A three-dimensional numerical model for large strain compression of nanofibrillar cellulose foams

Prashanth Srinivasa; Artem Kulachenko

doi:10.1515/npprj-2018-3023

Article

A three-dimensional numerical model for large strain compression of nanofibrillar cellulose foams

Prashanth Srinivasa and Artem Kulachenko

Published/Copyright: June 9, 2018

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From the journal Nordic Pulp & Paper Research Journal Volume 33 Issue 2

Abstract

We investigate the suitability of three-dimensional Voronoi structures in representing a large strain macroscopic compressive response of nanofibrillar cellulose foams and understanding the connection between the features of the response and details of the microstructure. We utilise Lloyd’s algorithm to generate centroidal tessellations to relax the Voronoi structures and have reduced polydispersity. We begin by validating these structures against simulations of structures recreated from microtomography scans. We show that by controlling the cell face curvature, it is possible to match the compressive response for a 96.02 % porous structure. For the structures of higher porosity (98.41 %), the compressive response can only be matched up to strain levels of 0.4 with the densification stresses being overestimated. We then ascertain the representative volume element (RVE) size based on the measures of relative elastic modulus and relative yield strength. The effects of cell face curvature and partially closed cells on the elastic modulus and plateau stress is then estimated. Finally, the large strain response is compared against the two-dimensional Voronoi model and available experimental data for NFC foams. The results show that compared to the two-dimensional model, the three-dimensional analysis provides a stiffer response at a given porosity due to earlier self-contact.

Keywords: large strain compression; meso-scale modelling; nanofibrillar cellulose foams; scaling laws; tomography; Voronoi tessellations

Funding statement: The authors wish to acknowledge BiMac Innovation for the financial support.

Conflict of interest: The authors wish to confirm that there is no conflict of interest to declare.

Elastic-plastic material model with bi-linear isotropic hardening

The implementation of the elasto-plastic material model in LS-DYNA® is based on the paper by (Krieg and Key 1976) and covered by the theory manual (LS-DYNA® Theory Manual 2017). We summarise these equations below.

The total strain tensor (ε) is the sum of the elastic ( ε e ) and plastic ( ε p ) strains. In terms of the strain and stress increments,

(A1) d ε = d ε e + d ε p

(A2) d σ = D ( d ε − d ε p )

where D is the elastic tangent stiffness matrix.

The flow rule provides

(A3) d ε p = d λ ∂ f ∂ s

where d λ ≥ 0 is the plastic multiplier and s is the deviatoric stress.

For the case of von Mises criterion as the yield criterion, the yield stress is pressure independent and the yield surface is a function of the deviatoric stress tensor ( s i j ) and thus the yield function is as given as below.

(A4) f = 1 2 s i j s i j − σ y 2 3 = 0

The yield stress ( σ y ) is a function of the effective plastic strain ( ε e f f p ) and the plastic tangent modulus ( E p ) and is given as below.

(A5) σ y ( ε e f f p ) = σ y 0 + E p ε e f f p where d ε e f f p = 2 3 d ε i j p d ε i j p 1 / 2 and E p = E E t E − E t

The consistency relations during the development of plastic strains provide,

(A6) d f = ∂ f ∂ s i j d s i j + ∂ f ∂ σ y d σ y where ∂ f ∂ s = s and d σ y = E p d ε e f f p

Using (A3) and s i j d s i j = s : d s = s : d σ, the plastic multiplier d λ is given by

(A7) d λ = s : D : d ε D : s : s + 2 3 σ y E p ( 2 3 s : s ) 1 / 2

Given a point X in the reference configuration which is mapped to a point x in the current configuration, using orthonormal basis vectors, the deformation gradient is defined by

(A8) F = ∂ x i ∂ X j e i ⊗ e j

The unique left and right polar decompositions are then given by the symmetric, positive definite left ( v) and right ( U) stretch tensors together with the proper orthogonal rotation tensor R as

(A9) F = R U = v R

The logarithmic (Hencky) strain follows from the right stretch tensor as

(A10) ε = l n U

The above is determined by spectral decomposition of U as

(A11) ε = ∑ i = 1 3 l n α i β i β i T

where α i and β i are the eigen values and eigen vectors respectively of the right stretch tensor U.

