From 28b51cff011f58a4a91e699893472ea2e37fb663 Mon Sep 17 00:00:00 2001 From: Prithivirajan Veerappan Date: Tue, 2 Jul 2024 21:21:18 -0600 Subject: [PATCH] Final cleanup and gold files update ref #184 Final cleanup and gold files update ref #184 --- .../materials/DamagePlasticityStressUpdate.md | 125 +++++++++--------- .../materials/DamagePlasticityStressUpdate.h | 53 -------- src/materials/DamagePlasticityStressUpdate.C | 9 -- .../gold/dilatancy_out.csv | 90 ++++++------- .../gold/shear_test_out.csv | 56 ++++---- .../gold/uniaxial_compression_out.csv | 90 ++++++------- .../gold/uniaxial_tension_2elem_out.csv | 25 ---- .../gold/uniaxial_tension_out.csv | 98 +++++++------- 8 files changed, 228 insertions(+), 318 deletions(-) delete mode 100644 test/tests/damage_plasticity_model/gold/uniaxial_tension_2elem_out.csv diff --git a/doc/content/source/materials/DamagePlasticityStressUpdate.md b/doc/content/source/materials/DamagePlasticityStressUpdate.md index 689cd492..714a149a 100644 --- a/doc/content/source/materials/DamagePlasticityStressUpdate.md +++ b/doc/content/source/materials/DamagePlasticityStressUpdate.md @@ -2,22 +2,24 @@ The [!cite](lee1996theory) model accounts for the independent damage in tension and compression. It also accounts for degradation of the elastic modulus of the concrete as the loading goes beyond yielding in either tension or compression. The model uses the incremental theory of plasticity and decomposes the total strain, $\boldsymbol{\varepsilon}$, into elastic strain, $\boldsymbol{\varepsilon}^{e}$, and plastic strain, $\boldsymbol{\varepsilon}^{p}$, as follows \begin{equation} - \boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}^{e} + \boldsymbol{\varepsilon}^{p} \label{eps_def} + \label{straindecomposition} + \boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}^{e} + \boldsymbol{\varepsilon}^{p} \end{equation} where bold symbol represents a vectoral or tensorial quantity. The relation between elastic strain and the stress, $\boldsymbol{\sigma}$, is given by \begin{equation} - \boldsymbol{\varepsilon}^{e} = \boldsymbol{\mathfrak{E}}^{-1}:\boldsymbol{\sigma} \label{eps_e_def} + \label{elasticstrain} + \boldsymbol{\varepsilon}^{e} = \boldsymbol{\mathfrak{E}}^{-1}:\boldsymbol{\sigma} \end{equation} -where $\boldsymbol{\mathfrak{E}}$ is the elasticity tensor. Using Eqs. \eqref{eps_def}-\eqref{eps_e_def}, the relation between $\boldsymbol{\sigma}$ and $\boldsymbol{\varepsilon}^{p}$ is expressed as +where $\boldsymbol{\mathfrak{E}}$ is the elasticity tensor. Using [straindecomposition] and [elasticstrain], the relation between $\boldsymbol{\sigma}$ and $\boldsymbol{\varepsilon}^{p}$ is expressed as \begin{equation} - \boldsymbol{\sigma} = \boldsymbol{\mathfrak{E}}:\left(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{e}\right) + \boldsymbol{\sigma} = \boldsymbol{\mathfrak{E}}:\left(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{p}\right) \end{equation} Since the model considers the effect of damage in elastic stiffness, an effective stress, $\boldsymbol{\sigma}^{e}$, is defined, where the stress for a given strain always corresponds to the undamaged elastic stiffness of the material, $\boldsymbol{\mathfrak{E}}_{0}$ The relation between $\boldsymbol{\sigma}^{e}$, $\boldsymbol{\varepsilon}$, and $\boldsymbol{\varepsilon}^{p}$ is given by \begin{equation} - \boldsymbol{\sigma}^e = \boldsymbol{\mathfrak{E}}_0:\left(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{e}\right) + \boldsymbol{\sigma}^e = \boldsymbol{\mathfrak{E}}_0:\left(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{p}\right) \end{equation} To consider the degradation of reinforced-concrete structures, an isotropic damage was considered in concrete material. Hence, the relation between $\boldsymbol{\sigma}^e$ and $\boldsymbol{\sigma}$ can be established by @@ -26,7 +28,8 @@ the isotropic scalar