A New Method to Investigate the Size Effect and Anisotropy of Mechanical Properties of Columnar Jointed Rock Mass

Abstract

As a special type of rock mass, the columnar jointed rock masses (CJRMs) present anisotropic mechanical properties and complicated failure modes due to the existence of columnar joints and micro-fractures. The determinations of mechanical properties and representative volume element (RVE) of CJRMs are important for the design, construction, and stability evaluation of hydropower projects. In this paper, the numerical simulations of CJRMs with different inclination angles and sizes under uniaxial and true triaxial compression conditions were conducted to investigate the size effect and anisotropic characteristics of mechanical properties. Results showed the failure modes presented anisotropic and size effect characteristics and the confining pressure influenced the failure modes by providing lateral enhancement. In addition, the mechanical properties first fluctuated and then reached stability as size increased, and the RVE were concentrated in 2 m to 3 m. The RVE were affected by the loading conditions and the CJRMs with inclination angles of 90° and 75° exhibited the smallest and largest RVE respectively. The anisotropy coefficients (ACs) of mechanical properties possessed a RVE of 2.5 m. Based on the simulations, a new method was proposed to investigate the correlation between the size effect and anisotropy of mechanical properties. Results showed the mechanical properties of CJRMs presented most stably at the inclination angle of 90°, and the inclination angles ranging from 40° to 75° were the most unfavorable. Based on this method, the anisotropy of size effect and the size effect of anisotropy, which were difficult to investigate in the past, could be discussed in detail through numerical simulations.

Highlights

• A new method aimed to acquire the values and representative elementary volume of mechanical properties of columnar jointed rock mass was proposed and testified.

• The failure modes of columnar jointed rock mass at different inclination angles presented anisotropic and size effect characteristics and they were influenced by the confining pressure.

• The representative elementary volume of compressive strength and elastic module were ranging from 2.5 to 3 m and presented anisotropic characteristics.

• The anisotropy coefficients of elastic module were stabilized at about 1.3 and the representative elementary volumes were both around 2 m.

Appendices

Appendix A. Code of Generating Voronoi diagram based on random points

[x,y] = meshgrid(0.0015:0.0015:1);

i = length(x);

x = x + (rand(i)-0.5)*randi([10,15],1)*0.01;

y = y + (rand(i)-0.5)*randi([10,15],1)*0.01;

p = randi(length(x(:)),1,round(length(x(:))/10));

x = x(:);

x(p) = [];

y = y(:);

y(p) = [];

data = [x(:),y(:)];

dots = delaunayTriangulation(data);

edge = dots.ConnectivityList;

num = [];

for i = 1:length(edge).

if n or m(dots. Points (edge (i, 1)) − dots. Points (edge (i, 2))) < 0.001.

num = [num,i];

end.

end.

edge(num,:) = [];

edge = [edge,edge];

edge = [edge(:,[1,2]);edge(:,[3,4]);edge(:,[5,6])];

edge = sortrows(edge,[1,2]);

result = tabulate(edge(:,1));

r1 = result;

result = tabulate(result(:,2));

r2 = result;

c = find(r2(:,1) =  = 3);

r3 = r2([c:end],[1,2]);

r3(:,2) = r3(:,2)./sum(r3(:,2));

sum_3456 = sum(r3(1:4,2)).

sum_triplus = sum(result([3,4,5,6],3));

delaunay_voronoi(x,y); %See Appendix B. Code of the Function of Voronoi diagram.

[v,c] = voronoin(data);

area = [];

for i = 1:length(c).

area_add = polyarea(v(c{i},1),v(c{i},2))*400,000,000;

if area_add <  = 100,000.

area = [area,area_add];

end

end

sum_area = [];

sum_area(1) = length(area(find(area <  = 2500)));

sum_area(2) = length(area(find(area > 2500&area < 10,000)));

sum_area(3) = length(area(find(area >  = 10,000&area < 22,500)));

sum_area(4) = length(area(find(area >  = 22,500)));

sum_area_portion = sum_area/sum(sum_area);

Appendix B. Code of the Function of Voronoi diagram (referenced from MATLAB)

Function [vxx,vy] = delaunay_voronoi(varargin).

if nargin > 0.

