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| In [[solid mechanics]], the '''Johnson–Holmquist damage model''' is used to model the mechanical behavior of damaged [[brittle]] materials, such as [[ceramic materials|ceramics]], [[rock (geology)|rock]]s, and [[concrete]], over a range of [[strain rate]]s. Such materials usually have high compressive strength but low tensile strength and tend to exhibit progressive damage under load due to the growth of [[microfracture]]s.
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| There are two variations of the Johnson-Holmquist model that are used to model the impact performance of ceramics under [[Terminal ballistics|ballistic]]ally delivered loads.<ref name=Walker>Walker, James D. ''Turning Bullets into Baseballs'', ''' SwRI Technology Today''', Spring 1998 http://www.swri.edu/3pubs/ttoday/spring98/bullet.htm</ref> These models were developed by Gordon R. Johnson and Timothy J. Holmquist in the 1990s with the aim of facilitating predictive numerical simulations of ballistic armor penetration. The first version of the model is called the 1992 Johnson-Holmquist 1 (JH-1) model.<ref name="JH1">Johnson, G. R. and Holmquist, T. J., 1992, ''A computational constitutive model for brittle materials subjected to large strains'', '''Shock-wave and High Strain-rate Phenomena in Materials''', ed. M. A. Meyers, L. E. Murr and K. P. Staudhammer, Marcel Dekker Inc. , New York, pp. 1075-1081.</ref> This original version was developed to account for large deformations but did not take into consideration progressive damage with increasing deformation; though the multi-segment stress-strain curves in the model can be interpreted as incorporating damage implicitly. The second version, developed in 1994, incorporated a damage evolution rule and is called the Johnson-Holmquist 2 (JH-2) model<ref name=JH2>Johnson, G. R. and Holmquist, T. J., 1994, ''An improved computational constitutive model for brittle materials'', '''High-Pressure Science and Technology''', American Institute of Physics.</ref> or, more accurately, the Johnson-Holmquist damage material model.
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| == Johnson-Holmquist 2 (JH-2) material model ==
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| The Johnson-Holmquist material model (JH-2), with damage, is useful when modeling brittle materials, such as ceramics,
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| subjected to large pressures, shear strain and high strain rates. The model attempts to include the phenomena encountered when brittle materials are subjected to load and damage, and is one of the most widely used models when dealing with ballistic impact on ceramics. The model simulates the increase in strength shown by ceramics subjected to hydrostatic pressure as well as the reduction in strength shown by damaged ceramics. This is done by basing the model on two sets
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| of curves that plot the yield stress against the pressure. The first set of curves accounts for the intact material, while the second one accounts for the failed material. Each curve set depends on the plastic strain and plastic strain rate. A damage variable D accounts for the level of fracture. | |
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| === Intact elastic behavior ===
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| The JH-2 material assumes that the material is initially elastic and isotropic and can be described by a relation of the form (summation is implied over repeated indices)
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| :<math>
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| \sigma_{ij} = -p(\epsilon_{kk})~\delta_{ij} + 2~\mu~\epsilon_{ij}
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| </math>
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| where <math>\sigma_{ij}</math> is a [[stress measures|stress measure]], <math>p(\epsilon_{kk})</math> is an [[equation of state]] for the pressure, <math>\delta_{ij}</math> is the [[Kronecker delta]], <math>\epsilon_{ij}</math> is a [[finite strain theory|strain measure]] that is energy conjugate to <math>\sigma_{ij}</math>, and <math>\mu</math> is a [[shear modulus]]. The quantity <math>\epsilon_{kk}</math> is frequently replaced by the hydrostatic compression <math>\xi</math> so that the equation of state is expressed as
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| :<math> | |
| p(\xi) = p(\xi(\epsilon_{kk})) = p\left(\cfrac{\rho}{\rho_0}-1\right) ~;~~ \xi := \cfrac{\rho}{\rho_0}-1
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| </math> | |
| where <math>\rho</math> is the current mass density ans <math>\rho_0</math> is the initial mass density.
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| The stress at the [[Hugoniot elastic limit]] is assumed to be given by a relation of the form
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| :<math>
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| \sigma_h = \mathcal{H}(\rho, \mu) = p_{\rm HEL}(\rho) + \cfrac{2}{3}~\sigma_{\rm HEL}(\rho, \mu)
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| </math> | |
| where <math>p_{\rm HEL}</math> is the pressure at the Hugoniot elastic limit and <math>\sigma_{\rm HEL}</math> is the stress at the Hugoniot elastic limit.
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| === Intact material strength ===
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| The uniaxial failure strength of the intact material is assumed to be given by an equation of the form
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| :<math>
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| \sigma^{*}_{\rm intact} = A~(p^* + T^*)^n~\left[1 + C~\ln\left(\cfrac{d\epsilon_p}{dt}\right)\right]
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| </math>
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| where <math>A, C, n</math> are material constants, <math>t</math> is the time, <math>\epsilon_p</math> is the inelastic strain. The inelastic strain rate is usually normalized by a reference strain rate to remove the time dependence. The reference strain rate is generally 1/s.
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| The quantities <math>\sigma^{*}</math> and <math>p^*</math> are normalized stresses and <math>T^*</math> is a normalized tensile strength, defined as
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| :<math>
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| \sigma^* = \cfrac{\sigma}{\sigma_{\rm HEL}} ~;~
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| p^* = \cfrac{p}{p_{\rm HEL}} ~;~~ T^* = \cfrac{T}{\sigma_h}
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| </math>
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| === Stress at complete fracture ===
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| The uniaxial stress at complete fracture is assumed to be given by
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| :<math>
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| \sigma^{*}_{\rm fracture} = B~(p^*)^m~\left[1 + C~\ln\left(\cfrac{d\epsilon_p}{dt}\right)\right]
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| </math>
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| where <math>B, C, M</math> are material constants.
