A new unified arc-length method for damage mechanics problems

Abstract

The numerical solution of continuum damage mechanics (CDM) problems suffers from convergence-related challenges during the material softening stage, and consequently existing iterative solvers are subject to a trade-off between computational expense and solution accuracy. In this work, we present a novel unified arc-length (UAL) method, and we derive the formulation of the analytical tangent matrix and governing system of equations for both local and non-local gradient damage problems. Unlike existing versions of arc-length solvers that monolithically scale the external force vector, the proposed method treats the latter as an independent variable and determines the position of the system on the equilibrium path based on all the nodal variations of the external force vector. This approach renders the proposed solver substantially more efficient and robust than existing solvers used in CDM problems. We demonstrate the considerable advantages of the proposed algorithm through several benchmark 1D problems with sharp snap-backs and 2D examples under various boundary conditions and loading scenarios. The proposed UAL approach exhibits a superior ability of overcoming critical increments along the equilibrium path. Moreover, in the presented examples, the proposed UAL method is 1–2 orders of magnitude faster than force-controlled arc-length and monolithic Newton–Raphson solvers.

Appendices

Appendix A: Coefficients used in PC derivation of non-local damage model

The following are the coefficients used in the definition of predictor and correcter values of Sect. 5.1.2. The coefficients $\varvec{A}$ to $\varvec{H}$ depends on the consistent tangent $\varvec{J}$ and the residuals $\varvec{r}$.

(A.1a)

(A.1b)

(A.1c)

(A.1d)

(A.1e)

(A.1f)

(A.1g)

(A.1h)

Note that proposed methods are implemented in MATLAB and the matrix inversion operations in the derivations are performed by using mldivide [83] to solve the system of equations.

Appendix B: Coefficients used in PNC derivation of local damage model

The following are the coefficients used in the definition of correcter values in Sect. 5.2.1.

(B.1a)

(B.1b)

(B.1c)

Appendix C: Unified arc-length Jacobian Matrix (local damage)

We begin the derivation of consistent tangent stiffness matrix ($\varvec{J}$) for the UAL Local damage law from the following residual equations,

(C.1a)

(C.1b)

(C.1c)

The linearized form of Eq. (C.1) expressed as $\varvec{J}\delta \varvec{x}=-\varvec{r}$ is presented below:

(C.2)

where:

(C.3)

(C.4)

(C.5)

(C.6)

(C.7)

(C.8)

(C.9)

(C.10)

(C.11)

(C.12)

Appendix D: Unified arc-length Jacobian Matrix (non-local gradient damage)

(D.1a)

(D.1b)

(D.1c)

(D.1d)

The linearized form of Eq. (D.1) expressed as $\varvec{J}\delta \varvec{x}=-\varvec{r}$ is presented below:

Thus, the final system of equations is:

(D.34)

Appendix E: Mazars damage model

In this work, a widely cited damage model first proposed by Mazars [105] is used. The following is the condition based on which damage is triggered in the model:

$$\begin{aligned} d(\varepsilon ^{*}_{eq}) = \left\{ \begin{array}{ll} 0 &{} \quad \text {if}\quad \varepsilon ^{*}_{eq} < \varepsilon _{D} \\ 1 - \frac{\varepsilon _{D}(1-\mathscr {A})}{\varepsilon ^{*}_{eq}} - \frac{\mathscr {A}}{\exp (\mathscr {B}(\varepsilon ^{*}_{eq} - \varepsilon _{D}))} &{} \quad \text {if}\quad \varepsilon ^{*}_{eq} \ge \varepsilon _{D} \end{array} \right. \end{aligned}$$

(E.1)

In the expression above, $\varepsilon ^{*}_{eq}$ is the local or non-local equivalent strain, $\varepsilon _{D}$ is the damage threshold strain at which damage initiates, while $\mathscr {A}$ and $\mathscr {B}$ are material properties. Two definitions of $\varepsilon ^{*}_{eq}$ are adopted in this work. In the problems with tensile loads, $\varepsilon ^{*}_{eq}$ is calculated following Mazars approach [106]:

$$\begin{aligned} \varepsilon ^{*}_{eq} = \sqrt{\sum _{I=1}^{3} {\langle \epsilon _I \rangle }^2} \end{aligned}$$

(E.2)

where, $\epsilon _I$, $I = 1,2,3$, are the principal strains, and the Macauley brackets denote the positive part ${\langle \varvec{\cdot } \rangle } = \frac{|\varvec{\cdot } |+ \varvec{\cdot } }{2}$. Equation (E.2) is used to calculate the equivalent strain in all the problems presented in Sect. 6 except SNS. In the Single Notch Shear (SNS) 2D problem, the $\varepsilon ^{*}_{eq}$ is based on the work of [107]; $I_{1}$ and $J_{2}$ are the strain invariants and $\mathbf {\varepsilon }$ is the strain tensor.

$$\begin{aligned} \varepsilon ^{*}_{eq} = \frac{k-1}{2k(1-2\nu )} + \frac{1}{2k} \sqrt{\frac{(k-1)^2}{(1-2\nu )^{2}}I_{1}^{2} + \frac{2k}{(1+\nu )^2}J_{2}} \nonumber \\ \end{aligned}$$

(E.3)

where:

$$\begin{aligned} I_{1}= & {} tr(\varvec{\varepsilon }) \end{aligned}$$

(E.4)

$$\begin{aligned} J_{2}= & {} 3tr(\varvec{\varepsilon } \cdot \varvec{\varepsilon }) - tr^{2}(\varvec{\varepsilon }) \end{aligned}$$

(E.5)