degradation damage variable, D, as follows \boldsymbol{\sigma} = \left(1-D\right)\boldsymbol{\sigma}^e \label{sigma_def} \end{equation} \begin{equation} - \boldsymbol{\sigma} = \left(1-D\right)\boldsymbol{\mathfrak{E}}_0:\left(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{e}\right)\label{sigma_def2} + \label{sigma_def2} + \boldsymbol{\sigma} = \left(1-D\right)\boldsymbol{\mathfrak{E}}_0:\left(\boldsymbol{\varepsilon} - \boldsymbol{\varepsilon}^{e}\right) \end{equation} The Damage Plasticity Model has various attributes to define the mechanical behavior of concrete in tension and compression such as the yield function, plastic potential, strength of material @@ -38,9 +41,10 @@ sections. ## Yield Function The yield function, $\mathfrak{F}$ is a function of $\boldsymbol{\sigma}$, the strength of the material in uniaxial tension, $f_t$, and the strength of the material in uniaxial compression, $f_c$. It was used to describe the admissible stress space. For this implementation, the yield function in stress space is defined as follows -\begin{equation} \label{yf} +\begin{equation} +\label{yf} \begin{gathered} - \mathfrak{F}\left(\boldsymbol{\sigma},f_t,f_c\right) = \frac{1}{1-\alpha} \\ + \mathfrak{F}\left(\boldsymbol{\sigma},f_t,f_c\right) = \frac{1}{1-\alpha} \left(\alpha I_1 + \sqrt{3J_2} + \beta\left(\boldsymbol{\kappa}\right)<{\hat{\boldsymbol{\sigma}}_{max}}>\right) - f_c\left(\boldsymbol{\kappa}\right) \end{gathered} \end{equation} @@ -53,26 +57,30 @@ relates tensile, $f_t\left(\boldsymbol{\kappa}\right)$, and compressive, $f_c\le function of a vector of damage variable, $\boldsymbol{\kappa} = \{\kappa_t, \kappa_c\}$ and $\kappa_t$ and $\kappa_c$ are the damage variables in tension and compression, respectively. -The implementation first solves the given problem in the effective stress space and then transform the effective stress to stress space using Eq. \eqref{sigma_def2}. Thus, the yield strength of the concrete under uniaxial loading is expressed as effective yield strength as follows +The implementation first solves the given problem in the effective stress space and then transform the effective stress to stress space using [sigma_def2]. Thus, the yield strength of the concrete under uniaxial loading is expressed as effective yield strength as follows \begin{equation} - f_t\left(\boldsymbol{\kappa}\right) = \left(1-D_t \left(\kappa_t\right)\right)f_{t}^{e}\left(\kappa_t\right) \label{ft} + \label{ft} + f_t\left(\boldsymbol{\kappa}\right) = \left(1-D_t \left(\kappa_t\right)\right)f_{t}^{e}\left(\kappa_t\right) \end{equation} \begin{equation} - f_c\left(\boldsymbol{\kappa}\right) = \left(1-D_c \left(\kappa_c\right)\right)f_{c}^{e}\left(\kappa_c\right) \label{fc} + \label{fc} + f_c\left(\boldsymbol{\kappa}\right) = \left(1-D_c \left(\kappa_c\right)\right)f_{c}^{e}\left(\kappa_c\right) \end{equation} where $f_{t}^{e}$ and $f_{c}^{e}$ are the yield strength of the concrete in tension and compression, respectively and $D_t$ and $D_c$ are the degradation damage variables in -tension and compression, respectively such that $0\leq D_t$\textless 1 and $0\leq D_c$\textless 1. +tension and compression, respectively such that $0\leq D_t\leq 1$ and $0\leq D_c\leq 1$. The scalar degradation damage variable is expressed in terms of $D_t$ and $D_c$ as follows \begin{equation} D\left(\boldsymbol{\kappa}\right) = 1-\left(1-D_t\left(\kappa_t\right)\right)\left(1-D_c\left(\kappa_c\right)\right) \label{D} \end{equation} -Hence, for uniaxial tension, $D=D_t$, while for uniaxial compression, $D=D_c$.The yield strength for multi-axial loading, i.e., Eqs. \eqref{ft}-\eqref{fc}, can be rewritten as +Hence, for uniaxial tension, $D=D_t$, while for uniaxial compression, $D=D_c$. The yield strength for multi-axial loading, i.e., [ft] and [fc], can be rewritten as \begin{equation} - f_t\left(\boldsymbol{\kappa}\right) = \left(1-D\left(\boldsymbol{\kappa}\right)\right)f_{t}^{e}\left(\kappa_t\right) \label{ft_new} + \label{ft_new} + f_t\left(\boldsymbol{\kappa}\right) = \left(1-D\left(\boldsymbol{\kappa}\right)\right)f_{t}^{e}\left(\kappa_t\right) \end{equation} \begin{equation} - f_c\left(\boldsymbol{\kappa}\right) = \left(1-D\left(\boldsymbol{\kappa}\right)\right)f_{c}^{e}\left(\kappa_c\right) \label{fc_new} + \label{fc_new} + f_c\left(\boldsymbol{\kappa}\right) = \left(1-D\left(\boldsymbol{\kappa}\right)\right)f_{c}^{e}\left(\kappa_c\right) \end{equation} Similarly, the first invariant of $\boldsymbol{\sigma}^e$, $I_1^e$, and second invariant of the deviatoric component of $\boldsymbol{\sigma}^e$, $J_2^e$, can be rewritten in terms of $I_1$ and $J_2$ as follows \begin{equation} @@ -81,13 +89,15 @@ Similarly, the first invariant of $\boldsymbol{\sigma}^e$, $I_1^e$, and second i \begin{equation} J_2^e = \left(1-D\left(\boldsymbol{\kappa}\right)\right)^2J_2 \label{J2e} \end{equation} -Since $D$ \textless 1, the maximum principal effective stress ${\hat{\boldsymbol{\sigma}}_{max}}^e$ is expressed in the terms of ${\hat{\boldsymbol{\sigma}}_{max}}$ as follows +The maximum principal effective stress ${\hat{\boldsymbol{\sigma}}_{max}}^e$ is expressed in the terms of ${\hat{\boldsymbol{\sigma}}_{max}}$ as follows \begin{equation} - {\hat{\boldsymbol{\sigma}}_{max}}^e = \left(1-D\left(\boldsymbol{\kappa}\right)\right){\hat{\boldsymbol{\sigma}}_{max}} \label{sig_max_e} + \label{sig_max_e} + {\hat{\boldsymbol{\sigma}}_{max}}^e = \left(1-D\left(\boldsymbol{\kappa}\right)\right){\hat{\boldsymbol{\sigma}}_{max}} \end{equation} Consequently, yield function $\left(\mathfrak{F}\left(\boldsymbol{\sigma},f_t,f_c\right)\right)$ is a homogenous -function, i.e., $x \mathfrak{F}\left(\boldsymbol{\sigma},f_t,f_c\right) = \mathfrak{F}\left(x \boldsymbol{\sigma},x f_t,x f_c\right)$ Hence, using Eqs. \eqref{ft_new}-\eqref{sig_max_e}, the yield function in the effective stress space was obtained by multiplying by a factor $\left(1-D\right)$ of both sides of Eq. \eqref{yf}, as follows -\begin{equation}\label{yf_e} +function, i.e., $x \mathfrak{F}\left(\boldsymbol{\sigma},f_t,f_c\right) = \mathfrak{F}\left(x \boldsymbol{\sigma},x f_t,x f_c\right)$ Hence, using [ft_new] and [sig_max_e], the yield function in the effective stress space was obtained by multiplying by a factor $\left(1-D\right)$ of both sides of [yf], as follows +\begin{equation} +\label{yf_e} \begin{gathered} \mathfrak{F}\left(\boldsymbol{\sigma}^e,f_t^e,f_c^e\right) = \frac{1}{1-\alpha} \\ \left(\alpha I_1^e + \sqrt{3J_2^e} + \beta\left(\boldsymbol{\kappa}\right)<{\hat{\boldsymbol{\sigma}}_{max}}^e>\right) - f_c^e\left(\boldsymbol{\kappa}\right) @@ -111,7 +121,8 @@ dilatancy of concrete, and $\dot{\gamma}$ is the plastic consistency parameter. Since the concrete shows strain-softening in tension and strain hardening and softening in compression, the concrete strength is expressed as a combination of two exponential functions as follows \begin{equation} - f_N = f_{N0} \left(\left(1+a_N\right) e^{-b_N \varepsilon^p}- a_N e^{-2b_N \varepsilon^p}\right) \label{fN} +\label{fN} + f_N = f_{N0} \left(\left(1+a_N\right) e^{-b_N \varepsilon^p}- a_N e^{-2b_N \varepsilon^p}\right) \end{equation} where $f_{N0}$ is the initial yield stress of the material, $N = t$, for the uniaxial tension, $N = c$, for uniaxial compression, $a_N$ and $b_N$, are the material constants @@ -119,12 +130,14 @@ that describe the softening and hardening behavior of the concrete. Similarly, t degradation of the elastic modulus is also expressed as another exponential function as follows \begin{equation} - D_N = 1 - e^{-d_N \varepsilon^p} \label{DN} +\label{DN} + D_N = 1 - e^{-d_N \varepsilon^p} \end{equation} where $d_N$ is a constant that determine the rate of degradation of $\boldsymbol{\mathfrak{E}}$ with the increase in