , $$

(73)

$$ \mathbb{G}_{j} = \mathcal{V} (\mathbb{T} \circ \mathbf{I}) \otimes \mathcal{V}(\mathbb{T}_{j} \circ \mathbf{I}), $$

(74)

$$ \mathbb{F}_{j} = \mathcal{V}(\mathbb{P} \circ \mathbf{I}) \otimes \mathcal{V}(\mathbb{P}_{j} \circ \mathbf{I}), $$

(75)

and

$$ \mathbb{G}_{j}^{\ast} = \mathcal{V} (\mathbb{T}^{\ast} \circ \mathbf{I}) \otimes \mathcal{V}(\mathbb{T}_{j} \circ \mathbf{I}). $$

(76)