MATVE

Bulk Data Entry Defines material properties for nonlinear viscoelastic materials.

Format A: Prony Series (Model = PRONY)

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
MATVE	`MID`	`Model`			`gD1`	`tD1`	`gB1`	`tB1`
	`gD2`	`tD2`	`gD3`	`tD3`	`gD4`	`tD4`	`gD5`	`tD5`
	`gB2`	`tB2`	`gB3`	`tB3`	`gB4`	`tB4`	`gB5`	`tB5`

Format B: Bergström-Boyce (Model = BBOYCE)

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
MATVE	`MID`	`Model`		`Sb`	`A`	`C`	`m`	`E`

Example A: Prony Series (Model = PRONY)

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
MATVE	2	PRONY			0.25	5e-2	0.25	5e-2

Example B: Bergström-Boyce (Model = BBOYCE)

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
MATVE	2	BBOYCE		2.0	0.1	-0.7	5.0	0.01

Definitions

Field	Contents	SI Unit Example
`MID`	Unique material identification number. No default (Integer > 0)
`Model`	Viscoelastic material model type. PRONY (Default) Linear viscoelastic model based on Prony series. BBOYCE Bergström-Boyce model.
`gDi`	Modulus ratio for the $i$ -th deviatoric Prony series. Default = Blank (Real > 0.0)
`tDi`	Relaxation time for the $i$ -th deviatoric Prony series. Default = Blank (Real > 0.0)
`gBi`	Modulus ratio for the $i$ -th bulk Prony series. Default = Blank (Real > 0.0)
`tBi`	Relaxation time for the $i$ -th bulk Prony series. Default = Blank (Real > 0.0)
`Sb`	Stress scaling factor that defines the ratio of the stress carried by network B to that carried by network A under identical elastic stretching. 7 No default (Real > 0.0)
`A`	Effective creep strain rate. 7 No default (Real > 0.0)
`C`	Negative exponent characterizes the creep strain dependence of the effective creep strain rate in network B. 7 No default (-1.0 ≤ Real ≤ 0.0)
`m`	Positive exponent characterizes the effective stress dependence of the effective creep strain rate in network B. 7 No default (Real ≥1.0)
`E`	Material parameter to regularize the creep strain rate in the vicinity of the undeformed state. 7 Default = 0.01 (Real ≥0.0)

Comments

The CHEXA, CTETRA, CPENTA, and CPYRA elements are currently supported.
The instantaneous response can be provided by MAT1, MAT9 or MATHE entries, which should have the same MID as the MATVE entry.
The linear viscoelastic material (Model = PRONY) is represented by the generalized Maxwell model. The material response is given by the following convolution representation.(1)
$σ = \int_{0}^{t} g (t - s) {\dot{σ}}_{0} d s$

Where,

$g (t) = γ_{\infty} + \sum_{i} γ_{i} e^{- \frac{t}{τ_{i}}}$

The subscript $i$ indicates the $i$ -th component in the Prony series. A maximum of 5 components are allowed.

$γ_{i}$

Modulus ratio

$τ_{i}$

Relaxation time

$σ_{0}$

Instantaneous stress response
For the isotropic model, the deviatoric and bulk responses can be specified separately. For the anisotropic model, only gDi and tDi are used and the bulk specifications are ignored.
The material relaxation response is controlled by the card VISCO. For example, if the user wants to simulate a physical relaxation test, the first subcase can omit the VISCO card so that material response is only the instantaneous elasticity in this subcase. In the next subcase, the user can add a VISCO card so that the material response is viscoelastic.
For Implicit Nonlinear Analysis, MATVE is supported for small displacement and large displacement nonlinear analysis.
The nonlinear viscoelastic material (Model = BBOYCE) is supported only for solid elements in Nonlinear Explicit Analysis.
The response of the material can be represented using an equilibrium hyperelastic network A, and a time-dependent hyperelastic - nonlinear viscoelastic network B. The Hyperelastic material models for network A and B can be selected from existing MATHE card.

The deformation gradient tensor, $F$ is assumed to act on both networks and is decomposed into elastic ( $F_{B}^{e}$ ) and inelastic ( $F_{B}^{c r}$ ) parts in network B as:(2)
$F = F_{A} = F_{B}^{e} . F_{B}^{c r}$

The evolution of inelastic deformation gradient on network B is governed by:(3)
$F_{B}^{e} . {\dot{F}}_{B}^{c r} . F_{B}^{c r - 1} . F_{B}^{e - 1} = {\dot{ε}}_{B}^{v} \frac{S_{B}}{{\bar{σ}}_{B}}$

The Bergström-Boyce hardening formulation is given by:(4)
${\dot{ε}}_{B}^{v} = A {(\tilde{λ} - 1 + E)}^{c} {\bar{σ}}_{B}^{m}$

Where,

${\bar{σ}}_{B} = \sqrt{S_{B} : S_{B}}$

$\tilde{λ} = \sqrt{\frac{1}{3} I : (F_{B}^{c r} . {(F_{B}^{c r})}^{T})}$

$S_{B}$

Deviatoric part of the Cauchy stress tensor in network B.

$F_{B}^{c r}$

Inelastic deformation gradient tensor in network B.