plastic strain. The strength of the material in the effective stress space was -obtained using Eqs. \eqref{ft_new}-\eqref{fc_new}, and \eqref{fN}-\eqref{DN}, as follows -\begin{equation}\label{fNe} +obtained using [ft_new], [fc_new], [fN], and [DN], as follows +\begin{equation} + \label{fNe} f_N^e = f_{N0} \left(\left(1+a_N\right) \left(e^{-b_N \varepsilon^p}\right)^{1-\frac{d_N}{b_N}}- a_N \left(e^{-b_N \varepsilon^p}\right)^{2-\frac{d_N}{b_N}}\right) \end{equation} @@ -144,7 +157,7 @@ Thus, the plastic strain can be presented in terms of damage variable as follows \begin{equation} \varepsilon^p = \frac{1}{b_N} \log{\frac{\sqrt{\Phi_N}}{a_N}} \label{eps_p} \end{equation} -where $\Phi_N = 1 + a_N \left(2+a_N \right)\kappa_N$. Using Eqs. \eqref{fN} and \eqref{eps_p}, the +where $\Phi_N = 1 + a_N \left(2+a_N \right)\kappa_N$. Using [fN] and [eps_p], the strength of the concrete can be expressed in terms of the damage variable as follows \begin{equation} f_N = f_{N0} \frac{1+a_N-\sqrt{\Phi_N\left(\kappa_N\right)}}{a_N}\sqrt{\Phi_N\left(\kappa_N\right)} \label{fN_new} @@ -160,7 +173,7 @@ Thus, the strength of the material and degradation damage variable in the effect where $a_N$, $b_N$,and $d_N$ are the modeling parameters, which are evaluated from the material properties. Since the maximum compressive strength of concrete, $f_{cm}$, was used as a material property, $f_{cm}$ was obtained in terms of $a_c$ by finding maximum value of -compressive strength in Eq. \eqref{fNe} as follows +compressive strength in [fNe] as follows \begin{equation} f_{cm} = \frac{f_{c0}\left(1+a_c\right)^2}{4a_c} \label{fcm} \end{equation} @@ -182,21 +195,25 @@ as follows \begin{equation} \left(\frac{d\sigma}{d\varepsilon^p}\right)_{\varepsilon^p=0} = f_{t0}b_t\left(a_t-1\right) \label{slope} \end{equation} -Thus, $a_t$ was obtained using Eqs. \eqref{bt}-\eqref{slope} as follows +Thus, $a_t$ was obtained using [bt]-[slope] as follows \begin{equation} - a_t = \sqrt{\frac{9}{4}+\frac{2\frac{G_t}{l_t} \left(\frac{d\sigma}{d\varepsilon^p}\right)_{\varepsilon^p=0}}{f_{t0}^2}}\label{at} + \label{a_t} + a_t = \sqrt{\frac{9}{4}+\frac{2\frac{G_t}{l_t} \left(\frac{d\sigma}{d\varepsilon^p}\right)_{\varepsilon^p=0}}{f_{t0}^2}} \end{equation} -The minimum slope of the $\sigma$ versus $\varepsilon^p$ curve is +To obtain a real value of $a_t$, the quantity inside the square root must be $\geq$ 0. Therefore, the minimum possible slope of the $\sigma$ versus $\varepsilon^p$ curve is $\left(\left(\frac{d\sigma}{d\varepsilon^p}\right)_{\varepsilon^p=0}\right)_{min}= -\frac{9}{8}\frac{f_{t0}^2}{\frac{G_t}{l_t}}$, which is a function of the characteristic length in tension. Therefore, a mesh independent slope parameter $\omega\in\left(0,1\right)$, is defined such that \begin{equation} \left(\frac{d\boldsymbol{\sigma}}{d\varepsilon^p}\right)_{\varepsilon^p=0} = \omega \left(\left(\frac{d\sigma}{d\varepsilon^p}\right)_{\varepsilon^p=0}\right)_{min} \label{slope_new} \end{equation} -Using Eqs. \eqref{at}-\eqref{slope_new}, $a_t$ is rewritten as follows +Using [a_t] and [slope_new], $a_t$ is rewritten as follows \begin{equation} a_t = \frac{3}{2}\sqrt{1-\omega}-\frac{1}{2}\label{at_new} \end{equation} + +Note that $\omega$ is a fitting parameter that must be provided by the user. + The ratio of $\frac{d_c}{b_c}$ was obtained by specifying degradation values for uniaxial compression case from experiments. If the degradation in the elastic modulus is known, denoted as $\widetilde{D}_c$, when the concrete is unloaded from $\sigma =f_{cm}$, then $\frac{d_c}{b_c}$ will be obtained using the following relation @@ -214,9 +231,7 @@ Similarly, if degradation in the elastic modulus is known, denoted as $\widetild \frac{d_t}{b_t} = \frac{\log\left(1-\widetilde{D}_t\right)}{\log\left(\frac{1+a_t-\sqrt{1+a_t^2}}{2a_t}\right)} \label{Dt_ft0} \end{equation} Thus, material