diff --git a/source/user/manual/material/ndMaterial.rst b/source/user/manual/material/ndMaterial.rst index 32fd14c7..9b92f723 100644 --- a/source/user/manual/material/ndMaterial.rst +++ b/source/user/manual/material/ndMaterial.rst @@ -27,6 +27,7 @@ The following contain information about matType? and the args required for each ndMaterials/ElasticIsotropic ndMaterials/ElasticOrthotropic + ndMaterials/TimberHoffman3D ndMaterials/J2Plasticity ndMaterials/DruckerPrager ndMaterials/ManzariDafalias diff --git a/source/user/manual/material/ndMaterials/TimberHoffman3D.rst b/source/user/manual/material/ndMaterials/TimberHoffman3D.rst new file mode 100644 index 00000000..2c63b707 --- /dev/null +++ b/source/user/manual/material/ndMaterials/TimberHoffman3D.rst @@ -0,0 +1,369 @@ +TimberHoffman3D Material +======================== + +This command is used to construct a nonlinear three-dimensional orthotropic timber material object based on the Hoffman failure criterion, isotropic hardening, and damage formulation. + +Tcl Version +----------- + +.. code-block:: tcl + + nDMaterial TimberHoffman3D $matTag $E1 $E2 $E3 $nu12 $nu13 $nu23 $G12 $G13 $G23 $fc1 $fc2 $fc3 $ft1 $ft2 $ft3 $f12 $f13 $f23 $h $Gf1t $Gf2t $Gf3t $eta + +Python Version +-------------- + +.. code-block:: python + +ops.nDMaterial('TimberHoffman3D', matTag, E1, E2, E3, nu12, nu13, nu23, G12, G13, G23, fc1, fc2, fc3, ft1, ft2, ft3, f12, f13, f23, h, Gf1t, Gf2t, Gf3t, eta) + ++----------------------+-------------+------------------------------------+ +| **Argument** | **Units / | **Description** | +| | type** | | ++======================+=============+====================================+ +| **matTag** | integer | Unique material tag. | ++----------------------+-------------+------------------------------------+ +| **E1, E2, E3** | Stress | Young’s moduli in orthotropic | +| | | material directions 1, 2, and 3. | ++----------------------+-------------+------------------------------------+ +| **nu12, nu13, nu23** | — | Independent Poisson ratios. | ++----------------------+-------------+------------------------------------+ +| **G12, G13, G23** | stress | Shear moduli in planes 1–2, 1–3, | +| | | and 2–3. | ++----------------------+-------------+------------------------------------+ +| **fc1, fc2, fc3** | stress | Compressive strengths in | +| | | directions 1, 2, and 3. | ++----------------------+-------------+------------------------------------+ +| **ft1, ft2, ft3** | stress | Tensile strengths in directions 1, | +| | | 2, and 3. | ++----------------------+-------------+------------------------------------+ +| **f12, f13, f23** | stress | Shear strengths in planes 1–2, | +| | | 1–3, and 2–3. | ++----------------------+-------------+------------------------------------+ +| **h** | stress | Linear isotropic hardening | +| | | modulus. | ++----------------------+-------------+------------------------------------+ +| **Gf1t, Gf2t, Gf3t** | energy/area | Tensile fracture energies in | +| | | directions 1, 2, and 3. | ++----------------------+-------------+------------------------------------+ +| **eta** | — | Viscous regularization parameter | +| | | for damage evolution. | ++----------------------+-------------+------------------------------------+ + +**Theory:** + +1. **Orthotropic linear-elastic constitutive behavior:** + +In the elastic range the orthotropic material stress-strain relationship +can be written in the form as: + + |image1| + + Where σ and ϵ are the applied stress and corresponding strain + tensors, respectively. E is the orthogonal elastic matrix which is + given as: + +|image2| + +Where *E\ i* represents the young’s moduli in the three orthotropic +directions, υ\ :sub:`ij` and G\ :sub:`ij` are the Poisson’s ratio and +shear moduli, respectively. Direction 1 is the direction parallel to +grain (longitudinal) and directions 2 and 3 are the radial and +tangential directions. + +2. **Elastoplastic constitutive behavior:** + +A constitutive relationship is adopted for timber material based on the +isotropic hardening elastoplastic model using the Hoffman yield +criterion, which is an extension of Hill’s criterion. **Hoffman yield +function** along with the yield criterion combined with isotropic +hardening is formulated as: + + |image3| + +|image4| + +|image5| + +Where; **P** and **Q** are mapping matrices with the variables as stated +in above formulas in which the **σ\ e** is the equivalent yield stress +and **σ\ ek** is the formulation stated as below: + +|image6| + +In which the h is the hardening modulus and k is the hardening variable. +“\ *k*\ ” depends on the equivalent plastic strain obtained by means +iterative method (Newton-Raphson) for each load increment. However, “h” +is an input parameter of the proposed model which can be directly +obtained from the uniaxial stress–strain diagram. Depending on the +chosen value of h, perfect plasticity, slight hardening or softening can +be modelled. “h” parameter can simply be computed from: + +|image7| + +In which the **T\ 2** is the plastic tangent modulus in the orthotropic +2 direction. In the hardening state, each yield value will change +depending on the corresponding hardening parameter ki (i =1,2,..,9). +Thus, the parameters