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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.
| | They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. The next step is to visit your Word - Press blog dashboard. These templates are professionally designed and are also Adsense ready. s ultimately easy to implement and virtually maintenance free. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. <br><br> |
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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.
| | The Internet is a vast open market where businesses and consumers congregate. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. Should you have any concerns about wherever in addition to how to employ [http://kedahan.net/backup_plugin_322839 wordpress backup], you can e mail us with the web site. This is the reason for the increased risk of Down Syndrome babies in women over age 35. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. 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. <br><br>Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. As of now, Pin2Press is getting ready to hit the market. Use this section to change many formatting elements. The first thing you need to do is to choose the right web hosting plan. After that the developer adds the unordered list for navigations. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. Word - Press is the most popular open source content management system (CMS) in the world today. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <br><br>A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. s ability to use different themes and skins known as Word - Press Templates or Themes. The days of spending a lot of time and money to have a website built are long gone. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. You can check out the statistics of page of views for your web pages using free tools that are available on the internet. |
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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>
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| 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>
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| 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 | |
| :<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>
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| 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 | |
| :<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
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| | 0
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| | 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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They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. The next step is to visit your Word - Press blog dashboard. These templates are professionally designed and are also Adsense ready. s ultimately easy to implement and virtually maintenance free. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting.
The Internet is a vast open market where businesses and consumers congregate. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. Should you have any concerns about wherever in addition to how to employ wordpress backup, you can e mail us with the web site. This is the reason for the increased risk of Down Syndrome babies in women over age 35. Being able to help with your customers can make a change in how a great deal work, repeat online business, and referrals you'll be given. 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.
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A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. s ability to use different themes and skins known as Word - Press Templates or Themes. The days of spending a lot of time and money to have a website built are long gone. Working with a Word - Press blog and the appropriate cost-free Word - Press theme, you can get a professional internet site up and published in no time at all. You can check out the statistics of page of views for your web pages using free tools that are available on the internet.