modeling parameters $a_N$,$b_N$, and $d_N$ were obtained, which were used in -defining the strength of concrete in both tension and compression as given in Eq. -\eqref{fNe_new}. These parameters are also used to define the degradation damage variable in -both tension and compression as indicated in Eq. \eqref{DN_new}. +defining the strength of concrete in both tension and compression as given in [fNe_new]. These parameters are also used to define the degradation damage variable in both tension and compression as indicated in [DN_new]. ## Hardening Potential @@ -226,7 +241,7 @@ The vector of two damage variables, $\boldsymbol{\kappa}=\{\kappa_t, \kappa_c\}$ \end{equation} The evolution of the damage variable is expressed in terms of the evolution of $\boldsymbol{\varepsilon}^p$ as follows \begin{equation} - \dot{\boldsymbol{\kappa}} = \frac{1}{g_N}f_N^e\left(\kappa_N\right)\dot{\boldsymbol{\varepsilon}^p} \label{kappa_ep} + \dot{\boldsymbol{\kappa}} = \frac{1}{g_N}f_N\left(\kappa_N\right)\dot{\boldsymbol{\varepsilon}^p} \label{kappa_ep} \end{equation} where $g_N$ is dissipated energy density during the process of cracking. The scalar $\dot{\boldsymbol{\varepsilon}^p}$, is extended to multi-dimensional case as follows \begin{equation} @@ -241,7 +256,7 @@ where $\delta_{ij}$ is the Dirac delta function and $\hat{\boldsymbol{\sigma}^e} \end{cases} \end{equation} $\dot{\varepsilon}^{p}_{max}$ and $\dot{\varepsilon}^{p}_{min}$ -are the maximum and minimum principal plastic strain, respectively. From Eqs. \eqref{kappa_ep} - \eqref{r_sige}, the evolution of $\boldsymbol{\kappa}$ was obtained as +are the maximum and minimum principal plastic strain, respectively. From [kappa_ep] - [r_sige], the evolution of $\boldsymbol{\kappa}$ was obtained as \begin{equation} \dot{\boldsymbol{\kappa}} = \boldsymbol{h}\left(\hat{\boldsymbol{\sigma}^e}\right):\dot{\boldsymbol{\varepsilon}}^{\hat{p}} \label{kappa_h_ep} \end{equation} @@ -249,12 +264,12 @@ where \begin{equation}\label{h} \boldsymbol{h}\left(\hat{\boldsymbol{\sigma}^e}\right)= \begin{bmatrix} - \frac{r\left(\hat{\boldsymbol{\sigma}^e}\right)}{g_t}f_t^e\left(\kappa_t\right)&0&0\\ + \frac{r\left(\hat{\boldsymbol{\sigma}^e}\right)}{g_t}f_t\left(\kappa_t\right)&0&0\\ 0&1&0\\ - 0&0&\frac{1-r\left(\hat{\boldsymbol{\sigma}^e}\right)}{g_c}f_c^e\left(\kappa_c\right)\\ + 0&0&\frac{1-r\left(\hat{\boldsymbol{\sigma}^e}\right)}{g_c}f_c\left(\kappa_c\right)\\ \end{bmatrix} \end{equation} -and ‘:’ represents products of two matrices. Hence, $H\left(\boldsymbol{\sigma}^e,\boldsymbol{\kappa}\right)$ in Eq. \eqref{kappa} was obtained as follows +and ‘:’ represents products of two matrices. Hence, $H\left(\boldsymbol{\sigma}^e,\boldsymbol{\kappa}\right)$ in [kappa] was obtained as follows \begin{equation} H\left(\boldsymbol{\sigma}^e, \boldsymbol{\kappa}\right) = \boldsymbol{h}\cdot \nabla_{\hat{\boldsymbol{\sigma}^e}}\Phi\left(\hat{\boldsymbol{\sigma}^e}\right) \label{H_def} \end{equation} @@ -292,11 +307,11 @@ During the plastic corrector step, the returned effective stress should satisfy \mathfrak{F}\left(\boldsymbol{\sigma}^e,f_t^e,f_c^e\right) = 0 \end{split} \end{equation} -As per flow rule in Eq. \eqref{flowRule}, the plastic corrector step, i.e., Eq. \eqref{plasticCorrector} can be rewritten as +As per flow rule in [flowRule], the plastic corrector step, i.e., [plasticCorrector] can be rewritten as \begin{equation} \boldsymbol{\sigma^e}_{n+1} = \boldsymbol{\sigma}_{n+1}^{e^{tr}}-\dot{\gamma}\left(2G\frac{\boldsymbol{s}_{n+1}^e}{\|\boldsymbol{s}_{n+1}^e\|} + 3K\alpha_p\boldsymbol{I}\right) \label{returnMap1} \end{equation} -where $G$ is shear modulus and $K$ is bulk modulus. After separating the volumetric and deviatoric components from Eq. \eqref{returnMap1} following relations can be obtained +where $G$ is shear modulus and $K$ is bulk modulus. After separating the volumetric