The material time derivative of the Cauchy stress tensor is given by

(A12) σ ˙ i j = σ i j ∇ + σ i k ω k j + σ j k ω k i

where ω is the spin tensor and σ i j ∇ = D i j k l ε ˙ k l ( σ i j ∇ is the Jaumann stress rate, ε ˙ k l is the strain rate tensor and D i j k l is the stress dependent constitutive matrix).

The strain updates maintaining objectivity are given by

(A13) ε i j n + 1 = ε i j n + ρ i j n + ε ˙ i j n + 1 / 2 Δ t n + 1 / 2 where ρ i j n = ( ε i p n ω p j n + 1 / 2 + ε j p n ω p i n + 1 / 2 ) Δ t n + 1 / 2

gives the rotational correction transforming the strain tensor from configuration at t n to t n + 1 .

The deviatoric stresses are updated elastically. For σ i j ∗ as the trial stress tensor, σ i j n as the stress tensor from the previous time step, D i j k l as the elastic tangent modulus matrix and Δ ε k l as the incremental strain tensor:

If there is no plastic yield,

(A14) σ i j ∗ = σ i j n + D i j k l d ε k l

If there is plastic yield (violation of the von Mises flow rule),

(A15) ε e f f p n + 1 = ε e f f p n + ( 1 2 s i j ∗ s i j ∗ ) 1 / 2 − σ y 3 G + E p = ε e f f p n + d ε e f f p

Compute the plastic strain increment using (A15)
d ε p = ( 1 2 s i j ∗ s i j ∗ ) 1 / 2 − σ y 3 G + E p
Update the plastic strain
ε p n + 1 = ε p n + d ε p
Update the yield stress using (A5)
σ y n + 1 = σ y n + E p d ε p
Compute the scale factor (m) using yield strength at time ( n + 1)
m = σ y n + 1 s ∗ where s ∗ = 3 2 s i j ′ n + 1 s i j ′ n + 1 1 / 2
where s ′ is the trial stress value.
Return the deviatoric stresses radially to the yield surface using
s i j n + 1 = m · s i j ′ n + 1

Plane stress plasticity for shells

Plane stress assumption used in shells requires normal stress σ 33 to be 0 and therefore needs additional treatment.

The stress and strain increments are computed in the lamina co-ordinate system in shells. Since the Jaumann rate allows for the possibility that normal stress σ 33 will not be zero, a trial plane stress update assuming that the incremental strains are elastic is carried out.

(A16) Δ ε 33 = − σ 33 + λ ( Δ ε 11 + Δ ε 22 ) λ + 2 μ

where λ and μ are Lame’s constants.

If the trial stress is within the yield surface, strain increment is elastic, and the stress update is completed. If it is not, then a secant iteration procedure is followed to find the normal strain increment required for σ 33 to be 0. The secant iteration for strain increment is given by

(A17) Δ ε 33 p i + 1 = Δ ε 33 p i − 1 − Δ ε 33 p i − Δ ε 33 p i − 1 σ 33 i − σ 33 i − 1 σ 33 i − 1

The initial values for the iteration are got from the elastic estimate and through the assumption of purely plastic increment. The convergence criterion for the iteration is based on the convergence of normal strains and the iterations are carried out until normal stress σ 33 i is sufficiently small.

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Received: 2017-09-15

Accepted: 2018-01-25

Published Online: 2018-06-09

Published in Print: 2018-07-26

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Articles in the same Issue

https://doi.org/10.1515/npprj-2018-3023

Keywords for this article

large strain compression; meso-scale modelling; nanofibrillar cellulose foams; scaling laws; tomography; Voronoi tessellations