in the mapping vector Q change also. Isotropic +hardening is considered in this paper, because it assumes that the +initial yield surface expands uniformly without translation and +distortion as plasticity occurs. Thus, all the yield values change in +the same order. Consequently, the parameters α11, α22 and α33 in the +mapping vector Q will change depending on the hardening parameter k. + +|image8| + +If the stress reaches the yield condition, the elastoplastic radial +mapping algorithm is used to constrain stress point on the yield +surface. According to the theory of plasticity, the total strain +increment *Δε* can be divided into elastic component *Δε\ e* and plastic +component *Δε\ p* as: + +|image9| + +Where the associated flow rule was used to determine the plastic strain +as: + +|image10| + +In which Δλ denotes the plastic multiplier and ∂f ∂σ is the gradient +vector of the plastic potential energy function of f(σ,k) which can be +written as: + +|image11| + +3. **Damage Evolution:** + +Anisotropic damage with non-uniform distribution of micro cracks and +voids in all directions of the timber matrix is considered. To consider +the anisotropic damage, three different damage factors must be +introduced. The reduction of the elastic stiffness matrix is simulated +here as a function of damage evolution in three orthogonal directions. +Therefore, the damaged stiffness matrix with the damage factors can be +written as: + +|image12| + +where d1, d2 and d3 are the damage factors defined in the longitudinal, +radial and tangential directions, respectively. The damage factor, di (i +=1,2,3), increases from 0 to 1 as damage grows from threshold value to +its ultimate value. + +Since the timber has different damage evolution for tensile and +compression failures, the damage models are used to define the tensile +and compression damage evolutions. In this study, different tensile +damages in three orthogonal directions and compressive damage in only +longitudinal direction are considered as: + +|image13| + +In which dit, and Gfit are the tensile damage factor and fracture energy +under tension in the i-th direction of the timber, respectively. d1c is +the compression damage factor in the longitudinal direction of the +timber. Lc is the element characteristic length. Fit and F1c are the +variables and “A” and “B” are the material constants. + +The stress-based continuum damage formulation with different failure +criteria in tension and compression is used in this study. It means, +damage starts to accumulate as soon as the stresses satisfy the failure +criterion. The failure criterion is described as: + +|image14| + +|image15| + +Damage models can encounter convergence problems in static analysis. To +improve the convergence, viscous regularisation method is used to define +the model. In the viscous regularisation, the response of damaged +material is evaluated using a viscous regularised damage variable dv +which is defined as: + +|image16| + +where η is the viscous parameter. A small value of the viscous parameter +usually helps to improve the convergence rate of the model without +significantly influencing the results. Thus, the value of viscous +parameter is chosen to be equal to 0.0001. + +**Responses/Backbones:** + +**Example: +** + ++-----------------+----------------+-----------------+----------------+ +| **Property** | **Value** | **Property** | **Value** | ++=================+================+=================+================+ +| **E1** | 2050.8 | **E2=E3** | 172.1 | ++-----------------+----------------+-----------------+----------------+ +| **G23** | 68 | **G12=G13** | 145.2 | ++-----------------+----------------+-----------------+----------------+ +| **ν23** | 0.5 | **ν12=ν13** | 0.45 | ++-----------------+----------------+-----------------+----------------+ +| **fc1** | 35 | **fc2=fc3** | 2.5 | ++-----------------+----------------+-----------------+----------------+ +| **ft1** | 20 | **ft2=ft3** | 0.7 | ++-----------------+----------------+-----------------+----------------+ +| **f23** | 0.5 | **f12=f13** | 5 | ++-----------------+----------------+-----------------+----------------+ +| **h** | 1200 | **Gf1t** | 60 | ++-----------------+----------------+-----------------+----------------+ +| **Gf2t=Gf3t** | 0.5 | **η** | 0.0001 | ++-----------------+----------------+-----------------+----------------+ + +**Without the Q evolution:** + +|image17| + +**With the Q evolution:** + +**Effect of damage and hardening on cyclic response.** The +perpendicular-to-grain cyclic response exhibited progressively narrower +hysteresis loops following the onset of damage. This behavior reflects +the combined effects of stiffness degradation and isotropic hardening in +the constitutive formulation. As damage accumulated, the effective +stiffness decreased substantially, resulting in reduced nominal stress +during subsequent loading cycles. Concurrently, evolution of the Hoffman +yield surface through the hardening variable Q(k) allowed a larger +portion of the unloading and reloading path to remain within the +expanded yield surface, thereby limiting additional plastic deformation +and associated hysteretic dissipation. Consequently, the combination of +damage-induced stiffness degradation and reduced repeated plastic +yielding produced thinner cyclic loops compared with the response +obtained using the initial, non-evolving Q0. In the perpendicular +specimen, this effect was particularly pronounced after localized damage +approached unity, while the volume-averaged damage remained +approximately 0.5, indicating a strongly nonuniform damage state. + +|image18| + +|image19| + +**Multi-Axial Tests:** + +**Combined compression–shear response.