and deviatoric components from [returnMap1] following relations can be obtained \begin{equation} I_{1|n+1} = I_{1|n+1}^{e^{tr}} - 9K\alpha \alpha_p \dot{\gamma} \label{stressRelation1} \end{equation} @@ -306,36 +321,18 @@ where $G$ is shear modulus and $K$ is bulk modulus. After separating the volumet {\|\boldsymbol{s}^{e}_{n+1}\|} = {\|\boldsymbol{s}_{n+1}^{e^{tr}}\|} - 2G\dot{\gamma} \end{gathered} \end{equation} -Using Eqs. \eqref{stressRelation1} and \eqref{stressRelation2}, Eq. \eqref{returnMap1} can be written as +Using [stressRelation1] and [stressRelation2], [returnMap1] can be written as \begin{equation} \boldsymbol{\sigma}_{n+1}^e = \boldsymbol{\sigma}_{n+1}^{e^{tr}}-\dot{\gamma}\left(2G\frac{\boldsymbol{s}^{e^{tr}}_{n+1}}{\|\boldsymbol{s}_{{n+1}}^{e^{tr}} \|}+ 3K\alpha_p\boldsymbol{I}\right) \label{returnMap2} \end{equation} -In case of plastic deformation, the returned state of stress should lie on the yield surface as per Kuhn-Tucker conditions (Eq. \eqref{khunTuckerConditions}, therefore $\mathfrak{F}\left(\boldsymbol{\sigma}_{n+1}^e,f_t^e,f_c^e\right) = 0$, i.e., -\begin{equation} \label{yfnext} - \begin{gathered} - \alpha I_{1|n+1}^e + \sqrt{3J_{2|n+1}^e} + \beta\left(\boldsymbol{\kappa}\right)<\hat{\boldsymbol{\sigma}}^e_{n+1|max}> \\ - \left(1-\alpha\right)f_c^e\left(\boldsymbol{\kappa}\right) = 0 - \end{gathered} -\end{equation} -Using Eq. \eqref{stressRelation1}, \eqref{stressRelation2}, and \eqref{returnMap2}, Eq. \eqref{yfnext} can be written as -\begin{equation} \label{yfzero} - \begin{gathered} - \alpha\left(I_{1|n+1}^{e^{tr}} - 9K\alpha \alpha_p \dot{\gamma}\right) + - \left(\sqrt{\frac{3}{2}}\|\boldsymbol{s}_{{n+1}}^{e^{tr}}\| - \sqrt{6}G\dot{\gamma}\right)\\+ - \beta\left(\boldsymbol{\kappa}\right)<\hat{\boldsymbol{\sigma}}^e_{n+1|max}> - \left(1-\alpha\right)f_c^e\left(\boldsymbol{\kappa}\right) = 0 - \end{gathered} -\end{equation} -Thus, the plastic multiplier can be by solving Eq. \eqref{yfzero} as -\begin{equation}\label{gammaDef} - \dot{\gamma} = - \begin{cases} - \frac{\alpha I_{1|n+1}^{e^{tr}}+\sqrt{\frac{3}{2}}\|\boldsymbol{s}_{{n+1}}^{e^{tr}}\|-\left(1-\alpha\right)f_c^e\left(\boldsymbol{\kappa}\right)} - {9K \alpha_p + \sqrt{6}G}, & \text{if $\sigma_{m|n+1}^e < 0$}\\ - \frac{\alpha I_{1|n+1}^{e^{tr}}+\sqrt{\frac{3}{2}}\|\boldsymbol{s}_{{n+1}}^{e^{tr}}\|+\beta\left(\boldsymbol{\kappa}\right) \sigma_{m|n+1}^{e^{tr}}-\left(1-\alpha\right)f_c^e\left(\boldsymbol{\kappa}\right)} - {9K \alpha_p + \sqrt{6}G + \beta\left(\boldsymbol{\kappa}\right)\left(2G\frac{s^{e^{tr}}_{m|n+1}}{\|\boldsymbol{s}_{{n+1}}^{e^{tr}} \|}+ 3K\alpha_p\right)}, & \text{otherwise}. - \end{cases} + +$\dot{\gamma}$ is calculated as: + +\begin{equation} \label{plasticparameter} + \dot{\gamma}=\frac{\|\boldsymbol{\sigma}_{n+1}^e - \boldsymbol{\sigma}_{n+1}^{e^{tr}}\|}{\|2G\frac{\boldsymbol{s}^{e^{tr}}_{n+1}}{\|\boldsymbol{s}_{{n+1}}^{e^{tr}} \|}+ 3K\alpha_p\boldsymbol{I}\|} \end{equation} -where $\sigma_{m|n+1}^e$, $\sigma_{m|n+1}^{e^{tr}}$, and $s^{e^{tr}}_{m|n+1}$ are the $m^{th}$ component of the $\hat{\boldsymbol{\sigma}}_{n+1}^e$, $\boldsymbol{\sigma}_{n+1}^{e^{tr}}$, and $\boldsymbol{s}^{e^{tr}}_{n+1}$, respectively, which corresponds to maximum principal effective stress in $\left(n+1\right)^{th}$ step. Eq. \eqref{gammaDef} is solved iteratively. +The plastic parameter, [plasticparameter], is evaluated during each iteration of the return mapping algorithm as the current stress is being updated. !syntax parameters /Materials/DamagePlasticityStressUpdate diff --git a/include/materials/DamagePlasticityStressUpdate.h b/include/materials/DamagePlasticityStressUpdate.h index ebb76b6e..a803e3b0 100644 --- a/include/materials/DamagePlasticityStressUpdate.h +++ b/include/materials/DamagePlasticityStressUpdate.h @@ -65,10 +65,6 @@ class DamagePlasticityStressUpdate : public MultiParameterPlasticityStressUpdate