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| === Current material strength ===
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| The uniaxial strength of the material at a given state of damage is then computed at a linear interpolation between the initial strength and the stress for complete failure, and is given by
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| :<math>
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| \sigma^{*} = \sigma^{*}_{\rm initial} - D~\left(\sigma^{*}_{\rm initial} - \sigma^{*}_{\rm fracture}\right)
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| </math>
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| The quantity <math>D</math> is a scalar variable that indicates damage accumulation.
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| === Damage evolution rule ===
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| The evolution of the damage variable <math>D</math> is given by
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| :<math>
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| \cfrac{dD}{dt} = \cfrac{1}{\epsilon_f}~\cfrac{d\epsilon_p}{dt}
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| </math>
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| where the strain to failure <math>\epsilon_f</math> is assumed to be
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| :<math>
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| \epsilon_f = D_1~(p^* + T^*)^{D_2}
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| </math>
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| where <math>D_1, D_2</math> are material constants.
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| === Material parameters for some ceramics ===
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| {| class="wikitable sortable" cellpadding="7" cellspacing="3" border="1" style="border-collapse: collapse"
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| |- bgcolor="#cccccc"
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| !material!!<math>\rho_0</math>!!<math>\mu</math>!!A!!B!!C!!m!!n!!<math>D_1</math>!!<math>D_2</math>!!<math>\sigma_h</math>!!Reference
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| |-
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| |(kg-m<sup>−3</sup>)
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| |(GPa)
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| | (GPa)
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| |-
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| | [[Boron carbide]] <math>B_4C</math>
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| | 2510
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| | 197
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| | 0.927
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| | 0.7
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| | 0.005
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| | 0.85
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| | 0.67
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| | 0.001
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| | 0.5
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| | 19
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| | <ref name=Cronin>Cronin, D. S., Bui, K., Kaufmann, C., 2003, ''Implementation and validation of the Johnson-Holmquist ceramic material model in LS-DYNA'', in '''Proc. 4th European LS-DYNA User Conference (DYNAmore)''', Ulm, Germany. http://www.dynamore.de/dynalook/eldc4/material/implementation-and-validation-of-the-johnson</ref>
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| |-
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| | [[Silicon carbide]] <math>SiC</math>
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| | 3163
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| | 183
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| | 0.96
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| | 0.35 | |
| | 0 | |
| | 1
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| | 0.65
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| | 0.48
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| | 0.48
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| | 14.6
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| | <ref name=Cronin/>
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| |-
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| | [[Aluminum nitride]] <math>AlN</math>
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| | 3226
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| | 127
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| | 0.85
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| | 0.31
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| | 0.013
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| | 0.21
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| | 0.29
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| | 0.02
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| | 1.85
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| | 9
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| | <ref name=Cronin/>
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| |-
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| | [[Alumina]] <math>Al_2O_3</math>
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| | 3700
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| | 90
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| | 0.93
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| | 0.31
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| | 0
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| | 0.6
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| | 0.6
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| | 0.005
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| | 1
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| | 2.8
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| | <ref name=Cronin/>
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| |-
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| | [[Silica]]float glass
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| | 2530
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| | 30
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| | 0.93
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| | 0.088
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| | 0.003
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| | 0.35
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| | 0.77
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| | 0.053
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| | 0.85
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| | 6
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| | <ref name=Cronin/>
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| |}
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| == Johnson–Holmquist equation of state ==
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| The function <math>p(\xi)</math> used in the Johnson–Holmquist material model is often called the '''Johnson–Holmquist equation of state''' and has the form
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| :<math>
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| p(\xi) = \begin{cases}
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| k_1~\xi + k_2~\xi^2 + k_3~\xi^3 + \Delta p & \qquad \text{Compression} \\
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| k_1~\xi & \qquad \text{Tension}
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| \end{cases}
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| </math>
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| where <math>\Delta p</math> is an increment in the pressure and <math>k_1, k_2, k_3</math> are material constants. The increment in pressure arises from the conversion of energy loss due to damage into internal energy. Frictional effects are neglected.
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| == Implementation in LS-DYNA ==
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| The Johnson-Holmquist material model is implemented in [[LS-DYNA]] as * MAT_JOHNSON_HOLMQUIST_CERAMICS.<ref>McIntosh, G., 1998, ''The Johnson-Holmquist ceramic model as used in ls-DYNA2D'', '''Report # DREV-TM-9822:19981216029''', Research and Development Branch, Department of National Defence, Canada, Valcartier, Quebec. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA357607&Location=U2&doc=GetTRDoc.pdf</ref>
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| == References ==
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| {{reflist}}
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| == See also ==
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| * [[Failure]]
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| * [[Material failure theory]]
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| {{DEFAULTSORT:Johnson-Holmquist Damage Model}}
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| [[Category:Solid mechanics]]
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| [[Category:Equations]]
| |
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Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. Wordpress have every reason with it which promote wordpress development. This is the reason for the increased risk of Down Syndrome babies in women over age 35. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. As soon as you start developing your Word - Press MLM website you'll see how straightforward and simple it is to create an online presence for you and the products and services you offer.
The entrepreneurs can easily captivate their readers by using these versatile themes. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. This platform can be customizedaccording to the requirements of the business. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait.
You can add keywords but it is best to leave this alone. High Quality Services: These companies help you in creating high quality Word - Press websites. Enterprise, when they plan to hire Word - Press developer resources still PHP, My - SQL and watch with great expertise in codebase. It's now become a great place to sell it thanks to Woo - Commerce. If you liked this article and you would like to obtain additional details relating to wordpress backup kindly browse through the web site. OSDI, a Wordpress Development Company based on ahmedabad, India.
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