M-RESS stands for Modified Reduced (in-plane) integration, Enhanced strain field, Solid-Shell element. This element is used in the major part of the analysis of laminated glass in this thesis. In this section, an overview of the theory of this element is provided. The element was originally presented in [13] for linear applications. Linear theory is used throughout this thesis and it is thus the linear version of the element that is presented.
The element has the geometry of a three-dimensional hexahadral solid element that has eight nodes and three degrees of freedom per node. The element geometry together with the involved coordinate systems and the integration point locations are displayed in Figure 9.
In the formulation of the element, the computational efficiency is increased through the use of a reduced integration scheme that has multiple integration points along the local ζ-axis only. As a downside, volumetric and Poisson locking problems as well as spurious
1
zero-energy modes may occur. As a remedy, the element applies the Enhanced Assumed Strain (EAS) approach, [45].
5.2.1 The EAS-method
The crucial point of the EAS method is to enlarge the strain field,ε, through adding a new field of enhanced strain parameters,α. It can be shown, [2], that only one enhancing parameter,α1, is enough in order to reduce the locking problems. This means that the locking problems can be reduced considerably, while maintaining high computational efficiency of the element formulation which is achieved through the reduced integration scheme. To overcome the hourglass modes that then may develop, hourglass stabilization is made by the Assumed Natural Strain (ANS) method, [18], for the transverse shear components whereas the membrane field were stabilized based on the stabilization vectors of [32].
In the local frame, the enhanced strain field is added to the ordinary strain field:
ε = ε + ε˜ α= [ ˆBuBˆα] [ u
α ]
= ˜B ˜u. (10)
Bˆuis the standard Finite Element Method (FEM) strain-displacement matrix,εαis the en-hanced part of the strain field and u is the displacement field. In the convective coordinate system, the enhanced strain field is chosen:
εαζζ=ζα1, (11)
which leads to the following enhanced strain-displacement matrix in the local coordinate system:
Bˆα= Q0[0 0 0 ζ 0 0]T. (12)
Q0is a transformation matrix, see [13] and references therein. The application of the EAS method leads to the following system of equations, [45]:
[ Kˆuu Kˆuα
Static condensation ofα can be performed on Equation (13) that leads to:
Kˆu+α= ˆKuu− ˆKuα( ˆKαα)−1Kˆαu. (14) The physical stabilization procedure adds an extra part, ˆKH, to the stiffness matrix as follows:
K = ˆˆ Ku+α+ ˆKH. (15)
The displacement field can now be obtained as:
u = ( ˆK)−1fext. (16)
5.2.2 Treatment of the Strain Field to Account for Stabilization
In order to apply the physical stabilization method, a division of the strain tensor into membrane, normal and transverse shear components is necessary. In the convective coor-dinate system the strain tensor can be written as:
ε = [εm...εn...εs]T = [εξξεηηεξη...εζζ...εξζεηζ]T, (17) where the strain components are defined as:
εab=1
2(J,au,b+ J,bu,a), (a, b =ξ,η,ζ), (18) where J,aare the lines of the Jacobian matrix J.
The strain tensor in the local coordinate system is given by
ε = Qˆ 0ε. (19)
It can be shown, [13], that the total strain field can be expanded to constant, linear and bilinear terms in the coordinatesξ, η and ζ. The constant membrane strain field is com-posed of a component evaluated at the center of the element and a component that depends only on theζ coordinate:
εCmI=ε0m+ζεζm. (20)
The constant membrane strain tensor must be transformed to the local coordinate system through the transformation of Equation (19). For a detailed description of the correspond-ing strain-displacement matrices, see [13].
The reduced integration scheme with integration points only along theζ-axis will lead to the cancellation of the contributions to the strain-displacement matrix that are correspond-ing to the non-constant terms of the strain field. Physical stabilization strain-displacement
relations are therefore required for those terms. The membrane part of the stabilization strain tensor is given by:
εHmI=ξεξm+ηεηm+ξηεξηm +ξζεξζm +ηζεηζm. (21) The strain tensor is transformed to the local coordinate system through the application of Equation (19). Explicit descriptions of the corresponding strain-displacement matrices are given in [13].
For the construction of strain-displacement stabilization matrices for the normal strain component,εζζ, and for the transverse shear strainsεξζandεηζ, the ANS-method is used.
For a description of the application of the ANS-method, we refer to [13].
A second stabilization method is applied to the membrane strain components as well as a method to remedy the volumetric locking that may occur to the stabilization. Details regarding these methods are out of scope of this presentation. More information is given in the first part of the thesis and in the references therein. To summarize, the resulting membrane strain tensor for the hourglass field is defined as
εˆHm=
where the nodal degrees of freedom, dI, are specified in the local coordinate system. The following transformation from global coordinates to local coordinates is used:
dˆI= ˆR0· dI. (23)
Rˆ0is defined in [13].
5.2.3 Stress Evaluation
The displacements obtained from Equation (16) are used together with Equation (10) in order to compute the strain field, ˜ε. Once the strain distribution has been determined, the stress distribution,σ, is given by:
σ = D˜ε = D · [ ˆBuBˆα] [ u
α ]
. (24)
Dis the constitutive matrix. The stresses are evaluated at the integration points. A stress smoothing procedure based on a quadratic least squares fit is used in order to extrapolate and average the stresses at the nodes, [16].