LS-DYNA Theory Manual Material Models

19.129

01

( )N

tm

m

g t e βα α −

== + (19.87.7)

given by,

1( ) tdg t E e β−= (19.87.8)

This model is effectively a Maxwell fluid which consists of a damper and spring in series. We characterize this in the input by a shear modulus, G , and decay constant, 1β . The Mooney-Rivlin rubber model is obtained by specifying n=1. In spite of the differences in formulations with Model 27, we find that the results obtained with this model are nearly identical with those of 27 as long as large values of Poisson’s ratio are used.

Figure 19.87.1. Cellular rubber with entrapped air. By setting the initial air pressure to zero, an open cell, cellular rubber can be simulated.

Material Model 88: MTS Model The Mechanical Threshhold Stress (MTS) model is due to Mauldin, Davidson, and Henninger [1990] and is available for applications involving large strains, high pressures and strain rates. As described in the foregoing reference, this model is based on dislocation mechanics and provides a better understanding of the plastic deformation process for ductile materials by using an internal state variable called the mechanical threshold stress. This kinematic quantity tracks the evolution of the material’s microstructure along some arbitrary strain, strain rate, and temperature-dependent path using a differential form that balances dislocation generation and recovery processes. Given a value for the mechanical threshold stress, the flow stress is determined using either a thermal-activation-controlled or a drag-controlled kinetics relationship. An equation-of-state is required for solid elements and a bulk modulus must be defined below for shell elements.

Material Models LS-DYNA Theory Manual

19.130

The flow stress σ is given by:

, ,0

ˆ ˆ ˆ ˆa th th i i th s s

Gs s s

Gσ σ σ σ σ= + + + (19.88.1)

The first product in the equation for σ contains a micro-structure evolution variable, i.e., σ̂ ,called the Mechanical Threshold Stress (MTS), that is multiplied by a constant-structure deformation variable ths : ths is a function of absolute temperature T and the plastic strain-rates

Pε . The evolution equation for σ̂ is a differential hardening law representing dislocation-dislocation interactions:

( )

ˆtanh

ˆˆ1

tanhs

op

ε

σασ∂σ

∂ε α≡ Θ − (19.88.2)

The term, ˆp

∂σ∂ε

, represents the hardening due to dislocation generation and the stress ratio, ˆ

ˆ sε

σσ

,

represents softening due to dislocation recovery. The threshold stress at zero strain-hardening ˆ sεσ is called the saturation threshold stress. Relationships for oΘ , ˆ sεσ are:

1 20 0

lnp p

o oa a aε εε ε

Θ = + + (19.88.3)

which contains the material constants oa , 1a , and 2a . The constant, ˆ sεσ , is given as:

3/

ˆ ˆkT Gb Ap

s soso

ε εε

εσ σε

= (19.88.4)

which contains the input constants: ˆ soεσ , soεε , b , A , and k . The shear modulus G appearing in

these equations is assumed to be a function of temperature and is given by the correlation.

2 /0 1 /( 1)b TG G b e= − − (19.88.5)

which contains the constants: 0G , 1b , and 2b . For thermal-activation controlled deformation ths

is evaluated via an Arrhenius rate equation of the form:

LS-DYNA Theory Manual Material Models

19.131

11

0

30

11

pq

p

th

kT ns

Gb g

εε= − (19.88.6)

The absolute temperature is given as:

ref pT T c Eρ= + (19.88.7)

where E in the internal energy density per unit initial volume.

Material Model 89: Plasticity Polymer Unlike other LS-DYNA material models, both the input stress-strain curve and the strain

to failure are defined as total true strain, not plastic strain. The input can be defined from uniaxial tensile tests; nominal stress and nominal strain from the tests must be converted to true stress and true strain. The elastic component of strain must not be subtracted out.

The stress-strain curve is permitted to have sections steeper (i.e. stiffer) than the elastic modulus. When these are encountered the elastic modulus is increased to prevent spurious energy generation.

Material Model 90: Acoustic This model is appropriate for tracking low-pressure stress waves in an acoustic media such as air or water and can be used only with the acoustic pressure element formulation. The acoustic pressure element requires only one unknown per node. This element is very cost effective.

Material Model 91: Soft Tissue The overall strain energy W is "uncoupled" and includes two isotropic deviatoric matrix

terms, a fiber term F, and a bulk term:

( ) ( ) 21

1 1 2 2 23 3 lnW C I C I F K Jλ= − + − + + (19.91.1)

Here, 1I and 2I are the deviatoric invariants of the right Cauchy deformation tensor, λ is the

deviatoric part of the stretch along the current fiber direction, and J = det F is the volume ratio. The material coefficients C1 and C2 are the Mooney-Rivlin coefficients, while K is the effective bulk modulus of the material (input parameter XK).

The derivatives of the fiber term F are defined to capture the behavior of crimped collagen. The fibers are assumed to be unable to resist compressive loading - thus the model is isotropic when λ <1. An exponential function describes the straightening of the fibers, while a linear function describes the behavior of the fibers once they are straightened past a critical fiber stretch level λ ≥ λ* (input parameter XLAM):