[varargin{:}] = convertStringsToChars(varargin{:});

end

[cax,args,nargs] = axescheck(varargin{:});

if nargs < 1.

error(message('error'));

elseif nargs > 4.

error(message('error'));

end

if isa(args{1}, 'DelaunayTri').

dt = args{1};

if dt.size(1) =  = 0.

error(message('error'));

elseif dt.size(2) ~  = 3.

error(message('error'));

end

x = dt.X(:,1);

y = dt.X(:,2);

tri = dt(:,:);

if nargs =  = 1.

ls = '';

else

ls = args{2};

end

elseif isa(args{1}, 'delaunayTriangulation').

dt = args{1};

if dt.size(1) =  = 0.

error(message('error'));

elseif dt.size(2) ~  = 3.

error(message('error'));

end

x = dt.Points(:,1);

y = dt.Points(:,2);

tri = dt(:,:);

if nargs =  = 1.

ls = '';

else.

ls = args{2};

end

else

x = args{1};

y = args{2};

if ~ isequal(size(x),size(y)).

error(message('error'));

end

if ~ ismatrix(x) ||~ ismatrix(y).

error(message('error'));

end

x = x(:);

y = y(:);

if nargs =  = 2.

tri = delaunay(x,y);

ls = '';

else

arg3 = args{3};

if nargs =  = 3.

ls = '';

else

arg4 = args{4};

ls = arg4;

end

if isempty(arg3).

tri = delaunay(x,y);

elseif ischar(arg3).

tri = delaunay(x,y);

ls = arg3;

elseif iscellstr(arg3) || isstring(arg3).

error(message('error'));

else

tri = arg3;

end

end

end

if isempty(tri).

return;

end

tr = triangulation(tri,x,y);

c = tr.circumcenter();

n = numel(x);

t = repmat((1:size(tri,1))',1,3);

T = sparse(tri,tri(:,[3 1 2]),t,n,n);

E = (T & T').*T;

F = xor(T, T').*T;

[~ , ~ ,v] = find(triu(E));

[~ , ~ ,vv] = find(triu(E'));

vx = [c(v,1) c(vv,1)]';

vy = [c(v,2) c(vv,2)]';

[i,j,z] = find(F);

dx = x(j)−x(i);

dy = y(j)−y(i);

rx = max(x)-min(x);

ry = max(y)-min(y);

cx = (max(x) + min(x))/2−c(z,1);

cy = (max(y) + min(y))/2−c(z,2);

nm = sqrt(rx.*rx + ry.*ry) + sqrt(cx.*cx + cy.*cy);

scale = nm./sqrt((dx.*dx + dy.*dy));

ex = [c(z,1) c(z,1)-dy.*scale]';

ey = [c(z,2) c(z,2) + dx.*scale]';

vx = [vx ex];

vy = [vy ey];

if nargout < 2.

if isempty(cax).

cax = gca;

end

if isempty(ls).

ls = '-';

end

[l,c,mp,msg] = colstyle(ls);

error(msg).

if isempty(mp).

mp = '.';

end

if isempty(l).

l = get(ancestor(cax,'figure'),'DefaultAxesLineStyleOrder');

end

if isempty(c).

co = get(ancestor(cax,'figure'),'DefaultAxesColorOrder');

c = co(1,:);

end

nume = size(vx,2);

vx = [vx; NaN(1,nume)];

vx = vx(:);

vy = [vy; NaN(1,nume)];

vy = vy(:);

line(vx,vy,'color','k','linestyle',l,'linewidth',0.1,'parent',cax,'yliminclude','off','xliminclude','off');

hold on

box on

set(gca,'xtick',[]);

set(gca,'ytick',[]);

set(gca,'position',[0 0 1 1]);

axis normal

axis square

hold off

print("picc","-dpng","-r1200");

if nargout =  = 1

vxx = [h1; h2];

end

else

vxx = vx;

end