const Real _ac; const Real _zt; const Real _zc; - const Real _dPhit; - const Real _dPhic; - const Real _sqrtPhit_max; - const Real _sqrtPhic_max; const Real _dt_bt; const Real _dc_bc; @@ -133,27 +129,6 @@ class DamagePlasticityStressUpdate : public MultiParameterPlasticityStressUpdate const Real & exponent, const Real & kappa) const; - // /** - // * Obtain the partial derivative of the undamaged tensile strength to the damage state - // * @param intnl (Array containing damage states in tension and compression, respectively) - // * @return value of dft (partial derivative of the tensile strength to the damage state) - // */ - // Real dftbar(const std::vector & intnl) const; - - // /** - // * Obtain the undamaged conpressive strength - // * @param intnl (Array containing damage states in tension and compression, respectively) - // * @return value of fc (compressive strength) - // */ - // Real fcbar(const std::vector & intnl) const; - - // /** - // * Obtain the partial derivative of the undamaged compressive strength to the damage state - // * @param intnl (Array containing damage states in tension and compression, respectively) - // * @return value of dfc - // */ - // Real dfcbar(const std::vector & intnl) const; - // /** // * Obtain the damaged tensile strength // * @param intnl (Array containing damage states in tension and compression, respectively) // * @return value of ft (tensile strength) @@ -162,27 +137,6 @@ class DamagePlasticityStressUpdate : public MultiParameterPlasticityStressUpdate const Real & a, const Real & kappa) const; - // /** - // * Obtain the partial derivative of the damaged tensile strength to the damage state - // * @param intnl (Array containing damage states in tension and compression, respectively) - // * @return value of dft (partial derivative of the tensile strength to the damage state) - // */ - // Real dft(const std::vector & intnl) const; - - // /** - // * Obtain the damaged compressive strength - // * @param intnl (Array containing damage states in tension and compression, respectively) - // * @return value of fc (compressive strength) - // */ - // Real fc(const std::vector & intnl) const; - - // /** - // * Obtain the partial derivative of the damaged compressive strength to the damage state - // * @param intnl (Array containing damage states in tension and compression, respectively) - // * @return value of dfc - // */ - // Real dfc(const std::vector & intnl) const; - /** * Obtain the partial derivative of the undamaged strength to the damage state * @param intnl (Array containing damage states in tension and compression, respectively) @@ -200,13 +154,6 @@ class DamagePlasticityStressUpdate : public MultiParameterPlasticityStressUpdate */ Real beta(const std::vector & intnl) const; - /** - * dbeta_dkappa is a derivative of beta wrt. kappa (plastic damage variable) - * @param intnl (Array containing damage states in tension and compression, respectively) - * @return value of dbeta_dkappa - */ - Real dbeta_dkappa(const std::vector & intnl) const; - /** * dbeta0 is a derivative of beta wrt. tensile strength (ft) * @param intnl (Array containing damage states in tension and compression, respectively) diff --git a/src/materials/DamagePlasticityStressUpdate.C b/src/materials/DamagePlasticityStressUpdate.C index 6a2f4d85..31255e42 100644 --- a/src/materials/DamagePlasticityStressUpdate.C +++ b/src/materials/DamagePlasticityStressUpdate.C @@ -130,15 +130,6 @@ DamagePlasticityStressUpdate::initQpStatefulProperties() used, the following commented lines show several different options. Some other options are still being considered. In this code, we define the element length as the cube root of the element volume */ - - // if (_current_elem->n_vertices() < 3) - // _ele_len[_qp] = _current_elem->length(0, 1); - // else if (_current_elem->n_vertices() < 5) - // _ele_len[_qp] = (_current_elem->length(0, 