Explosions become a very attractive research area in the last decades. This is due to the increase of accidental and intentional explosions. Historical structure were not designed and built against the extreme loading events. Homogenization Techniques were developed to assess the masonry response.

1. Department of Structural Engineering
Faculty of Civil Engineering
Budapest University of Technology and Economics
Examples of Homogenization Techniques in the Material
Modeling Under the Effect of Blast Loading
Lecturer:
Prof. Dr. Imre Bojtar
by:
Kezh Sardasht (BKE6RW)
Research paper
Material Models in Mechanics
(BMEEOTMDT72)
2019-12-02

2. General Overview
 Explosions become a very attractive research area in
the last decades.
 This is due to the increase of accidental and
intentional explosions.
 Historical structure were not designed and built
against the extreme loading events.
 Homogenization Techniques were developed to assess
the masonry response.
Vs. The importance and application of homogenization
techniques are elaborated based on the previous
scientific works.
 Representative volume element (RVE) response under
various stress conditions has been simulated
numerically.
 For which equivalent elastic properties, strength
envelope, and different failure patterns of masonry
material are homogenized,2D and 3D.
In this Report:
 Be able to measure the structural response, predict the deterioration pattern, and analyze
large scale structures numerically.
Finally:

3. Cont’…
 Two methods, namely discrete and continuum model, were developed to understand the
mechanical behavior of masonry structures.
Diagonal strut model
1. The Discrete Element Method (EDM).
Refined continuum model
1. The Continuum Model.
 Masonry structure is periodic structure:

4. Homogenization of Masonry
Using Numerical Simulations
(Hong H. et
al. (2001))

5. Summary
 The examples given below are derived from the works of Hong et al. in 2001 and 2005.
 Both in-plane and out-of-plane were incorporated in the analysis.
 The first work was 2D, and then the approach is extended to 3D in the second study.
 The response of masonry structures was evaluated against blast loading.
 Mode I: Mortar Tension Failure
 Mode II: Shear Failure of Mortar and Brick
 Mode III: Brick Compressive Failure
Failure Modes:

6. Representative Volume Element
(RVE)
 There are some abstract measures should be taken to select the best fitted RVE, namely
 It includes all the existing materials in the model,
 Constitute the entire structure by periodic and continuous distribution,
 And it has to be the smallest volume which is able to achieve the first two conditions
 The average stress and strain can be derived as follows
 The constitutive relation of mortar joint can be as following:

7. Mode I failure
 The strength degradation can be expressed as following.
 Total fracture release energy per unit thickness of masonry:
 Tensile damage failure mode (Mode I)
Mode II failure
 The current failure mode is associated with compressive normal stress p > 0. Mohr-
Coulomb yield criterion can be used in this region.
 According to the traditional plastic flow rule, the complete elasto-plastic incremental
stress-strain relationship is presented as follows:

8.  Then incremental plastic work is
 The damage value at the compressive-shear region can be written as following:
Mode III failure
 Damage of brick is composed of compressive crushing and tensile splitting due to high
compression
 The damage scalar consists of two parts

9.  The weights At and Ac can be defined as followings:
 The current material model is encoded into finite element program to generate numerical
analysis.
Numerical Simulation

10.  Two different configuration of RVE
 The utilization of continuum model requires strength properties of equivalent materials.
Based on the various stress-strain conditions, the equivalent strength envelope can be
derived.
Strength Envelope

11. Derivation of 3D Masonry
Properties Using Numerical
Homogenization
Technique (Hong H. et al.(2005))

12. Summary
 Equivalent elastic properties, strength envelope, and failure characteristics of 3D masonry
model was numerically simulated based on homogenization technique and damage
mechanics theory.
 The stress-strain relations of masonry material under various conditions were obtained
from the simulation.
 The accuracy and reliability of the model was verified in order to use the homogenized
material properties in the study of large scale masonry structure.
 In order to derive the homogenized inelastic material properties of masonry Representative
Volume Element (RVE), a reliable material model for masonry components (brick and
mortar) is important.
Numerical Model for Mortar and Brick
 Uniaxial Damage Variables in Tension and Compression

13.  Based on the continuum damage theorem,
damage elastic constitutive is as following
 The degradation damage variable is not
directly calculated by the simple
summation of fracture energy density in
tension and compression, but instead it can
be expressed as
 The yield strength of quasi-brittle
materials is of non-linearity and can be
modeled by Drucker-Prager strength
criterion
 The inelastic strain which is expressed as a non-associated
flow rule can be calculated as described in the 2D
homogenization section.

14.  It can be characterized by two damage scalars, namely, Dt and Dc corresponding,
respectively, to the damage measured in tension and compression states. They are defined
as
Numerical Results
 The masonry wall model is encoded into a finite element program to evaluate the stress-
strain relations of the RVE in the numerical analysis.
 The size effect, and existence of openings were investigated to determine the capability of
the proposed material model

15.  The stress-strain relation have been analyzed under the various stress status, this is to
have a general picture of the material properties.

16.  The equivalent elastic moduli and Poissons ratio of the masonry material can be calculated
using the stress-strain relations corresponding to the uniaxial compressive conditions in the
three directions as
 The out-of-plane elastic moduli is the greatest while the vertical one is the smallest.
 It is also seen that the in-plane results are the same of the ones in the 2D model.
 The strength properties of quasi-brittle materials such as masonry are important
parameters needed in masonry structure failure modelling.

17. The 3D Homogenized Model
 A numerical model for homogenized masonry material can be developed by using the
threshold tensile strains and the yield surface of masonry presented in the above section.
 The constitutive relations of the homogenized masonry material at macro level can be

18.  The applicability of the proposed material model is examined by simulation the 1.92m by
1.92m masonry wall which includes 8x8 basic cells under the effect of air-blast loading.
Numerical Application of the
Homogenized Masonry Model
 The air blast overpressure is applied. The predicted pressure time histories are applied on
the surface of the masonry panel as input for the analysis.
 The masonry units of both continuum and homogenized models are exposed to two different
types of blast loading.

19.  Type I, is the response of the models at time 0.025s
 Type I, is the response of the models at time 0.105s

20. Recommendation
 It is well known, that the blast over-pressure is strain rate dependent loading type. It is
estimated that the high strain rate of blast-over pressure affects the material properties
and it is a must to incorporate the effect of the high strain rate ~ 10000m/s.
 The second points is, in case of blown of situation which it is common phenomenon in the
extreme events such as explosions, the blown off of bricks might be seen clearly in the
continuum model but it is not accurately expected in the homogenized model because of the
uniform material properties used in the homogenized model. Therefore, if fragmentation is
of concern, more detailed masonry model needs be developed.
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References
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