PSI - Issue 35

E.A. Dizman et al. / Procedia Structural Integrity 35 (2022) 91–97 Author name / Structural Integrity Procedia 00 (2021) 000–000

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4

where E f is the longitudinal modulus of elasticity along the fiber direction, E m 2 and E m 3 are the transverse moduli in the other two orthogonal directions. It is to be noted that due to transverse isotropy G 23 = 0 . 5 E m / (1 + ν 23 ) holds, and all the components are given with respect to local coordinate system. The corresponding Kirchho ff stress tensor is obtained by τ = F e S F T e which in turn is used to obtain the slip driving Schmid stress on slip system η through τ η = τ : ( s η ⊗ m η ) = s η · τ m η . For the evolution of slip on a specific system, the following rate dependent power type relationship Asaro and Needleman (1985),

sgn( τ α )

α | τ

1 / m

α |

˙ γ α = ˙ γ 0

(4)

g α

is used where g α is the yield strength and ˙ γ α denotes the reference shear strain rate. m is the rate sensitivity exponent and as m tends to zero, the rate independent behaviour is retrieved. As opposed to metallic crystals, confining pres sure influences plastic flow considered here since the yield strength of matrix material is pressure sensitive. Friction coe ffi cient µ is used to inject pressure dependency as, g α =   τ α y + µ p if p ≥ 0 τ α y otherwise (5) where τ y is the initial yield strength (yield strength in the absence of pressure) p is the confining pressure and defined as p = − 1 / 2 ( S 22 + S 33 ), in terms of the components of S . 3. Stress Update Algorithm and Implementation In a displacement driven incremental set-up, all the quantities associated with time t n are known and the stress update algorithm is basically driven by an estimate for F n + 1 . Furthermore it is supposed that estimates for slip increments ∆ γ η = γ η n + 1 − γ n n are available. Using the exponential map algorithm de Souza Neto et al. (2008), ( F p ) n + 1 can be expressed as ( F p ) n + 1 = exp 6 η = 1 ∆ γ η s η 0 ⊗ m η 0 ( F p ) n with which it is trivial to obtain ( F e ) n + 1 = ( F e ) trial n + 1 exp 6 η = 1 − ∆ γ η s η 0 ⊗ m η 0 where ( F e ) trial = F n + 1 ( F p ) − 1 n . Exponent of a tensor is evaluated using the truncated series expansion given in de Souza Neto et al. (2008). Calculation of stress tensors S n + 1 and τ n + 1 is straight forward and followed by the calculation of Schmid stresses. Schmid stresses and the current values of yield strength are used to check whether time discretized slip equations written in residual form

0   | τ η g η

n + 1 | n + 1   1 / m

r η = ∆ γ η − ∆ t ˙ γ η

sgn( τ η

n + 1 ) = 0

(6)

are satisfied or not. This set of nonlinear equations is solved for slip increments iteratively using Newton-Raphson method. For the solution of system of equilibrium equations, the sensitivity of Kirchho ff stresses to total deformation gradient has to be obtained. This however requires the sensitivity of slips to total deformation gradient which is evaluated through the persistency condition enforcing the steady fulfillment of residual equations r α , i.e. d r α d F n + 1 = 0.

4. Numerical Examples

Summarized material model is implemented in finite element software Abaqus through user subroutines. Although it could be introduced as a user defined material model (through UMAT subroutine), a user element is implemented due