1) + _current_elem->length(1, 2)) / 2.; - // else - // _ele_len[_qp] = - // (_current_elem->length(0, 1) + _current_elem->length(1, 2) + _current_elem->length(0, 4)) - // / 3.; _ele_len[_qp] = std::cbrt(_current_elem->volume()); _gt[_qp] = _FEt / _ele_len[_qp]; diff --git a/test/tests/damage_plasticity_model/gold/dilatancy_out.csv b/test/tests/damage_plasticity_model/gold/dilatancy_out.csv index 77c7ca92..83b8a04c 100644 --- a/test/tests/damage_plasticity_model/gold/dilatancy_out.csv +++ b/test/tests/damage_plasticity_model/gold/dilatancy_out.csv @@ -5,48 +5,48 @@ time,displacement_x,e_xx,ep_xx,react_x,s_xx,volumetric_strain 30,-0.0003,-0.0003,0,9.5100000000004,-9.5100000000004,-0.00019198156917977 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+500,-0.005,-0.005,-0.0042249056959746,3.2242571376929,-3.2242571376929,0.0068895004943903 diff --git a/test/tests/damage_plasticity_model/gold/shear_test_out.csv b/test/tests/damage_plasticity_model/gold/shear_test_out.csv index 671ae84a..e73d2d32 100644 --- a/test/tests/damage_plasticity_model/gold/shear_test_out.csv +++ b/test/tests/damage_plasticity_model/gold/shear_test_out.csv @@ -2,31 +2,31 @@ time,e_xy,ep_xy,s_xy 0,0,0,0 5,7.8182642983415e-06,0,0.66498115657902 10,1.5636528591598e-05,0,1.3299623130426 -15,2.3452074951896e-05,4.8983821438774e-07,1.8597699615448 -20,3.1171096317802e-05,2.3402390354366e-06,1.9957411207585 -25,3.8953715744496e-05,4.3144052368041e-06,2.1193699480148 -30,4.6800867741694e-05,6.3955307618849e-06,2.2428277546278 -35,5.5655695121865e-05,1.2095885208344e-05,2.2884800606222 -40,6.4887705610769e-05,1.8474922611195e-05,2.3236101673253 -45,7.440502468223e-05,2.5198091613366e-05,2.356566968123 -50,8.4187054613461e-05,3.2241992754993e-05,2.3882556131892 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a/test/tests/damage_plasticity_model/gold/uniaxial_compression_out.csv b/test/tests/damage_plasticity_model/gold/uniaxial_compression_out.csv index 93bb413f..77b90094 100644 --- a/test/tests/damage_plasticity_model/gold/uniaxial_compression_out.csv +++ b/test/tests/damage_plasticity_model/gold/uniaxial_compression_out.csv @@ -5,48 +5,48 @@ time,displacement_x,e_xx,ep_xx,react_x,s_xx,volumetric_strain 30,-0.0003,-0.0003,0,9.5100000000004,-9.5100000000004,-0.00019198156917977 40,-0.0004,-0.0004,0,12.68,-12.68,-0.00025596723479604 50,-0.0005,-0.0005,0,15.85,-15.85,-0.00031994880546093 -60,-0.0006,-0.0006,-1.1804722007126e-05,18.64579033546,-18.64579033546,-0.00036249125782661 -70,-0.0007,-0.0007,-6.2975427612293e-05,20.122808012171,-20.122808012171,-0.00033355073055652 -80,-0.0008,-0.0008,-0.0001154167741537,21.277335768097,-21.277335768097,-0.0003023022102745 -90,-0.0009,-0.0009,-0.00016916425553668,22.340882112175,-22.340882112175,-0.00026868076341113 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-1,5e-05,5.0000000001209e-05,0,-1.5850000000321,1.5850000000321,3.2000512005526e-05 -2,0.0001,0.00010000000000003,0,-3.1700000000012,3.1700000000012,6.4002048043743e-05 -3,0.00015,0.00014999999727429,4.0024734223106e-05,-3.4862156892916,3.4862156892916,9.6779051148882e-05 -4,0.0002,0.00017862734293029,6.8093756626783e-05,-3.477471609198,3.477471609198,0.00011216165953351 -5,0.00025,0.00024835041190431,0.000137933469697,-3.4652098234512,3.4652098234512,0.0001616852321851 -6,0.0003,0.00026993896006456,0.00015888007847208,-3.4291886168407,3.4291886168407,0.00017133095578839 -7,0.00035,0.00034485935625532,0.00023401341905145,-3.3921352348948,3.3921352348948,0.00022512601060176 -8,0.0004,0.00035654059152294,0.0002450169275363,-3.35653185387,3.35653185387,0.0002265865480974 -9,0.00045,0.00043833198349069,0.00032705678256184,-3.3201883999425,3.3201883999425,0.00028619595227733 -10,0.0005,0.00044651293525115,0.00033495631513795,-3.2792710308472,3.2792710308472,0.00028450762708399 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