** The constitutive response under +combined longitudinal compression and 1-2 shear was examined using a +prescribed non-monotonic strain path involving shear reversal. During +the initial loading stage, the equivalent plastic strain (k) increased +together with the isotropic hardening variables Qi(k), while +longitudinal damage (d1) progressively developed. During subsequent +shear reversal, the normalized Hoffman yield function became negative, +and both k and Qi(k) remained approximately constant, indicating elastic +unloading/reloading within the expanded yield surface. Continued reverse +loading brought the effective stress state back to the yield surface, +producing a second increase in equivalent plastic strain and hardening. +The accumulated directional damage remained irreversible throughout the +reversal, with substantial damage developing only in the longitudinal +direction. Overall, the response demonstrates the interaction of Hoffman +plasticity, isotropic hardening, directional damage, and loading-path +dependence under combined normal and shear deformation. The results are +interpreted as constitutive material-point diagnostics rather than +specimen-level or experimental validation. + +|image20| + +**Transverse compression–shear response.** The constitutive response +under combined transverse compression and 2-3 shear was examined using a +non-monotonic loading path involving shear reversal. During the initial +loading stage, the equivalent plastic strain (k) increased together with +the isotropic hardening variables Qi(k), indicating activation of the +Hoffman plasticity model and expansion of the yield surface. During +subsequent shear reversal, the normalized Hoffman yield function became +negative, while k and Qi(k) remained approximately constant, indicating +elastic unloading/reloading within the expanded yield surface. Continued +reverse loading brought the effective stress state back to the yield +surface, resulting in a second increase in equivalent plastic strain and +isotropic hardening. In contrast to the longitudinal compression and 1-2 +shear case, no appreciable directional damage developed during this +loading history, demonstrating that plastic yielding and damage +initiation remain distinct mechanisms within the constitutive +formulation. Overall, the response demonstrates the interaction of +anisotropic Hoffman plasticity, isotropic hardening, and loading-path +reversal under combined transverse compression and shear. The results +are interpreted as constitutive material-point diagnostics rather than +specimen-level or experimental validation.\ |image21| **Biaxial +compression response.** The constitutive response under biaxial +compression was examined by simultaneously prescribing compressive +strains in the longitudinal and transverse material directions, followed +by unloading. During the loading stage, plastic yielding resulted in an +increase in the equivalent plastic strain (k) together with the +isotropic hardening variables Qi(k), indicating progressive expansion of +the Hoffman yield surface. Directional damage developed predominantly in +the longitudinal direction, with d1 increasing substantially while d2 +and d3 remained negligible over most of the loading history. During +unloading, the normalized Hoffman yield function became negative and +moved into the interior of the evolved yield surface, while k and Qi(k) +remained approximately constant, indicating that no significant +additional plastic evolution occurred. The resulting nonlinear and +path-dependent stress response reflects the combined effects of +multiaxial Hoffman plasticity, isotropic hardening, and directional +damage. The results are interpreted as constitutive material-point +diagnostics rather than specimen-level or experimental validation. + +|image22| + +.. |image1| image:: media/image1.png + :width: 0.83341in + :height: 0.29169in +.. |image2| image:: media/image2.png + :width: 6.5in + :height: 2.87083in +.. |image3| image:: media/image3.png + :width: 2.87525in + :height: 0.58338in +.. |image4| image:: media/image4.png + :width: 4.95876in + :height: 1.47513in +.. |image5| image:: media/image5.png + :width: 4.00833in + :height: 5.56876in +.. |image6| image:: media/image6.png + :width: 1.56712in + :height: 0.40833in +.. |image7| image:: media/image7.png + :width: 1.42417in + :height: 0.57222in +.. |image8| image:: media/image8.png + :width: 2.26686in + :height: 0.66672in +.. |image9| image:: media/image9.png + :width: 1.20844in + :height: 0.37503in +.. |image10| image:: media/image10.png + :width: 1.35012in + :height: 0.58338in +.. |image11| image:: media/image11.png + :width: 3.892in + :height: 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